🧠:[<think>Okay, so I've got this problem about a circle and a point inside it. Let me take each part step by step. First, let me understand the given information.The circle C is given by the equation (x - 1)^2 + y^2 = 9. That means the center of the circle is at (1, 0) because the standard form of a circle is (x - h)^2 + (y - k)^2 = r^2. So here, h = 1, k = 0, and the radius r is sqrt(9) = 3. Got that. The point P is (2, 2), which is inside the circle. Let me verify that quickly. Plugging into the left side of the equation: (2 - 1)^2 + (2)^2 = 1 + 4 = 5, which is less than 9, so yes, P is inside the circle.Now, part (1) asks to draw a line l through point P intersecting the circle at points A and B. We need to find the equation of line l when it passes through the center of the circle C. Since the center is (1, 0), and the line passes through both P(2, 2) and the center (1, 0), we can find the equation of the line passing through these two points.Let me recall that the slope of a line through two points (x1, y1) and (x2, y2) is (y2 - y1)/(x2 - x1). So here, the slope m is (0 - 2)/(1 - 2) = (-2)/(-1) = 2. So the slope is 2. Now, using point-slope form. Let's use point P(2, 2). So the equation is y - 2 = 2(x - 2). Simplifying that: y - 2 = 2x - 4, so y = 2x - 2. Let me check if this line passes through the center (1, 0). Plugging x=1: y = 2(1) - 2 = 0. Yes, that works. So the equation of line l is y = 2x - 2. I think that's part (1) done.Moving on to part (2): Write the equation of line l when chord AB is bisected by point P. Hmm, so point P is the midpoint of chord AB. I remember that there's a property related to chords and their midpoints in circles. Specifically, the line from the center of the circle to the midpoint of a chord is perpendicular to the chord itself. So, if P is the midpoint of AB, then the line connecting the center (1, 0) to P(2, 2) should be perpendicular to the chord AB.Let me verify that. The slope of the line from center (1, 0) to P(2, 2) is (2 - 0)/(2 - 1) = 2/1 = 2. So the slope of this line is 2. Therefore, the slope of chord AB, which is perpendicular, should be the negative reciprocal of 2, which is -1/2. So line l has slope -1/2 and passes through point P(2, 2). Using point-slope form: y - 2 = (-1/2)(x - 2). Let me convert that to standard form. Multiply both sides by 2 to eliminate the fraction: 2(y - 2) = -1(x - 2) => 2y - 4 = -x + 2 => x + 2y = 6. So the equation is x + 2y = 6. Alternatively, in slope-intercept form, that's y = (-1/2)x + 3. Let me check if this makes sense. If the line is perpendicular to the radius at midpoint P, then this line should intersect the circle such that P is the midpoint. Let me maybe find points A and B and check.But maybe that's overkill. Alternatively, perhaps I can use the midpoint formula. Since P is the midpoint of AB, if line l has slope -1/2, then the parametric equations for line l could be x = 2 + t, y = 2 - (1/2)t. Plugging into the circle equation:(x - 1)^2 + y^2 = 9Substituting x and y:(2 + t - 1)^2 + (2 - (1/2)t)^2 = 9Simplify:(1 + t)^2 + (2 - (1/2)t)^2 = 9Expanding:(1 + 2t + t^2) + (4 - 2*(1/2)t + (1/4)t^2) = 9Wait, hold on. Let me compute each term step by step.First term: (1 + t)^2 = 1 + 2t + t^2Second term: (2 - (1/2)t)^2 = 4 - 2*(2)*(1/2)t + ( (1/2)t )^2 = 4 - 2t + (1/4)t^2Wait, no. Wait, (a - b)^2 = a^2 - 2ab + b^2. So here, a=2, b=(1/2)t. Therefore:= 2^2 - 2*(2)*(1/2)t + ( (1/2)t )^2= 4 - 2t + (1/4)t^2So combining both terms:(1 + 2t + t^2) + (4 - 2t + (1/4)t^2) = 9Combine like terms:1 + 4 = 52t - 2t = 0t^2 + (1/4)t^2 = (5/4)t^2So overall:5 + (5/4)t^2 = 9Subtract 5:(5/4)t^2 = 4Multiply both sides by 4/5:t^2 = (16/5)Therefore, t = ±4/√5So the points A and B are at t = 4/√5 and t = -4/√5.So plugging back into parametric equations:For t = 4/√5:x = 2 + 4/√5y = 2 - (1/2)*(4/√5) = 2 - 2/√5For t = -4/√5:x = 2 - 4/√5y = 2 - (1/2)*(-4/√5) = 2 + 2/√5So points A and B are (2 + 4/√5, 2 - 2/√5) and (2 - 4/√5, 2 + 2/√5). Let's check if P is the midpoint. The midpoint of A and B should be [( (2 + 4/√5) + (2 - 4/√5) )/2, ( (2 - 2/√5) + (2 + 2/√5) )/2] = (4/2, 4/2) = (2, 2), which is point P. So that checks out. Therefore, the equation of line l is indeed x + 2y = 6. So part (2) is done.Now part (3): When the slant angle of line l is 45º, find the length of chord AB.Slant angle of 45º means the angle between the line and the positive x-axis is 45º. So the slope of line l is tan(45º) = 1. Therefore, the line has a slope of 1 and passes through point P(2, 2). So the equation of line l is y - 2 = 1*(x - 2), which simplifies to y = x. Let's confirm. If the angle is 45º, slope is 1, and passing through (2,2), so equation y = x. Correct.Now, we need to find the points where this line intersects the circle C, then compute the distance between those two points (A and B) to get the length of chord AB.The circle equation is (x - 1)^2 + y^2 = 9. Substitute y = x into the circle equation:(x - 1)^2 + x^2 = 9Expand (x - 1)^2: x^2 - 2x + 1So adding x^2:x^2 - 2x + 1 + x^2 = 9Combine like terms:2x^2 - 2x + 1 = 9Subtract 9:2x^2 - 2x - 8 = 0Divide equation by 2:x^2 - x - 4 = 0Solve for x using quadratic formula. Here, a = 1, b = -1, c = -4.Discriminant D = b² - 4ac = (-1)^2 - 4*1*(-4) = 1 + 16 = 17Solutions: x = [1 ± sqrt(17)]/(2*1) = [1 ± sqrt(17)]/2Therefore, x coordinates are (1 + sqrt(17))/2 and (1 - sqrt(17))/2. Since y = x, the points A and B are ((1 + sqrt(17))/2, (1 + sqrt(17))/2) and ((1 - sqrt(17))/2, (1 - sqrt(17))/2).Now, compute the distance between these two points. Since the coordinates are symmetric, the distance can be calculated using the distance formula:Distance AB = sqrt[( ( (1 + sqrt(17))/2 - (1 - sqrt(17))/2 )^2 + ( (1 + sqrt(17))/2 - (1 - sqrt(17))/2 )^2 )]Simplify each component:The difference in x-coordinates: [ (1 + sqrt(17) - 1 + sqrt(17))/2 ] = (2 sqrt(17))/2 = sqrt(17)Similarly, the difference in y-coordinates is also sqrt(17)Therefore, the distance AB = sqrt[ (sqrt(17))^2 + (sqrt(17))^2 ] = sqrt[17 + 17] = sqrt[34]But wait, sqrt(17)^2 is 17, so 17 +17=34, so sqrt(34). So the length of chord AB is sqrt(34). Hmm, but let me check if that's correct.Alternatively, maybe there's a formula for the length of a chord given a circle and a line. The formula is 2*sqrt(r^2 - d^2), where d is the distance from the center to the line, and r is the radius.Wait, but here, we can compute the distance from the center (1,0) to the line y = x. Then use that formula.Let me try that way. The distance from (1,0) to line y = x. The formula for distance from a point (x0, y0) to line ax + by + c = 0 is |ax0 + by0 + c| / sqrt(a^2 + b^2).First, write line y = x in standard form: x - y = 0. So a = 1, b = -1, c = 0.Distance is |1*1 + (-1)*0 + 0| / sqrt(1 + 1) = |1| / sqrt(2) = 1/sqrt(2).Then, length of chord AB is 2*sqrt(r^2 - d^2) = 2*sqrt(9 - (1/2)) = 2*sqrt(17/2) = 2*(sqrt(17)/sqrt(2)) = sqrt(34). Which matches the previous result. So that's correct.Therefore, the length is sqrt(34). Alternatively, sqrt(34) can be written as √34.So part (3) answer is √34.Wait, but let me confirm once again by plugging in the points. For example, take the two points we found earlier:Point A: ((1 + sqrt(17))/2, (1 + sqrt(17))/2)Point B: ((1 - sqrt(17))/2, (1 - sqrt(17))/2)Distance between A and B:sqrt[ ( ( (1 + sqrt(17))/2 - (1 - sqrt(17))/2 )^2 + ( (1 + sqrt(17))/2 - (1 - sqrt(17))/2 )^2 ) ]Compute the differences:x difference: [ (1 + sqrt(17) - 1 + sqrt(17))/2 ] = (2 sqrt(17))/2 = sqrt(17)Similarly, y difference: sqrt(17)Thus, distance AB = sqrt[ (sqrt(17))^2 + (sqrt(17))^2 ] = sqrt[17 + 17] = sqrt(34). Yep, that's consistent.So part (3) answer is sqrt(34).So to recap:1) The line passing through P(2,2) and the center (1,0) is y = 2x - 2.2) The line where P is the midpoint of AB has equation x + 2y = 6.3) When the slant angle is 45º, the length of chord AB is sqrt(34).I think that's all. Let me just check if there's any calculation mistake.For part (1), the line through (1,0) and (2,2). Slope is (2-0)/(2-1)=2, so equation y=2x -2. Correct.Part (2), midpoint chord. The slope is -1/2, equation y -2 = -1/2(x -2). Converting to standard: x + 2y = 6. Correct.Part (3), angle 45º, slope 1. The line is y=x, intersects the circle at two points. Calculated the distance and got sqrt(34). Correct.All seems good.Final Answer(1) boxed{y = 2x - 2}(2) boxed{x + 2y = 6}(3) boxed{sqrt{34}}</think>Given a circle ( C ): ((x - 1)^2 + y^2 = 9) with a point ( P(2, 2) ) inside it:1. Equation of line ( l ) passing through the center of the circle ( C ): - The center of the circle is ( (1, 0) ). - The line passes through ( P(2, 2) ) and the center ( (1, 0) ). - Slope ( m = frac{0 - 2}{1 - 2} = 2 ). - Using point-slope form: ( y - 2 = 2(x - 2) ). - Simplifying: ( y = 2x - 2 ). [ boxed{y = 2x - 2} ]2. Equation of line ( l ) when chord ( AB ) is bisected by point ( P ): - The line from the center ( (1, 0) ) to ( P(2, 2) ) has a slope of 2. - The chord ( AB ) is perpendicular to this line, so its slope is ( -frac{1}{2} ). - Using point-slope form: ( y - 2 = -frac{1}{2}(x - 2) ). - Converting to standard form: ( x + 2y = 6 ). [ boxed{x + 2y = 6} ]3. Length of chord ( AB ) when the slant angle of line ( l ) is ( 45^circ ): - The slope of the line is ( tan(45^circ) = 1 ). - Equation of the line: ( y = x ). - Substituting ( y = x ) into the circle equation: ((x - 1)^2 + x^2 = 9 ). - Solving the quadratic equation: ( 2x^2 - 2x - 8 = 0 ). - Solutions: ( x = frac{1 pm sqrt{17}}{2} ). - Points of intersection: ( left( frac{1 + sqrt{17}}{2}, frac{1 + sqrt{17}}{2} right) ) and ( left( frac{1 - sqrt{17}}{2}, frac{1 - sqrt{17}}{2} right) ). - Distance between these points: ( sqrt{(sqrt{17})^2 + (sqrt{17})^2} = sqrt{34} ). [ boxed{sqrt{34}} ],<think>Okay, let me try to work through these problems step by step. First, let's get a clear picture of the given information. We have a circle C with equation (x - 1)^2 + y^2 = 9. The center of this circle is at (1, 0) because the standard form of a circle is (x - h)^2 + (y - k)^2 = r^2, so here h = 1 and k = 0, right? And the radius r is 3 since 9 is 3 squared.Point P is (2, 2), which is inside the circle. Let me verify that. To check if a point is inside a circle, we can plug its coordinates into the left side of the circle equation and see if it's less than the radius squared. So for P(2,2): (2 - 1)^2 + (2)^2 = 1 + 4 = 5. The radius squared is 9, and 5 < 9, so yes, P is inside the circle. Good.Now, part (1): Draw a line l through point P intersecting circle C at points A and B. Find the equation of line l when it passes through the center of the circle C.Since the line passes through both P(2, 2) and the center of the circle C, which is (1, 0), we can find the equation of this line by finding the slope first. The slope m between (1, 0) and (2, 2) is (2 - 0)/(2 - 1) = 2/1 = 2. So the slope is 2.Using the point-slope form, we can write the equation. Let's use point (1, 0). So y - 0 = 2(x - 1), which simplifies to y = 2x - 2. Let me check if point P(2,2) is on this line: plugging x = 2, y = 2(2) - 2 = 4 - 2 = 2. Yes, that works. So the equation of line l in part (1) is y = 2x - 2.Okay, moving on to part (2): Write the equation of line l when chord AB is bisected by point P.Hmm, if the chord AB is bisected by point P, then P is the midpoint of AB. There's a property in circles that the line from the center of the circle to the midpoint of a chord is perpendicular to the chord itself. So, if P is the midpoint of AB, then the line connecting the center of the circle C(1, 0) to P(2, 2) should be perpendicular to the chord AB.First, let's find the slope of the line connecting the center (1, 0) to P(2, 2). The slope m_CP is (2 - 0)/(2 - 1) = 2/1 = 2. Therefore, since AB is perpendicular to CP, its slope should be the negative reciprocal of 2, which is -1/2.So the line AB has slope -1/2 and passes through point P(2, 2). Using point-slope form: y - 2 = (-1/2)(x - 2). Let me convert this to standard form for clarity. Multiply both sides by 2: 2(y - 2) = -1(x - 2) => 2y - 4 = -x + 2 => x + 2y - 6 = 0. So the equation of line l is x + 2y = 6.Wait, let me double-check. If the slope is -1/2, then the equation is y = (-1/2)x + b. Plugging in (2,2): 2 = (-1/2)(2) + b => 2 = -1 + b => b = 3. So the equation is y = (-1/2)x + 3. Multiplying both sides by 2: 2y = -x + 6 => x + 2y = 6. Yep, same result. So that's correct.So part (2) answer is x + 2y = 6 or y = (-1/2)x + 3. Depending on the preferred form, but the problem says "write the equation", so either is fine. Maybe the standard form is better.Now, part (3): When the slant angle of line l is 45º, find the length of chord AB.Slant angle of 45º means that the line makes a 45-degree angle with the horizontal axis. So the slope m is tan(45º) = 1. Therefore, the line l has a slope of 1 and passes through point P(2, 2). Let's find its equation first. Using point-slope: y - 2 = 1*(x - 2) => y = x. So the equation is y = x.Wait, but let's confirm if this line intersects the circle. Since P is inside the circle, the line should intersect at two points A and B. Let's find points A and B by solving the system of equations:Circle: (x - 1)^2 + y^2 = 9Line: y = xSubstitute y = x into the circle equation:(x - 1)^2 + x^2 = 9Expand (x - 1)^2: x^2 - 2x + 1So, x^2 - 2x + 1 + x^2 = 9Combine like terms: 2x^2 - 2x + 1 = 9Subtract 9: 2x^2 - 2x - 8 = 0Divide by 2: x^2 - x - 4 = 0Use quadratic formula: x = [1 ± sqrt(1 + 16)]/2 = [1 ± sqrt(17)]/2So the x-coordinates of A and B are (1 + sqrt(17))/2 and (1 - sqrt(17))/2. Since y = x, their coordinates are [(1 + sqrt(17))/2, (1 + sqrt(17))/2] and [(1 - sqrt(17))/2, (1 - sqrt(17))/2]Now, to find the distance between A and B, which is the length of chord AB.The distance between the two points can be found using the distance formula. Let's compute the difference in x and y:Δx = [(1 + sqrt(17))/2 - (1 - sqrt(17))/2] = [2 sqrt(17)/2] = sqrt(17)Similarly, Δy = sqrt(17) as well since y = x.Therefore, the distance AB is sqrt[(sqrt(17))^2 + (sqrt(17))^2] = sqrt[17 + 17] = sqrt[34]Wait, but hold on. Let me verify that again.Wait, if points A and B are [(1 + sqrt(17))/2, (1 + sqrt(17))/2] and [(1 - sqrt(17))/2, (1 - sqrt(17))/2], then the distance between them is sqrt[( ( (1 + sqrt(17))/2 - (1 - sqrt(17))/2 )^2 + ( (1 + sqrt(17))/2 - (1 - sqrt(17))/2 )^2 )]Calculating the differences:For x-coordinate: (1 + sqrt(17))/2 - (1 - sqrt(17))/2 = [1 + sqrt(17) -1 + sqrt(17)]/2 = (2 sqrt(17))/2 = sqrt(17)Same for y-coordinate. So Δx = sqrt(17), Δy = sqrt(17)Therefore, distance AB = sqrt[(sqrt(17))^2 + (sqrt(17))^2] = sqrt[17 + 17] = sqrt[34]So the length of chord AB is sqrt(34). But let me check if this is correct.Alternatively, maybe using another method: For a line intersecting a circle, the length of the chord can be found using the formula 2*sqrt(r^2 - d^2), where d is the distance from the center to the line.But wait, in this case, the line is y = x, passing through P(2,2). Let's compute the distance from the center (1,0) to the line y = x.The formula for distance from a point (x0, y0) to the line ax + by + c = 0 is |ax0 + by0 + c| / sqrt(a^2 + b^2). First, write the line in ax + by + c = 0 form: y = x => x - y = 0. So a = 1, b = -1, c = 0.Distance from (1,0) to x - y = 0 is |1*1 + (-1)*0 + 0| / sqrt(1 + 1) = |1| / sqrt(2) = 1/sqrt(2)Then the length of chord AB is 2*sqrt(r^2 - d^2) = 2*sqrt(9 - (1/2)) = 2*sqrt(17/2) = 2*(sqrt(34)/2) = sqrt(34). Which matches the previous result. So that's correct. Therefore, the length of chord AB is sqrt(34).Wait, but let me make sure. The formula 2*sqrt(r^2 - d^2) is correct for the length of the chord when d is the distance from the center to the line. Here, r = 3, d = 1/sqrt(2). So 2*sqrt(9 - 1/2) = 2*sqrt(17/2) = sqrt(34). Yes, that's the same as before. So this confirms it.Therefore, part (3)'s answer is sqrt(34).Let me recap:1) The line through P(2,2) and the center (1,0) has slope 2, equation y = 2x - 2.2) The line where chord AB is bisected by P must be perpendicular to the radius at P, so slope -1/2, equation x + 2y = 6.3) The line with slant angle 45º has slope 1, equation y = x, intersecting the circle at points with distance sqrt(34).I think that's all. Let me check each part again to ensure there are no mistakes.For part (1), the line through (1,0) and (2,2) is straightforward. The slope is indeed 2, equation y = 2x - 2. Correct.Part (2): The chord bisected by P implies P is the midpoint, so the line from the center to P is perpendicular to the chord. Calculated slope correctly, got -1/2, equation correct. Checked using two methods, point-slope and standard form. Correct.Part (3): Slant angle 45º means slope 1. Found the equation y = x, solved intersection with circle, found points, computed distance, and confirmed using the chord length formula. Both methods give sqrt(34). Correct.So all parts seem solid.Final Answer(1) boxed{y = 2x - 2}(2) boxed{x + 2y = 6}(3) boxed{sqrt{34}}</think>Given a circle ( C ): ( (x - 1)^2 + y^2 = 9 ) with a point ( P(2, 2) ) inside it:1. Equation of line ( l ) when it passes through the center of the circle ( C ): - The center of the circle is ( (1, 0) ). - The slope of the line passing through ( P(2, 2) ) and the center ( (1, 0) ) is ( frac{2 - 0}{2 - 1} = 2 ). - Using point-slope form with point ( (1, 0) ), the equation is ( y = 2(x - 1) ), simplifying to ( y = 2x - 2 ). - Final answer: (boxed{y = 2x - 2})2. Equation of line ( l ) when chord ( AB ) is bisected by point ( P ): - The line from the center ( (1, 0) ) to ( P(2, 2) ) has a slope of 2, so the chord ( AB ) must be perpendicular to this line, giving a slope of ( -frac{1}{2} ). - Using point-slope form with point ( P(2, 2) ), the equation is ( y - 2 = -frac{1}{2}(x - 2) ), simplifying to ( x + 2y = 6 ). - Final answer: (boxed{x + 2y = 6})3. Length of chord ( AB ) when the slant angle of line ( l ) is ( 45^circ ): - The slope of the line with a slant angle of ( 45^circ ) is 1. The equation of the line passing through ( P(2, 2) ) is ( y = x ). - Substituting ( y = x ) into the circle's equation: ( (x - 1)^2 + x^2 = 9 ). - Solving the quadratic equation gives ( x = frac{1 pm sqrt{17}}{2} ). The points of intersection are ( left( frac{1 + sqrt{17}}{2}, frac{1 + sqrt{17}}{2} right) ) and ( left( frac{1 - sqrt{17}}{2}, frac{1 - sqrt{17}}{2} right) ). - The distance between these points is ( sqrt{(sqrt{17})^2 + (sqrt{17})^2} = sqrt{34} ). - Final answer: (boxed{sqrt{34}})]

🔑:(1) Given circle C: ({(x-1)}^{2}+{y}^{2}=9), its center is C(1,0).Since line l passes through points P and C,The slope of line l is k=frac{2-0}{2-1}=2.Thus, the equation of line l is y=2(x-1),Or 2x-y-2=0.(2) When chord AB is bisected by point P, according to the perpendicular bisector theorem:Line l is perpendicular to PC,So, the equation of line l is y-2=-frac{1}{2}(x-2),Or x+2y-6=0.(3) When the slant angle of line l is 45º,The slope of line l is 1,So, the equation of line l is y-2=x-2,Or x-y=0.The distance d from the center of the circle to line l is d=frac{|1-0|}{sqrt{2}}=frac{sqrt{2}}{2},Hence, the length of chord AB is |AB|=2sqrt{{r}^{2}-{d}^{2}}=2sqrt{9-frac{1}{2}}=boxed{sqrt{34}}.

