🧠:[<think>Okay, so I need to figure out how many extreme values the function f(x) = x³ + 3x² + 3x - a has. The options are 2, 1, 0, or it depends on a. Hmm, let me start by recalling what extreme values are. They are the maximum or minimum points on the function, right? So, to find those, I should take the derivative of the function and find the critical points where the derivative is zero or undefined. Then determine if those points are maxima or minima.First, let's compute the derivative of f(x). The derivative of x³ is 3x², the derivative of 3x² is 6x, the derivative of 3x is 3, and the derivative of -a is 0 since a is a constant. So, f'(x) = 3x² + 6x + 3. Now, to find critical points, set f'(x) = 0. So:3x² + 6x + 3 = 0Let me divide both sides by 3 to simplify:x² + 2x + 1 = 0Wait a second, this simplifies to x² + 2x + 1 = 0. That factors into (x + 1)² = 0. So, the only solution is x = -1. So, there is only one critical point at x = -1. But to determine if that critical point is a maximum or minimum, I can use the second derivative test. Let's compute the second derivative f''(x). The derivative of 3x² is 6x, the derivative of 6x is 6, and the derivative of 3 is 0. So, f''(x) = 6x + 6.Plugging x = -1 into the second derivative:f''(-1) = 6*(-1) + 6 = -6 + 6 = 0Hmm, the second derivative is zero. That means the test is inconclusive. So, I need another way to determine if x = -1 is a maximum, minimum, or a saddle point (inflection point). Maybe I can look at the sign changes in the first derivative around x = -1.Let's pick a value slightly less than -1, say x = -2. Plugging into f'(x):f'(-2) = 3*(-2)^2 + 6*(-2) + 3 = 3*4 - 12 + 3 = 12 - 12 + 3 = 3. So, positive.Now, pick a value slightly more than -1, say x = 0. Plugging into f'(x):f'(0) = 3*0 + 6*0 + 3 = 3. Also positive. Wait, so the derivative is positive on both sides of x = -1. That means the function is increasing both before and after x = -1, so the critical point at x = -1 is not a maximum or minimum—it's a horizontal inflection point. Therefore, there are no extreme values?But hold on, the original function is a cubic. Cubic functions usually have one inflection point and can have a local maximum and minimum if the derivative has two real roots. But in this case, the derivative is a quadratic equation that only has one real root (a double root). So, the function doesn't have any local maxima or minima. So, the number of extreme values is 0? But wait, the answer options include 0 (C), but let me double-check.Wait, another way to think about it: if the first derivative has a double root, that means the graph of the derivative just touches the x-axis at x = -1. Since the leading coefficient of the derivative (which is a quadratic) is positive, the parabola opens upwards. So, the derivative is always positive except at x = -1 where it's zero. So, the function f(x) is always increasing, but with a horizontal tangent at x = -1. Therefore, there are no maxima or minima. So, the function is monotonically increasing with an inflection point at x = -1. Hence, the number of extreme values is zero.But wait, the original question is about "extreme values". In calculus, extreme values can refer to local or global extrema. For a cubic function like this, since it tends to infinity as x approaches infinity and negative infinity as x approaches negative infinity (since the leading term is x³ with positive coefficient), it doesn't have global extrema. But local extrema would require the function to change direction. But since the derivative doesn't cross zero (only touches), the function doesn't have local maxima or minima either. So, yes, the number of extreme values is zero, regardless of the value of a. But wait, the function is given as f(x) = x³ + 3x² + 3x - a. The parameter a is just a vertical shift. So, shifting the graph up or down by a doesn't affect the number of critical points or the shape of the graph. So, regardless of a, the derivative is the same, so the critical points are the same, and as we saw, there are no local maxima or minima. Therefore, the answer should be C: 0.But wait, let me check again. Suppose a is such that the function crosses the x-axis three times or once. But no, the number of real roots depends on a, but the question isn't about roots, it's about extreme values, which are maxima and minima. So, the existence of maxima and minima is determined by the derivative, which doesn't depend on a. Therefore, regardless of a, there are no extreme values. Therefore, the answer is C:0.Wait, but the options are A:2, B:1, C:0, D:Depends on a. Since the derivative doesn't depend on a, the critical points are the same, and there's only one critical point which is not an extremum. Therefore, the answer is C:0. So, the answer is C.Wait, but maybe I made a mistake in the derivative. Let me check again. f(x) = x³ + 3x² + 3x - a. Then f'(x) is 3x² + 6x + 3. Correct. Then setting to zero, dividing by 3: x² + 2x +1=0, which is (x+1)^2=0. So, only one critical point at x=-1. Then, since the derivative doesn't change sign around x=-1 (remains positive on both sides), there's no extremum. So, regardless of a, the function has zero extreme values. Therefore, the answer is C:0. So, the answer should be C. So, the correct option is C.Final Answerboxed{C}</think>To determine the number of extreme values of the function ( f(x) = x^3 + 3x^2 + 3x - a ), we start by finding its first derivative:[f'(x) = 3x^2 + 6x + 3]Setting the first derivative to zero to find critical points:[3x^2 + 6x + 3 = 0 implies x^2 + 2x + 1 = 0 implies (x + 1)^2 = 0 implies x = -1]This gives a single critical point at ( x = -1 ). To determine if this critical point is a maximum, minimum, or neither, we use the second derivative test. The second derivative is:[f''(x) = 6x + 6]Evaluating the second derivative at ( x = -1 ):[f''(-1) = 6(-1) + 6 = -6 + 6 = 0]Since the second derivative is zero, the test is inconclusive. We then examine the sign of the first derivative around ( x = -1 ). For values slightly less than (-1) (e.g., ( x = -2 )) and slightly more than (-1) (e.g., ( x = 0 )):[f'(-2) = 3(-2)^2 + 6(-2) + 3 = 12 - 12 + 3 = 3 quad (text{positive})][f'(0) = 3(0)^2 + 6(0) + 3 = 3 quad (text{positive})]The first derivative is positive on both sides of ( x = -1 ), indicating the function is increasing on both sides. Therefore, ( x = -1 ) is a point of inflection, not an extremum. Since the function ( f(x) ) is a cubic polynomial with a positive leading coefficient, it does not have any local or global maxima or minima. The parameter ( a ) only shifts the graph vertically and does not affect the critical points or the shape of the function. Thus, the number of extreme values is always zero.[boxed{C}],<think>Okay, so I have this problem here: I need to determine the number of extreme values of the function f(x) = x³ + 3x² + 3x - a. The options are A: 2, B: 1, C: 0, D: Depends on the value of a. Hmm, let's think through this step by step.First, I remember that extreme values of a function occur where the first derivative is zero or undefined, right? Since this is a polynomial function, it's differentiable everywhere, so I only need to check where the derivative is zero. Then, depending on the critical points, we can determine if they are maxima or minima by using the second derivative or the first derivative test.So, let's start by finding the first derivative of f(x). The function is f(x) = x³ + 3x² + 3x - a. Taking the derivative term by term:The derivative of x³ is 3x², the derivative of 3x² is 6x, the derivative of 3x is 3, and the derivative of -a is 0 since a is a constant. So, f'(x) = 3x² + 6x + 3.Now, to find critical points, we set f'(x) = 0:3x² + 6x + 3 = 0.Let me simplify this equation. First, divide both sides by 3 to make it easier:x² + 2x + 1 = 0.Wait, x² + 2x + 1 is a perfect square trinomial. That factors into (x + 1)² = 0. So, the solution is x = -1. So, there's only one critical point at x = -1. But hold on, critical points are where the derivative is zero. Since the derivative is a quadratic, normally a quadratic equation can have 0, 1, or 2 real roots. However, in this case, after simplifying, we got a repeated root at x = -1. So, does this mean there's only one critical point?Yes, exactly. So, the first derivative is zero only at x = -1, and that's the only critical point. Therefore, there's only one critical point. But does that