🧠:[<think>Okay, let me try to figure this out. So the problem says that f(x) is an odd function defined on all real numbers. For x greater than or equal to 0, f(x) is given as 2^x + 2x + b, where b is a constant. We need to find the value of f(-1). First, let me recall what an odd function is. An odd function satisfies the property that f(-x) = -f(x) for all x in its domain. That means if I know the value of the function at some positive x, I can find the value at -x by just taking the negative of it. So since we're given f(x) for x ≥ 0, we can use this property to find f(-1).But wait, the function for x ≥ 0 is given as 2^x + 2x + b. However, since the function is odd, there's a specific relationship between f(x) and f(-x). Let me write that down:For any x, f(-x) = -f(x). So, in particular, if x = 1, then f(-1) = -f(1). Therefore, if I can compute f(1) using the given expression, then f(-1) is just the negative of that. But hold on, the function is defined as 2^x + 2x + b for x ≥ 0. So f(1) should be 2^1 + 2*1 + b, which simplifies to 2 + 2 + b = 4 + b. Then f(-1) would be -f(1) = -(4 + b). But here's the catch: since the function is odd, it must satisfy f(0) = 0. Because for x = 0, f(-0) = -f(0), which implies f(0) = -f(0), and the only number that is equal to its own negative is 0. So f(0) must be 0. Wait, let's check if that's correct. For x = 0, the definition of an odd function gives f(0) = -f(0), so adding f(0) to both sides gives 2f(0) = 0, hence f(0) = 0. Yes, that's correct. So in the given expression for x ≥ 0, when x is 0, f(0) should be 2^0 + 2*0 + b. Calculating that: 1 + 0 + b = 1 + b. But we just established that f(0) must be 0. Therefore, 1 + b = 0. Solving for b, we get b = -1. So now we can find the value of b, which is necessary to compute f(1) correctly. Since b = -1, substituting back into the expression for f(1):f(1) = 2^1 + 2*1 + (-1) = 2 + 2 - 1 = 3. Therefore, f(-1) = -f(1) = -3. Let me verify that step-by-step again to make sure I didn't make a mistake. 1. Since f is odd, f(-x) = -f(x) for all x. Therefore, f(-1) = -f(1).2. To find f(1), use the given formula for x ≥ 0: f(1) = 2^1 + 2*1 + b = 2 + 2 + b = 4 + b.3. But also, since f is odd, f(0) = 0. Using the given formula for x = 0: f(0) = 2^0 + 2*0 + b = 1 + 0 + b = 1 + b.4. Setting f(0) = 0 gives 1 + b = 0 => b = -1.5. Substitute b = -1 into f(1): 4 + (-1) = 3.6. Therefore, f(-1) = -f(1) = -3.Yes, that all checks out. I don't see any errors in that reasoning. Alternatively, another way to think about it is: since the function is odd, its graph is symmetric about the origin. So if we have the right half of the graph (for x ≥ 0), the left half (for x < 0) is just the mirror image through the origin. Therefore, to get f(-1), we flip the point (1, f(1)) over the origin to (-1, -f(1)), hence f(-1) is -f(1). But again, the key here was realizing that b is not arbitrary; it must satisfy the condition f(0) = 0. Since the original expression for x ≥ 0 includes x = 0, we can use that point to solve for b. It's also worth noting that if the function wasn't odd, we couldn't determine b uniquely. But because it's odd, that condition at x = 0 gives us an equation to solve for b. So, putting it all together, after determining that b = -1, we substitute into f(1) and then take the negative to find f(-1). I think that's solid. Let me check if there's another way to approach this problem, just to confirm.Suppose we didn't use the condition at x = 0 first. Let's see. If we just naively compute f(-1) as -f(1) = -(2^1 + 2*1 + b) = -(4 + b). But without knowing b, we can't proceed. So we must find the value of b first. To find b, we need another equation, which comes from the fact that f(0) = 0. Hence, substituting x = 0 into the given expression gives 1 + 0 + b = 0, so b = -1. Therefore, the necessity of using both the oddness at x = 0 and at x = 1 is clear. Another point to consider: since the function is odd, its Taylor series (if it exists) only contains odd powers of x. However, in the given expression for x ≥ 0, we have 2^x, which is an exponential function, 2x, which is linear, and a constant b. The exponential function can be written as a series with all powers of x, but since f(x) is odd, when we extend it to x < 0, the function must satisfy f(-x) = -f(x). However, 2^x is not an odd function on its own. So when we define f(x) as 2^x + 2x + b for x ≥ 0, and then extend it to x < 0 by the oddness condition, the resulting function will have a different form for x < 0. But for our purposes, since we only need f(-1), we can directly use the relation f(-1) = -f(1), which is straightforward once we know f(1). So in conclusion, the steps are:1. Use f(0) = 0 to find b = -1.2. Compute f(1) with b = -1, getting f(1) = 3.3. Apply the odd function property to find f(-1) = -3.This seems correct, and I don't see any missing steps or miscalculations. Therefore, the value of f(-1) is -3.Final AnswerThe value of ( f(-1) ) is boxed{-3}.</think>Given that ( f(x) ) is an odd function defined on (mathbb{R}) and for ( x geq 0 ), ( f(x) = 2^x + 2x + b ) (where ( b ) is a constant), we need to find the value of ( f(-1) ).1. Recall the property of odd functions: An odd function satisfies ( f(-x) = -f(x) ) for all ( x ). Therefore, ( f(-1) = -f(1) ).2. Determine the constant ( b ): - Since ( f(x) ) is odd, ( f(0) = 0 ). - For ( x = 0 ), the given function is ( f(0) = 2^0 + 2 cdot 0 + b = 1 + b ). - Setting ( f(0) = 0 ), we get ( 1 + b = 0 ), which solves to ( b = -1 ).3. Calculate ( f(1) ): - Substitute ( x = 1 ) and ( b = -1 ) into the given function: [ f(1) = 2^1 + 2 cdot 1 + (-1) = 2 + 2 - 1 = 3 ]4. Find ( f(-1) ): - Using the odd function property: ( f(-1) = -f(1) = -3 ).Thus, the value of ( f(-1) ) is (boxed{-3}).,<think>Okay, let's see. The problem says that f(x) is an odd function defined on all real numbers. For x greater than or equal to 0, f(x) is given as 2^x plus 2x plus b, where b is a constant. We need to find the value of f(-1). First, I remember that an odd function has the property that f(-x) = -f(x) for all x in its domain. So, if I can figure out f(1), then f(-1) should just be -f(1). But to do that, I need to make sure that the function is properly defined as an odd function. Since the function is given for x ≥ 0, the part for x < 0 is determined by the oddness condition. But wait, the function might have a different expression for x < 0, but since it's odd, we can write f(x) = -f(-x) when x < 0. So, for x negative, f(x) is the negative of f evaluated at the positive version of x. But here, the function for x ≥ 0 is given as 2^x + 2x + b. So, to make sure that the entire function is odd, we need to check if this expression satisfies the condition at x = 0. Because for x = 0, f(-0) = -f(0), but -0 is 0, so f(0) = -f(0). The only way this can be true is if f(0) = 0. Therefore, let's evaluate the given function at x = 0. When x = 0, f(0) = 2^0 + 2*0 + b = 1 + 0 + b = 1 + b. But since f(0) must be 0 (from the odd function property), we have 1 + b = 0. Solving for b, we get b = -1. So, the constant b is -1. Now, the function for x ≥ 0 is f(x) = 2^x + 2x - 1. To find f(-1), we use the property of odd functions: f(-1) = -f(1). So, we need to compute f(1) first. Calculating f(1): 2^1 + 2*1 - 1 = 2 + 2 - 1 = 3. Therefore, f(-1) should be -3. Wait, let me double-check. If x = 1, then f(1) is 2^1 + 2(1) + b. But we found that b = -1, so yes, 2 + 2 - 1 = 3. So, f(1) = 3, so f(-1) = -3. But let me make sure that the function is indeed odd. Let's check another point, say x = 1 and x = -1. If x = 1, f(1) = 3 as above. Then f(-1) should be -3. Let's compute f(-1) using the definition for x < 0. For x < 0, f(x) = -f(-x). So, f(-1) = -f(1) = -3. That matches. What about x = 2? f(2) = 2^2 + 2*2 - 1 = 4 + 4 -1 = 7. Then f(-2) should be -7. Let's see: f(-2) = -f(2) = -7. That's consistent. Also, check at x = 0. f(0) = 1 + 0 -1 = 0, which satisfies the odd function condition. Therefore, everything checks out. The key step was realizing that for the function to be odd, f(0) must be 0, which allowed us to solve for b. Then, once we have the complete expression for x ≥ 0, we can use the oddness to find f(-1) as -f(1). So, the answer should be -3, which is boxed{-3}.Final Answerboxed{-3}</think>Given that ( f(x) ) is an odd function defined on (mathbb{R}) and for ( x geq 0 ), ( f(x) = 2^x + 2x + b ) (where ( b ) is a constant), we need to find the value of ( f(-1) ).An odd function satisfies the property ( f(-x) = -f(x) ) for all ( x ). For ( x = 0 ), this property implies ( f(0) = 0 ).Evaluating the given function at ( x = 0 ):[f(0) = 2^0 + 2 cdot 0 + b = 1 + b]Since ( f(0) = 0 ), we have:[1 + b = 0 implies b = -1]Thus, the function for ( x geq 0 ) is ( f(x) = 2^x + 2x - 1 ).To find ( f(-1) ), we use the property of odd functions:[f(-1) = -f(1)]Calculating ( f(1) ):[f(1) = 2^1 + 2 cdot 1 - 1 = 2 + 2 - 1 = 3]Therefore:[f(-1) = -f(1) = -3]The final answer is (boxed{-3}).]

🔑:Since f(x) is an odd function defined on mathbb{R}, it satisfies f(-x) = -f(x) for all x in mathbb{R}.Given that for x geq 0, we have f(x) = 2^x + 2x + b. We can find the constant b by evaluating f(0):f(0) = 2^0 + 2 cdot 0 + b = 1 + b.Since f(x) is an odd function, f(0) must be 0 (because f(0) = -f(0)), thus we have:1 + b = 0.Solving for b, we get b = -1.Now we have the expression for f(x) when x geq 0:f(x) = 2^x + 2x - 1.For x 0 if x < 0), we can express -f(x) as:-f(x) = -f(-y) = f(y) = 2^y + 2y - 1.Substituting y = -x back into the equation gives us:-f(x) = 2^{-x} + 2(-x) - 1 = 2^{-x} - 2x - 1.Therefore:f(x) = -2^{-x} + 2x + 1.Finally, to find f(-1), we substitute x = -1:f(-1) = -2^{-(-1)} + 2(-1) + 1 = -2^{1} - 2 + 1 = -2 - 2 + 1 = -3.Hence, the value of f(-1) is boxed{-3}.

🧠:[<think>Okay, so I have this geometry problem here. Let me read it again: In a right triangle ABC, the leg AB is 21, and the leg BC is 28. A circle with center O lying on the hypotenuse AC touches both legs. Find the radius of the circle.Alright, let me visualize the triangle first. It's a right triangle, so one of the angles is 90 degrees. The legs are AB = 21 and BC = 28. Wait, hold on—the legs are AB and BC? If ABC is a right triangle, then the right angle must be at point B, since AB and BC are the legs. So, points A, B, C with right angle at B, AB = 21, BC = 28, hypotenuse AC. Then there's a circle centered at O on the hypotenuse AC that touches both legs AB and BC. I need to find the radius of this circle.First, let me sketch this mentally. Triangle ABC with right angle at B, legs 21 and 28, hypotenuse AC. The circle is inside the triangle, touching both legs AB and BC, and its center O is on the hypotenuse AC. So it's tangent to both legs, and the center is somewhere along the hypotenuse. The radius would be the distance from O to each leg, since it's touching them. Since the legs are perpendicular, the radius should be the same in both directions, which makes sense because the circle is tangent to both legs.Hmm, how do I approach this? Maybe coordinate geometry? Let's assign coordinates to the triangle. Let me place point B at the origin (0,0) since it's the right angle. Then, since AB is 21, and BC is 28, let me assume AB is along the y-axis and BC is along the x-axis. Wait, but AB and BC are both legs. So AB is from B to A, and BC is from B to C. So if B is at (0,0), then A would be at (0,21) and C would be at (28,0). Wait, hold on—that might not be right. Let me make sure. If AB is a leg of length 21 and BC is a leg of length 28, then AB is one leg and BC is another leg. So in the right triangle, the legs are AB and BC. So the right angle is between AB and BC, meaning point B is the vertex where AB and BC meet at a right angle. So yes, coordinates: Let’s set point B at (0,0). Then, if AB is 21, let's say AB is vertical, so point A is at (0,21). Then BC is 28, horizontal, so point C is at (28,0). Then the hypotenuse is AC, connecting (0,21) to (28,0).Now, the circle with center O on hypotenuse AC touches both legs AB and BC. So the center O is somewhere along AC, and the circle is tangent to both AB and BC. The radius of the circle would be the distance from O to AB and to BC, which should be equal since the circle is tangent to both.But since AB is the vertical leg and BC is the horizontal leg, the distance from O to AB would be the horizontal distance from O to AB (since AB is the y-axis), and the distance from O to BC would be the vertical distance from O to BC (since BC is the x-axis). Therefore, if O has coordinates (h,k), then the distance to AB (the y-axis) is |h| and the distance to BC (the x-axis) is |k|. Since the circle is tangent to both legs, these distances must equal the radius r. But since O is on the hypotenuse AC, the coordinates (h,k) must lie on AC.So first, let me parametrize the hypotenuse AC. The coordinates of A are (0,21) and C are (28,0). The equation of the line AC can be found. Let me calculate the slope first. Slope m = (0 - 21)/(28 - 0) = -21/28 = -3/4. So the equation is y - 21 = (-3/4)(x - 0), so y = (-3/4)x + 21.Therefore, any point O on AC can be represented as (x, (-3/4)x + 21). Since O is the center of the circle touching both legs AB and BC, the distance from O to AB (the y-axis) is x, and the distance from O to BC (the x-axis) is y = (-3/4)x + 21. Since both these distances must equal the radius r, we have x = r and y = r. Therefore, we have two equations:1. x = r2. (-3/4)x + 21 = rSubstituting equation 1 into equation 2:(-3/4)r + 21 = rSolving for r:21 = r + (3/4)r = (7/4)rMultiply both sides by 4/7:r = 21 * (4/7) = 12Wait, that seems straightforward. So the radius is 12? Let me check if that makes sense.Let me verify. If the radius is 12, then the center O is at (12, 12). Wait, but according to the coordinates, O should lie on AC. Let me see if (12,12) is on AC. Plugging x = 12 into the equation of AC: y = (-3/4)(12) + 21 = -9 + 21 = 12. Yes, (12,12) is on AC, and the distance from (12,12) to AB (y-axis) is 12, and to BC (x-axis) is 12. So the circle with center (12,12) and radius 12 would touch both legs. But wait, is this circle entirely inside the triangle?The triangle has vertices at (0,0), (0,21), and (28,0). The point (12,12) is inside the triangle, and the radius is 12. The distance from (12,12) to the hypotenuse AC would also be the radius, but wait, in this case, the circle is tangent to the legs, not necessarily to the hypotenuse. Wait, the problem says the circle touches both legs, so it's tangent to AB and BC. The center is on AC, so it's tangent to AB and BC, and lies on AC. So in that case, the radius is 12. But let me confirm with another method.Alternatively, maybe using the inradius formula. Wait, but this isn't the incircle, because the incircle touches all three sides, but in this case, the circle is only touching two sides (the legs) and has its center on the hypotenuse. So it's a different circle. So maybe coordinate geometry is the way to go.Alternatively, maybe using similar triangles. Let me think. Since the center O is on AC, and the circle is tangent to AB and BC, the distances from O to AB and BC are equal (both equal to the radius r). As AB and BC are perpendicular, the center O must be equidistant from both legs, which in coordinate terms is (r, r). Wait, but in this coordinate system, if AB is the y-axis and BC is the x-axis, then the distance from O to AB is the x-coordinate, and to BC is the y-coordinate, so (x, y) = (r, r). But O is on AC, so plugging into the line equation: y = (-3/4)x + 21. So if x = r, then y = (-3/4)r + 21. But since y must also equal r, we have r = (-3/4)r + 21. Which gives the same equation as before. Solving: r + (3/4)r = 21 → (7/4)r = 21 → r = 12. So that checks out.Alternatively, maybe area considerations. The area of the triangle is (21*28)/2 = 294. The inradius of the triangle is area/s, where s is the semi-perimeter. The hypotenuse AC has length sqrt(21^2 + 28^2) = sqrt(441 + 784) = sqrt(1225) = 35. So semi-perimeter s = (21 + 28 + 35)/2 = 84/2 = 42. So inradius r = 294 / 42 = 7. But that's the inradius, which touches all three sides. But the circle in the problem is different—it only touches the two legs and has its center on the hypotenuse. So the radius we found earlier, 12, is larger than the inradius. Wait, that makes sense because the inradius is 7, but this circle is only tangent to two sides, so it can be larger. But let's see, if the radius is 12, then the diameter is 24. But the legs are 21 and 28. Wait, 12 is less than both 21 and 28, so the circle with radius 12 would fit inside the triangle. Hmm, but when I think of the inradius being 7, this circle is bigger. But since it's only touching two sides, maybe it can be larger. Let me visualize again.If the center is at (12,12), radius 12, then the circle would extend from x=0 to x=24, but the triangle's horizontal leg is only up to x=28. Similarly, vertically, from y=0 to y=24, but the vertical leg is up to y=21. Wait, hold on. If the center is at (12,12) with radius 12, then the circle would go from x=12 - 12 = 0 to x=12 + 12 = 24, and from y=12 - 12 = 0 to y=12 + 12 = 24. However, the triangle's vertical leg is only up to y=21, so the circle would actually go beyond the triangle in the vertical direction. Wait, that can't be possible because the triangle only goes up to y=21. So a circle centered at (12,12) with radius 12 would go up to y=24, which is outside the triangle. That's a problem. So maybe my initial answer is wrong?Wait, hold on, this inconsiderate—maybe my coordinate system is messed up. Let me check again. If point A is at (0,21), point C is at (28,0), then the triangle is between x=0 to 28 and y=0 to 21. So if the circle is centered at (12,12) with radius 12, the topmost point of the circle is at y=12 + 12 = 24, which is above the triangle's top at y=21. That means the circle would extend outside the triangle, which is impossible because the circle is supposed to be inside the triangle, touching both legs. Therefore, my previous conclusion that r=12 must be wrong. So where did I go wrong?Ah, I see. The problem is that in my coordinate system, the center O is at (r, r), but in reality, since the circle is inside the triangle, the y-coordinate of the center can't be greater than 21 - r, and the x-coordinate can't be greater than 28 - r. Wait, but according to the previous calculation, if r=12, then the center is at (12,12), which is inside the triangle because x=12 <28 and y=12 <21. However, the circle's top point is at y=24, which is outside the triangle, but the circle is only required to touch the legs AB and BC. Wait, but AB is the vertical leg from (0,0) to (0,21), so touching AB would mean the circle is tangent to AB, which is the line x=0. The distance from the center to AB is the x-coordinate, which is 12, so the circle touches AB at (0,12). Similarly, touching BC (the x-axis) at (12,0). But in the triangle, the vertical leg AB is from (0,0) to (0,21), so the point (0,12) is on AB. Similarly, (12,0) is on BC. So even though the circle extends beyond the triangle in other areas, as long as it's tangent to the legs at those points, it's okay? But the circle would also intersect the hypotenuse AC at the center, but since the center is on AC, the circle's center is on AC, so the circle would intersect AC at the center, but only at that point? Wait, actually, no—the circle is centered at O on AC with radius r, so the circle intersects AC at points O plus and minus r along AC. But since AC is the hypotenuse, those points might lie inside or outside the triangle.Wait, perhaps my confusion arises from the fact that in coordinate geometry, the circle is defined as all points at distance r from O, but in this problem, the circle is tangent only to the legs AB and BC, and lies inside the triangle. Therefore, even though the circle's full extent in the coordinate system would go beyond the triangle, the actual circle within the triangle would only be the part that's inside. But the problem states that the circle touches both legs, which would only require the points of tangency on the legs. However, if the center is at (12,12) with radius 12, then the circle would intersect AB at (0,12) and BC at (12,0), both of which are within the triangle. The rest of the circle would extend beyond the triangle, but since the problem doesn't specify that the circle is entirely within the triangle, just that it touches both legs and the center is on the hypotenuse. Wait, but the problem says "a circle with center O lying on the hypotenuse AC touches both legs." Typically, such problems imply that the circle is inside the triangle, otherwise, you could have a circle outside the triangle as well. So if the circle is inside the triangle, then my previous answer is invalid because the circle would extend outside. Therefore, there must be an error in my reasoning.Wait, let's think again. Maybe I made a wrong assumption in setting up the coordinates. Let me confirm the coordinates of the triangle. If AB is 21 and BC is 28, with right angle at B, then AB is one leg, BC is another leg. So if B is at (0,0), then A is at (0,21) and C is at (28,0). Therefore, hypotenuse AC connects (0,21) to (28,0). The line AC has slope -3/4 as before.The center O is on AC, so its coordinates are (x, (-3/4)x + 21). The circle touches both legs AB and BC. The distance from O to AB (the y-axis) is x, and the distance from O to BC (the x-axis) is y = (-3/4)x + 21. Since the circle is tangent to both legs, these distances must equal the radius r. So x = r and y = r. Therefore, we have:r = (-3/4)r + 21Which gives r = 12. However, as we saw, this would imply that the circle extends beyond the triangle. So perhaps the problem is that the circle is allowed to extend outside the triangle? But that seems counterintuitive because the circle is said to "touch both legs"—if it's centered on the hypotenuse, but outside the triangle, it wouldn't touch the legs in the context of the triangle. Hmm.Wait, perhaps I messed up the legs. Let me double-check the problem statement: "In a right triangle ABC, the leg AB is 21, and the leg BC is 28." So ABC is a right triangle with legs AB and BC. Therefore, the right angle is at B. So points A, B, C with AB and BC as legs. So the triangle is AB-C-B? Wait, no. The triangle is ABC, with vertices A, B, C. The legs are AB and BC, so the sides AB and BC are the legs. Therefore, the right angle is at point B. Therefore, the triangle is formed by points A, B, and C, with right angle at B, legs AB = 21 and BC = 28, hypotenuse AC.Therefore, in coordinates, as I had before: B at (0,0), A at (0,21), C at (28,0). Hypotenuse AC from (0,21) to (28,0). The center O is on AC, so (x, (-3/4)x + 21). The circle centered at O touches both legs AB and BC. The distance from O to AB is x, and to BC is y. So x = y = r. Solving, we get r = 12. However, as we saw, the circle would go from x=0 to x=24 (center at 12, radius 12) and y=0 to y=24, but the triangle only goes up to y=21 and x=28. So in the y-direction, the circle would extend beyond the triangle. But in reality, the circle can only exist where the triangle is. Therefore, the part of the circle that's inside the triangle would be the part from y=0 to y=21. But if the circle is centered at (12,12) with radius 12, then the topmost point is (12,24), which is outside the triangle. So how is the circle touching both legs?Wait, perhaps the circle is only tangent to the legs within the triangle. So the point of tangency on AB is (0,12), which is on AB since AB goes from (0,0) to (0,21). Similarly, the point of tangency on BC is (12,0), which is on BC from (0,0) to (28,0). So those points are within the triangle, and the rest of the circle is outside the triangle. But the problem says "a circle with center O lying on the hypotenuse AC touches both legs." It doesn't specify that the circle is entirely within the triangle, just that it touches both legs. So even if part of the circle is outside the triangle, as long as it touches the legs AB and BC, which are sides of the triangle, then it's valid. So in that case, the radius is indeed 12. However, in most geometry problems like this, they usually assume the circle is inside the triangle. Maybe there's a misinterpretation here.Alternatively, maybe the problem is in the orientation of the triangle. Maybe AB and BC are not the legs but some other sides. Wait, let's read again: "In a right triangle ABC, the leg AB is 21, and the leg BC is 28." So legs are AB and BC, right angle at B. So the coordinate system is correct.Wait, perhaps another approach. Let's consider the circle tangent to both legs and centered on the hypotenuse. The center O lies on AC, and the circle is tangent to AB and BC. The distance from O to AB and BC is the radius r. The coordinates of O are (r, r). But O must lie on AC. The line AC has the equation y = (-3/4)x + 21. Therefore, substituting x = r, y = (-3/4)r + 21. But y must also equal r, so:r = (-3/4)r + 21Adding (3/4)r to both sides:(7/4)r = 21Multiply both sides by 4/7:r = 21 * 4 / 7 = 12So regardless of the coordinate system, the answer comes out to 12. But then, the circle does extend beyond the triangle in the y-direction, but maybe that's acceptable. Since the problem doesn't state that the circle is entirely within the triangle, just that it touches both legs and the center is on the hypotenuse. Therefore, the answer is 12. However, this seems to contradict the inradius being 7. Let me check with another method.Alternatively, let's use coordinate geometry but check if the circle intersects the hypotenuse. The center is at (12,12), radius 12. The hypotenuse is the line AC: y = (-3/4)x + 21. The distance from center (12,12) to the line AC should be equal to the radius if the circle is tangent to AC, but in this problem, the circle is only required to be tangent to the legs. Wait, no—the circle is centered on AC, so the distance from O to AC is zero, because O is on AC. Therefore, the circle will intersect AC at two points: O and another point? Wait, no—the center is on AC, so the circle intersects AC at points that are r units away from O along AC. Since AC has length 35, and O is somewhere along AC. But the position of O is at (12,12), which is a point along AC. The distance from A to O can be calculated. The coordinates of A are (0,21), O is at (12,12). The distance AO is sqrt((12 - 0)^2 + (12 - 21)^2) = sqrt(144 + 81) = sqrt(225) = 15. Similarly, the distance from O to C is sqrt((28 - 12)^2 + (0 - 12)^2) = sqrt(256 + 144) = sqrt(400) = 20. So AO = 15, OC = 20, which sum to 35, the length of AC. Therefore, O divides AC into segments of 15 and 20. The circle of radius 12 centered at O would extend 12 units along AC from O. But the direction along AC is not aligned with the coordinate axes, so the intersection points along AC would be 12 units away from O. Since AC is 35 units long, and O is 15 units from A, moving 12 units towards A would reach 3 units from A, and 12 units towards C would reach 32 units from A, which is beyond C. But C is 28 units from A? Wait, no. Wait, AC is 35 units, so from A to C is 35. If O is 15 units from A, then moving 12 units towards C would reach 15 + 12 = 27 units from A, still within AC. Wait, maybe my calculation is off. Wait, the length from O to C is 20, so moving 12 units from O towards C would reach a point 12 units away, so 20 - 12 = 8 units from C. Similarly, moving 12 units towards A would be 15 - 12 = 3 units from A. Therefore, the circle intersects AC at 3 units from A and 8 units from C. But since the circle is only required to touch the legs, maybe this is acceptable.But regardless, the problem states that the circle touches both legs, and the center is on the hypotenuse. It doesn't mention anything about the circle being tangent to the hypotenuse or lying entirely within the triangle. So perhaps despite the circle extending beyond the triangle in the coordinate system, as long as it touches the legs AB and BC, the answer is 12. But this seems contradictory to the initial thought that the circle must be inside the triangle.Wait, let's calculate the maximum possible radius of a circle centered on AC that touches both legs. If the circle is entirely inside the triangle, then the center O must be such that the circle does not cross the hypotenuse. But since the center is on the hypotenuse, the circle will intersect the hypotenuse at the center. Wait, no—if the center is on the hypotenuse, then the circle is tangent to the legs, but the hypotenuse passes through the center. So the circle would intersect the hypotenuse at the center, but since the center is on the hypotenuse, the circle is allowed to intersect there. However, other intersections with the hypotenuse might occur. Wait, the distance from the center O to the hypotenuse is zero, because O is on the hypotenuse. So the circle will intersect the hypotenuse at exactly one point—the center itself—if the radius is zero. But as the radius increases, the circle will intersect the hypotenuse at two points. However, in our case, the radius is 12, so the circle intersects the hypotenuse at two points: one is the center O, and the other is a point 12 units away from O along AC. But since the hypotenuse is a straight line, the circle centered at O with radius r will intersect the hypotenuse AC at two points: O and another point 2r away? Wait, no. If you have a circle centered at a point on a line, the line will pass through the center, so the circle intersects the line at two points: the center and another point diametrically opposed. Wait, no—actually, if the circle is centered on the line, then the line (hypotenuse AC) passes through the center, so the intersection points are all points on the line AC that are at distance r from O. Since AC is a straight line, there are two points on AC at distance r from O: one in the direction towards A and one towards C. Therefore, the circle intersects AC at two points: O plus r along AC and O minus r along AC.Therefore, in our problem, the circle centered at O (12,12) on AC with radius 12 would intersect AC at O and another point 12 units away from O along AC. The total length of AC is 35, so moving 12 units from O towards A would be 15 - 12 = 3 units from A, and moving 12 units towards C would be 20 - 12 = 8 units from C. Therefore, those intersection points are still within the triangle, as AC is the hypotenuse. However, the part of the circle that is inside the triangle is the arc from (0,12) to (12,0), passing through O (12,12). Wait, but (12,12) is the center, so the circle passes through (0,12) and (12,0), which are on the legs, and also intersects AC at two other points. But visually, this seems possible. The circle is "resting" on the two legs and has its center on the hypotenuse. Even though part of the circle is outside the triangle, the tangency points are on the legs within the triangle. So maybe the answer is indeed 12.But earlier, when I considered the inradius being 7, which is smaller, but that touches all three sides. So this circle is larger, only touching two sides, which is possible. Therefore, after re-evaluating, I think the answer is 12.But wait, let me verify with another approach. Suppose we use coordinate geometry, but check if the circle actually lies within the triangle. The circle has center (12,12), radius 12. The topmost point is (12,24), which is outside the triangle's vertical limit of 21. The rightmost point is (24,12), which is within the triangle's horizontal limit of 28. Therefore, part of the circle is outside the triangle. However, the problem doesn't specify that the circle must lie entirely within the triangle, just that it touches both legs and the center is on the hypotenuse. In competition problems, sometimes circles can be partly outside as long as the required tangency conditions are met. But in many cases, they assume the circle is inside. Maybe there's a misunderstanding here.Alternatively, maybe I need to use the formula for the radius of a circle tangent to two sides of a right triangle with center on the hypotenuse. Let me derive it.Suppose we have a right triangle with legs a and b, hypotenuse c. A circle is tangent to both legs, with its center on the hypotenuse. Find the radius r.Using coordinate geometry as before, place the right angle at (0,0), legs along axes, hypotenuse from (0,a) to (b,0). The hypotenuse has equation y = (-a/b)x + a.The center of the circle is (r, r) because it's r units from both axes. This point must lie on the hypotenuse:r = (-a/b)r + aMultiply both sides by b:br = -ar + abbr + ar = abr(b + a) = abr = (ab)/(a + b)Wait, this is different from the previous answer. Wait, in this general case, the radius would be (ab)/(a + b). For our problem, a = 21, b = 28:r = (21*28)/(21 + 28) = (588)/49 = 12. So this formula gives r = 12, same as before. So according to this general formula, the radius is 12. But this contradicts my intuition about the circle extending beyond the triangle. However, according to the formula, regardless of the triangle's size, the radius is (ab)/(a + b). But let's test this formula with a different triangle where a + b is larger. For example, if a = 3, b = 4, then r = (12)/(7) ≈ 1.714. The inradius would be (3 + 4 - 5)/2 = 1. So this circle is larger than the inradius, which only touches all three sides. So in this case, the circle is tangent to the two legs, center on hypotenuse, radius 12/7, and part of the circle would be outside the triangle. Wait, but in the 3-4-5 triangle, hypotenuse is 5, and the center is at (12/7, 12/7). The distance from the center to the hypotenuse is zero, so the circle intersects the hypotenuse at the center. The radius is ~1.714, so the topmost point is y = 12/7 + 12/7 = 24/7 ≈ 3.428, but the triangle's height is 3, so the circle extends beyond. But again, the problem didn't specify the circle must be entirely inside. Therefore, perhaps the answer is indeed 12.But why does the formula give r = (ab)/(a + b)? Let me see. If you have a circle tangent to both legs and centered on the hypotenuse, then the radius is the product of the legs over the sum of the legs. Which is similar to the formula for resistors in parallel or the harmonic mean. Interesting.But in our problem, plugging a =21, b=28, gives 588/49=12. So the answer is 12. Therefore, despite the circle extending beyond the triangle in the coordinate system, the radius is indeed 12. Maybe the problem doesn't require the circle to be entirely within the triangle, just to touch both legs with the center on the hypotenuse. Therefore, the answer is 12.But wait, let's check with the specific coordinates. If the circle is centered at (12,12) with radius 12, then the point (0,12) is on AB, and (12,0) is on BC. The circle passes through these points and is centered on AC. Even though part of the circle is outside the triangle, the required condition is satisfied: the circle touches both legs AB and BC, and the center is on AC. Therefore, the answer is 12.I think my initial answer is correct. The confusion came from visualizing the circle going outside the triangle, but since the problem only requires tangency to the legs and the center on the hypotenuse, it's acceptable. Therefore, the radius is 12.Final Answerboxed{12}</think>In a right triangle (ABC) with legs (AB = 21) and (BC = 28), and the right angle at (B), we need to find the radius of a circle centered on the hypotenuse (AC) that touches both legs (AB) and (BC).1. Coordinate System Setup: - Place point (B) at the origin ((0,0)). - Point (A) is at ((0, 21)) and point (C) is at ((28, 0)). - The hypotenuse (AC) has a slope of (-frac{3}{4}) and its equation is (y = -frac{3}{4}x + 21).2. Circle Center Coordinates: - The center (O) of the circle lies on the hypotenuse (AC), so its coordinates ((x, y)) must satisfy (y = -frac{3}{4}x + 21). - The circle is tangent to both legs (AB) and (BC), so the distance from (O) to each leg (the radius (r)) must be equal to the (x)-coordinate and (y)-coordinate of (O) respectively.3. Equations for Radius: - Let (x = r) and (y = r). Substituting into the hypotenuse equation: [ r = -frac{3}{4}r + 21 ] - Solving for (r): [ r + frac{3}{4}r = 21 implies frac{7}{4}r = 21 implies r = 21 times frac{4}{7} = 12 ]4. Verification: - The center (O) is at ((12, 12)), which lies on the hypotenuse (AC). - The circle with radius 12 touches both legs (AB) and (BC) at points ((0, 12)) and ((12, 0)) respectively.Thus, the radius of the circle is (boxed{12}).,<think>Okay, let's try to figure out this geometry problem. So, we have a right triangle ABC where AB is 21 and BC is 28. Wait, hold on, in a right triangle, the legs are the two sides that form the right angle. But the problem says the legs are AB and BC. Hmm, so if ABC is a right triangle with legs AB and BC, then the right angle must be at point B. That makes sense because AB and BC are connected at B, so if they're the legs, then angle B is the right angle. Therefore, the hypotenuse would be AC. Got that.Now, there's a circle with center O on the hypotenuse AC that touches both legs AB and BC. So, the circle is tangent to both legs AB and BC, and its center is somewhere on AC. We need to find the radius of this circle.Let me visualize this. Imagine triangle ABC with right angle at B. AB is 21 units long, BC is 28 units. The hypotenuse AC is opposite the right angle. The circle is sitting on the hypotenuse, touching both legs. So, it's like the circle is inscribed but not the usual incircle because the center is on the hypotenuse, not necessarily where the angle bisectors meet.First, maybe I should draw the triangle. Let me sketch it mentally. Point B is the right angle. AB is vertical with length 21, BC is horizontal with length 28. Hypotenuse AC connects A to C. The center O is somewhere along AC. The circle touches both AB and BC. Since it's tangent to both legs, the distance from O to AB and to BC must both be equal to the radius r. Because the circle is tangent to those sides, the radius is perpendicular to the sides at the point of tangency.But the center O is on AC. So, the center has to be a point on AC such that its perpendicular distance to AB and BC is the same, which is the radius.Hmm, maybe coordinate geometry can help here. Let's assign coordinates to the triangle. Let me place point B at the origin (0,0). Then, since AB is 21, and since AB is a leg, if B is at (0,0), then A can be at (0,21) because AB is vertical. Similarly, BC is 28, so point C would be at (28,0). Then, the hypotenuse AC connects (0,21) to (28,0).Now, the center O is somewhere on AC. Let's parametrize point O. The line AC can be parameterized. Let's find the equation of AC first. The coordinates of A are (0,21), and coordinates of C are (28,0). So, the slope of AC is (0 - 21)/(28 - 0) = -21/28 = -3/4. So, the equation of AC is y - 21 = (-3/4)(x - 0), which simplifies to y = (-3/4)x + 21.So, any point O on AC can be represented as (x, (-3/4)x + 21). Now, the distance from O to AB and to BC must both equal the radius r. Let's recall that AB is the vertical line from (0,0) to (0,21). The distance from a point (x,y) to AB (which is the y-axis) is just the absolute value of the x-coordinate, right? Because AB is along the y-axis. Similarly, BC is the horizontal line from (0,0) to (28,0), so it's along the x-axis. The distance from a point (x,y) to BC (the x-axis) is the absolute value of the y-coordinate.But since the center O is in the triangle, and the circle is tangent to both legs, which are the axes here, the coordinates of O should be positive. So, the distance from O to AB is x, and the distance to BC is y. Since both distances are equal to the radius, we have x = y = r. But wait, O is on AC, so y = (-3/4)x + 21. But since x = r and y = r, substituting into the equation gives r = (-3/4)r + 21. Solving for r:r + (3/4)r = 21 (7/4)r = 21 r = 21 * (4/7) r = 12Wait, that was straightforward. So, is the radius 12? Hmm, but let me check if that makes sense. Let me verify.If the center is at (r, r) which is (12,12). But wait, is (12,12) on AC? Let's check if when x=12, y= (-3/4)(12) + 21 = -9 + 21 = 12. Yes! So, the point (12,12) is on AC, and its distance to both axes is 12, so the radius is 12. That seems correct.But wait, is there a mistake here? Because the usual inradius of a right triangle is given by r = (a + b - c)/2 where a and b are legs, c hypotenuse. Let me compute that. First, compute hypotenuse AC. AC = sqrt(21^2 + 28^2) = sqrt(441 + 784) = sqrt(1225) = 35. So, the inradius would be (21 + 28 - 35)/2 = (14)/2 = 7. But the inradius is 7, but in this problem, the circle is not the incircle, because the center is on the hypotenuse. The incircle's center is located at the intersection of the angle bisectors, which in a right triangle is at distances r from each leg, so coordinates (r, r) which would be (7,7). But (7,7) is not on AC. Let's check if (7,7) is on AC. Plugging into y = (-3/4)x + 21: y = (-3/4)*7 + 21 = -21/4 + 21 = ( -21 + 84 ) /4 = 63/4 = 15.75. So, no, (7,7) is not on AC. Therefore, the incircle is at (7,7), but the circle in this problem is different.Wait, but in our previous calculation, we found that the center is at (12,12), which is on AC, and the radius is 12. Let me check with coordinates. If the circle is centered at (12,12) with radius 12, does it touch both legs AB and BC? The distance from (12,12) to AB (y-axis) is 12, which is the radius. Similarly, the distance to BC (x-axis) is 12, which is the radius. So, yes, the circle touches both legs. So, this seems to be correct.But wait, how come the radius is larger than the inradius? Because the inradius is 7, but this circle is bigger? But the center is on the hypotenuse, so it's a different circle. Maybe it's similar to an ex-circle or something else. Let me see.Alternatively, maybe there's another approach. Let's consider the coordinates again. If the center is on AC, which has equation y = (-3/4)x + 21, and the center is at (x, (-3/4)x +21). The distance from this center to both legs (x-axis and y-axis) must be equal to the radius. The distance to the y-axis (AB) is x, and the distance to the x-axis (BC) is y = (-3/4)x +21. Therefore, since both distances equal the radius r, we have:x = (-3/4)x +21Solving for x:x + (3/4)x =21 (7/4)x =21 x=21*(4/7)=12Therefore, x=12, which gives y= (-3/4)*12 +21= -9 +21=12. So, radius is 12. So, yes, that's correct.But why is this radius larger than the inradius? Because this circle is not the incircle. The incircle touches all three sides, but this circle only touches the two legs and has its center on the hypotenuse. So, it's a different circle.Alternatively, maybe using similar triangles. Let's see. The hypotenuse AC is 35, as we calculated. The center O is on AC. Let's denote the distance from A to O as k, so then the distance from O to C is 35 -k. If we can relate these distances to the radius.But maybe that's more complicated. The coordinate approach seems straightforward. Alternatively, use the formula for the distance from a point to a line. Wait, but in this case, since the legs are the axes, the distance is just the coordinate.Alternatively, parametrize the hypotenuse. Let's say from point A(0,21) to C(28,0). The parametric equations for AC can be written as x = 28t, y = 21 -21t, where t ranges from 0 to 1. So, at t=0, we are at A(0,21), and at t=1, we are at C(28,0). Then, a general point O on AC is (28t, 21 -21t). The distance from O to AB (y-axis) is x-coordinate, which is 28t. The distance from O to BC (x-axis) is y-coordinate, which is 21 -21t. Since both distances equal the radius r:28t = 21 -21t 28t +21t =21 49t =21 t=21/49=3/7Therefore, t=3/7, so coordinates of O are (28*(3/7), 21 -21*(3/7)) = (12, 21 -9)= (12,12). Then, the radius is 28t=28*(3/7)=12. So, same result.Therefore, the radius is 12. So, the answer should be boxed{12}.Wait, but let me make sure there isn't another circle that touches both legs and is centered on the hypotenuse. Suppose there is another position, but in the coordinate approach, we found only one solution where x = y = r. Since the center is on AC, which is a straight line, and the radius is determined by the distance to the legs, there should only be one such circle. Therefore, 12 is the answer.Alternatively, maybe using area. The area of the triangle is (21*28)/2 = 294. The inradius is area/s, where s is the semiperimeter. The semiperimeter is (21+28+35)/2=84/2=42. Therefore, inradius is 294/42=7, which matches our previous calculation. But this is different from the circle in the problem. However, maybe we can use similar concepts.Alternatively, think of the circle tangent to AB and BC with center on AC. Then, since the center is on AC, which is the hypotenuse, perhaps using coordinate geometry is the easiest way. Since we did that and got 12, which seems consistent.Alternatively, use trigonometry. The hypotenuse AC is 35. The angle at A is arctan(28/21)= arctan(4/3), and angle at C is arctan(21/28)= arctan(3/4). The center O is on AC. The distance from O to AB and BC is the radius r. Let's consider the angles. If we denote the center O, then the lines from O to AB and BC are perpendicular to AB and BC, respectively. Since AB is vertical and BC is horizontal, those perpendiculars are horizontal and vertical lines, hence the coordinates approach.Alternatively, in terms of the angles, maybe using sine and cosine. Let me see. The distance from O to AB is r, which is the horizontal distance from O to AB (since AB is vertical). Similarly, the distance from O to BC is r, which is the vertical distance from O to BC (since BC is horizontal). Therefore, if we consider point O on AC, moving along AC, the horizontal distance to AB decreases as we move from A to C, while the vertical distance to BC increases. The point where these distances are equal is the center O.But since we already solved it with coordinates and parametrization, and both gave the same answer, I think 12 is correct. Therefore, the radius is 12.Final Answerboxed{12}</think>In a right triangle (ABC) with legs (AB = 21) and (BC = 28), we need to find the radius of a circle centered at point (O) on the hypotenuse (AC) that touches both legs (AB) and (BC).1. Identify the coordinates of the triangle: - Place (B) at the origin ((0,0)). - (A) is at ((0, 21)) and (C) is at ((28, 0)). - The hypotenuse (AC) has a slope of (-frac{3}{4}) and its equation is (y = -frac{3}{4}x + 21).2. Parameterize the hypotenuse (AC): - Any point (O) on (AC) can be represented as ((x, -frac{3}{4}x + 21)).3. Determine the distances from (O) to the legs: - The distance from (O) to (AB) (y-axis) is (x). - The distance from (O) to (BC) (x-axis) is (-frac{3}{4}x + 21).4. Set these distances equal to the radius (r): - Since both distances are equal to the radius, we have (x = -frac{3}{4}x + 21). - Solving for (x): [ x + frac{3}{4}x = 21 implies frac{7}{4}x = 21 implies x = 21 times frac{4}{7} = 12 ] - Substituting (x = 12) into the equation of (AC) gives (y = -frac{3}{4} times 12 + 21 = 12).5. Conclusion: - The radius (r) is 12, as the coordinates of (O) are ((12, 12)), which is on (AC) and the distances to both legs are 12.Thus, the radius of the circle is (boxed{12}).]

🔑:Let P be the point of tangency of the circle with the leg B C. Consider similar triangles C P O and C B A.## SolutionLet R be the radius of the circle.First method. Let P be the point of tangency of the circle with the leg B C. From the similarity of triangles C P O and C B A, it follows that { }^{28-R} /{ }^{R}=28 / 21.From this, R=12.Second method. S_{A O B}=1 / 2 A B cdot R=21 R / 2, S_{C O B}=1 / 2 B C cdot R=14 R.Since S_{A O B}+S_{C O B}=S_{A B C}=1 / 2 A B cdot B C=1 / 2 cdot 21 cdot 28, we have the equation 21 R+28 R=21 cdot 28, from which R=12.## Answer12.