necessarily mean there's only one extreme value?Well, in general, for a cubic function like this, the graph should have a local maximum and a local minimum if there are two critical points. However, if there's only one critical point, that would mean the graph has a point where the slope is zero, but it doesn't change direction from increasing to decreasing or vice versa. Wait, actually, for a cubic function with leading coefficient positive (since the x³ term is positive), the function tends to negative infinity as x approaches negative infinity and positive infinity as x approaches positive infinity. So, typically, a cubic function has one inflection point and two critical points (a local maximum and a local minimum). But in this case, since the derivative is a perfect square, leading to only one critical point, does that mean the graph has a horizontal tangent at that point but no maximum or minimum?Wait, maybe I need to check the second derivative to see if it's a maximum, minimum, or a point of inflection.Let me compute the second derivative. The first derivative is 3x² + 6x + 3, so the second derivative f''(x) is 6x + 6.Now, substitute x = -1 into the second derivative:f''(-1) = 6*(-1) + 6 = -6 + 6 = 0.Hmm, the second derivative is zero at x = -1. That means the test is inconclusive here. So, the second derivative test doesn't tell us whether it's a maximum or minimum. In such cases, we might need to use the first derivative test.Let's apply the first derivative test around x = -1. Since the critical point is at x = -1, let's check the sign of f'(x) just to the left and right of x = -1.Take a number slightly less than -1, say x = -2. Plug into f'(x):f'(-2) = 3*(-2)^2 + 6*(-2) + 3 = 3*4 - 12 + 3 = 12 - 12 + 3 = 3. That's positive.Take a number slightly more than -1, say x = 0. Plug into f'(x):f'(0) = 3*0 + 6*0 + 3 = 3. Also positive.So, the derivative is positive both to the left and right of x = -1. That means the function is increasing on both sides of x = -1. So, the critical point at x = -1 is neither a local maximum nor a local minimum; it's a saddle point or a point of inflection with a horizontal tangent.Therefore, the function f(x) has no local extreme values. Wait, but that seems counterintuitive. If the derivative is always positive except at x = -1 where it's zero, then the function is always increasing. So, there are no local maxima or minima. Hence, the function has no extreme values. So, the answer should be C: 0. But let me confirm this.Wait, if the function is strictly increasing (since the derivative is non-negative everywhere and only zero at a single point), then it doesn't have any local maxima or minima. That's correct. So, even though there's a critical point, it's not an extremum. So, the number of extreme values is zero. Therefore, the answer should be C: 0.But let me check again. Maybe I made a mistake in the derivative?Original function: x³ + 3x² + 3x - aFirst derivative: 3x² + 6x + 3. Correct.Set to zero: 3x² + 6x + 3 = 0 => x² + 2x + 1 = 0 => (x+1)^2 = 0. Correct, so only one critical point.Second derivative: 6x + 6. At x = -1, 6*(-1) + 6 = 0. Correct. So, second derivative test inconclusive.First derivative test: check the sign around x = -1. For x < -1, f'(x) positive, for x > -1, f'(x) positive. So, no change in sign. Hence, no extremum. Therefore, the function is monotonically increasing everywhere, with a horizontal tangent at x = -1. Therefore, there are no extreme values. So, the answer is C: 0. So, the correct answer should be C.But wait, looking at the answer choices: A:2, B:1, C:0, D: Depends on a.But wait, the function is f(x) = x³ + 3x² + 3x - a. The "-a" term is a vertical shift. But when we take the derivative, the "-a" disappears. So, the derivative is independent of a. Therefore, the critical points do not depend on a. So, regardless of the value of a, the function will have the same critical points, which is x = -1. And as we saw, there are no extreme values regardless of a. Therefore, the answer should be C:0. So, the answer is C.Wait, but maybe the question is about extreme values in the sense of global maxima or minima? For a cubic function, since it tends to infinity in both directions, there are no global maxima or minima. But local extrema would require the function to change direction. But in this case, the function is strictly increasing, so there are no local extrema. Therefore, regardless of a, the function has zero extreme values. So, answer C.But let me check with a graph. If a is varied, the graph shifts up or down, but its shape remains the same. So, shifting up or down doesn't affect the presence of local maxima or minima. Since the original function without the "-a" term is x³ + 3x² + 3x, which has a critical point at x = -1. Let's compute the original function at x = -1:f(-1) = (-1)^3 + 3*(-1)^2 + 3*(-1) - a = -1 + 3*1 + (-3) - a = -1 + 3 -3 - a = -1 - a.So, the point (-1, -1 -a) is the critical point. But since the function is strictly increasing, this point is just a horizontal tangent, not a maximum or minimum.Therefore, regardless of the value of a, the function doesn't have any extreme values. So, the answer is C:0.But the options include D: Depends on a. But according to this, it's C. So, why is D an option? Maybe I made a mistake.Wait, let me check if the function can have extreme values for certain a. Suppose a is chosen such that the horizontal tangent point coincides with a root or something. Wait, but extreme values are about maxima or minima, regardless of the function's value. Even if the function crosses the x-axis at that point, it doesn't affect whether it's a maximum or minimum. The vertical shift doesn't influence the derivative, so the critical points remain the same, and the nature of those critical points (maxima, minima, or neither) also remains the same. Therefore, the number of extreme values is independent of a.Therefore, the answer is C:0.Wait, but just to confirm, let's take specific values of a. Let's set a = 0. Then the function is x³ + 3x² + 3x. Let's see its graph. Since the derivative is always positive except at x=-1, the function is increasing everywhere. So, no local maxima or minima. If a = 1, the function is shifted down by 1 unit. Still, the derivative is the same, so the function is still increasing everywhere, just shifted vertically. So, still no maxima or minima. Similarly for any a. Therefore, regardless of a, there are no extreme values. Therefore, the answer is C:0. So, the correct choice is C.But the problem says "extreme values". In some contexts, extreme values can refer to global maxima or minima, but for a cubic function, which goes to infinity in both directions, there are no global maxima or minima. However, if the function had local maxima or minima, those would be considered extreme values. But in this case, since there are no local maxima or minima, the number of extreme values is zero. Therefore, the answer is C:0.Final Answerboxed{C}</think>To determine the number of extreme values of the function ( f(x) = x^3 + 3x^2 + 3x - a ), we start by finding the first derivative:[f'(x) = 3x^2 + 6x + 3]Setting the first derivative to zero to find critical points:[3x^2 + 6x + 3 = 0 implies x^2 + 2x + 1 = 0 implies (x + 1)^2 = 0 implies x = -1]This gives a single critical point at ( x = -1 ). Next, we check the second derivative to determine the nature of this critical point:[f''(x) = 6x + 6]Evaluating the second derivative at ( x = -1 ):[f''(-1) = 6(-1) + 6 = 0]Since the second derivative is zero, the test is inconclusive. We then use the first derivative test by checking the sign of ( f'(x) ) around ( x = -1 ):- For ( x < -1 ) (e.g., ( x = -2 )): ( f'(-2) = 3 ), which is positive.- For ( x > -1 ) (e.g., ( x = 0 )): ( f'(0) = 3 ), which is positive.The first derivative is positive on both sides of ( x = -1 ), indicating the function is increasing on both sides. Therefore, ( x = -1 ) is neither a local maximum nor a local minimum but a point of inflection with a horizontal tangent.Since the function ( f(x) ) is strictly increasing everywhere (despite the horizontal tangent at ( x = -1 )), it has no local extreme values. The vertical shift caused by the constant ( -a ) does not affect the derivative or the critical points, so the number of extreme values remains zero regardless of ( a ).Thus, the number of extreme values of the function is (boxed{C}).]