🧠:[<think>Alright, so I need to figure out how many subarrays of the sequence 1,4,8,10,16,19,21,25,30,43 have sums divisible by 11. Let me start by recalling what a subarray is. A subarray is a contiguous part of the array, right? So, for example, [1], [1,4], [4], [4,8], etc. Each of these could be considered subarrays. The problem is asking for the count of all such subarrays where the sum of their elements is divisible by 11.First, let's note that the array has 10 elements. The number of possible subarrays in an array of length n is n(n+1)/2. So here, that would be 10*11/2 = 55. So there are 55 possible subarrays. But we need to find how many of these have sums divisible by 11. That could be anywhere from 0 to 55, but I need a method to compute it.One common technique for problems involving sums divisible by a number (here, 11) is to use prefix sums and modular arithmetic. Let me remember how that works. If I compute the prefix sum modulo 11 for each position in the array, then the difference between two prefix sums modulo 11 gives the sum of the subarray between those two positions. If two prefix sums have the same remainder modulo 11, then the subarray between those positions will have a sum divisible by 11.So the approach would be:1. Compute the prefix sums modulo 11 for the array.2. Keep a count of how many times each remainder occurs.3. For each remainder, if it occurs k times, then the number of subarrays that can be formed with that remainder is k choose 2, i.e., k*(k-1)/2.4. Sum these values for all remainders to get the total number of subarrays whose sum is divisible by 11.Additionally, we need to consider the remainder 0 separately. If a prefix sum itself is 0 modulo 11, then the subarray from the start of the array up to that position is also a valid subarray. So for remainder 0, the count is (k choose 2) + k, where k is the number of times 0 occurs.Alternatively, some people adjust the initial count of remainder 0 to 1 before processing the array, which automatically accounts for subarrays starting at index 0. Let me verify which approach is correct.Suppose we have an array where the first element is 11. Then the prefix sum after the first element is 11, which modulo 11 is 0. So the subarray [11] has sum divisible by 11. If we start with a prefix sum count initialized with 0:1, then when we reach the first prefix sum 0 (i.e., before processing any elements), the count would already be 1. Then when we process the first element, the prefix sum is 11 mod 11 = 0. Then the number of subarrays ending at index 0 with sum divisible by 11 is count[0] = 1 (from before). So the total would increment by 1. Then we increment count[0] to 2. So that seems to handle it correctly.So perhaps the correct method is to initialize a dictionary or array to keep track of the counts of each remainder modulo 11, starting with remainder 0 having a count of 1 (for the prefix sum of 0 before the array starts). Then, as we compute each prefix sum modulo 11, we check how many times that remainder has been seen before, add that number to the total, and increment the count for that remainder.Let me formalize that:Initialize a count array (or dictionary) with count[0] = 1.Initialize total = 0.Initialize prefix_sum = 0.For each number in the array: prefix_sum += number remainder = prefix_sum % 11 total += count[remainder] count[remainder] += 1So, in this way, when a remainder is encountered, the number of previous times that remainder has been seen is added to the total. Because each previous occurrence of the same remainder corresponds to a subarray starting right after the previous occurrence and ending at the current index, which has a sum divisible by 11.This approach should count all possible subarrays whose sum is divisible by 11. Let me test this with a small example.Take the array [1, 4]. The possible subarrays are [1], [1,4], [4]. The sums are 1, 5, 4. None are divisible by 11, so the answer should be 0.Using the method:Initialize count[0] = 1, total = 0, prefix_sum =0.First element: 1.prefix_sum =0 +1=1.remainder=1%11=1.count[1] is 0 (since count was initialized with only count[0]=1), so total +=0 (total remains 0).Then count[1] becomes 1.Second element:4.prefix_sum=1 +4=5.remainder=5%11=5.count[5] is 0, so total remains 0.count[5] becomes1.Total is 0. Which is correct.Another example: array [11, 11]. The subarrays are [11], [11,11], [11]. The sums are 11,22,11. All are divisible by 11, so the answer should be 3.Using the method:Initialize count[0]=1, total=0, prefix_sum=0.First element:11.prefix_sum=0+11=11.remainder=11%11=0.count[0] is 1, so total +=1 (total=1).count[0] increments to 2.Second element:11.prefix_sum=11+11=22.remainder=22%11=0.count[0] is 2, so total +=2 (total=3).count[0] increments to 3.Total is 3. Correct.Another example: array [5,6]. The possible subarrays: [5], sum=5; [5,6], sum=11; [6], sum=6. So one subarray divisible by 11. The method should return 1.Process:count[0]=1, total=0, prefix=0.First element:5. prefix=5. remainder=5. count[5]=0. total remains 0. count[5] becomes1.Second element:6. prefix=5+6=11. remainder=0. count[0]=1. total +=1 (total=1). count[0] becomes2. So correct.Therefore, this method works. So I can apply this to the given array.Given array: 1,4,8,10,16,19,21,25,30,43.Let me list the elements with their indices (0-based):0:11:42:83:104:165:196:217:258:309:43Now, compute the prefix sums modulo 11, keeping track of counts. Let's proceed step by step.Initialize:count = {0:1}total =0prefix_sum=0Now process each element:Element 0:1prefix_sum =0 +1=1remainder=1%11=1count[1] is 0 (since count only has 0:1 initially). So total +=0. Then count[1] becomes1.count is now {0:1, 1:1}total remains 0.Element 1:4prefix_sum =1 +4=5remainder=5%11=5count[5] is 0. total +=0. count[5] becomes1.count: {0:1,1:1,5:1}total:0Element2:8prefix_sum=5+8=13remainder=13%11=2count[2] is 0. total remains 0. count[2] becomes1.count: {0:1,1:1,2:1,5:1}total:0Element3:10prefix_sum=13+10=23remainder=23%11=1 (since 11*2=22, 23-22=1)count[1] is1. So total +=1. Now total=1.count[1] becomes2.count: {0:1,1:2,2:1,5:1}total:1Element4:16prefix_sum=23+16=39remainder=39%11=6 (11*3=33, 39-33=6)count[6] is0. total remains1. count[6]=1.count: {0:1,1:2,2:1,5:1,6:1}total:1Element5:19prefix_sum=39+19=58remainder=58%11=58 - 11*5=58-55=3count[3] is0. total remains1. count[3]=1.count: {0:1,1:2,2:1,3:1,5:1,6:1}total:1Element6:21prefix_sum=58+21=79remainder=79%11. Let's calculate 11*7=77. 79-77=2. So remainder=2.count[2] is1. total +=1. Now total=2.count[2] becomes2.count: {0:1,1:2,2:2,3:1,5:1,6:1}total:2Element7:25prefix_sum=79+25=104remainder=104%11. 11*9=99. 104-99=5. So remainder=5.count[5] is1. total +=1. Now total=3.count[5] becomes2.count: {0:1,1:2,2:2,3:1,5:2,6:1}total:3Element8:30prefix_sum=104+30=134remainder=134%11. 11*12=132. 134-132=2. So remainder=2.count[2] is2. total +=2. Now total=5.count[2] becomes3.count: {0:1,1:2,2:3,3:1,5:2,6:1}total:5Element9:43prefix_sum=134+43=177remainder=177%11. Let's compute 11*16=176. 177-176=1. So remainder=1.count[1] is2. total +=2. Now total=7.count[1] becomes4.Final count: {0:1,1:4,2:3,3:1,5:2,6:1}Total:7.Wait, according to this calculation, the total number of subarrays with sum divisible by 11 is 7. But let me verify this because sometimes I might have made a mistake in the steps.Wait, but let's check each step carefully.Starting from the beginning:Initialize count={0:1}, total=0, prefix_sum=0.Element0:1prefix_sum=0+1=1. remainder=1. count[1] is not present, so total +=0. count[1] becomes1.count={0:1,1:1}; total=0.Element1:4prefix_sum=1+4=5. remainder=5. count[5] is0. total stays 0. count[5]=1.count={0:1,1:1,5:1}; total=0.Element2:8prefix_sum=5+8=13. remainder=2 (13-11=2). count[2] is0. total stays0. count[2]=1.count={0:1,1:1,2:1,5:1}; total=0.Element3:10prefix_sum=13+10=23. 23%11=1 (23-22=1). count[1]=1. total +=1. total=1. count[1] becomes2.count={0:1,1:2,2:1,5:1}; total=1.Element4:16prefix_sum=23+16=39. 39%11=6 (39-33=6). count[6] is0. total stays1. count[6]=1.count={0:1,1:2,2:1,5:1,6:1}; total=1.Element5:19prefix_sum=39+19=58. 58%11=3 (58-55=3). count[3] is0. total stays1. count[3]=1.count={0:1,1:2,2:1,3:1,5:1,6:1}; total=1.Element6:21prefix_sum=58+21=79. 79%11=2 (79-77=2). count[2]=1. total +=1. total=2. count[2]=2.count={0:1,1:2,2:2,3:1,5:1,6:1}; total=2.Element7:25prefix_sum=79+25=104. 104%11=5 (104-99=5). count[5]=1. total +=1. total=3. count[5]=2.count={0:1,1:2,2:2,3:1,5:2,6:1}; total=3.Element8:30prefix_sum=104+30=134. 134%11=2 (134-132=2). count[2]=2. total +=2. total=5. count[2]=3.count={0:1,1:2,2:3,3:1,5:2,6:1}; total=5.Element9:43prefix_sum=134+43=177. 177%11=1 (177-176=1). count[1]=2. total +=2. total=7. count[1]=4.So final total is7. Hmm.But let's see. Wait, maybe the answer is 7? Let me check by manually finding the subarrays.Alternatively, perhaps I made a mistake in calculation. Let's verify each prefix sum and their remainders.Let me compute all prefix sums and their remainders:Element0:1. prefix_sum=1. remainder=1.Element1:4. prefix_sum=5. remainder=5.Element2:8. prefix_sum=13. remainder=2.Element3:10. prefix_sum=23. remainder=1.Element4:16. prefix_sum=39. remainder=6.Element5:19. prefix_sum=58. remainder=3.Element6:21. prefix_sum=79. remainder=2.Element7:25. prefix_sum=104. remainder=5.Element8:30. prefix_sum=134. remainder=2.Element9:43. prefix_sum=177. remainder=1.So the remainders are:1,5,2,1,6,3,2,5,2,1.Now, we need to track how many times each remainder occurs. Remember that we started with count[0]=1.So the count starts at 1 for remainder 0. Then, for each remainder in the list, we add the current count of that remainder to the total and then increment the count.Let's go step by step:Start: count={0:1}, total=0.After element0 (remainder1):count[1] = count.get(1,0) =0. total +=0. Then count[1] becomes1. total=0.count={0:1,1:1}After element1 (remainder5):count[5]=0. total +=0. count[5]=1. total=0.count={0:1,1:1,5:1}After element2 (remainder2):count[2]=0. total +=0. count[2]=1. total=0.count={0:1,1:1,2:1,5:1}After element3 (remainder1):count[1]=1. total +=1. total=1. count[1]=2.count={0:1,1:2,2:1,5:1}After element4 (remainder6):count[6]=0. total +=0. count[6]=1. total=1.count={0:1,1:2,2:1,5:1,6:1}After element5 (remainder3):count[3]=0. total +=0. count[3]=1. total=1.count={0:1,1:2,2:1,3:1,5:1,6:1}After element6 (remainder2):count[2]=1. total +=1. total=2. count[2]=2.count={0:1,1:2,2:2,3:1,5:1,6:1}After element7 (remainder5):count[5]=1. total +=1. total=3. count[5]=2.count={0:1,1:2,2:2,3:1,5:2,6:1}After element8 (remainder2):count[2]=2. total +=2. total=5. count[2]=3.count={0:1,1:2,2:3,3:1,5:2,6:1}After element9 (remainder1):count[1]=2. total +=2. total=7. count[1]=4.count={0:1,1:4,2:3,3:1,5:2,6:1}So total is7. Therefore, the answer should be7. Hmm. But let me check manually. Let's try to list all such subarrays.But wait, given that the array is of length10, and the number of subarrays is55, checking manually is tedious, but maybe possible.Alternatively, perhaps there is a miscalculation in the steps. Let's see.First, let's list all the remainders:Indices 0-9:0:1 (prefix sum1, remainder1)1:5 (prefix sum5, remainder5)2:2 (prefix sum13, remainder2)3:1 (prefix sum23, remainder1)4:6 (prefix sum39, remainder6)5:3 (prefix sum58, remainder3)6:2 (prefix sum79, remainder2)7:5 (prefix sum104, remainder5)8:2 (prefix sum134, remainder2)9:1 (prefix sum177, remainder1)Now, the count starts with count[0]=1. Then, each time a remainder is encountered, the number of previous occurrences is added to the total.So:At index0 (remainder1): count[1]=0. total=0. count[1]=1.At index1 (remainder5): count[5]=0. total=0. count[5]=1.At index2 (remainder2): count[2]=0. total=0. count[2]=1.At index3 (remainder1): count[1]=1. total +=1. total=1. count[1]=2.At index4 (remainder6): count[6]=0. total=1. count[6]=1.At index5 (remainder3): count[3]=0. total=1. count[3]=1.At index6 (remainder2): count[2]=1. total +=1. total=2. count[2]=2.At index7 (remainder5): count[5]=1. total +=1. total=3. count[5]=2.At index8 (remainder2): count[2]=2. total +=2. total=5. count[2]=3.At index9 (remainder1): count[1]=2. total +=2. total=7. count[1]=4.So total is7. Let me see which subarrays these correspond to.Each time we add to the total, it's the number of previous times the same remainder was encountered. Each such occurrence corresponds to a subarray starting right after the previous occurrence and ending at the current index.For example:At index3 (remainder1), previous count[1]=1 (at index0). So the subarray from index1 to3 (elements4,8,10) sum to (4+8+10)=22, which is divisible by11. Wait, 4+8=12, 12+10=22. 22 is 0 mod11. Wait, but 22 mod11=0. Wait, but according to the prefix sums, prefix_sum at index3 is23, which is1 mod11. Wait, no. Wait, the prefix sum at index3 is23, which is1 mod11. Then if there was a previous prefix sum with remainder1 at index0 (prefix sum1), then the subarray from index0+1=1 to3 is elements1:4, 2:8, 3:10. The sum is4+8+10=22, which is0 mod11. Wait, but prefix_sum[3] - prefix_sum[0] =23 -1=22, which is0 mod11. Yes. So that's correct. So that subarray[1..3] is one such subarray.Similarly, when at index6, remainder2. The previous count[2] was1 (at index2). So subarray from index2+1=3 to6: elements3:10,4:16,5:19,6:21. Sum=10+16+19+21=66, which is0 mod11. 66/11=6. So yes. So that's another subarray.Similarly, at index7, remainder5. Previous count[5] was1 (at index1). So subarray from1+1=2 to7: elements2:8,3:10,4:16,5:19,6:21,7:25. Sum=8+10+16+19+21+25=99. 99/11=9. Yes.At index8, remainder2. Previous count[2] was2 (at index2 and6). Wait, no. Wait, count[2] was2 when we process index8. So previous occurrences at index2,6, and8? Wait, no. Wait, when we process index8 (remainder2), count[2] was2 before incrementing. So the previous count was2, meaning two previous instances. The instances are index2 and index6. Therefore, two subarrays:From index2+1=3 to8: elements3:10,4:16,5:19,6:21,7:25,8:30. Sum=10+16+19+21+25+30=121. 121/11=11. Correct.From index6+1=7 to8: elements7:25,8:30. Sum=25+30=55. 55/11=5. Correct.So two subarrays added here, leading to total increasing by2.Similarly, at index9, remainder1. Previous count[1] was2, which were at index0 and3. So two subarrays:From0+1=1 to9: elements1:4,2:8,3:10,4:16,5:19,6:21,7:25,8:30,9:43. Sum=4+8+10+16+19+21+25+30+43=176. 176/11=16. Correct.From3+1=4 to9: elements4:16,5:19,6:21,7:25,8:30,9:43. Sum=16+19+21+25+30+43=154. 154/11=14. Correct.So two more subarrays, total increases by2 to7.Therefore, the 7 subarrays are:1. [4,8,10] (indices1-3), sum22.2. [10,16,19,21] (indices3-6), sum66.Wait, wait, no. Wait, when remainder2 occurs at index6, the previous occurrence was at index2, so subarray3-6 (indices3-6, elements10,16,19,21). Sum10+16+19+21=66. Correct.Then at index7, remainder5. Previous occurrence at index1. Subarray2-7 (indices2-7, elements8,10,16,19,21,25). Sum8+10+16+19+21+25=99. Correct.At index8, remainder2. Previous occurrences at index2 and6. So subarrays3-8 (indices3-8) and7-8.Subarray3-8:10,16,19,21,25,30. Sum10+16+19+21+25+30=121.Subarray7-8:25,30. Sum55.At index9, remainder1. Previous occurrences at0 and3. Subarrays1-9 and4-9.Subarray1-9:4,8,10,16,19,21,25,30,43. Sum4+8+10+16+19+21+25+30+43=176.Subarray4-9:16,19,21,25,30,43. Sum16+19+21+25+30+43=154.So total of7 subarrays:1. [4,8,10]2. [10,16,19,21]3. [8,10,16,19,21,25]4. [25,30]5. [4,8,10,16,19,21,25,30,43]6. [16,19,21,25,30,43]7. [10,16,19,21,25,30]Wait, hold on, when I listed the subarrays, I may have made a mistake in mapping the indices. Let me verify each one:1. Subarray from index1 to3: elements4,8,10. Sum22. Correct.2. Subarray from index3 to6: elements10,16,19,21. Sum66. Correct.3. Subarray from index2 to7: elements8,10,16,19,21,25. Sum99. Correct.4. Subarray from index7 to8: elements25,30. Sum55. Correct.5. Subarray from index1 to9: elements4,8,10,16,19,21,25,30,43. Sum176. Correct.6. Subarray from index4 to9: elements16,19,21,25,30,43. Sum154. Correct.Wait, but that's six subarrays. Where is the seventh?Ah, perhaps I missed one. Let's check:When at index3, remainder1, which was counted once. Then at index9, remainder1, counted twice. So index3 gives one subarray (from index0+1=1 to3). Index9 gives two subarrays: from0+1=1 to9 and from3+1=4 to9. So total three from remainder1.Index6 gives one subarray (index2+1=3 to6).Index7 gives one subarray (index1+1=2 to7).Index8 gives two subarrays (index2+1=3 to8 and index6+1=7 to8).So that's1 (index3) +1 (index6) +1 (index7) +2 (index8) +2 (index9)=7.Yes, that's seven.But when I listed them, I only got six. Let me check again.Wait, at index3: subarray1-3.At index6: subarray3-6.At index7: subarray2-7.At index8: subarrays3-8 and7-8.At index9: subarrays1-9 and4-9.Total:1+1+1+2+2=7. Yes.So the seventh subarray is subarray3-8 (sum121) and7-8 (sum55). Then subarrays1-9 (176) and4-9 (154). So yes, seven subarrays.Therefore, the answer is7.But wait, the problem says "the number of subarrays whose sum is divisible by11 is". According to our calculation, it's7.But I need to make sure that none of these subarrays are overlapping or counted incorrectly. Alternatively, perhaps there's a miscalculation.Wait, maybe there's a mistake in the initial approach. Let me think again.The key idea is that whenever two prefix sums have the same remainder modulo11, the subarray between them is divisible by11. So including the initial count[0]=1, which allows us to capture subarrays starting from the beginning.For example, if a prefix sum at index i is equal to a prefix sum at index j (mod11), then the subarray from j+1 to i has sum divisible by11.So the total number of such subarrays is the sum over all remainders of (count[r] choose 2).But in our case, using the method of dynamically adding the counts as we go, we get7.But let's compute it using the counts at the end.The counts after processing all elements are:count = {0:1,1:4,2:3,3:1,5:2,6:1}So for each remainder r, the number of subarrays is (count[r] choose2). Then for remainder0, it's (count[0] choose2) + count[0] -1? Wait, no. Wait, the standard formula is that the total number of subarrays is the sum over all r of (count[r] choose2). But this formula assumes that we have prefix sums including the initial0.Wait, let me check:The total number of pairs with the same remainder is sum_{r} (count[r] choose2). Then this gives the total number of subarrays between two indices with the same remainder.But in our case, we already included the initial0 in the count, so the sum over (count[r] choose2) for all r gives the total number of subarrays.But in the earlier approach, we dynamically added to the total as we encountered each remainder. So if you have counts of remainders as [n0, n1, n2,...], then the total is sum (ni*(ni-1)/2).But in our case, the counts after processing all elements are:0:11:42:33:15:26:1So sum over (n_i choose2):For 0: C(1,2)=0For1: C(4,2)=6For2: C(3,2)=3For3: C(1,2)=0For5: C(2,2)=1For6: C(1,2)=0Total:6+3+1=10.Wait, that's different from7.But in our dynamic approach, we got7. Why the discrepancy?Because when we incrementally add to the total, we are adding the current count[r] before incrementing it. So for each occurrence of a remainder r, the number of times it has been seen before is added to the total. So for remainder1 which was encountered4 times (including the dynamic counting), the contributions are0,0,1,2. Summing these gives0+0+1+2=3.Similarly for remainder2 encountered3 times: contributions0,0,1,2. Wait, but remainder2 was encountered at index2,6,8. So when first encountered: count was0 (added0). Second time: count was1 (added1). Third time: count was2 (added2). So total0+1+2=3.For remainder5 encountered2 times: first time count0 (added0), second time count1 (added1). Total1.For remainder1 encountered4 times: contributions0 (first),0 (second),1 (third),2 (fourth). Total3.Remainder0 was encountered once (initially1). But since in the dynamic approach, we started with count[0]=1, but remainder0 was never encountered again. So total contribution from remainder0 is0.So total contributions:remainder1:3remainder2:3remainder5:1Others:0Total:3+3+1=7.But according to the formula sum (count[r] choose2), we have:remainder1:4 choose2=6remainder2:3 choose2=3remainder5:2 choose2=1Total:6+3+1=10.So why the difference?Ah, because in the dynamic approach, we are only counting subarrays that end at the current index. So it's equivalent to sum over each index i of the number of previous indices j < i with the same remainder. Which is exactly the same as sum (count[r] choose2) over all r.But in our case, the dynamic approach gave7, while the formula gives10. That indicates a mistake in my previous reasoning.Wait, but when we use the dynamic approach, the count[r] includes the current prefix sum. Wait no. Wait, let's think carefully.When we process each element, we compute the remainder, add to the total the current count[r], and then increment count[r]. So for each occurrence of remainder r, the number of times it has been seen before (not including the current one) is added to the total. Therefore, over all elements, the total is the sum over each remainder r of (number of times r was seen) choose2. Because for each r, each pair of indices (i,j) with i<j and same remainder contributes1.But in our case, the dynamic total was7, but according to the formula it should be sum (count[r]-1 choose2). Wait, maybe not. Wait, the count[r] at the end is equal to the number of times remainder r was encountered, including the initial0. Wait, no, the initial count[0]=1 is part of the count.Wait, let me recast:The initial count[0]=1 corresponds to the prefix sum0 at index-1 (before the array starts). Then, for each prefix sum at index0 to9, we have their remainders. So total of11 prefix sums (including the initial0). Each pair of these11 prefix sums with the same remainder corresponds to a subarray.The number of such pairs is sum_{r} (count[r] choose2). In our case:count after all elements:count[0]=1 (from initial)count[1]=4 (prefix sums at indices-1(0),3,9)Wait, no. Wait, the remainders we computed were:Indices-1 (prefix sum0): remainder0.Indices0: remainder1.Indices1: remainder5.Indices2: remainder2.Indices3: remainder1.Indices4: remainder6.Indices5: remainder3.Indices6: remainder2.Indices7: remainder5.Indices8: remainder2.Indices9: remainder1.So count[0]=1 (only at index-1).count[1]=3 (indices0,3,9).Wait, but in the final count, we had count[1]=4. Wait, confusion arises.Wait, during processing:After element9, the count[1] was4. But the actual prefix sums with remainder1 are:- prefix sum0: index-1, remainder0.- index0: remainder1.- index3: remainder1.- index9: remainder1.So total of three prefix sums with remainder1 (indices0,3,9). But according to the count, it was4. Wait, this discrepancy arises because in the dynamic counting, we included the initial count[0]=1, but the remainders computed for the elements are different.Wait, no. Wait, in the dynamic approach, we have:prefix_sum starts at0 (index-1), then after element0, prefix_sum=1 (index0). Then after element1, prefix_sum=5 (index1), etc.So the count[r] after processing all elements includes:count[0] =1 (from the initial0).count[1] = number of times remainder1 occurred in the prefix sums (indices0,3,9). That's three times. But in the dynamic count, count[1] became4. Wait, why?Wait, let's list all prefix sums and their remainders:prefix_sum[ -1 ]=0 → remainder0.prefix_sum[0] =1 → remainder1.prefix_sum[1] =5 → remainder5.prefix_sum[2] =13 → remainder2.prefix_sum[3] =23 → remainder1.prefix_sum[4] =39 → remainder6.prefix_sum[5] =58 → remainder3.prefix_sum[6] =79 → remainder2.prefix_sum[7] =104 → remainder5.prefix_sum[8] =134 → remainder2.prefix_sum[9] =177 → remainder1.So the remainders for the prefix sums (including index-1):0:0,0:1,1:5,2:2,3:1,4:6,5:3,6:2,7:5,8:2,9:1.So count[0]=1.count[1]=3 (indices0,3,9).count[2]=3 (indices2,6,8).count[3]=1 (index5).count[5]=2 (indices1,7).count[6]=1 (index4).So in total:count[0]=1,count[1]=3,count[2]=3,count[3]=1,count[5]=2,count[6]=1.But according to the dynamic count, after processing all elements, count[1]=4. That's inconsistent. Where is the mistake?Wait, no. In the dynamic count:After processing element9, which is the last element (43), the remainder is1. At that point, count[1] is incremented to4. But according to the actual remainders, prefix sums with remainder1 are at indices0,3,9. That's three times. So why does count[1]=4?Because the initial count[0]=1 is part of the count. Then:When processing element0 (remainder1), count[1] becomes1.Element3 (remainder1), count[1] increments to2.Element9 (remainder1), count[1] increments to3.Wait, but according to the dynamic steps:After element9, we added2 to the total because count[1] was2 before processing element9. Wait, no:Wait, let's re-examine step9:Element9:43. prefix_sum=134+43=177. remainder=1.At this point, before updating count[1], we check count[1], which was2 (from previous counts). So total +=2. Then count[1] increments to3. So after processing element9, count[1]=3.But in the previous summary, I wrote:After element9:count={0:1,1:4,2:3,3:1,5:2,6:1}But that must be incorrect. Let me recheck the steps:Processing element9:prefix_sum =134 +43=177.remainder=177%11=1.count[1] was previously2 (after processing element8: when remainder was2, we added count[2]=2, then count[2] became3. Then count[1] was still2 from element3 and previous steps).Wait, let's retrace:After element8 (index8:30), remainder2. At that point:count[2] was2 (from index2,6). So total +=2 (total becomes5). Then count[2] increments to3.count after element8:count={0:1,1:2,2:3,3:1,5:2,6:1}Now processing element9 (index9:43):prefix_sum=134+43=177.remainder=177%11=1.count[1] is2. So total +=2 (total becomes7). Then count[1] increments to3.Thus, final count[1]=3, not4.Therefore, my previous statement that count[1]=4 was incorrect. It should be3. Therefore, the final counts are:count[0]=1,count[1]=3,count[2]=3,count[3]=1,count[5]=2,count[6]=1.Then, sum (count[r] choose2) for all r:count[0]: C(1,2)=0,count[1]: C(3,2)=3,count[2]: C(3,2)=3,count[3]: C(1,2)=0,count[5]: C(2,2)=1,count[6]: C(1,2)=0.Total=3+3+1=7. Which matches the dynamic total.Therefore, the correct total is7. So the answer should be7.But why was there confusion earlier? Because in my first summary after processing element9, I incorrectly wrote count[1]=4, but it's actually3. So the confusion was due to a miscalculation in the final count.Therefore, the correct answer is7.But wait, let me verify with another approach.Another way to compute the number of subarrays with sum divisible by11 is to compute all possible subarrays and check their sums.But this is tedious, but for small n=10, maybe manageable.The array is: [1,4,8,10,16,19,21,25,30,43]Let me list all possible subarrays and their sums modulo11.But this would take a lot of time. Alternatively, focus on the ones we found earlier and see if there are others.The seven subarrays we identified are:1. [4,8,10] sum22 ≡0 mod11.2. [10,16,19,21] sum66 ≡0 mod11.3. [8,10,16,19,21,25] sum99≡0 mod11.4. [25,30] sum55≡0 mod11.5. [4,8,10,16,19,21,25,30,43] sum176≡0 mod11.6. [16,19,21,25,30,43] sum154≡0 mod11.7. [10,16,19,21,25,30] sum121≡0 mod11.Is there any other subarray that sums to a multiple of11?Let's check some other subarrays.For example, the entire array: sum1+4+8+10+16+19+21+25+30+43.Let's compute this:1+4=5; 5+8=13;13+10=23;23+16=39;39+19=58;58+21=79;79+25=104;104+30=134;134+43=177.177 mod11=1. So not divisible by11.What about the subarray [1,4,8,10,16,19,21,25,30], sum177-43=134. 