🔑:Since f'(x) = 3x^2 + 6x + 3 = 3(x + 1)^2 geq 0,∴ f(x) is an increasing function and has no extreme points.Therefore, the correct choice is boxed{C}.

🧠:[<think>Okay, so I need to find the last digit of 2002 raised to the power of 2002. That is, the units digit of 2002^2002. The options are A) 4, B) 2, C) 8, D) 0, E) 6. Hmm. Let me think. First, I remember that when you're dealing with the units digit of a number raised to a power, you only need to consider the units digit of the original number. Because the units digit of a product depends only on the units digits of the numbers being multiplied. So, for example, 12 * 13 = 156, and the units digit is 6, which comes from 2*3=6. So, in this case, the base is 2002. The units digit of 2002 is 2. So, the problem simplifies to finding the units digit of 2^2002. That should make things easier.So now, instead of dealing with 2002^2002, I can just focus on 2^2002. But 2^2002 is still a huge number. Obviously, I can't compute that directly. But maybe there's a pattern in the units digits of powers of 2. Let me recall. The units digits of powers of 2 cycle in a pattern. Let's write them out:2^1 = 2 → units digit 22^2 = 4 → units digit 42^3 = 8 → units digit 82^4 = 16 → units digit 62^5 = 32 → units digit 22^6 = 64 → units digit 42^7 = 128 → units digit 82^8 = 256 → units digit 6So, the pattern here is 2, 4, 8, 6, and then repeats every 4 powers. So the cycle length is 4. That means that every 4 exponents, the units digit cycles through 2, 4, 8, 6. So, to find the units digit of 2^2002, I need to figure out where 2002 falls in this cycle.To do that, I can divide 2002 by 4 and find the remainder. The remainder will tell me which position in the cycle it is. Let me recall that in modular arithmetic terms, we can write 2002 mod 4. If the remainder is 1, it's the first position (2); remainder 2, second position (4); remainder 3, third position (8); and if it's divisible by 4 (remainder 0), then it's the fourth position (6).So, let's compute 2002 divided by 4. 4*500 = 2000. So 2000 is divisible by 4, and 2002 - 2000 = 2. Therefore, 2002 divided by 4 is 500 with a remainder of 2. So, 2002 ≡ 2 mod 4.That means the exponent 2002 corresponds to the second position in the cycle. Looking back at the cycle: 1st position remainder 1 → 2, 2nd position remainder 2 → 4, 3rd position remainder 3 → 8, 4th position remainder 0 → 6. So remainder 2 corresponds to the second position, which is 4. Therefore, the units digit of 2^2002 is 4. Therefore, the answer should be A) 4.Wait, but let me double-check my reasoning here. So, the cycle is 4 numbers: 2,4,8,6. So exponents 1 to 4 give these units digits. Then exponent 5 would be the same as exponent 1, since 5 divided by 4 is 1 with a remainder of 1, right? So 2^5 has units digit 2 again. So, the remainder when divided by 4 tells you the position in the cycle. So, 2002 divided by 4 is 500 with remainder 2, so the remainder is 2, so that's the second position in the cycle, which is 4. So, that seems correct.But let me check with another exponent. For example, 2^6. 6 divided by 4 is 1 with a remainder of 2. So according to this, the units digit should be 4. But 2^6 is 64, which has a units digit of 4. Correct. Similarly, 2^7. 7 divided by 4 is 1 with a remainder of 3. So third position, which is 8. 2^7 is 128, units digit 8. Correct. 2^8. 8 divided by 4 is 2 with remainder 0. So fourth position, units digit 6. 2^8 is 256, correct. So the pattern holds.Therefore, since 2002 divided by 4 gives a remainder of 2, the units digit of 2^2002 is the second in the cycle, which is 4. Therefore, the answer should be A) 4.But wait, let me check another example to be thorough. Let's take 2^10. 10 divided by 4 is 2 with a remainder of 2. So the units digit should be 4. 2^10 is 1024, units digit 4. Correct. So that works. So seems like the logic is solid.Alternatively, maybe I can think of 2^2002 as (2^4)^500 * 2^2. Since 2002 = 4*500 + 2. So, 2^2002 = (2^4)^500 * 2^2. Now, 2^4 is 16, which has a units digit of 6. Then, (16)^500. The units digit of 16 raised to any power is 6, because 6*6=36, which still ends with 6. So, (16)^500 has a units digit of 6. Then, multiply by 2^2, which is 4. So, 6*4=24, which has a units digit of 4. So that gives the same result. Therefore, confirms that the units digit is 4.So, this method also leads to the same conclusion, which is good. So, both reasoning approaches confirm that the answer is 4. So, A) 4.But let me check once again with actual calculation for a smaller exponent. For example, 2^6, which we saw is 64. Let's use the exponent as 6. 6 divided by 4 gives a remainder of 2. So, according to the cycle, the units digit should be 4. 2^6 is 64, which is 4. Correct. Similarly, 2^2. 2 divided by 4 is 0 remainder 2. Wait, 2 divided by 4? Wait, no. Wait, when we do the exponent divided by 4, if the exponent is less than 4, we just take the remainder. So, 2 divided by 4 is 0 with remainder 2. So, remainder 2, so the units digit should be 4. But 2^2 is 4, correct. So, that's right.Wait, but here's a confusion. When the exponent is 2, the remainder is 2. But in the cycle, 2^1 is remainder 1, 2^2 is remainder 2, 2^3 is remainder 3, 2^4 is remainder 0. So, for exponent n, the remainder when divided by 4 is r, where r = n mod 4, except that if r is 0, then it's the fourth position.Wait, perhaps the confusion is about the mapping of remainder to the position. Let me clarify. For exponent n:- If n ≡ 1 mod 4, units digit is 2.- If n ≡ 2 mod 4, units digit is 4.- If n ≡ 3 mod 4, units digit is 8.- If n ≡ 0 mod 4 (i.e., divisible by 4), units digit is 6.So, 2002 mod 4 is 2002 divided by 4. 4*500 = 2000, so 2002 - 2000 = 2. Therefore, 2002 ≡ 2 mod 4. Therefore, units digit is 4. So, that's consistent.Alternatively, if we have exponent 4, which is 0 mod 4, units digit 6. Which is correct, 2^4=16.So, the mapping is clear. Therefore, 2002 mod 4 is 2, so units digit is 4. So, answer A.Therefore, I feel confident that the answer is 4, which is option A.Final Answerboxed{A}</think>To find the units digit of (2002^{2002}), we only need to consider the units digit of the base, which is 2. Therefore, we need to determine the units digit of (2^{2002}).The units digits of powers of 2 follow a repeating cycle: 2, 4, 8, 6. This cycle repeats every 4 exponents. To determine where (2002) falls in this cycle, we compute (2002 mod 4):[2002 div 4 = 500 text{ remainder } 2]Thus, (2002 equiv 2 mod 4). This means the units digit of (2^{2002}) is the second element in the cycle, which is 4.Therefore, the units digit of (2002^{2002}) is (boxed{A}).,<think>Okay, so I need to find the last digit of 2002 raised to the power of 2002. Hmm, the question is asking for the units digit of that huge number. I remember that when dealing with units digits, especially for exponents, there's a pattern that repeats every certain number of powers. Maybe I can use that to figure it out without having to compute the entire number. Let me think.First, the base here is 2002. The units digit of the base is 2, right? Because 2002 ends with a 2. So actually, the problem simplifies to finding the units digit of 2 raised to the power of 2002. That makes sense because the units digit of a product only depends on the units digits of the numbers being multiplied. So 2002^2002 will have the same units digit as 2^2002. Cool, that simplifies things a lot.Now, I need to find the pattern in the units digit of powers of 2. Let me recall the units digits of the first few powers of 2:2^1 = 2 --> units digit 22^2 = 4 --> units digit 42^3 = 8 --> units digit 82^4 = 16 --> units digit 62^5 = 32 --> units digit 22^6 = 64 --> units digit 42^7 = 128 --> units digit 82^8 = 256 --> units digit 6Hmm, okay, so the pattern for the units digit of 2^n is 2, 4, 8, 6, and then it repeats every 4 powers. So the cycle is 2, 4, 8, 6, 2, 4, 8, 6, etc., with a cycle length of 4. That means every time the exponent increases by 4, the units digit cycles through these four numbers.Therefore, to find the units digit of 2^2002, I need to determine where 2002 falls in this 4-cycle. In other words, I need to find the remainder when 2002 is divided by 4. The remainder will tell me which position in the cycle it is, and thus what the units digit is.Let me compute 2002 divided by 4. Let's do the division:4 goes into 20 five times (5*4=20), remainder 0.Bring down the 0: 0 divided by 4 is 0.Bring down the 2: 2 divided by 4 is 0 with a remainder of 2.So 2002 divided by 4 is 500 with a remainder of 2. Wait, 4*500 = 2000, and 2002 - 2000 = 2. Yep, the remainder is 2.So, the remainder is 2. That means 2002 is equivalent to 2 modulo 4. Therefore, 2^2002 will have the same units digit as 2^2, which is 4. So according to this, the answer should be 4, which is option A.Wait, but let me double-check to make sure I didn't make a mistake. Sometimes when working with cycles, it's easy to get confused about the correspondence between the remainder and the position in the cycle.So, if the cycle is 2, 4, 8, 6, then the first position (remainder 1) is 2, remainder 2 is 4, remainder 3 is 8, and remainder 0 (since 4 divides into the exponent) would correspond to 6. Let me verify that.For example:2^1: exponent 1, remainder 1 when divided by 4: units digit 2 (correct)2^2: exponent 2, remainder 2: units digit 4 (correct)2^3: exponent 3, remainder 3: units digit 8 (correct)2^4: exponent 4, remainder 0: units digit 6 (correct)2^5: exponent 5, remainder 1: units digit 2 (correct)So yes, if the exponent divided by 4 leaves a remainder of r, then the units digit is the r-th element in the cycle [2, 4, 8, 6]. But wait, if the remainder is 0, that's like the 4th position, which is 6. So in general, for exponent n:If n mod 4 = 1: 2If n mod 4 = 2: 4If n mod 4 = 3: 8If n mod 4 = 0: 6So in our case, 2002 mod 4 = 2, so the units digit is the second element in the cycle, which is 4. That still checks out. Therefore, the answer should be 4, which is option A.But just to make sure, let me test this with another exponent. Let's take exponent 6. 6 mod 4 is 2, so according to the cycle, units digit should be 4. 2^6 is 64, which has units digit 4. Correct. Exponent 7: 7 mod 4 is 3, units digit 8. 2^7 is 128, which is 8. Correct. Exponent 8: 8 mod 4 is 0, units digit 6. 2^8 is 256, which is 6. Correct. Exponent 9: 9 mod 4 is 1, units digit 2. 2^9 is 512, which is 2. Correct. So the pattern holds. Therefore, my reasoning seems solid.Therefore, the units digit of 2002^2002 is the same as that of 2^2002, which is 4, so the correct answer is A.Final Answerboxed{A}</think>To find the units digit of (2002^{2002}), we focus on the units digit of the base, which is 2. Therefore, we need to determine the units digit of (2^{2002}).The units digits of powers of 2 follow a repeating cycle: 2, 4, 8, 6. This cycle repeats every 4 exponents. To determine the position of (2^{2002}) in this cycle, we compute the remainder of (2002) when divided by 4.[2002 div 4 = 500 text{ remainder } 2]This means (2002 equiv 2 pmod{4}). The units digit of (2^{2002}) corresponds to the second number in the cycle [2, 4, 8, 6], which is 4.Thus, the units digit of (2002^{2002}) is (boxed{A}).]

🔑:The last digit (that is, the units digit) of (2002)^{2002} is(A) 4(B) 2(C) 8(D) 0(E) 6## SolutionThe units digit of (2002)^{2002} is the same as the units digit of 2^{2002}, since the first three digits of 2002 do not affect the units digit.We write out the first few powers of 2 and check for the units digit.| n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 || :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: || 2^{n} | 2 | 4 | 8 | 16 | 32 | 64 | 128 | 256 | 512 |From this table, we see that the units digits repeat every 4 powers. So the units digit of 2^{2000} will be 6 , and thus the units digit of 2^{2002} (and so also of (2002)^{2002} ) will be 4 .