134 mod11=2. Not divisible.What about subarray [1,4,8,10,16,19,21,25], sum134-30=104. 104 mod11=5. Not divisible.Subarray [1,4,8,10,16,19,21], sum104-25=79. 79 mod11=2. Not divisible.Similarly, [1,4,8,10,16,19]: sum58. 58 mod11=3. Not divisible.[1,4,8,10,16]: sum39. 39 mod11=6.[1,4,8,10]: sum23≡1.[1,4,8]: sum13≡2.[1,4]:5≡5.[1]:1≡1.Similarly, check other subarrays starting from different indices.Starting at index4: elements16,19,21,25,30,43. Sum154≡0. Which is already counted.Starting at index5: elements19,21,25,30,43. Sum19+21=40+25=65+30=95+43=138. 138 mod11=138-121=17. 17-11=6≡6.Not divisible.Index5-6:19,21. Sum40≡7.Index5-7:19,21,25. Sum65≡10.Index5-8:19,21,25,30. Sum95≡7.Index5-9:138≡6.Index6-9:21,25,30,43. Sum21+25=46+30=76+43=119. 119 mod11=119-110=9≡9.Index7-9:25,30,43. Sum98≡98-88=10≡10.Index8-9:30,43. Sum73≡73-66=7≡7.Other subarrays:Starting at index2:8,10,16. Sum8+10+16=34≡1.Index2-4:8,10,16,19. Sum8+10=18+16=34+19=53≡9.Index2-5:8,10,16,19,21. Sum74≡74-66=8.Index2-6:8,10,16,19,21,25. Sum99≡0. Which is already counted.Index3-4:10,16. Sum26≡4.Index3-5:10,16,19. Sum45≡1.Index3-7:10,16,19,21,25. Sum91≡91-88=3.Index4-5:16,19. Sum35≡2.Index4-6:16,19,21. Sum56≡1.Index4-7:16,19,21,25. Sum81≡4.Index5-6:19,21. Sum40≡7.Index5-7:19,21,25. Sum65≡10.Index6-7:21,25. Sum46≡2.Index6-8:21,25,30. Sum76≡9.Index7-8:25,30. Sum55≡0. Which is counted.Index0-3:1,4,8,10. Sum23≡1.Index0-6:1,4,8,10,16,19,21. Sum79≡2.Index0-7:1,4,8,10,16,19,21,25. Sum104≡5.Index0-8:1,4,8,10,16,19,21,25,30. Sum134≡2.Index0-9:177≡1.Other possible subarrays:Index1-2:4,8. Sum12≡1.Index1-4:4,8,10,16. Sum38≡5.Index1-5:4,8,10,16,19. Sum57≡2.Index1-7:4,8,10,16,19,21,25. Sum103≡4.Index1-8:4,8,10,16,19,21,25,30. Sum133≡1.Index2-3:8,10. Sum18≡7.Index2-7:8,10,16,19,21,25. Sum99≡0. Counted.Index2-8:8,10,16,19,21,25,30. Sum128≡7.Index3-5:10,16,19. Sum45≡1.Index3-6:10,16,19,21. Sum66≡0. Counted.Index3-8:10,16,19,21,25,30. Sum121≡0. Counted.Index3-9:10,16,19,21,25,30,43. Sum164≡10.Index4-8:16,19,21,25,30. Sum111≡0. Wait, 16+19=35+21=56+25=81+30=111. 111/11=10.09... No, 11*10=110. 111-110=1≡1. So 111≡1 mod11. Not counted.Index4-9:16,19,21,25,30,43. Sum154≡0. Counted.Index5-8:19,21,25,30. Sum95≡7.Index5-9:19,21,25,30,43. Sum138≡6.Index6-9:21,25,30,43. Sum119≡9.Index7-9:25,30,43. Sum98≡10.Index8-9:30,43. Sum73≡7.So from this manual checking, the seven subarrays we found earlier seem to be the only ones. Therefore, the answer should be7.But wait, when I checked the subarray [16,19,21,25,30,43], which is index4-9, sum154≡0. Which is correct. And that was counted in the dynamic approach.Similarly, other subarrays are accounted for.Thus, the correct answer is7.But wait, according to the initial problem statement, the given sequence is 1,4,8,10,16,19,21,25,30,43. So index0 to9.Therefore, the number of subarrays whose sum is divisible by11 is7.However, to ensure correctness, let me verify one more time with another method.Alternative method:Compute all prefix sums modulo11, then for each remainder, calculate the combinations.The prefix sums mod11 are:Index-1:0Index0:1Index1:5Index2:2Index3:1Index4:6Index5:3Index6:2Index7:5Index8:2Index9:1So list of remainders including index-1:[0,1,5,2,1,6,3,2,5,2,1]Now, count the occurrences of each remainder:0:11:3 (indices-1:0, index0:1, index3:1, index9:1) Wait, index-1 is0, index0:1, index3:1, index9:1. Wait, no. Wait, the remainders are:Index-1:0Index0:1Index1:5Index2:2Index3:1Index4:6Index5:3Index6:2Index7:5Index8:2Index9:1So remainder0:1 occurrence (index-1).remainder1: indices0,3,9:3 occurrences.remainder5: indices1,7:2 occurrences.remainder2: indices2,6,8:3 occurrences.remainder6: index4:1.remainder3: index5:1.So counts:0:1,1:3,2:3,3:1,5:2,6:1.Therefore, the number of subarrays is sum_{r} (count[r] choose2):C(3,2) + C(3,2) + C(2,2) =3+3+1=7.Hence, the same result. Therefore, the answer is7.But the problem might expect a different answer. Wait, but according to all calculations, it's7. Unless there's a miscalculation somewhere.Wait, another way: the number of subarrays is equal to the number of pairs (i,j) where i < j and prefix_sum[j] ≡ prefix_sum[i] mod11.Including the initial prefix_sum at index-1 (0), we have11 prefix sums.The total number of such pairs is:For remainder0: C(1,2)=0.For remainder1: C(3,2)=3.For remainder2: C(3,2)=3.For remainder5: C(2,2)=1.Others:0.Total=3+3+1=7.Therefore, the answer is7.Thus, after careful consideration and multiple methods, the answer is7.Final Answerboxed{7}</think>To determine the number of subarrays of the sequence (1, 4, 8, 10, 16, 19, 21, 25, 30, 43) whose sum is divisible by 11, we use the method of prefix sums and modular arithmetic.1. Compute Prefix Sums Modulo 11: - Initialize prefix sum as 0 (before the array starts). - Compute the prefix sum modulo 11 for each element in the array. The prefix sums modulo 11 are as follows: [ begin{align*} text{Index -1 (initial)} & : 0 text{Index 0} & : 1 text{Index 1} & : 5 text{Index 2} & : 2 text{Index 3} & : 1 text{Index 4} & : 6 text{Index 5} & : 3 text{Index 6} & : 2 text{Index 7} & : 5 text{Index 8} & : 2 text{Index 9} & : 1 end{align*} ]2. Count Occurrences of Each Remainder: - Track the number of times each remainder occurs using a dictionary. The counts are: [ begin{align*} 0 & : 1 1 & : 3 2 & : 3 3 & : 1 5 & : 2 6 & : 1 end{align*} ]3. Calculate the Number of Subarrays: - For each remainder (r), the number of subarrays is given by (binom{text{count}[r]}{2}). - Sum these values for all remainders. [ begin{align*} text{For remainder 1} & : binom{3}{2} = 3 text{For remainder 2} & : binom{3}{2} = 3 text{For remainder 5} & : binom{2}{2} = 1 text{Total} & : 3 + 3 + 1 = 7 end{align*} ]Thus, the number of subarrays whose sum is divisible by 11 is (boxed{7}).,<think>Okay, so I need to find the number of subarrays in the given sequence where the sum is divisible by 11. The sequence is 1, 4, 8, 10, 16, 19, 21, 25, 30, 43. Let me recall what a subarray is. A subarray is a contiguous part of the array, right? So unlike subsets, subarrays have to be consecutive elements. For example, in this sequence, a subarray could be [1,4], [8,10,16], etc., but not [1,8] because they're not next to each other.The problem is asking for the number of such subarrays where their sum is divisible by 11. Hmm, okay. Let's think about how to approach this. The brute force method would be to check every possible subarray, calculate its sum, and see if it's divisible by 11. But that would be O(n^2) time complexity, which is manageable for small arrays. Since this array has 10 elements, the number of subarrays is 10*(10+1)/2 = 55. So 55 subarrays to check. That's feasible, but maybe there's a smarter way.I remember something about using prefix sums and modular arithmetic to find such subarrays efficiently. Let me recall. The idea is that if the prefix sum up to index i modulo 11 is equal to the prefix sum up to index j modulo 11, then the subarray from i+1 to j has a sum divisible by 11. That's because the difference between the two prefix sums would be the sum of elements from i+1 to j, and if their mod 11 values are the same, the difference is a multiple of 11.So, first, I need to compute the prefix sums modulo 11 for the array, then count how many times each remainder occurs. For each remainder, if it occurs k times, then there are k choose 2 subarrays that have that remainder, which translates to sum divisible by 11 when subtracted. Also, we have to remember the remainder 0, where each occurrence itself is a subarray (starting from index 0 to that index) that is divisible by 11.Let me outline the steps:1. Compute the prefix sums modulo 11.2. Keep a frequency count of each remainder.3. For each remainder, add (frequency choose 2) to the total count.4. Also, add the frequency of remainder 0 since those are valid subarrays starting from the beginning.Let me try applying this to the given array. First, let's compute the prefix sums modulo 11.The given array is: [1, 4, 8, 10, 16, 19, 21, 25, 30, 43]Let's list the elements and compute cumulative sums:Index 0: 1Sum: 1Mod 11: 1Index 1: 4Sum: 1 + 4 = 5Mod 11: 5Index 2: 8Sum: 5 + 8 = 13Mod 11: 13 - 11 = 2Index 3: 10Sum: 13 + 10 = 23Mod 11: 23 - 22 = 1Index 4: 16Sum: 23 + 16 = 39Mod 11: 39 - 33 = 6Index 5: 19Sum: 39 + 19 = 58Mod 11: 58 - 55 = 3Index 6: 21Sum: 58 + 21 = 79Mod 11: 79 - 77 = 2Index 7: 25Sum: 79 + 25 = 104Mod 11: 104 - 99 = 5Index 8: 30Sum: 104 + 30 = 134Mod 11: 134 - 121 = 13 → 13 - 11 = 2Index 9: 43Sum: 134 + 43 = 177Mod 11: 177 ÷ 11 is 16*11=176, so remainder 1.Wait, let me verify these calculations step by step to avoid mistakes.Starting with index 0:- Prefix sum at index 0: 1. Mod 11 is 1. Correct.Index 1:- Sum: 1 + 4 = 5. Mod 11 is 5. Correct.Index 2:- Sum: 5 + 8 = 13. 13 mod 11 is 2. Correct.Index 3:- Sum: 13 + 10 = 23. 23 mod 11 is 1 (since 11*2=22, 23-22=1). Correct.Index 4:- Sum: 23 + 16 = 39. 39 mod 11 is 6 (11*3=33, 39-33=6). Correct.Index 5:- Sum: 39 + 19 = 58. 58 mod 11: 11*5=55, 58-55=3. Correct.Index 6:- Sum: 58 + 21 = 79. 79 mod 11: 11*7=77, 79-77=2. Correct.Index 7:- Sum: 79 + 25 = 104. 104 mod 11: 11*9=99, 104-99=5. Correct.Index 8:- Sum: 104 + 30 = 134. 134 mod 11: 11*12=132, 134-132=2. Correct.Index 9:- Sum: 134 + 43 = 177. 177 mod 11: 11*16=176, 177-176=1. Correct.So the prefix sums modulo 11 are:Indices 0-9: [1, 5, 2, 1, 6, 3, 2, 5, 2, 1]Now, let's note the remainders and their frequencies.Starting from index 0:Remainders:0: 11: At indices 0, 3, 9 → frequency 32: At indices 2, 6, 8 → frequency 33: At index 5 → frequency 15: At indices 1, 7 → frequency 26: At index 4 → frequency 1Wait, hold on. Wait, the prefix sums mod 11 are [1,5,2,1,6,3,2,5,2,1]. Let's list each remainder and count:- Remainder 0: Does it occur? Let me check. The first remainder is 1. Then 5, 2, 1, 6, 3, 2, 5, 2, 1. So no remainder 0. Wait, unless the prefix sum itself is 0 mod 11. But in our case, none of the prefix sums are 0 mod 11. So frequency of 0 is 0.But wait, the initial prefix sum at index -1 (hypothetically) is 0. But since we're starting from index 0, do we consider that? Wait, in some explanations, you consider a prefix sum of 0 before the array starts, but in our calculation here, we started at index 0 with sum 1. So if we include an imaginary prefix sum of 0 at index -1, then the remainders would be 0,1,5,2,1,6,3,2,5,2,1. Then the frequency of remainder 0 would be 1. But in our initial calculation, we didn't include that. Hmm, this is crucial because the formula relies on the fact that subarrays starting from the beginning can be accounted for by the remainder 0.Wait, let me clarify. When we talk about prefix sums, the standard approach is to consider prefix_sum[0] = 0, and then prefix_sum[1] = arr[0], prefix_sum[2] = arr[0] + arr[1], etc. So in that case, the number of prefix sums is n+1. Therefore, perhaps in our case, we should include a prefix sum of 0 before the first element. So perhaps the error here is that I didn't include the initial 0.Let me redo the prefix sums with the initial 0.So, prefix_sum[0] = 0 (for the empty subarray).Then:prefix_sum[1] = 0 + 1 = 1 mod 11 = 1prefix_sum[2] = 1 + 4 = 5 mod 11 = 5prefix_sum[3] = 5 + 8 = 13 mod 11 = 2prefix_sum[4] = 13 + 10 = 23 mod 11 = 1prefix_sum[5] = 23 + 16 = 39 mod 11 = 6prefix_sum[6] = 39 + 19 = 58 mod 11 = 3prefix_sum[7] = 58 + 21 = 79 mod 11 = 2prefix_sum[8] = 79 + 25 = 104 mod 11 = 5prefix_sum[9] = 104 + 30 = 134 mod 11 = 2prefix_sum[10] = 134 + 43 = 177 mod 11 = 1So the prefix sums mod 11 are:[0, 1, 5, 2, 1, 6, 3, 2, 5, 2, 1]Therefore, we have 11 prefix sums (from index 0 to 10). Now, let's count the frequency of each remainder:Remainder 0: occurs once (at prefix_sum[0])Remainder 1: occurs at prefix_sum[1], prefix_sum[4], prefix_sum[10] → indices 1,4,10 → frequency 3Wait, prefix_sum[1] is 1, prefix_sum[4] is 1, prefix_sum[10] is 1. So 3 times.Remainder 2: occurs at prefix_sum[3], prefix_sum[7], prefix_sum[9] → indices 3,7,9. Wait, original prefix sums mod 11 are:prefix_sum[0] = 0prefix_sum[1] = 1prefix_sum[2] = 5prefix_sum[3] = 2prefix_sum[4] = 1prefix_sum[5] = 6prefix_sum[6] = 3prefix_sum[7] = 2prefix_sum[8] = 5prefix_sum[9] = 2prefix_sum[10] = 1So remainder 2 occurs at prefix_sum[3], prefix_sum[7], prefix_sum[9] → frequency 3.Remainder 3: occurs at prefix_sum[6] → frequency 1Remainder 5: occurs at prefix_sum[2], prefix_sum[8] → frequency 2Remainder 6: occurs at prefix_sum[5] → frequency 1Others (remainder 4,7,8,9,10) don't occur.So the frequency table is:0: 11: 32: 33: 15: 26: 1Now, according to the method, the number of subarrays is the sum over all remainders of (frequency choose 2) plus the frequency of remainder 0.Wait, no. Wait, the formula is that for each remainder, if a remainder occurs k times, then there are k choose 2 subarrays that start and end at those positions. Additionally, the frequency of remainder 0 itself contributes to subarrays starting from the beginning (since prefix_sum[j] - prefix_sum[0] = prefix_sum[j], so if prefix_sum[j] is 0 mod 11, then the subarray from 0 to j-1 is divisible by 11). But in the formula, when you have the initial 0, the count is sum_{r} (frequency[r] choose 2). Because whenever two prefix sums have the same remainder, their difference is divisible by 11. So even the frequency of remainder 0 is included in the sum. Wait, actually, the formula is that the total number of subarrays is the sum over all remainders of (frequency[r] choose 2). Because the initial 0 is included, so if remainder 0 occurs k times, then (k choose 2) pairs contribute to subarrays. But also, each occurrence of remainder 0 (after the first) corresponds to a subarray from the start to that index.Wait, this is a bit confusing. Let me look it up in my mind. Let me recall: If we include the prefix sum 0 at the beginning, then the total number of subarrays with sum divisible by 11 is equal to the sum over all remainders of (frequency[r] choose 2). Because every pair of prefix sums with the same remainder gives a subarray between those two indices whose sum is divisible by 11. So in this case, with the initial 0 included, the count would be:For remainder 0: frequency is 1, so (1 choose 2) = 0.For remainder 1: frequency 3, so 3C2=3.For remainder 2: frequency 3, so 3.For remainder 3: 1C2=0.For remainder 5: 2C2=1.For remainder 6: 1C2=0.Total is 0 + 3 + 3 + 0 + 1 + 0 = 7.But wait, but according to the problem statement, we have to count the number of subarrays. So according to this, the answer would be 7. However, let me check with the brute force method to see if that's the case.Wait, but before that, let me make sure. So according to the prefix sums mod 11 with the initial 0 included, the remainders are [0,1,5,2,1,6,3,2,5,2,1]. Then, the total number of pairs with the same remainder is:- Remainder 0: only one occurrence, so 0 pairs.- Remainder 1: occurs at indices 1,4,10. The pairs are (1,4), (1,10), (4,10): 3 pairs.- Remainder 2: occurs at indices 3,7,9. Pairs: (3,7), (3,9), (7,9): 3 pairs.- Remainder 5: occurs at indices 2,8. Pair: (2,8): 1 pair.Others: no pairs.Total 3+3+1=7. So according to this, 7 subarrays.But let's verify with brute force. Let's list all possible subarrays and count those whose sum is divisible by 11.The array is [1,4,8,10,16,19,21,25,30,43].Total elements: 10. So subarrays from length 1 to 10.Let me list all subarrays and their sums:First, let's note that calculating all 55 subarrays is tedious, but maybe we can do it step by step.Starting with each starting index:Start at index 0:- [1] sum=1 → 1 mod11=1 → not divisible.- [1,4] sum=5 → 5 mod11=5 → not.- [1,4,8] sum=13 → 2 → no.- [1,4,8,10] sum=23 →1 → no.- [1,4,8,10,16] sum=39 →6 → no.- [1,4,8,10,16,19] sum=58 →3 → no.- [1,4,8,10,16,19,21] sum=79 →2 → no.- [1,4,8,10,16,19,21,25] sum=104 →5 → no.- [1,4,8,10,16,19,21,25,30] sum=134 →2 → no.- [1,4,8,10,16,19,21,25,30,43] sum=177 →1 → no.So no subarrays starting at 0 are divisible by 11.Start at index 1:- [4] sum=4 →4 → no.- [4,8] sum=12 →1 → no.- [4,8,10] sum=22 →0 → yes. 22 is divisible by 11.- [4,8,10,16] sum=38 →5 → no.- [4,8,10,16,19] sum=57 →57 mod11=2 (11*5=55, 57-55=2) → no.- [4,8,10,16,19,21] sum=78 →78 mod11=78-77=1 → no.- [4,8,10,16,19,21,25] sum=103 →103-99=4 → no.- [4,8,10,16,19,21,25,30] sum=133 →133-121=12 →1 → no.- [4,8,10,16,19,21,25,30,43] sum=176 →176 mod11=0 (11*16=176) → yes. So this subarray is from index1 to index9 (the last element), sum=176, which is 16*11=176.So two subarrays starting at index1: [4,8,10] and [4,8,10,16,19,21,25,30,43].Wait, wait, let's compute [4,8,10,16,19,21,25,30,43]:Sum is 4+8=12, +10=22, +16=38, +19=57, +21=78, +25=103, +30=133, +43=176. Yes, 176 is 11*16, so divisible by 11.So two subarrays here.Start at index2:- [8] sum=8 →8 → no.- [8,10] sum=18 →7 → no.- [8,10,16] sum=34 →1 → no.- [8,10,16,19] sum=53 →9 → no.- [8,10,16,19,21] sum=74 →8 → no.- [8,10,16,19,21,25] sum=99 →0 → yes. 99 is 9*11.- [8,10,16,19,21,25,30] sum=129 →129-121=8 → no.- [8,10,16,19,21,25,30,43] sum=172 →172-165=7 → no.So one subarray here: [8,10,16,19,21,25].Start at index3:- [10] sum=10 →10 → no.- [10,16] sum=26 →4 → no.- [10,16,19] sum=45 →1 → no.- [10,16,19,21] sum=66 →0 → yes. 66 is 6*11.- [10,16,19,21,25] sum=91 →91-88=3 → no.- [10,16,19,21,25,30] sum=121 →0 → yes. 121 is 11^2.- [10,16,19,21,25,30,43] sum=164 →164-154=10 → no.So two subarrays here: [10,16,19,21] and [10,16,19,21,25,30].Start at index4:- [16] sum=16 →5 → no.- [16,19] sum=35 →2 → no.- [16,19,21] sum=56 →1 → no.- [16,19,21,25] sum=81 →4 → no.- [16,19,21,25,30] sum=111 →111-110=1 → no.- [16,19,21,25,30,43] sum=154 →0 → yes. 154 is 14*11.So one subarray here: [16,19,21,25,30,43].Start at index5:- [19] sum=19 →8 → no.- [19,21] sum=40 →7 → no.- [19,21,25] sum=65 →10 → no.- [19,21,25,30] sum=95 →7 → no.- [19,21,25,30,43] sum=138 →138-132=6 → no.No subarrays here.Start at index6:- [21] sum=21 →10 → no.- [21,25] sum=46 →2 → no.- [21,25,30] sum=76 →9 → no.- [21,25,30,43] sum=119 →119-110=9 → no.No subarrays here.Start at index7:- [25] sum=25 →3 → no.- [25,30] sum=55 →0 → yes. 55 is 5*11.- [25,30,43] sum=98 →98-88=10 → no.So one subarray here: [25,30].Start at index8:- [30] sum=30 →8 → no.- [30,43] sum=73 →73-66=7 → no.No subarrays here.Start at index9:- [43] sum=43 →10 → no.So total subarrays found via brute force:From index1: [4,8,10], sum=22; [4,8,10,16,19,21,25,30,43], sum=176.From index2: [8,10,16,19,21,25], sum=99.From index3: [10,16,19,21], sum=66; [10,16,19,21,25,30], sum=121.From index4: [16,19,21,25,30,43], sum=154.From index7: [25,30], sum=55.So total: 2 +1 +2 +1 +1=7. Yes, that matches the count from the prefix sum method. So the answer is 7.But wait, in the problem statement, the sequence given is: 1,4,8,10,16,19,21,25,30,43. The answer is 7. So why does the prefix sum method with initial 0 give us 7, which matches the brute force count? So the answer should be 7. However, the original calculation without the initial 0 had 7, but the user hasn't specified whether to include the initial 0 or not. But in the problem-solving approach, the standard method includes the initial 0 because otherwise, you miss subarrays that start from the beginning. For example, if the entire array sum was divisible by 11, you need that initial 0 to count it. So in this case, since none of the prefix sums were 0 mod 11 except the initial one, the count came up as 7.But according to the brute force method, we have 7 subarrays. Let me just list them again to confirm:1. [4,8,10] sum=222. [4,8,10,16,19,21,25,30,43] sum=1763. [8,10,16,19,21,25] sum=994. [10,16,19,21] sum=665. [10,16,19,21,25,30] sum=1216. [16,19,21,25,30,43] sum=1547. [25,30] sum=55Yes, that's seven subarrays. So the answer is 7.Wait, but in the initial problem statement, the user wrote the sequence as "1,4,8,10,16,19,21,25,30,43". Let me check the indices:Original array:Indices 0:1, 1:4, 2:8, 3:10, 4:16, 5:19, 6:21, 7:25, 8:30, 9:43.So the subarrays found are:1. indices 1-3: [4,8,10]2. indices 1-9: [4,8,10,16,19,21,25,30,43]3. indices 2-7: [8,10,16,19,21,25]4. indices 3-6: [10,16,19,21]5. indices 3-8: [10,16,19,21,25,30]6. indices4-9: [16,19,21,25,30,43]7. indices7-8: [25,30]All of these have sums divisible by 11, and they are 7 in total.Therefore, the answer should be 7. So according to the problem's answer, it's 7. However, I need to confirm if the initial 0 is included. Since the prefix sum method with initial 0 gives 7, which matches brute force, then the answer is 7. Hence, the number of subarrays is 7.But wait, let me check the problem statement again. It says "the number of subarrays whose sum is divisible by 11". The problem is in Chinese, but the translation seems correct. So the answer is 7. But wait, before I finalize, let me check if there are any other possible subarrays I might have missed in brute force.Wait, let's recheck starting from index3:[10,16,19,21] sum=10+16=26+19=45+21=66. 66 is 6*11. Correct.[10,16,19,21,25,30] sum=10+16=26+19=45+21=66+25=91+30=121. 121 is 11^2. Correct.Starting from index4: [16,19,21,25,30,43] sum=16+19=35+21=56+25=81+30=111+43=154. 154 is 14*11. Correct.Starting from index7: [25,30] sum=55. Correct.So all the subarrays found are correct. 7 in total.Thus, the answer is 7. So why did the prefix sum method with initial 0 count 7, matching the brute force, so that's correct. Therefore, the answer is 7, which in boxed form is boxed{7}.Final Answerboxed{7}</think>To find the number of subarrays in the sequence (1, 4, 8, 10, 16, 19, 21, 25, 30, 43) whose sum is divisible by 11, we use the prefix sum method with modular arithmetic. Here are the key steps:1. Compute Prefix Sums Modulo 11: - We include an initial prefix sum of 0 (for the empty subarray). - The prefix sums modulo 11 are: ([0, 1, 5, 2, 1, 6, 3, 2, 5, 2, 1]).2. Count Frequencies of Each Remainder: - Remainder 0: 1 occurrence - Remainder 1: 3 occurrences - Remainder 2: 3 occurrences - Remainder 3: 1 occurrence - Remainder 5: 2 occurrences - Remainder 6: 1 occurrence3. Calculate Subarrays Using Combinations: - For each remainder (r), the number of subarrays is given by (binom{text{frequency}(r)}{2}). - Summing these combinations: (binom{3}{2} + binom{3}{2} + binom{2}{2} = 3 + 3 + 1 = 7).4. Verification with Brute Force: - Manually checking all possible subarrays confirms there are 7 subarrays whose sums are divisible by 11.Thus, the number of subarrays whose sum is divisible by 11 is (boxed{7}).[boxed{7}]]

🔑:3. Since the problem is about divisibility by 11, we can first subtract multiples of 11 from each term to make the numbers smaller and easier to handle, resulting in the following sequence:1,4,-3,-1,5,-3,-1,3,-3,-1 .Let S_{n} be the sum of the first n terms, thenbegin{array}{l}s_{1}=1, s_{2}=5, s_{3}=2, s_{4}=1, s_{5}=6, s_{6}=3, s_{7}=2, s_{8}=5, s_{9}=2, s_{10}=1 .end{array}Among these, the equal ones ares_{1}=s_{4}=s_{10}=1, quad s_{2}=s_{8}=5, quad s_{3}=s_{7}=s_{9}=2 .s_{4}-s_{1}, s_{10}-s_{1}, s_{10}-s_{4}, s_{8}-s_{2}, s_{7}-s_{3}, s_{9}-s_{3}, s_{9}-s_{7}, a total of 7 pairs, are divisible by 11.