🧠:[<think>Okay, so I have this problem here about vectors. Let me try to understand it step by step. The problem says:Given that the magnitude of vector a is 2, the magnitude of vector b is 3, and the angle between them is 60 degrees. Then we have two vectors c and d defined as c = 12a + 3b and d = 3a + k b. We need to find the value of the real number k for two cases: (1) when c is parallel to d, and (2) when c is perpendicular to d.Alright, so first, I need to recall what it means for two vectors to be parallel or perpendicular. If two vectors are parallel, one is a scalar multiple of the other. That is, if c is parallel to d, then there exists some scalar λ such that c = λ d. On the other hand, if two vectors are perpendicular, their dot product is zero. So, c · d = 0.Given that, let's tackle each part one by one.Starting with part (1): c is parallel to d.So, if c is parallel to d, then c = λ d for some scalar λ. Let's write that out:12a + 3b = λ (3a + k b)Expanding the right-hand side, we get:12a + 3b = 3λ a + λ k bSince vectors a and b are not necessarily in the same direction (they have an angle of 60 degrees between them), they are linearly independent. Therefore, the coefficients of a and b on both sides must be equal. So, we can set up the equations:For the coefficients of a:12 = 3λFor the coefficients of b:3 = λ kSo, solving the first equation: 12 = 3λ => λ = 12 / 3 = 4.Then plug λ = 4 into the second equation: 3 = 4k => k = 3 / 4 = 0.75.So, for part (1), k should be 3/4. That seems straightforward.But wait, let me double-check. The reasoning is that since a and b are not colinear (since the angle between them is 60 degrees, not 0 or 180), they form a basis in the plane, so any vector can be uniquely expressed as a linear combination of a and b. Therefore, the coefficients must indeed be equal. So, yes, the value of k is 3/4. That seems correct.Now moving on to part (2): c is perpendicular to d.For this, the dot product of c and d should be zero. So:c · d = 0Let me compute c · d. Since c = 12a + 3b and d = 3a + k b, their dot product is:(12a + 3b) · (3a + k b) = 12*3 (a · a) + 12*k (a · b) + 3*3 (b · a) + 3*k (b · b)Simplify this expression:= 36 (a · a) + 12k (a · b) + 9 (b · a) + 3k (b · b)But remember that the dot product is commutative, so a · b = b · a. Therefore, we can combine the cross terms:= 36 |a|² + (12k + 9)(a · b) + 3k |b|²Now, we know |a| = 2, so |a|² = 4. Similarly, |b| = 3, so |b|² = 9. The angle between a and b is 60 degrees, so a · b = |a||b|cos(60°) = 2*3*(0.5) = 3.Substituting these values into the equation:= 36*4 + (12k + 9)*3 + 3k*9Let me compute each term:36*4 = 144(12k + 9)*3 = 36k + 273k*9 = 27kAdding them all together:144 + 36k + 27 + 27k = 144 + 27 + (36k + 27k) = 171 + 63kSo, the dot product c · d = 171 + 63k. For them to be perpendicular, this must equal zero:171 + 63k = 0Solving for k:63k = -171k = -171 / 63Simplify the fraction. Let's divide numerator and denominator by 9:-171 ÷ 9 = -1963 ÷ 9 = 7So, k = -19/7 ≈ -2.714...Therefore, for part (2), k is -19/7.Let me check my calculations again to make sure I didn't make any errors.First, computing the dot product:c · d = (12a + 3b) · (3a + k b) = 12*3 (a · a) + 12*k (a · b) + 3*3 (b · a) + 3*k (b · b)Which is 36(a · a) + 12k(a · b) + 9(a · b) + 3k(b · b)Yes, then combining terms:36|a|² + (12k + 9)(a · b) + 3k|b|²Substituting the known values:36*(4) + (12k + 9)*(3) + 3k*(9)Calculates to:144 + 36k + 27 + 27k = 144 + 27 + 63k = 171 + 63kSet to zero:171 + 63k = 0 => k = -171/63 = -19/7Yes, that seems correct.Wait, but let me just check if I made a mistake in combining coefficients. Let's break it down again step by step.Original expansion:12a · 3a = 36 a · a12a · k b = 12k a · b3b · 3a = 9 b · a3b · k b = 3k b · bSo total:36(a · a) + 12k(a · b) + 9(a · b) + 3k(b · b)Then combining the middle terms: (12k + 9)(a · b)So yes, that's right. Then coefficients:a · a = |a|² = 4a · b = 3 as calculated beforeb · b = |b|² = 9Therefore, substituting:36*4 = 144(12k + 9)*3 = 36k + 273k*9 = 27kAdding 144 + 36k + 27 + 27k = 171 + 63k, correct.So, yes, k is -19/7. So that should be the answer for part (2).Therefore, summarizing:(1) For c parallel to d, k = 3/4(2) For c perpendicular to d, k = -19/7But just to make sure, let me check if these answers make sense dimensionally or if there's any inconsistency.First, for the parallel case, since c is a combination of a and b with coefficients 12 and 3, and d is 3a + k b. To be parallel, the ratio of the coefficients of a and b in c and d should be the same. That is, the ratio 12/3 (coefficient of a in c over coefficient of a in d) should equal 3/k (coefficient of b in c over coefficient of b in d). So:12/3 = 3/k => 4 = 3/k => k = 3/4. Exactly what we found. So that checks out.For the perpendicular case, another way to check could be plugging the value back into the original equation.Let me compute c · d with k = -19/7.Compute 171 + 63k = 171 + 63*(-19/7) = 171 - (63/7)*19 = 171 - 9*19 = 171 - 171 = 0. Yes, that works. So that's correct.Therefore, I feel confident that the answers are correct.Final AnswerFor (1): boxed{dfrac{3}{4}} For (2): boxed{-dfrac{19}{7}}</think>Given ( |vec{a}| = 2 ), ( |vec{b}| = 3 ), and the angle between (vec{a}) and (vec{b}) is (60^circ). Vectors (vec{c}) and (vec{d}) are defined as (vec{c} = 12vec{a} + 3vec{b}) and (vec{d} = 3vec{a} + kvec{b}). We need to find the value of (k) for which (1) (vec{c} parallel vec{d}) and (2) (vec{c} perp vec{d}). (1) For (vec{c} parallel vec{d}):If (vec{c}) is parallel to (vec{d}), there exists a scalar (lambda) such that (vec{c} = lambda vec{d}). This gives:[12vec{a} + 3vec{b} = lambda (3vec{a} + kvec{b})]Equating coefficients of (vec{a}) and (vec{b}):[12 = 3lambda quad text{and} quad 3 = lambda k]Solving for (lambda) from the first equation:[lambda = frac{12}{3} = 4]Substituting (lambda = 4) into the second equation:[3 = 4k implies k = frac{3}{4}] (2) For (vec{c} perp vec{d}):If (vec{c}) is perpendicular to (vec{d}), their dot product is zero:[(12vec{a} + 3vec{b}) cdot (3vec{a} + kvec{b}) = 0]Expanding the dot product:[36 (vec{a} cdot vec{a}) + (12k + 9) (vec{a} cdot vec{b}) + 27k (vec{b} cdot vec{b}) = 0]Using (|vec{a}| = 2), (|vec{b}| = 3), and (vec{a} cdot vec{b} = 2 cdot 3 cdot cos 60^circ = 3):[36 cdot 4 + (12k + 9) cdot 3 + 27k cdot 9 = 0]Simplifying each term:[144 + 36k + 27 + 243k = 0]Combining like terms:[171 + 63k = 0 implies 63k = -171 implies k = -frac{171}{63} = -frac{19}{7}] Final AnswerFor (1): (boxed{dfrac{3}{4}}) For (2): (boxed{-dfrac{19}{7}}),<think>Okay, so I have this problem here about vectors. Let me try to understand it step by step. The problem says:Given that the magnitude of vector a is 2, and the magnitude of vector b is 3. The angle between them is 60 degrees. Then we have vectors c and d defined as c = 12a + 3b and d = 3a + k b. We need to find the real number k such that (1) c is parallel to d, and (2) c is perpendicular to d. Alright, first, let me recall some vector properties. If two vectors are parallel, one must be a scalar multiple of the other. So, if c is parallel to d, there must exist some scalar λ such that c = λ d. On the other hand, if they are perpendicular, their dot product must be zero. Let me tackle part (1) first: finding k such that c is parallel to d.So, c = 12a + 3b and d = 3a + k b. If they are parallel, then 12a + 3b = λ (3a + k b). That should hold for some scalar λ. Let's write that equation:12a + 3b = λ * 3a + λ * k b.This implies that the coefficients of a and b on both sides must be equal. So:For the a component: 12 = 3λFor the b component: 3 = λ kFrom the first equation, solving for λ: 12 = 3λ ⇒ λ = 4.Then plug λ = 4 into the second equation: 3 = 4k ⇒ k = 3/4.Hmm, that seems straightforward, but maybe I should check if there's any condition based on the vectors a and b not being colinear themselves. Since the angle between them is 60 degrees, they are not parallel or antiparallel, so they form a basis in the plane. Therefore, the coefficients must match as above. So k = 3/4 should be the answer for part (1).Now part (2): c is perpendicular to d. That means their dot product is zero. So, c · d = 0.Let me compute c · d. Since c = 12a + 3b and d = 3a + k b, their dot product is:(12a + 3b) · (3a + k b) = 12*3 (a · a) + 12*k (a · b) + 3*3 (b · a) + 3*k (b · b)But dot product is commutative, so (a · b) = (b · a). So:= 36 (a · a) + (12k + 9) (a · b) + 3k (b · b)We know |a| = 2, so a · a = |a|² = 4. Similarly, |b| = 3, so b · b = 9. The angle between a and b is 60 degrees, so a · b = |a||b|cos60° = 2*3*(1/2) = 3.Plugging these values in:= 36*4 + (12k + 9)*3 + 3k*9Calculate each term:36*4 = 144(12k + 9)*3 = 36k + 273k*9 = 27kSo total:144 + 36k + 27 + 27k = 144 + 27 + (36k + 27k) = 171 + 63kSet this equal to zero for perpendicularity:171 + 63k = 0Solving for k:63k = -171k = -171 / 63Simplify the fraction. Both numerator and denominator are divisible by 9:171 ÷ 9 = 1963 ÷ 9 = 7So k = -19/7.Let me check my calculations again to make sure I didn't make a mistake.First, computing the dot product:(12a + 3b) · (3a + k b) = 12*3 (a·a) + 12*k (a·b) + 3*3 (b·a) + 3*k (b·b)Yes, that's 36(a·a) + (12k + 9)(a·b) + 3k (b·b). Then substituting:a·a = 4, a·b = 3, b·b =9.So 36*4 = 144, (12k +9)*3 = 36k + 27, 3k*9=27k. Adding together: 144 +36k +27 +27k = 171 +63k. So 171 +63k =0 => k = -171/63 = -19/7. That seems correct.Alternatively, maybe there's another way to think about this. For perpendicular vectors, the dot product must be zero. So another approach is to compute each term step by step:First, compute each component:c · d = (12a + 3b) · (3a + k b) = 12*3(a·a) + 12*k(a·b) + 3*3(b·a) + 3*k(b·b)Which is 36(a·a) + 12k(a·b) +9(a·b) +3k(b·b). Then grouping terms:36(a·a) + (12k +9)(a·b) + 3k(b·b). Which is same as before.Yes, so the same result. So k = -19/7. So that's the answer for part (2).Wait, but let me check the arithmetic once more for part (2):171 divided by 63 is 2.714..., which is 19/7 ≈ 2.714, so negative of that is approximately -2.714. So -19/7 is indeed correct.Alternatively, 171 ÷ 63: 63*2=126, 171-126=45; 63*0.7=44.1, so 2.7 is 2 + 0.7, but 0.7*63=44.1, which is close to 45, but not exact. However, since 171 and 63 are both divisible by 9, we simplified to -19/7. So that's the exact answer.So to recap:For part (1), k=3/4, and for part (2), k=-19/7.I think that's all. Let me check if the problem had any other conditions, but it just says "for what value of the real number k" for each case. So separate answers for each. So the two answers are 3/4 and -19/7.Yes, that seems correct.Final AnswerFor (1) boxed{dfrac{3}{4}} and for (2) boxed{-dfrac{19}{7}}</think>Given ( |vec{a}| = 2 ), ( |vec{b}| = 3 ), and the angle between ( vec{a} ) and ( vec{b} ) is ( 60^circ ), we need to find the value of ( k ) such that (1) ( vec{c} parallel vec{d} ) and (2) ( vec{c} perp vec{d} ), where ( vec{c} = 12vec{a} + 3vec{b} ) and ( vec{d} = 3vec{a} + kvec{b} ). Part (1): (vec{c} parallel vec{d})If ( vec{c} ) is parallel to ( vec{d} ), then there exists a scalar ( lambda ) such that ( vec{c} = lambda vec{d} ). This gives:[12vec{a} + 3vec{b} = lambda (3vec{a} + kvec{b})]Equating the coefficients of ( vec{a} ) and ( vec{b} ):[12 = 3lambda quad text{and} quad 3 = lambda k]Solving for ( lambda ):[lambda = frac{12}{3} = 4]Substituting ( lambda = 4 ) into the second equation:[3 = 4k implies k = frac{3}{4}] Part (2): (vec{c} perp vec{d})If ( vec{c} ) is perpendicular to ( vec{d} ), their dot product must be zero:[(12vec{a} + 3vec{b}) cdot (3vec{a} + kvec{b}) = 0]Calculating the dot product using ( vec{a} cdot vec{a} = 4 ), ( vec{b} cdot vec{b} = 9 ), and ( vec{a} cdot vec{b} = 3 ):[36(vec{a} cdot vec{a}) + (12k + 9)(vec{a} cdot vec{b}) + 3k(vec{b} cdot vec{b}) = 0]Substituting the values:[36 cdot 4 + (12k + 9) cdot 3 + 3k cdot 9 = 0]Simplifying:[144 + 36k + 27 + 27k = 0 implies 171 + 63k = 0]Solving for ( k ):[63k = -171 implies k = -frac{171}{63} = -frac{19}{7}] Final AnswerFor (1) (boxed{dfrac{3}{4}}) and for (2) (boxed{-dfrac{19}{7}}).]

🔑:Solution: (1) Since |vec{a}|=2, |vec{b}|=3, and the angle between vec{a} and vec{b} is 60^{circ}, we have vec{a}cdotvec{b}=2cdot3cdotcos 60^{circ}=3. Since vec{c}=12vec{a}+3vec{b} and vec{d}=3vec{a}+kvec{b}, if vec{c}parallel vec{d}, then frac{3}{12}=frac{k}{3}, which gives k=frac{3}{4}. (2) If vec{c}perp vec{d}, then vec{c}cdotvec{d}=36vec{a}^2+(12k+9)vec{a}cdotvec{b}+3k|vec{b}|^2=36cdot4+(12k+9)cdot3+3kcdot9=0, solving for k gives k=-frac{19}{7}.Therefore, the answers are k=boxed{frac{3}{4}} for vec{c}parallel vec{d} and k=boxed{-frac{19}{7}} for vec{c}perp vec{d}.