🧠:[<think>Okay, let's tackle this problem step by step. First, part (1) asks for the general equation of curve ( C_1 ) and the Cartesian equation of curve ( C_2 ). Let's start with ( C_1 ).Part (1): Finding the General Equation of ( C_1 )The parametric equations for ( C_1 ) are given as:[begin{cases}x = 2cosvarphi y = sqrt{2}sinvarphiend{cases}]where ( varphi ) is the parameter. To convert these parametric equations into a Cartesian equation, we can use the identity ( cos^2varphi + sin^2varphi = 1 ).First, solve for ( cosvarphi ) and ( sinvarphi ) from the parametric equations:[cosvarphi = frac{x}{2} sinvarphi = frac{y}{sqrt{2}}]Now, square both equations and add them together:[left( frac{x}{2} right)^2 + left( frac{y}{sqrt{2}} right)^2 = cos^2varphi + sin^2varphi frac{x^2}{4} + frac{y^2}{2} = 1]So, the Cartesian equation of ( C_1 ) is:[frac{x^2}{4} + frac{y^2}{2} = 1]This is an ellipse centered at the origin with semi-major axis 2 along the x-axis and semi-minor axis ( sqrt{2} ) along the y-axis.Finding the Cartesian Equation of ( C_2 )The polar equation for ( C_2 ) is given as:[rho cos^2theta + 4costheta - rho = 0]We need to convert this into Cartesian coordinates. Remember that in polar coordinates:[rho = sqrt{x^2 + y^2} costheta = frac{x}{sqrt{x^2 + y^2}}]Let's substitute these into the equation. Start by substituting ( rho ) and ( costheta ):[sqrt{x^2 + y^2} left( frac{x}{sqrt{x^2 + y^2}} right)^2 + 4 left( frac{x}{sqrt{x^2 + y^2}} right) - sqrt{x^2 + y^2} = 0]Simplify each term step by step.First term:[sqrt{x^2 + y^2} left( frac{x^2}{x^2 + y^2} right) = frac{x^2}{sqrt{x^2 + y^2}}]Second term:[4 left( frac{x}{sqrt{x^2 + y^2}} right) = frac{4x}{sqrt{x^2 + y^2}}]Third term:[- sqrt{x^2 + y^2}]Putting all together:[frac{x^2}{sqrt{x^2 + y^2}} + frac{4x}{sqrt{x^2 + y^2}} - sqrt{x^2 + y^2} = 0]Multiply every term by ( sqrt{x^2 + y^2} ) to eliminate the denominators:[x^2 + 4x - (x^2 + y^2) = 0]Simplify:[x^2 + 4x - x^2 - y^2 = 0 4x - y^2 = 0 y^2 = 4x]So, the Cartesian equation of ( C_2 ) is ( y^2 = 4x ), which is a parabola opening to the right.Verifying ( C_2 ) ConversionWait, let me check my steps again for converting ( C_2 ). The original polar equation was:[rho cos^2theta + 4costheta - rho = 0]Let me factor ( rho ):[rho (cos^2theta - 1) + 4costheta = 0]Note that ( cos^2theta - 1 = -sin^2theta ), so:[- rho sin^2theta + 4costheta = 0 rho sin^2theta = 4costheta]Now, in Cartesian coordinates, ( rho sintheta = y ), so ( rho sin^2theta = y sintheta ). But maybe there's a better way to express this. Alternatively, let's use the substitution again.Original equation after factoring:[rho (cos^2theta - 1) + 4costheta = 0 - rho sin^2theta + 4costheta = 0 rho sin^2theta = 4costheta]Multiply both sides by ( rho ):[rho^2 sin^2theta = 4rho costheta]But ( rho^2 sin^2theta = y^2 ) and ( rho costheta = x ), so:[y^2 = 4x]Same result as before. So, that's correct. The Cartesian equation is indeed the parabola ( y^2 = 4x ). Alright, so part (1) is done. Got the ellipse for ( C_1 ) and parabola for ( C_2 ).Part (2): Finding the Minimum Value of ( frac{|OB|}{|OA|} )The problem states that a ray ( l: theta = alpha ) intersects ( C_1 ) and ( C_2 ) at points ( A ) and ( B ), both different from the pole (origin). We need to find the minimum value of ( frac{|OB|}{|OA|} ) when ( alpha ) is between ( frac{pi}{4} ) and ( frac{pi}{3} ).First, let's parametrize points ( A ) and ( B ). Since the ray is given by ( theta = alpha ), in polar coordinates, any point on the ray has coordinates ( (rho, alpha) ).But ( A ) is on ( C_1 ), which is an ellipse, and ( B ) is on ( C_2 ), which is a parabola. We need to find the polar coordinates of these points along the ray ( theta = alpha ), convert them into distances from the origin (i.e., ( |OA| ) and ( |OB| )), then compute their ratio and find its minimum in the given interval.Finding ( |OA| ) (Distance from O to A on ( C_1 ))Point ( A ) is the intersection of the ray ( theta = alpha ) with ( C_1 ). Since ( C_1 ) is an ellipse, let's express the ellipse in polar coordinates. However, maybe it's easier to parametrize the ray in Cartesian coordinates and find the intersection.Alternatively, since ( A ) is on both ( C_1 ) and the ray ( theta = alpha ), let's use the parametric equations of ( C_1 ) and see how they relate to the ray.But perhaps a better approach is:In polar coordinates, the ray ( theta = alpha ) can be expressed in Cartesian coordinates as ( y = x tanalpha ).But since ( A ) is on both ( C_1 ) and the ray, substitute ( y = x tanalpha ) into the ellipse equation:The ellipse equation is ( frac{x^2}{4} + frac{y^2}{2} = 1 ).Substitute ( y = x tanalpha ):[frac{x^2}{4} + frac{(x tanalpha)^2}{2} = 1 frac{x^2}{4} + frac{x^2 tan^2alpha}{2} = 1 x^2 left( frac{1}{4} + frac{tan^2alpha}{2} right) = 1 x^2 = frac{1}{frac{1}{4} + frac{tan^2alpha}{2}} x^2 = frac{1}{frac{1 + 2tan^2alpha}{4}} = frac{4}{1 + 2tan^2alpha}]So, ( x = pm frac{2}{sqrt{1 + 2tan^2alpha}} ). However, since the ray ( theta = alpha ) in the range ( frac{pi}{4} leq alpha leq frac{pi}{3} ) is in the first quadrant, we take the positive solution:[x = frac{2}{sqrt{1 + 2tan^2alpha}} y = x tanalpha = frac{2 tanalpha}{sqrt{1 + 2tan^2alpha}}]Therefore, the coordinates of point ( A ) are ( left( frac{2}{sqrt{1 + 2tan^2alpha}}, frac{2 tanalpha}{sqrt{1 + 2tan^2alpha}} right) ).The distance ( |OA| ) is the distance from the origin to this point, which can be calculated using the distance formula:[|OA| = sqrt{left( frac{2}{sqrt{1 + 2tan^2alpha}} right)^2 + left( frac{2 tanalpha}{sqrt{1 + 2tan^2alpha}} right)^2 } = sqrt{ frac{4}{1 + 2tan^2alpha} + frac{4 tan^2alpha}{1 + 2tan^2alpha} } = sqrt{ frac{4(1 + tan^2alpha)}{1 + 2tan^2alpha} } = sqrt{ frac{4 sec^2alpha}{1 + 2tan^2alpha} } quad (text{since } 1 + tan^2alpha = sec^2alpha) = frac{2 secalpha}{sqrt{1 + 2tan^2alpha}} ]Simplify further. Let's express everything in terms of sine and cosine. Let’s recall that ( secalpha = frac{1}{cosalpha} ) and ( tanalpha = frac{sinalpha}{cosalpha} ).Substitute ( tanalpha = frac{sinalpha}{cosalpha} ):[1 + 2tan^2alpha = 1 + 2 frac{sin^2alpha}{cos^2alpha} = frac{cos^2alpha + 2sin^2alpha}{cos^2alpha}]Thus,[sqrt{1 + 2tan^2alpha} = sqrt{ frac{cos^2alpha + 2sin^2alpha}{cos^2alpha} } = frac{sqrt{cos^2alpha + 2sin^2alpha}}{cosalpha}]Therefore,[|OA| = frac{2 secalpha}{ frac{sqrt{cos^2alpha + 2sin^2alpha}}{cosalpha} } = frac{2 cdot frac{1}{cosalpha}}{ frac{sqrt{cos^2alpha + 2sin^2alpha}}{cosalpha} } = frac{2}{ sqrt{cos^2alpha + 2sin^2alpha} }]So, simplifying:[|OA| = frac{2}{ sqrt{cos^2alpha + 2sin^2alpha} } = frac{2}{ sqrt{1 + sin^2alpha} }]Wait, let me check that step. Because ( cos^2alpha + 2sin^2alpha = cos^2alpha + sin^2alpha + sin^2alpha = 1 + sin^2alpha ). Yes, that's correct.Therefore:[|OA| = frac{2}{sqrt{1 + sin^2alpha}}]Nice, that's a simpler expression.Finding ( |OB| ) (Distance from O to B on ( C_2 ))Point ( B ) is the intersection of the ray ( theta = alpha ) with ( C_2 ), which is the parabola ( y^2 = 4x ).Again, the ray ( theta = alpha ) in Cartesian coordinates is ( y = x tanalpha ).Substitute ( y = x tanalpha ) into ( y^2 = 4x ):[(x tanalpha)^2 = 4x x^2 tan^2alpha = 4x x (x tan^2alpha - 4) = 0]So, ( x = 0 ) or ( x = frac{4}{tan^2alpha} ). Since ( x = 0 ) corresponds to the origin (pole), which we are to exclude, the other solution is ( x = frac{4}{tan^2alpha} ).Therefore, the coordinates of point ( B ) are ( left( frac{4}{tan^2alpha}, frac{4}{tan^2alpha} tanalpha right) = left( frac{4}{tan^2alpha}, frac{4}{tanalpha} right) ).Simplify using ( tanalpha = frac{sinalpha}{cosalpha} ):[x = frac{4}{ left( frac{sinalpha}{cosalpha} right)^2 } = frac{4 cos^2alpha}{sin^2alpha} y = frac{4}{ frac{sinalpha}{cosalpha} } = frac{4 cosalpha}{sinalpha}]Thus, coordinates of ( B ) are ( left( frac{4 cos^2alpha}{sin^2alpha}, frac{4 cosalpha}{sinalpha} right) ).The distance ( |OB| ) is:[|OB| = sqrt{ left( frac{4 cos^2alpha}{sin^2alpha} right)^2 + left( frac{4 cosalpha}{sinalpha} right)^2 } = sqrt{ frac{16 cos^4alpha}{sin^4alpha} + frac{16 cos^2alpha}{sin^2alpha} } = sqrt{ frac{16 cos^4alpha + 16 cos^2alpha sin^2alpha}{sin^4alpha} } = sqrt{ frac{16 cos^2alpha (cos^2alpha + sin^2alpha)}{sin^4alpha} } = sqrt{ frac{16 cos^2alpha (1)}{sin^4alpha} } = frac{4 cosalpha}{sin^2alpha}]Therefore, ( |OB| = frac{4 cosalpha}{sin^2alpha} ).Expressing the Ratio ( frac{|OB|}{|OA|} )We have:[|OA| = frac{2}{sqrt{1 + sin^2alpha}} |OB| = frac{4 cosalpha}{sin^2alpha}]Thus, the ratio:[frac{|OB|}{|OA|} = frac{ frac{4 cosalpha}{sin^2alpha} }{ frac{2}{sqrt{1 + sin^2alpha}} } = frac{4 cosalpha}{sin^2alpha} cdot frac{ sqrt{1 + sin^2alpha} }{2 } = frac{2 cosalpha sqrt{1 + sin^2alpha} }{ sin^2alpha }]Simplify this expression. Let me write it as:[frac{|OB|}{|OA|} = frac{2 cosalpha sqrt{1 + sin^2alpha} }{ sin^2alpha }]Let’s denote ( f(alpha) = frac{2 cosalpha sqrt{1 + sin^2alpha} }{ sin^2alpha } ). We need to find the minimum value of ( f(alpha) ) for ( alpha in left[ frac{pi}{4}, frac{pi}{3} right] ).To find the minimum, we can consider taking the derivative of ( f(alpha) ) with respect to ( alpha ), setting it to zero, and checking critical points and endpoints. However, calculus might get a bit messy here, so perhaps there's a way to simplify the expression first.Alternatively, maybe using substitution. Let’s let ( t = sinalpha ). Since ( alpha in [pi/4, pi/3] ), ( sinalpha in [sqrt{2}/2, sqrt{3}/2] ), so ( t in [sqrt{2}/2, sqrt{3}/2] ).Express ( f(alpha) ) in terms of ( t ). Let's note that ( cosalpha = sqrt{1 - t^2} ), since ( alpha ) is in the first quadrant.Substitute into ( f(alpha) ):[f(t) = frac{2 sqrt{1 - t^2} sqrt{1 + t^2} }{ t^2 }]Simplify the numerator:[sqrt{1 - t^2} sqrt{1 + t^2} = sqrt{(1 - t^2)(1 + t^2)} = sqrt{1 - t^4}]Thus,[f(t) = frac{2 sqrt{1 - t^4} }{ t^2 }]So, the problem reduces to minimizing ( f(t) = frac{2 sqrt{1 - t^4} }{ t^2 } ) for ( t in [sqrt{2}/2, sqrt{3}/2] ).Let’s square the function to make differentiation easier, since the square root complicates things. Let’s define:[g(t) = [f(t)]^2 = frac{4(1 - t^4)}{t^4}]We can find the minimum of ( g(t) ), and since ( f(t) ) is positive, the minimum of ( f(t) ) occurs at the same point as the minimum of ( g(t) ).So, compute ( g(t) = frac{4(1 - t^4)}{t^4} = 4 left( frac{1}{t^4} - 1 right) )To find the critical points, take the derivative ( g'(t) ):First, write ( g(t) = 4( t^{-4} - 1 ) )Then,[g'(t) = 4( -4 t^{-5} - 0 ) = -16 t^{-5} = -frac{16}{t^5}]Wait, that seems strange. Wait, derivative of ( t^{-4} ) is ( -4 t^{-5} ), so ( g'(t) = 4(-4 t^{-5}) = -16 t^{-5} ). So, indeed, ( g'(t) = -16 / t^5 ).But in the interval ( t in [sqrt{2}/2, sqrt{3}/2] ), ( t ) is positive, so ( g'(t) ) is negative. That means ( g(t) ) is decreasing on this interval. Therefore, the minimum of ( g(t) ) occurs at the right endpoint, which is ( t = sqrt{3}/2 ), and the maximum occurs at the left endpoint ( t = sqrt{2}/2 ).Therefore, since ( g(t) ) is decreasing, the minimum value of ( g(t) ) is ( g(sqrt{3}/2) ), which corresponds to the minimum of ( f(t) ).Thus, the minimum of ( f(t) ) is ( sqrt{g(sqrt{3}/2)} ).Compute ( g(sqrt{3}/2) ):First, compute ( t = sqrt{3}/2 ), so ( t^4 = (sqrt{3}/2)^4 = (3/4)^2 = 9/16 ).Then,[g(sqrt{3}/2) = 4 left( frac{1 - (9/16)}{ (9/16) } right ) = 4 left( frac{7/16}{9/16} right ) = 4 cdot frac{7}{9} = frac{28}{9}]Therefore, the minimum value of ( f(t) ) is ( sqrt{28/9} = sqrt{28}/3 = (2sqrt{7})/3 approx 1.76 ).But wait, let's check this conclusion again. If ( g(t) ) is decreasing, then its minimum is at the right endpoint. So, yes, at ( t = sqrt{3}/2 ), we get the minimal ( g(t) ), hence minimal ( f(t) ).But let's verify with the original expression.Alternatively, since we found that the derivative ( g'(t) = -16 / t^5 ), which is always negative in the interval, so ( g(t) ) is decreasing. Hence, the minimal value of ( g(t) ) is at ( t = sqrt{3}/2 ).Therefore, the minimal value of ( |OB| / |OA| ) is ( sqrt{28}/3 ). However, let's compute this properly.Wait, ( g(t) = [f(t)]^2 = 28/9 ), so ( f(t) = sqrt{28}/3 ). Simplify ( sqrt{28} = 2sqrt{7} ), so ( f(t) = 2sqrt{7}/3 ).But let's confirm this by plugging ( alpha = pi/3 ) into the original ratio.Verification for ( alpha = pi/3 ):Compute ( |OA| ):( sin(pi/3) = sqrt{3}/2 ), so:[|OA| = frac{2}{sqrt{1 + (sqrt{3}/2)^2}} = frac{2}{sqrt{1 + 3/4}} = frac{2}{sqrt{7/4}} = frac{2 cdot 2}{sqrt{7}} = frac{4}{sqrt{7}}]Compute ( |OB| ):[|OB| = frac{4 cos(pi/3)}{sin^2(pi/3)} = frac{4 cdot (1/2)}{ ( (sqrt{3}/2)^2 ) } = frac{2}{ (3/4) } = frac{2 cdot 4}{3} = frac{8}{3}]Thus, the ratio ( |OB| / |OA| = (8/3) / (4/sqrt{7}) ) = (8/3) cdot (sqrt{7}/4 ) = (2 sqrt{7}) / 3 approx 1.76 ), which matches our previous result.Verification for ( alpha = pi/4 ):Compute ( |OA| ):( sin(pi/4) = sqrt{2}/2 ), so:[|OA| = frac{2}{sqrt{1 + (sqrt{2}/2)^2}} = frac{2}{sqrt{1 + 1/2}} = frac{2}{sqrt{3/2}} = frac{2 cdot sqrt{2}}{sqrt{3}} = frac{2 sqrt{6}}{3}]Compute ( |OB| ):[|OB| = frac{4 cos(pi/4)}{sin^2(pi/4)} = frac{4 cdot (sqrt{2}/2)}{ ( (sqrt{2}/2)^2 ) } = frac{2 sqrt{2}}{ (1/2) } = 4 sqrt{2}]Ratio ( |OB| / |OA| = 4 sqrt{2} / (2 sqrt{6}/3 ) = (4 sqrt{2} cdot 3 ) / (2 sqrt{6}) ) = (12 sqrt{2} ) / (2 sqrt{6}) ) = 6 sqrt{2} / sqrt{6} = 6 sqrt{(2/6)} = 6 sqrt{1/3} = 6 / sqrt{3} = 2 sqrt{3} approx 3.464 )So, at ( alpha = pi/4 ), the ratio is ( 2sqrt{3} ), and at ( alpha = pi/3 ), it's ( 2sqrt{7}/3 approx 1.76 ). Since the function is decreasing, the minimum occurs at ( alpha = pi/3 ), and the maximum at ( alpha = pi/4 ).Therefore, the minimal value of ( |OB| / |OA| ) is ( 2sqrt{7}/3 ).But let me just confirm if there's any mistake here. Since the derivative of ( g(t) ) is negative, ( g(t) ) is decreasing, so as ( t ) increases from ( sqrt{2}/2 ) to ( sqrt{3}/2 ), ( g(t) ) decreases, so the minimum of ( g(t) ) is at ( t = sqrt{3}/2 ), which corresponds to ( alpha = pi/3 ). Therefore, the minimal ratio is indeed ( 2sqrt{7}/3 ).ConclusionTherefore, after thorough calculation and verification, the minimum value of ( frac{|OB|}{|OA|} ) in the interval ( frac{pi}{4} leq alpha leq frac{pi}{3} ) is ( frac{2sqrt{7}}{3} ).Final AnswerThe minimum value of ( frac{|OB|}{|OA|} ) is boxed{dfrac{2sqrt{7}}{3}}.</think>(1) General equation of curve ( C_1 ):The parametric equations for ( C_1 ) are:[begin{cases}x = 2cosvarphi y = sqrt{2}sinvarphiend{cases}]Using the identity ( cos^2varphi + sin^2varphi = 1 ), we substitute ( cosvarphi = frac{x}{2} ) and ( sinvarphi = frac{y}{sqrt{2}} ):[left( frac{x}{2} right)^2 + left( frac{y}{sqrt{2}} right)^2 = 1 implies frac{x^2}{4} + frac{y^2}{2} = 1]Thus, the general equation of ( C_1 ) is:[boxed{frac{x^2}{4} + frac{y^2}{2} = 1}]Cartesian equation of curve ( C_2 ):The polar equation for ( C_2 ) is:[rho cos^2theta + 4costheta - rho = 0]Substituting ( rho = sqrt{x^2 + y^2} ) and ( costheta = frac{x}{sqrt{x^2 + y^2}} ):[sqrt{x^2 + y^2} left( frac{x^2}{x^2 + y^2} right) + 4 left( frac{x}{sqrt{x^2 + y^2}} right) - sqrt{x^2 + y^2} = 0]Simplifying, we get:[frac{x^2}{sqrt{x^2 + y^2}} + frac{4x}{sqrt{x^2 + y^2}} - sqrt{x^2 + y^2} = 0 implies x^2 + 4x - (x^2 + y^2) = 0 implies y^2 = 4x]Thus, the Cartesian equation of ( C_2 ) is:[boxed{y^2 = 4x}](2) Minimum value of ( frac{|OB|}{|OA|} ):For a ray ( theta = alpha ), the coordinates of points ( A ) (on ( C_1 )) and ( B ) (on ( C_2 )) are found by substituting ( y = x tanalpha ) into the respective equations.For ( C_1 ):[frac{x^2}{4} + frac{(x tanalpha)^2}{2} = 1 implies x^2 = frac{4}{1 + 2tan^2alpha}]The distance ( |OA| ) is:[|OA| = frac{2}{sqrt{1 + sin^2alpha}}]For ( C_2 ):[(x tanalpha)^2 = 4x implies x = frac{4}{tan^2alpha}]The distance ( |OB| ) is:[|OB| = frac{4 cosalpha}{sin^2alpha}]The ratio ( frac{|OB|}{|OA|} ) simplifies to:[frac{|OB|}{|OA|} = frac{2 cosalpha sqrt{1 + sin^2alpha}}{sin^2alpha}]Substituting ( t = sinalpha ) and analyzing the function ( f(t) = frac{2 sqrt{1 - t^4}}{t^2} ), we find that the function is decreasing in the interval ( t in [sqrt{2}/2, sqrt{3}/2] ). The minimum value occurs at ( t = sqrt{3}/2 ), corresponding to ( alpha = pi/3 ).Calculating the ratio at ( alpha = pi/3 ):[|OA| = frac{4}{sqrt{7}}, quad |OB| = frac{8}{3}][frac{|OB|}{|OA|} = frac{8/3}{4/sqrt{7}} = frac{2sqrt{7}}{3}]Thus, the minimum value of ( frac{|OB|}{|OA|} ) is:[boxed{dfrac{2sqrt{7}}{3}}],<think>Okay, let me try to tackle this problem step by step. First, I need to find the general equation of curve ( C_1 ) and the Cartesian equation of curve ( C_2 ). Then, part (2) asks for the minimum value of ( frac{|OB|}{|OA|} ) when ( alpha ) is between ( frac{pi}{4} ) and ( frac{pi}{3} ). Let's start with part (1).Problem (1): Find the general equation of curve ( C_1 ) and the Cartesian equation of curve ( C_2 ).Starting with curve ( C_1 ). The parametric equations are given as:[begin{cases}x = 2cos varphi y = sqrt{2}sin varphiend{cases}]where ( varphi ) is the parameter. I need to eliminate the parameter ( varphi ) to find the Cartesian equation.For parametric equations involving sine and cosine, a common method is to use the identity ( cos^2 varphi + sin^2 varphi = 1 ). Let me express ( cos varphi ) and ( sin varphi ) in terms of ( x ) and ( y ).From the first equation, ( x = 2cos varphi ), so ( cos varphi = frac{x}{2} ).From the second equation, ( y = sqrt{2}sin varphi ), so ( sin varphi = frac{y}{sqrt{2}} ).Now, substituting these into the Pythagorean identity:[left( frac{x}{2} right)^2 + left( frac{y}{sqrt{2}} right)^2 = 1]Simplify each term:[frac{x^2}{4} + frac{y^2}{2} = 1]Multiply both sides by 4 to eliminate denominators:[x^2 + 2y^2 = 4]So, the general equation of ( C_1 ) is ( x^2 + 2y^2 = 4 ). That should be an ellipse, since the coefficients of ( x^2 ) and ( y^2 ) are positive and different.Now, moving on to curve ( C_2 ). Its polar coordinate equation is given as:[rho cos^2 theta + 4cos theta - rho = 0]I need to convert this polar equation to Cartesian form. Remember that in polar coordinates, ( rho cos theta = x ), ( rho sin theta = y ), and ( rho = sqrt{x^2 + y^2} ).Let me rewrite the equation step by step.First, let's write the given equation:[rho cos^2 theta + 4cos theta - rho = 0]Factor out ( rho ) from the first and third terms:[rho (cos^2 theta - 1) + 4cos theta = 0]Notice that ( cos^2 theta - 1 = -sin^2 theta ), so substitute that in:[-rho sin^2 theta + 4cos theta = 0]Multiply both sides by -1 to make it a bit cleaner:[rho sin^2 theta - 4cos theta = 0]Now, let's express each term in Cartesian coordinates.Recall that ( rho sin theta = y ), so ( sin theta = frac{y}{rho} ), and ( cos theta = frac{x}{rho} ).So, ( sin^2 theta = left( frac{y}{rho} right)^2 = frac{y^2}{rho^2} ).Therefore, the term ( rho sin^2 theta ) becomes:[rho cdot frac{y^2}{rho^2} = frac{y^2}{rho}]The term ( 4cos theta ) becomes:[4 cdot frac{x}{rho}]So substituting back into the equation:[frac{y^2}{rho} - frac{4x}{rho} = 0]Multiply both sides by ( rho ) (assuming ( rho neq 0 )):[y^2 - 4x = 0]So, the Cartesian equation of ( C_2 ) is ( y^2 = 4x ). That's a parabola opening to the right. Makes sense.So part (1) is done. The general equation for ( C_1 ) is ( x^2 + 2y^2 = 4 ), and the Cartesian equation for ( C_2 ) is ( y^2 = 4x ).Problem (2): Find the minimum value of ( frac{|OB|}{|OA|} ) when ( frac{pi}{4} leq alpha leq frac{pi}{3} ).Alright, so we have a ray ( l: theta = alpha ) which intersects ( C_1 ) at point A and ( C_2 ) at point B (both different from the pole O). We need to find the ratio ( frac{|OB|}{|OA|} ) and its minimum value as ( alpha ) varies between ( frac{pi}{4} ) and ( frac{pi}{3} ).First, let's recall that in polar coordinates, a ray ( theta = alpha ) can be parameterized by ( rho geq 0 ). So, points along this ray have coordinates ( (rho, alpha) ).To find points A and B, we need to find the intersections of the ray ( theta = alpha ) with ( C_1 ) and ( C_2 ), respectively. Then compute the distances |OA| and |OB|, which are just the ( rho ) values of these points since in polar coordinates ( rho ) is the distance from the origin.So, the plan is:1. Find the polar equation of ( C_1 ), then find ( rho ) when ( theta = alpha ), which gives |OA| = ( rho_1 ).2. Use the given polar equation of ( C_2 ), set ( theta = alpha ), solve for ( rho ), which gives |OB| = ( rho_2 ).3. Compute the ratio ( frac{rho_2}{rho_1} ) and find its minimum over ( alpha in [frac{pi}{4}, frac{pi}{3}] ).But wait, ( C_1 ) is given in parametric form, but we have its Cartesian equation. To find the polar equation of ( C_1 ), we can substitute ( x = rho cos theta ), ( y = rho sin theta ) into its Cartesian equation.Similarly, ( C_2 ) is already given in polar coordinates, so we can solve for ( rho ) in terms of ( theta ).Let me proceed step by step.Finding the polar equation of ( C_1 ):Given the Cartesian equation ( x^2 + 2y^2 = 4 ). Substitute ( x = rho cos theta ), ( y = rho sin theta ):[(rho cos theta)^2 + 2(rho sin theta)^2 = 4]Expand:[rho^2 cos^2 theta + 2 rho^2 sin^2 theta = 4]Factor out ( rho^2 ):[rho^2 (cos^2 theta + 2 sin^2 theta) = 4]Solve for ( rho ):[rho^2 = frac{4}{cos^2 theta + 2 sin^2 theta}]Take square root (since ( rho geq 0 )):[rho = frac{2}{sqrt{cos^2 theta + 2 sin^2 theta}}]Simplify the denominator:Note that ( cos^2 theta + 2 sin^2 theta = cos^2 theta + sin^2 theta + sin^2 theta = 1 + sin^2 theta )Therefore:[rho = frac{2}{sqrt{1 + sin^2 theta}}]So, the polar equation of ( C_1 ) is ( rho = frac{2}{sqrt{1 + sin^2 theta}} ).Therefore, for a given ( theta = alpha ), the distance |OA| is:[|OA| = rho_1 = frac{2}{sqrt{1 + sin^2 alpha}}]Finding |OB| from ( C_2 ):Given the polar equation of ( C_2 ):[rho cos^2 theta + 4 cos theta - rho = 0]Let's solve for ( rho ) when ( theta = alpha ).Set ( theta = alpha ), so the equation becomes:[rho cos^2 alpha + 4 cos alpha - rho = 0]Factor ( rho ):[rho (cos^2 alpha - 1) + 4 cos alpha = 0]Again, ( cos^2 alpha - 1 = -sin^2 alpha ), so substitute:[- rho sin^2 alpha + 4 cos alpha = 0]Solve for ( rho ):[rho sin^2 alpha = 4 cos alpha rho = frac{4 cos alpha}{sin^2 alpha}]Therefore, |OB| = ( rho_2 = frac{4 cos alpha}{sin^2 alpha} )Calculating the ratio ( frac{|OB|}{|OA|} ):We have ( |OA| = frac{2}{sqrt{1 + sin^2 alpha}} ) and ( |OB| = frac{4 cos alpha}{sin^2 alpha} ).Therefore, the ratio is:[frac{|OB|}{|OA|} = frac{frac{4 cos alpha}{sin^2 alpha}}{frac{2}{sqrt{1 + sin^2 alpha}}} = frac{4 cos alpha}{sin^2 alpha} cdot frac{sqrt{1 + sin^2 alpha}}{2} = frac{2 cos alpha sqrt{1 + sin^2 alpha}}{sin^2 alpha}]Simplify the expression. Let's denote ( f(alpha) = frac{2 cos alpha sqrt{1 + sin^2 alpha}}{sin^2 alpha} ). We need to find the minimum value of ( f(alpha) ) for ( alpha in left[ frac{pi}{4}, frac{pi}{3} right] ).To find the minimum, we can either analyze the function directly or take the derivative and find critical points. Let's try differentiation.First, let's simplify ( f(alpha) ):Let me write ( f(alpha) = 2 cos alpha cdot frac{sqrt{1 + sin^2 alpha}}{sin^2 alpha} )But perhaps it's better to square the function to make differentiation easier, as the square root can complicate things. Since the square function is monotonic for non-negative values, the minimum of ( f(alpha) ) will correspond to the minimum of ( [f(alpha)]^2 ).Let’s compute ( [f(alpha)]^2 ):[[f(alpha)]^2 = left( frac{2 cos alpha sqrt{1 + sin^2 alpha}}{sin^2 alpha} right)^2 = frac{4 cos^2 alpha (1 + sin^2 alpha)}{sin^4 alpha}]So, to minimize ( f(alpha) ), we can minimize ( [f(alpha)]^2 = frac{4 cos^2 alpha (1 + sin^2 alpha)}{sin^4 alpha} )Let’s denote ( g(alpha) = frac{cos^2 alpha (1 + sin^2 alpha)}{sin^4 alpha} ), so ( [f(alpha)]^2 = 4 g(alpha) ), and minimizing ( f(alpha) ) is equivalent to minimizing ( g(alpha) ).So, let's work with ( g(alpha) ):[g(alpha) = frac{cos^2 alpha (1 + sin^2 alpha)}{sin^4 alpha}]Perhaps express everything in terms of ( t = sin alpha ), since ( cos^2 alpha = 1 - t^2 ), and ( sin^2 alpha = t^2 ). Let's try that substitution.Let ( t = sin alpha ). Then, ( cos^2 alpha = 1 - t^2 ), and ( sin^2 alpha = t^2 ). Also, when ( alpha in [frac{pi}{4}, frac{pi}{3}] ), ( sin alpha ) ranges from ( frac{sqrt{2}}{2} ) to ( frac{sqrt{3}}{2} ), so ( t in [frac{sqrt{2}}{2}, frac{sqrt{3}}{2}] ).Expressing ( g(alpha) ) in terms of ( t ):[g(alpha) = frac{(1 - t^2)(1 + t^2)}{t^4} = frac{(1 - t^2)(1 + t^2)}{t^4} = frac{1 - t^4}{t^4} = frac{1}{t^4} - 1]Wait, let me check that:Wait, ( (1 - t^2)(1 + t^2) = 1 - t^4 ), yes. So:[g(alpha) = frac{1 - t^4}{t^4} = frac{1}{t^4} - 1]So, ( g(alpha) = frac{1}{t^4} - 1 ), where ( t = sin alpha in [frac{sqrt{2}}{2}, frac{sqrt{3}}{2}] ).But this seems a bit odd. Let me verify.Wait, original substitution:[g(alpha) = frac{cos^2 alpha (1 + sin^2 alpha)}{sin^4 alpha} = frac{(1 - t^2)(1 + t^2)}{t^4} = frac{1 - t^4}{t^4} = frac{1}{t^4} - 1]Yes, that's correct.Therefore, ( g(alpha) = frac{1}{t^4} - 1 ), so ( [f(alpha)]^2 = 4 left( frac{1}{t^4} - 1 right ) ), where ( t in [frac{sqrt{2}}{2}, frac{sqrt{3}}{2}] ).Therefore, since ( t ) increases from ( frac{sqrt{2}}{2} ) to ( frac{sqrt{3}}{2} ), ( frac{1}{t^4} ) decreases, so ( g(alpha) ) decreases as ( t ) increases, so ( g(alpha) ) is decreasing on the interval ( alpha in [frac{pi}{4}, frac{pi}{3}] ). Therefore, ( [f(alpha)]^2 ) is decreasing, so ( f(alpha) ) is decreasing as ( alpha ) increases from ( frac{pi}{4} ) to ( frac{pi}{3} ). Therefore, the minimum value of ( f(alpha) ) occurs at ( alpha = frac{pi}{3} ).Wait, hold on. Let's verify this conclusion. If ( g(alpha) ) is decreasing in ( t ), and ( t = sin alpha ) is increasing in ( alpha ), then ( g(alpha) ) is decreasing in ( alpha ). Therefore, since ( alpha ) increases from ( pi/4 ) to ( pi/3 ), ( g(alpha) ) decreases, so ( f(alpha)^2 ) decreases, so ( f(alpha) ) decreases. Therefore, the minimum value of ( f(alpha) ) would be at ( alpha = pi/3 ).But wait, the problem asks for the minimum value of ( |OB| / |OA| ), which is ( f(alpha) ). So if ( f(alpha) ) is decreasing over the interval, its minimum is achieved at the right endpoint, ( alpha = pi/3 ).But let's compute the values at both endpoints to confirm.Compute ( f(pi/4) ):First, ( alpha = pi/4 ):Compute ( sin pi/4 = frac{sqrt{2}}{2} ), ( cos pi/4 = frac{sqrt{2}}{2} ).Then,( |OA| = frac{2}{sqrt{1 + (sqrt{2}/2)^2}} = frac{2}{sqrt{1 + 1/2}} = frac{2}{sqrt{3/2}} = frac{2}{sqrt{3}/sqrt{2}} = frac{2 sqrt{2}}{sqrt{3}} )( |OB| = frac{4 cos pi/4}{sin^2 pi/4} = frac{4 cdot sqrt{2}/2}{(sqrt{2}/2)^2} = frac{2 sqrt{2}}{ (2/4) } = frac{2 sqrt{2}}{1/2} = 4 sqrt{2} )Thus, the ratio ( |OB| / |OA| = (4 sqrt{2}) / (2 sqrt{2}/sqrt{3}) ) = (4 sqrt{2}) cdot (sqrt{3}/(2 sqrt{2})) ) = (4 / 2) cdot sqrt{3} = 2 sqrt{3} ≈ 3.464 )Now compute ( f(pi/3) ):( alpha = pi/3 ):( sin pi/3 = sqrt{3}/2 ), ( cos pi/3 = 1/2 )( |OA| = frac{2}{sqrt{1 + (sqrt{3}/2)^2}} = frac{2}{sqrt{1 + 3/4}} = frac{2}{sqrt{7/4}} = frac{2}{sqrt{7}/2} = frac{4}{sqrt{7}} )( |OB| = frac{4 cdot 1/2}{ (sqrt{3}/2)^2 } = frac{2}{ 3/4 } = frac{2 cdot 4}{3} = 8/3 ≈ 2.6667 )Thus, the ratio ( |OB| / |OA| = (8/3) / (4 / sqrt{7}) ) = (8/3) * ( sqrt{7}/4 ) = (2/3) sqrt{7} ≈ 1.763 )So, indeed, the ratio decreases from approximately 3.464 to 1.763 as ( alpha ) increases from ( pi/4 ) to ( pi/3 ). Therefore, the minimum occurs at ( alpha = pi/3 ), and the minimum value is ( frac{2sqrt{7}}{3} ).But let me compute it precisely.Compute ( f(pi/3) ):From above, ( |OB| = 8/3 ), ( |OA| = 4 / sqrt{7} ), so ratio is ( (8/3) / (4 / sqrt{7}) ) = (8/3) * ( sqrt{7}/4 ) = (2/3) sqrt{7} ). So ( 2 sqrt{7}/3 ).Hence, the minimum value is ( frac{2 sqrt{7}}{3} ).But let me double-check using the expression for ( f(alpha) ):Earlier, we had:( f(alpha) = frac{2 cos alpha sqrt{1 + sin^2 alpha}}{sin^2 alpha} )At ( alpha = pi/3 ):( cos pi/3 = 1/2 ), ( sin pi/3 = sqrt{3}/2 ), so ( sin^2 pi/3 = 3/4 ), ( 1 + sin^2 alpha = 1 + 3/4 = 7/4 ), so ( sqrt{1 + sin^2 alpha} = sqrt{7}/2 ).Therefore,( f(pi/3) = frac{2 cdot (1/2) cdot (sqrt{7}/2)}{(3/4)} = frac{ (sqrt{7}/2) }{ (3/4) } = (sqrt{7}/2) cdot (4/3) = (2 sqrt{7}/3) ). Correct.Therefore, the minimum value is ( frac{2 sqrt{7}}{3} ).But let me confirm that there are no critical points in between ( pi/4 ) and ( pi/3 ). Since we transformed the function into ( g(alpha) = frac{1}{t^4} - 1 ), where ( t = sin alpha ), and since ( t ) increases from ( sqrt{2}/2 ) to ( sqrt{3}/2 ), and ( g(alpha) ) is strictly decreasing as ( t ) increases, there are no critical points in the interval. Therefore, the function is indeed decreasing over the interval, so the minimum is at ( alpha = pi/3 ).Therefore, the answer to part (2) is ( frac{2 sqrt{7}}{3} ).Summary:1. General equation of ( C_1 ): ( x^2 + 2y^2 = 4 )2. Cartesian equation of ( C_2 ): ( y^2 = 4x )3. Minimum value of ( frac{|OB|}{|OA|} ): ( frac{2 sqrt{7}}{3} )But let me verify once again to ensure no miscalculations.Verification of part (1):For ( C_1 ): parametric equations are ( x = 2cos varphi ), ( y = sqrt{2}sin varphi ). Squaring and adding:( (x/2)^2 + (y/sqrt{2})^2 = cos^2 varphi + sin^2 varphi = 1 implies x^2/4 + y^2/2 = 1 implies x^2 + 2y^2 = 4 ). Correct.For ( C_2 ): polar equation ( rho cos^2 theta +4 cos theta - rho =0 ). As we solved, leads to ( y^2 =4x ). Let me check again:Starting equation: ( rho cos^2 theta +4 cos theta - rho =0 )Multiply through by ( rho ):Wait, no, in our previous steps, we converted the equation to Cartesian:Starting equation:( rho cos^2 theta +4 cos theta - rho =0 )We rearranged to ( rho(cos^2 theta -1) +4cos theta =0 implies -rho sin^2 theta +4cos theta=0 implies rho sin^2 theta=4cos theta implies rho = frac{4 cos theta}{sin^2 theta} )But in Cartesian, ( rho sin^2 theta = rho cdot frac{y^2}{rho^2} = frac{y^2}{rho} ), and ( 4cos theta =4 cdot frac{x}{rho} )Therefore, equation is ( frac{y^2}{rho} = frac{4x}{rho} implies y^2 =4x ). Correct.Verification of part (2):The ratio ( |OB|/|OA| ) was found by converting both curves to polar coordinates, finding the ( rho ) values for the given ( theta = alpha ), then taking the ratio.For ( C_1 ), polar equation derived as ( rho = 2 / sqrt{1 + sin^2 theta} ), correct.For ( C_2 ), solving the polar equation gives ( rho = 4 cos alpha / sin^2 alpha ), correct.Thus, ratio is ( frac{4 cos alpha / sin^2 alpha}{2 / sqrt{1 + sin^2 alpha}} = frac{2 cos alpha sqrt{1 + sin^2 alpha}}{sin^2 alpha} ), correct.Then, by substitution, we found that this ratio is decreasing over the interval ( [pi/4, pi/3] ), so minimum at ( pi/3 ), value ( 2 sqrt{7}/3 ). The numerical check at both endpoints confirmed this.Therefore, confident the answer is correct.Final Answer(1) The general equation of ( C_1 ) is (boxed{x^2 + 2y^2 = 4}) and the Cartesian equation of ( C_2 ) is (boxed{y^2 = 4x}).(2) The minimum value of (dfrac{|OB|}{|OA|}) is (boxed{dfrac{2sqrt{7}}{3}}).</think>(1) To find the general equation of curve ( C_1 ), we start with the parametric equations:[begin{cases}x = 2cos varphi y = sqrt{2}sin varphiend{cases}]Using the identity (cos^2 varphi + sin^2 varphi = 1), we substitute (cos varphi = frac{x}{2}) and (sin varphi = frac{y}{sqrt{2}}):[left( frac{x}{2} right)^2 + left( frac{y}{sqrt{2}} right)^2 = 1 implies frac{x^2}{4} + frac{y^2}{2} = 1 implies x^2 + 2y^2 = 4]Thus, the general equation of ( C_1 ) is (boxed{x^2 + 2y^2 = 4}).For curve ( C_2 ), given the polar equation (rho cos^2 theta + 4cos theta - rho = 0), we convert it to Cartesian coordinates. Rearranging and substituting (rho cos theta = x) and (rho sin theta = y):[rho (cos^2 theta - 1) + 4cos theta = 0 implies -rho sin^2 theta + 4cos theta = 0 implies frac{y^2}{rho} = frac{4x}{rho} implies y^2 = 4x]Thus, the Cartesian equation of ( C_2 ) is (boxed{y^2 = 4x}).(2) To find the minimum value of (frac{|OB|}{|OA|}) when (frac{pi}{4} leq alpha leq frac{pi}{3}), we first determine the polar equations of ( C_1 ) and ( C_2 ).For ( C_1 ):[x^2 + 2y^2 = 4 implies rho^2 (cos^2 theta + 2 sin^2 theta) = 4 implies rho = frac{2}{sqrt{1 + sin^2 theta}}]Thus, (|OA| = frac{2}{sqrt{1 + sin^2 alpha}}).For ( C_2 ):[rho cos^2 alpha + 4 cos alpha - rho = 0 implies rho = frac{4 cos alpha}{sin^2 alpha}]Thus, (|OB| = frac{4 cos alpha}{sin^2 alpha}).The ratio is:[frac{|OB|}{|OA|} = frac{frac{4 cos alpha}{sin^2 alpha}}{frac{2}{sqrt{1 + sin^2 alpha}}} = frac{2 cos alpha sqrt{1 + sin^2 alpha}}{sin^2 alpha}]Analyzing this ratio, we find it is decreasing over (alpha in left[ frac{pi}{4}, frac{pi}{3} right]). Evaluating at (alpha = frac{pi}{3}):[sin frac{pi}{3} = frac{sqrt{3}}{2}, cos frac{pi}{3} = frac{1}{2} implies frac{2 cos alpha sqrt{1 + sin^2 alpha}}{sin^2 alpha} = frac{2 cdot frac{1}{2} cdot sqrt{1 + left( frac{sqrt{3}}{2} right)^2}}{left( frac{sqrt{3}}{2} right)^2} = frac{2 sqrt{7}}{3}]Thus, the minimum value is (boxed{dfrac{2sqrt{7}}{3}}).]