🧠:[<think>Okay, let's tackle this problem step by step. It's about finding the minimal number of elements in a set of numbers that satisfy certain conditions. The given solution for part b suggests that the minimal number is 11. I need to understand why that is the case and verify if the example provided actually meets all the conditions.First, let me list out the conditions mentioned in the problem. Although they aren't explicitly stated here, from the solutions given, I can infer the conditions:1. The arithmetic mean of the set is 7.2. The median of the set is 2.3. The maximum number in the set is 13.4. The set must contain natural numbers (since the example uses integers like 2 and 13).Wait, the problem statement says "conditions 1-4", but the user hasn't provided them. Hmm, maybe the original problem had those conditions, and this is part b of the solution. Let me check the original problem again. Oh, looking at the user's message, they provided parts a and b of the solution. So, perhaps part a shows that a set of 9 numbers can't satisfy the conditions, and part b finds that 11 is the minimal number.Let me parse part a first. It says that if there are 9 numbers, the median would be the 5th number when sorted. Since the median is 2, that means five numbers are less than or equal to 2. The remaining four numbers can't exceed 13 (the maximum), so the total sum is at most 5*2 + 4*13 = 10 + 52 = 62. But the required sum for an arithmetic mean of 7 is 7*9 = 63. 62 < 63, which is a contradiction. Therefore, 9 numbers are impossible.Now, part b generalizes this. Let the number of numbers be 2n + 1 (so it's an odd number, which makes the median well-defined as the middle number). In such a set, there are n + 1 numbers that don't exceed the median (which is 2). The remaining n numbers don't exceed 13. So the maximum possible sum is 13n + 2(n + 1) = 15n + 2. The required sum for a mean of 7 is 7*(2n + 1) = 14n + 7. Setting 14n + 7 ≤ 15n + 2 gives n ≥ 5. Hence, the minimal n is 5, leading to 2*5 +1 =11 numbers. The example set has six 2s and five 13s. Let me verify this example.The example set: six 2s and five 13s. Let's count the numbers: 6 + 5 =11. The median is the 6th number when sorted (since 11 is odd). Sorting the set: [2,2,2,2,2,2,13,13,13,13,13]. The 6th number is 2, so the median is 2. Good. The maximum is 13. The arithmetic mean is (6*2 + 5*13)/11 = (12 + 65)/11 = 77/11 =7. Perfect, that's the required mean. So this set satisfies all conditions.But the question was about the minimal number. Since part a ruled out 9, part b shows that 11 works and n must be at least 5 (so 11 is minimal). Therefore, the answer is 11.Wait, but why can't we have n=4? Then the number of elements would be 2*4 +1=9. But part a already showed that 9 is impossible. So n must be at least 5. Hence, 11 is minimal. That makes sense.Let me check with n=5. The maximum sum allowed is 15*5 +2=75+2=77. The required sum is 14*5 +7=70+7=77. So exactly 77. Therefore, the example set reaches exactly the maximum sum allowed. So it's tight. If we tried n=4, the maximum sum would be 15*4 +2=62, but required sum is 14*4 +7=63, which is more. Hence, impossible. Therefore, n=5 is the smallest possible, leading to 11 numbers.Therefore, the minimal number is 11. The example set proves it's possible, and the earlier parts show that smaller numbers (like 9) are impossible. So the answer is 11.Final AnswerThe minimal number of elements in the set is boxed{11}.</think>To determine the minimal number of elements in a set that satisfies the given conditions (arithmetic mean of 7, median of 2, maximum number of 13, and natural numbers), we analyze the problem as follows:1. Part a Analysis: - For a set of 9 numbers, the median is 2, meaning five numbers do not exceed 2. - The remaining four numbers do not exceed 13. - The maximum possible sum is (5 cdot 2 + 4 cdot 13 = 62), but the required sum for a mean of 7 is (7 cdot 9 = 63), leading to a contradiction. Thus, 9 numbers are impossible.2. Part b Analysis: - Let the number of elements be (2n + 1) (an odd number). - There are (n + 1) numbers not exceeding the median (2) and (n) numbers not exceeding 13. - The maximum possible sum is (13n + 2(n + 1) = 15n + 2). - The required sum for a mean of 7 is (7(2n + 1) = 14n + 7). - Solving (14n + 7 leq 15n + 2) gives (n geq 5). Thus, the minimal (n) is 5, leading to (2 cdot 5 + 1 = 11) numbers.3. Verification: - An example set with six 2s and five 13s has 11 numbers. - The median is 2, the maximum is 13, and the arithmetic mean is (frac{6 cdot 2 + 5 cdot 13}{11} = 7).Thus, the minimal number of elements in the set is (boxed{11}).,<think>Okay, so I need to understand this problem where there's a set of numbers with certain conditions. The problem is divided into parts a) and b). Let me start with part a).Part a) says: Suppose there are 9 numbers in the set. Then five of them do not exceed the median, which is 2. Another four numbers do not exceed 13. The sum is calculated as 5*2 + 4*13 = 62. But since the arithmetic mean is 7, the total sum should be 7*9 = 63. This is a contradiction, so the set can't have 9 numbers. Hmm, let me make sure I get this.First, if the set has 9 numbers, the median is the 5th number when sorted. Since the median is 2, that means five numbers are less than or equal to 2, and the remaining four are greater than or equal to 2. But the problem says "another four numbers do not exceed 13." Wait, that might mean the remaining four numbers are each at most 13. So the maximum possible sum would be 5 times 2 plus 4 times 13, which is 10 + 52 = 62. But the required sum for a mean of 7 is 63. Since 62 < 63, that's impossible. Therefore, a set of 9 numbers can't satisfy all the conditions. That makes sense.Now part b). The number of numbers is 2n + 1, so it's an odd number. The median is 2, so n + 1 numbers are ≤ 2, and the remaining n numbers are ≤13. The maximum sum would then be 2*(n + 1) + 13*n. That's 2n + 2 + 13n = 15n + 2. The actual sum needed is 7*(2n +1) = 14n +7. So setting 14n +7 ≤ 15n +2. Subtract 14n: 7 ≤ n + 2. Subtract 2: 5 ≤ n. So n ≥5. Therefore, the smallest number of elements is 2*5 +1 =11.They provide an example set: six 2s and five 13s. Let me check that. The sorted set would be [2,2,2,2,2,2,13,13,13,13,13]. The median is the 6th number, which is 2. That's good. The mean is (6*2 +5*13)/11. Let's compute that: 12 + 65 =77. 77/11=7. Perfect. So that set works. Therefore, 11 is the minimal number.Wait, but why can't we have a set with 11 numbers where some numbers are higher than 13? The problem says "the remaining numbers do not exceed 13," so maybe the maximum allowed for the numbers above the median is 13. So in the problem statement, is there a condition that all numbers above the median are ≤13? Let me check.Looking back at the problem statement: It says "Another four numbers do not exceed the number 13." Wait, in part a), with 9 numbers, four numbers do not exceed 13. Wait, actually, if the median is 2, then in a set of 9 numbers, five numbers are ≤2, and the remaining four are ≥2. But the problem says "another four numbers do not exceed 13." So maybe the problem has an additional condition that all numbers are ≤13? Or is that part of the given?Wait, the problem might have had some initial conditions that aren't fully restated here. Since the user only provided the solution text, maybe the original problem included conditions like:1. All numbers are integers.2. The median is 2.3. The arithmetic mean is 7.4. All numbers are at most 13.Therefore, in part a), trying with 9 numbers, which leads to a contradiction, so 9 is impossible, and part b) generalizes it to find the minimal odd number of elements, which is 11.So, the key is that the maximum sum you can get with 2n +1 numbers is 2*(n +1) +13*n, which has to be at least 7*(2n +1). Solving the inequality gives n ≥5, so minimal n is5, leading to 11 numbers. The example set is given to confirm.Therefore, the answer is 11, and it's possible by having six 2s and five 13s.Final AnswerThe minimal number of elements is boxed{11}.</think>a) Suppose there are 9 numbers in the set. The median is 2, so five numbers do not exceed 2. The remaining four numbers do not exceed 13. The maximum possible sum is calculated as:5 cdot 2 + 4 cdot 13 = 10 + 52 = 62However, the required sum for an arithmetic mean of 7 is:7 cdot 9 = 63Since 62 < 63, this is a contradiction. Therefore, the set cannot consist of 9 numbers.b) Let the number of numbers in the set be (2n + 1) (where (n) is a natural number). The median is 2, so (n + 1) numbers are ≤ 2, and the remaining (n) numbers are ≤ 13. The maximum possible sum is:2(n + 1) + 13n = 15n + 2The required sum for an arithmetic mean of 7 is:7(2n + 1) = 14n + 7Solving the inequality (14n + 7 leq 15n + 2) gives (n geq 5). Therefore, the minimal number of elements is:2 cdot 5 + 1 = 11An example set is:{2, 2, 2, 2, 2, 2, 13, 13, 13, 13, 13}This set has a median of 2, an arithmetic mean of 7, and all numbers ≤ 13. The minimal number of elements is boxed{11}.]