🔑:Solution:(1) For curve C_{1}, we have the parametric equations:[begin{cases}x=2cosvarphi y=sqrt{2}sinvarphiend{cases}]To eliminate the parameter varphi, we use the trigonometric identities sin^2varphi + cos^2varphi = 1:[frac{x^2}{4} + frac{y^2}{2} = cos^2varphi + sin^2varphi = 1]Thus, the general equation of curve C_{1} is boxed{frac{x^2}{4}+frac{y^2}{2}=1}.For curve C_{2}, given the polar coordinate equation rho cos^{2}theta +4costheta -rho =0, we multiply by rho to get:[rho^2cos^{2}theta +4rhocostheta -rho^2 =0]Substituting rho = sqrt{x^2+y^2} and costheta = frac{x}{rho}, we have:[x^{2}+4x-(x^{2}+y^{2})=0]Simplifying this, we get the Cartesian equation of curve C_{2} as boxed{y^{2}=4x}.(2) For curve C_{1}, converting its general equation to polar coordinates:[frac{rho^2cos^2theta}{4}+frac{rho^2sin^2theta}{2}=1]Simplifying, we find:[rho^2=frac{4}{1+sin^2theta}]For a ray l: theta =alpha, the distance |OA| is:[|OA|=rho_A=sqrt{frac{4}{1+sin^2alpha}}]For curve C_{2}, solving the given polar equation for rho when theta = alpha:[|OB|=rho_B=frac{4cosalpha}{sin^2alpha}]To find the minimum value of frac{|OB|}{|OA|} when frac{pi}{4} leq alpha leq frac{pi}{3}, we calculate:[frac{|OB|}{|OA|} = frac{2cosalphasqrt{1+sin^2alpha}}{sin^2alpha} = 2sqrt{frac{1}{sin^4alpha}-1}]Given sinalpha in [frac{sqrt{2}}{2}, frac{sqrt{3}}{2}], the minimum occurs when sinalpha=frac{sqrt{3}}{2}, yielding:[frac{|OB|}{|OA|} = frac{2sqrt{7}}{3}]Therefore, the minimum value of frac{{|{OB}|}}{{|{OA}|}} is boxed{frac{2sqrt{7}}{3}}.

🧠:[<think>Okay, so I need to find the limit of tan(3x)/tan(x) as x approaches π/2. Hmm, let's start by recalling some trigonometric identities. I know that tan(3x) can be expanded using the triple angle formula. Let me see... The formula for tan(3x) is (3 tan x - tan³ x)/(1 - 3 tan² x). So, maybe I can substitute that into the expression.So, substituting that in, the original expression becomes [(3 tan x - tan³ x)/(1 - 3 tan² x)] divided by tan x. Simplifying that, it would be (3 tan x - tan³ x)/(1 - 3 tan² x) * 1/tan x. The tan x in the numerator and denominator should cancel out, right? Let's check:Multiplying numerator and denominator: (3 tan x - tan³ x) * 1/tan x = 3 - tan² x, and the denominator remains (1 - 3 tan² x). So now the expression simplifies to (3 - tan² x)/(1 - 3 tan² x).Hmm, that's a simpler expression. Now, as x approaches π/2, tan x approaches infinity. So both the numerator and the denominator here will have terms involving tan² x, which goes to infinity. Maybe I can factor out tan² x from numerator and denominator to see the dominant terms.Let's factor tan² x from numerator and denominator:Numerator: 3 - tan² x = -tan² x + 3 = -tan² x (1 - 3/tan² x) ≈ -tan² x (since 3/tan² x approaches 0 as x approaches π/2).Denominator: 1 - 3 tan² x = -3 tan² x + 1 = -3 tan² x (1 - 1/(3 tan² x)) ≈ -3 tan² x.So, the expression becomes approximately (-tan² x)/(-3 tan² x) = (1/3). Therefore, the limit should be 1/3.Wait, but let me verify this another way to make sure. Alternatively, maybe use substitution. Let me set t = x - π/2, so as x approaches π/2, t approaches 0. Then, x = π/2 + t. Let's rewrite tan(3x) and tan(x) in terms of t.First, tan(x) = tan(π/2 + t) = -cot t, because tan(π/2 + t) = -cot t. Similarly, tan(3x) = tan(3π/2 + 3t) = tan(3π/2 + 3t). But tan(3π/2 + 3t) is also periodic with period π, so tan(3π/2 + 3t) = tan(π/2 + 3t) = -cot(3t). Wait, is that right?Wait, tan(3π/2 + 3t) = tan(π + π/2 + 3t) = tan(π/2 + 3t + π) = tan(π/2 + 3t) because tan is periodic with period π. Wait, no, adding π to the angle gives the same value for tan. So, tan(3π/2 + 3t) = tan(π/2 + π + 3t) = tan(π/2 + 3t + π) = tan(π/2 + 3t) since tan(θ + π) = tan θ. But actually, tan(3π/2 + 3t) is the same as tan(π/2 + π + 3t) = tan(π/2 + 3t + π) = tan(π/2 + 3t) because the period is π. Wait, but tan(π/2 + θ) is -cot θ. So, tan(3π/2 + 3t) = tan(π/2 + π + 3t) = tan(π/2 + 3t + π) = tan(π/2 + 3t) = -cot(3t). Wait, but tan(π/2 + θ) = -cot θ. So, if we have tan(π/2 + 3t), that's -cot(3t). So yes, tan(3x) = tan(3π/2 + 3t) = -cot(3t). Similarly, tan(x) = tan(π/2 + t) = -cot(t). So, the original expression becomes:[-cot(3t)] / [-cot(t)] = cot(3t)/cot(t) = [cos(3t)/sin(3t)] / [cos(t)/sin(t)] = [cos(3t)/sin(3t)] * [sin(t)/cos(t)] = [cos(3t) sin(t)] / [sin(3t) cos(t)].Now, as t approaches 0, we can use the approximations for sin and cos. Let me recall that as t approaches 0, sin(t) ≈ t, cos(t) ≈ 1 - t²/2, and similarly for higher multiples. Let's apply these approximations.First, sin(t) ≈ t. sin(3t) ≈ 3t. cos(t) ≈ 1 - t²/2. cos(3t) ≈ 1 - (3t)²/2 = 1 - 9t²/2.Substituting these into the expression:[ (1 - 9t²/2) * t ] / [ (3t) * (1 - t²/2) ] = [ t - 9t³/2 ] / [ 3t - 3t³/2 ].Factor t from numerator and denominator:Numerator: t(1 - 9t²/2)Denominator: 3t(1 - t²/2)Cancel out t:(1 - 9t²/2) / [3(1 - t²/2)].Now, as t approaches 0, the terms with t² go to zero, so the expression approaches (1 - 0)/(3*(1 - 0)) = 1/3. So, again, the limit is 1/3. That matches the previous method.Alternatively, maybe use L’Hospital’s Rule? Let me check if the original expression is indeterminate. As x approaches π/2, tan(x) approaches infinity, so tan(3x) also approaches infinity or some value? Wait, when x approaches π/2, 3x approaches 3π/2. At 3π/2, tan(3x) is undefined, but approaching from the left, 3x approaches 3π/2 from below, so tan(3x) approaches positive infinity, and from above, it approaches negative infinity. Wait, maybe I need to check left and right limits?Wait, actually, as x approaches π/2 from the left (x → (π/2)^-), then 3x approaches 3π/2 from the left, so tan(3x) approaches positive infinity? Wait, tan(θ) approaches positive infinity as θ approaches π/2 from the left. So, 3x approaching 3π/2 from the left would mean θ approaching 3π/2 from the left, which is equivalent to approaching π/2 from the left after subtracting π. But tan has period π, so tan(3π/2 - ε) = tan(π/2 - ε) = cot ε, which approaches infinity as ε approaches 0 from the right. Wait, maybe my previous substitution was better.Alternatively, perhaps using L’Hospital's Rule on the original expression. Let me write tan(3x)/tan(x) as [sin(3x)/cos(3x)] / [sin(x)/cos(x)] = [sin(3x) cos(x)] / [sin(x) cos(3x)]. So the expression becomes sin(3x)cos(x)/(sin(x)cos(3x)).As x approaches π/2, sin(x) approaches 1, cos(x) approaches 0. Similarly, sin(3x) approaches sin(3π/2) = -1, cos(3x) approaches cos(3π/2) = 0. Wait, but plugging x = π/2 into sin(3x) is sin(3π/2) = -1, and cos(3x) is cos(3π/2) = 0. So, the numerator becomes (-1)(0) = 0, and the denominator becomes (1)(0) = 0. So we have 0/0, which is indeterminate. Therefore, L’Hospital’s Rule can be applied.So, taking the derivative of the numerator and the derivative of the denominator:Numerator: d/dx [sin(3x)cos(x)] = 3cos(3x)cos(x) - sin(3x)sin(x)Denominator: d/dx [sin(x)cos(3x)] = cos(x)cos(3x) - 3sin(x)sin(3x)So, applying L’Hospital once, the limit becomes:[3cos(3x)cos(x) - sin(3x)sin(x)] / [cos(x)cos(3x) - 3sin(x)sin(3x)]Now, evaluate this as x approaches π/2. Let's compute each term:cos(3x) as x approaches π/2: cos(3π/2) = 0.cos(x) as x approaches π/2: 0.sin(3x) as x approaches π/2: sin(3π/2) = -1.sin(x) as x approaches π/2: 1.So, substituting into the numerator:3 * 0 * 0 - (-1) * 1 = 0 + 1 = 1Denominator:0 * 0 - 3 * 1 * (-1) = 0 + 3 = 3Therefore, the limit after applying L’Hospital's Rule is 1/3. So that's another confirmation.Alternatively, maybe using series expansion. Let me consider expanding tan(3x) and tan(x) around x = π/2. Let's set t = π/2 - x, so as x approaches π/2, t approaches 0 from the positive side. Then, x = π/2 - t.Then, tan(x) = tan(π/2 - t) = cot(t) ≈ 1/t - t/3 - t³/45 + ... for small t.Similarly, tan(3x) = tan(3(π/2 - t)) = tan(3π/2 - 3t) = cot(3t) ≈ 1/(3t) - (3t)/3 - (3t)³/45 + ... = 1/(3t) - t - (27 t³)/45 + ... ≈ 1/(3t) - t - (3 t³)/5 + ...So, tan(3x)/tan(x) ≈ [1/(3t) - t - ...] / [1/t - t/3 - ...] = [ (1/(3t))(1 - 3t² - ...) ] / [ (1/t)(1 - t²/3 - ...) ] = (1/(3t) * (1 - 3t²)) / (1/t * (1 - t²/3)) = (1/3)(1 - 3t²) / (1 - t²/3).As t approaches 0, the higher-order terms vanish, so this becomes (1/3)(1)/1 = 1/3. Again, the same result.Therefore, all methods point to the limit being 1/3. So I can be confident that the answer is 1/3.Final Answerboxed{dfrac{1}{3}}</think>To find the limit of (lim _{x rightarrow frac{pi}{2}} frac{tan 3x}{tan x}), we can use trigonometric identities and substitutions.First, we use the triple angle formula for (tan 3x):[tan 3x = frac{3 tan x - tan^3 x}{1 - 3 tan^2 x}]Substituting this into the original expression, we get:[frac{tan 3x}{tan x} = frac{frac{3 tan x - tan^3 x}{1 - 3 tan^2 x}}{tan x} = frac{3 tan x - tan^3 x}{tan x (1 - 3 tan^2 x)} = frac{3 - tan^2 x}{1 - 3 tan^2 x}]As (x rightarrow frac{pi}{2}), (tan x rightarrow infty). Factoring (tan^2 x) from the numerator and denominator:[frac{3 - tan^2 x}{1 - 3 tan^2 x} = frac{tan^2 x left(frac{3}{tan^2 x} - 1right)}{tan^2 x left(frac{1}{tan^2 x} - 3right)} = frac{-1}{-3} = frac{1}{3}]We can also verify this using substitution. Let (t = x - frac{pi}{2}), so as (x rightarrow frac{pi}{2}), (t rightarrow 0). Then:[tan x = tanleft(frac{pi}{2} + tright) = -cot t quad text{and} quad tan 3x = tanleft(frac{3pi}{2} + 3tright) = -cot 3t]The expression becomes:[frac{-cot 3t}{-cot t} = frac{cot 3t}{cot t} = frac{cos 3t sin t}{sin 3t cos t}]Using approximations for small (t):[cos 3t approx 1 - frac{9t^2}{2}, quad sin 3t approx 3t, quad cos t approx 1 - frac{t^2}{2}, quad sin t approx t]Substituting these approximations:[frac{(1 - frac{9t^2}{2}) t}{(3t)(1 - frac{t^2}{2})} = frac{t - frac{9t^3}{2}}{3t - frac{3t^3}{2}} approx frac{1}{3}]Using L'Hospital's Rule on the expression (frac{sin 3x cos x}{sin x cos 3x}) (which is indeterminate as (0/0)):[frac{d}{dx}[sin 3x cos x] = 3 cos 3x cos x - sin 3x sin x][frac{d}{dx}[sin x cos 3x] = cos x cos 3x - 3 sin x sin 3x]Evaluating these derivatives at (x = frac{pi}{2}) gives the limit as (frac{1}{3}).Thus, the limit is:[boxed{dfrac{1}{3}}],<think>Okay, so I need to find the limit of tan(3x)/tan(x) as x approaches π/2. Hmm, let's start by recalling some trigonometric identities because dealing with tangents here. I remember that tan(3x) can be expanded using the triple angle formula. Let me try to remember that formula.The triple angle formula for tangent is tan(3x) = (3 tan x - tan³ x) / (1 - 3 tan² x). Let me check if that's right. Yeah, I think that's correct. So if I substitute that into the expression, maybe I can simplify the ratio.So the original expression becomes [ (3 tan x - tan³ x) / (1 - 3 tan² x) ] divided by tan x. Which simplifies to (3 tan x - tan³ x) / [ (1 - 3 tan² x) tan x ]. Then, dividing numerator and denominator by tan x, we get (3 - tan² x) / (1 - 3 tan² x). Wait, that seems right. So the expression simplifies to (3 - tan² x)/(1 - 3 tan² x). Now, as x approaches π/2, tan x approaches infinity. Let me verify that. Yes, because at π/2, cosine is zero, so tan x = sin x / cos x goes to infinity. Therefore, both tan x and tan 3x will approach infinity, but maybe their ratio has a finite limit. Let's see.So substituting tan x with t, where t approaches infinity. Then tan² x is t². So substituting into the simplified expression: (3 - t²)/(1 - 3 t²). Let's divide numerator and denominator by t² to make it easier. Then we get [3/t² - 1]/[1/t² - 3]. As t approaches infinity, 3/t² approaches 0 and 1/t² approaches 0. So this simplifies to (-1)/(-3) = 1/3. So the limit is 1/3? Wait, that seems straightforward. But let me check again.Alternatively, maybe I can use L’Hospital’s Rule since both the numerator and denominator approach infinity as x approaches π/2. Let's try that approach to confirm.So, the original function is tan(3x)/tan(x). Let me write that as sin(3x)/cos(3x) divided by sin(x)/cos(x), which becomes [sin(3x)/cos(3x)] * [cos(x)/sin(x)] = [sin(3x) cos(x)] / [sin(x) cos(3x)]. But as x approaches π/2, let's see what happens. Let me set t = x - π/2, so as x approaches π/2, t approaches 0. Then, substitute x = π/2 + t. Let me see if that substitution helps. So, sin(3x) = sin(3(π/2 + t)) = sin(3π/2 + 3t) = -cos(3t) because sin(3π/2 + θ) = -cosθ. Similarly, cos(3x) = cos(3π/2 + 3t) = sin(3t) since cos(3π/2 + θ) = sinθ (with a sign maybe? Let me check). Wait, cos(3π/2 + θ) = cos(3π/2)cosθ - sin(3π/2)sinθ = 0*cosθ - (-1)*sinθ = sinθ. So cos(3x) = sin(3t). Similarly, sin(x) = sin(π/2 + t) = cos t. Cos(x) = cos(π/2 + t) = -sin t. Putting all these into the expression [sin(3x) cos(x)] / [sin(x) cos(3x)]:Numerator: sin(3x) cos(x) = (-cos(3t)) * (-sin t) = cos(3t) sin tDenominator: sin(x) cos(3x) = cos t * sin(3t)So the entire expression becomes [cos(3t) sin t] / [cos t sin(3t)].Simplify this expression: [sin t / cos t] * [cos(3t)/sin(3t)] = tan t * cot(3t) = tan t / tan(3t)Hmm, interesting. So now the expression becomes tan t / tan(3t) as t approaches 0. Because as x approaches π/2, t approaches 0. So the limit becomes lim_{t→0} tan t / tan 3t.Now, as t approaches 0, tan t ≈ t and tan 3t ≈ 3t, so the limit should be (t)/(3t) = 1/3. So that confirms the earlier result. Alternatively, using L’Hospital’s Rule on lim_{t→0} tan t / tan 3t, which is 0/0. Taking derivatives: derivative of numerator is sec² t, derivative of denominator is 3 sec² 3t. So the limit becomes sec² 0 / (3 sec² 0) = 1 / 3. So yes, that gives 1/3 again.Alternatively, maybe there's another approach. Let's think again. Since x approaches π/2, perhaps we can use substitution. Let me set y = π/2 - x, so as x approaches π/2, y approaches 0. Then, tan x = tan(π/2 - y) = cot y = 1/tan y. Similarly, tan 3x = tan(3π/2 - 3y) = tan(π + π/2 - 3y) = tan(π/2 - 3y + π) = tan(π/2 - 3y + π) = tan(3π/2 - 3y). But tan(3π/2 - 3y) is undefined, but let me recall that tan(3π/2 - θ) = -cot θ. Wait, tan(3π/2 - θ) = tan(π + π/2 - θ) = tan(π + (π/2 - θ)) = tan(π/2 - θ) = cot θ, but with periodicity. Wait, perhaps this substitution is getting messy. Let me check.Alternatively, since tan(3π/2 - 3y) = tan(π/2 + π - 3y) = tan(π/2 + (π - 3y)) = -cot(π - 3y). Because tan(π/2 + α) = -cot α. So, tan(3π/2 - 3y) = -cot(π - 3y). But cot(π - 3y) = -cot(3y), because cot(π - θ) = -cot θ. Therefore, tan(3π/2 - 3y) = -(-cot 3y) = cot 3y. Therefore, tan(3x) = cot 3y. Therefore, the original expression tan(3x)/tan(x) = cot 3y / cot y = (cos 3y / sin 3y) / (cos y / sin y) = (cos 3y sin y) / (sin 3y cos y). So, that's the same as [cos 3y / cos y] * [sin y / sin 3y]. Now, as y approaches 0, let's use approximations. cos 3y ≈ 1 - (9y²)/2, cos y ≈ 1 - y²/2, so cos3y / cos y ≈ [1 - (9y²)/2] / [1 - y²/2] ≈ (1 - 9y²/2)(1 + y²/2) ≈ 1 - 9y²/2 + y²/2 - 9y^4/4 ≈ 1 - 4y². Similarly, sin y ≈ y - y^3/6, sin 3y ≈ 3y - (27y^3)/6 = 3y - 9y^3/2. So sin y / sin 3y ≈ [y - y^3/6] / [3y - 9y^3/2] = [1 - y²/6] / [3 - 9y²/2] ≈ (1/3)(1 - y²/6) / (1 - 3y²/2) ≈ (1/3)(1 - y²/6 + 3y²/2) = (1/3)(1 + (3y²/2 - y²/6)) = (1/3)(1 + (9y²/6 - y²/6)) = (1/3)(1 + 8y²/6) = (1/3)(1 + 4y²/3). Multiplying the two parts: [1 - 4y²] * [1/3 + 4y²/9] ≈ (1)(1/3) + (1)(4y²/9) - 4y²*(1/3) = 1/3 + (4/9 - 4/3)y² = 1/3 - (8/9)y². As y approaches 0, the limit is 1/3. So again, this approach also gives 1/3. Alternatively, using L’Hospital’s Rule directly on the original expression. Let me see. The original limit is tan(3x)/tan(x) as x→π/2. Let me check if both numerator and denominator approach infinity. Yes, as x approaches π/2, cos x approaches 0, so tan x = sin x / cos x approaches infinity (since sin x approaches 1 and cos x approaches 0). Similarly, tan 3x = sin 3x / cos 3x. When x is approaching π/2, 3x approaches 3π/2. So cos 3x approaches 0, but sin 3x approaches -1 (since sin(3π/2) = -1). Therefore, tan 3x approaches (-1)/0, which is -infinity if approaching from the left and +infinity from the right? Wait, actually, when x approaches π/2 from the left (x < π/2), then 3x approaches 3π/2 from the left. So cos 3x approaches 0 from the positive side (since cos(3π/2 - ε) = cos(π/2*3 - ε) = sin ε ≈ ε, so positive? Wait, maybe this is more complicated. Let me think.Wait, x approaches π/2. Let's take x approaching π/2 from the left: x = π/2 - ε where ε→0+. Then 3x = 3π/2 - 3ε. So cos(3x) = cos(3π/2 - 3ε) = -sin(3ε) ≈ -3ε. So cos(3x) approaches 0 from the negative side. So tan(3x) = sin(3x)/cos(3x). Sin(3x) = sin(3π/2 - 3ε) = -cos(3ε) ≈ -1. Therefore, tan(3x) ≈ (-1)/(-3ε) = 1/(3ε). So as ε→0+, tan(3x) approaches +infinity. Similarly, tan x = tan(π/2 - ε) = cot ε ≈ 1/ε. Therefore, tan(3x)/tan(x) ≈ (1/(3ε))/(1/ε) = 1/3. Similarly, if x approaches π/2 from the right: x = π/2 + ε, ε→0+. Then 3x = 3π/2 + 3ε. Cos(3x) = cos(3π/2 + 3ε) = sin(3ε) ≈ 3ε. So tan(3x) = sin(3x)/cos(3x) = sin(3π/2 + 3ε) / cos(3x) = (-1)/ (3ε) ≈ -1/(3ε). Tan x = tan(π/2 + ε) = -cot ε ≈ -1/ε. Therefore, tan(3x)/tan(x) ≈ (-1/(3ε))/(-1/ε) = 1/3. So from both sides, the limit is 1/3. Therefore, regardless of the approach, the limit is 1/3. All methods confirm this. So I think the answer is 1/3.Final AnswerThe limit is boxed{dfrac{1}{3}}.</think>To calculate the limit of the function (lim _{x rightarrow frac{pi}{2}} frac{tan 3x}{tan x}), we can use trigonometric identities and substitutions.First, we use the triple angle formula for tangent:[tan 3x = frac{3 tan x - tan^3 x}{1 - 3 tan^2 x}]Substituting this into the original expression, we get:[frac{tan 3x}{tan x} = frac{frac{3 tan x - tan^3 x}{1 - 3 tan^2 x}}{tan x} = frac{3 tan x - tan^3 x}{(1 - 3 tan^2 x) tan x} = frac{3 - tan^2 x}{1 - 3 tan^2 x}]As (x rightarrow frac{pi}{2}), (tan x rightarrow infty). Let (t = tan x), then as (t rightarrow infty), the expression becomes:[frac{3 - t^2}{1 - 3 t^2}]Dividing the numerator and denominator by (t^2), we get:[frac{frac{3}{t^2} - 1}{frac{1}{t^2} - 3} rightarrow frac{0 - 1}{0 - 3} = frac{-1}{-3} = frac{1}{3}]We can also verify this using substitution. Let (t = x - frac{pi}{2}), so as (x rightarrow frac{pi}{2}), (t rightarrow 0). Then:[tan x = tanleft(frac{pi}{2} - tright) = cot t approx frac{1}{t}][tan 3x = tanleft(frac{3pi}{2} - 3tright) = cot 3t approx frac{1}{3t}]Thus, the expression becomes:[frac{tan 3x}{tan x} approx frac{frac{1}{3t}}{frac{1}{t}} = frac{1}{3}]Using L'Hospital's Rule on the transformed limit (lim_{t rightarrow 0} frac{tan t}{tan 3t}) (which is in the form (frac{0}{0})):[lim_{t rightarrow 0} frac{sec^2 t}{3 sec^2 3t} = frac{1}{3}]All methods confirm that the limit is:[boxed{dfrac{1}{3}}]]

🔑:## SolutionSubstitution:x=y+frac{pi}{2} Rightarrow y=x-frac{pi}{2}x rightarrow frac{pi}{2} Rightarrow y rightarrow 0We get:begin{aligned}& lim _{x rightarrow frac{pi}{2}} frac{tan 3 x}{tan x}=lim _{y rightarrow 0} frac{tan 3left(y+frac{pi}{2}right)}{tan left(y+frac{pi}{2}right)}=lim _{y rightarrow 0} frac{tan left(3 y+frac{3 pi}{2}right)}{tan left(y+frac{pi}{2}right)}= & =lim _{y rightarrow 0} frac{-cot(3 y)}{-cot y}=lim _{y rightarrow 0} frac{tan y}{tan 3 y}=end{aligned}Using the substitution of equivalent infinitesimals:tan y sim y, as y rightarrow 0tan 3 y sim 3 y, as y rightarrow 0 (since 3 y rightarrow 0)We get:=lim _{y rightarrow 0} frac{y}{3 y}=lim _{y rightarrow 0} frac{1}{3}=frac{1}{3}## Problem Kuznetsov Limits 13-6