🔑:Answer: a) no; b) 11.Scoring criteria| Both parts solved correctly or only part (b) | 3 points || :--- | :---: || The correct estimate of the number of numbers in part (b) is found, but no example is given | 2 points || Part (a) is solved correctly | 1 point || The solution is incorrect or missing (in particular, only the answer is given) | 0 points |

🧠:[<think>Okay, let's tackle this problem step by step. It seems a bit complex at first glance, but if I break it down, maybe it'll make more sense. First, let's parse the problem statement again to ensure I understand all the constraints and what's being asked.We have a party with some gentlemen and ladies. After the party, we know the following:1. For Gentlemen: Each danced with at least 10 and at most 12 ladies. So, every gentleman danced between 10 and 12 times.2. For Ladies: Each danced with at least 12 and at most 14 gentlemen. So, every lady danced between 12 and 14 times.3. No repeats: No gentleman danced with the same lady more than once. So, each pair can dance at most once.4. Total dances if everyone danced with everyone: If every gentleman had danced with every lady exactly once, there would have been 480 dances. But since they didn't, the actual number of dances is somewhere less, but the maximum possible is 480.The problem has two parts:a) Find all possible numbers of gentlemen (G) and ladies (L) that fit these constraints.b) Provide a possible arrangement (like a rectangular scheme) of dance pairs that meets the conditions.Starting with part a). Let's focus on that first.First, the key piece of information is that if every gentleman danced with every lady once, there would be 480 dances. Since each dance is counted as one dance per pair, the total number of possible dances is G * L = 480. So, G * L = 480. That's equation one.But in reality, each gentleman danced between 10 and 12 ladies. So, the total number of dances that actually occurred is somewhere between 10G and 12G. Similarly, each lady danced between 12 and 14 gentlemen, so the total number of dances is also between 12L and 14L. Since the total number of dances must be the same whether we count from the gentlemen's side or the ladies' side, we have:10G ≤ Total dances ≤ 12Gand12L ≤ Total dances ≤ 14LTherefore, these intervals must overlap. So, the total number of dances must satisfy both:10G ≤ 14Land12G ≥ 12LWait, let me think. Since the total dances are the same, let's denote T as the total number of dances. Then:From gentlemen: T ≥ 10G and T ≤ 12GFrom ladies: T ≥ 12L and T ≤ 14LTherefore, combining both:12L ≤ T ≤ 12Gand10G ≤ T ≤ 14LWait, that's conflicting. Maybe I need to approach it differently.Since T must satisfy both upper and lower bounds from both perspectives, the maximum of the lower bounds and the minimum of the upper bounds must have an overlap. So:The lower bound for T is max(10G, 12L)The upper bound for T is min(12G, 14L)So, for there to be a valid T, we must have:max(10G, 12L) ≤ min(12G, 14L)Which implies:10G ≤ 14L and 12L ≤ 12GSimplify:10G ≤ 14L ⇒ 5G ≤ 7L ⇒ G ≤ (7/5)Land 12L ≤ 12G ⇒ L ≤ GSo combining these, we have L ≤ G ≤ (7/5)LBut also, we know that G * L = 480 from the maximum possible dances. Wait, hold on. The problem says "If every gentleman had danced with every lady exactly once, 480 dances would have taken place." So that means the total number of possible pairs is G * L = 480. Therefore, G * L = 480. Therefore, our variables G and L must be positive integers such that G * L = 480. Also, L ≤ G ≤ (7/5)L. So, we can model this as:Find pairs (G, L) where G * L = 480, and L ≤ G ≤ (7/5)L.So, first, let's list all possible pairs (G, L) such that G * L = 480. Then, among those, we need to find the ones that satisfy L ≤ G ≤ (7/5)L.But let's first factor 480 to find all possible integer pairs (G, L):480 factors: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 32, 40, 48, 60, 80, 96, 120, 160, 240, 480.So the possible pairs (G, L) are all pairs where G and L are factors of 480, and G * L = 480. So L = 480 / G.But since L and G must be positive integers, we can generate all possible pairs:For example, G=1, L=480; G=2, L=240; G=3, L=160; G=4, L=120; G=5, L=96; G=6, L=80; G=8, L=60; G=10, L=48; G=12, L=40; G=15, L=32; G=16, L=30; G=20, L=24; G=24, L=20; G=30, L=16; etc., up to G=480, L=1.But considering that G and L must be in the constraints L ≤ G ≤ (7/5)L. So first, L ≤ G. So we can ignore pairs where L > G. So starting from G ≥ L.So possible pairs are:(1,480) – but L=480 > G=1, so discard.(2,240) – same, discard....Wait, perhaps starting from the other side. Let's list all pairs where G ≥ L:So starting from G=480, L=1 (G=480, L=1)But 480 and 1: but since L must be at least 12? Wait, hold on. Wait, the problem states that each lady danced with at least 12 gentlemen. So L is the number of ladies, and each lady danced with at least 12 gentlemen. So the number of dances per lady is between 12 and 14. But if there are G gentlemen, then the maximum number of dances a lady can have is G. So for each lady, the number of dances (which is between 12 and 14) must be ≤ G. Therefore, G ≥ 12.Similarly, each gentleman danced with at least 10 ladies, so L ≥ 10 (since each gentleman can dance with up to L ladies). Wait, no: a gentleman dances with between 10 and 12 ladies. So the number of ladies must be at least 10, because otherwise a gentleman couldn't have danced with 10 ladies. So L ≥ 10. Similarly, G ≥ 12.Wait, hold on. Let's check:For gentlemen: each danced with at least 10 ladies, so since there are L ladies, each gentleman must have danced with at least 10, so L ≥ 10. Similarly, each gentleman danced with at most 12 ladies, so L ≥ 10 (if L was less than 10, they couldn't have danced with 10). Wait, but L is the total number of ladies. So if L is the total number of ladies, then each gentleman danced with between 10 and 12 ladies. Therefore, L must be at least 10. Otherwise, if there were fewer than 10 ladies, a gentleman couldn't have danced with 10 ladies. Therefore, L ≥ 10.Similarly, for ladies: each danced with at least 12 gentlemen. So since there are G gentlemen, each lady must have danced with at least 12, so G ≥ 12. So G must be at least 12.Additionally, the problem states that if every gentleman had danced with every lady exactly once, there would have been 480 dances. So G * L = 480.Therefore, we need to find pairs (G, L) with G ≥ 12, L ≥ 10, G * L = 480, and L ≤ G ≤ (7/5)L.So let's list all possible factor pairs of 480 where G ≥ L, G ≥ 12, L ≥ 10.Starting with L=10: Then G=480/10=48. Check if 48 ≥ 12: yes. Then check if 48 ≤ (7/5)*10=14. 48 ≤14? No. So discard.Wait, that's a problem. Wait, 48 is way bigger than 14. So (G=48, L=10) is a factor pair but doesn't satisfy G ≤ (7/5)L.Similarly, next L=12: G=480/12=40. Check G=40, L=12. Check if 12 ≤40 and 40 ≤ (7/5)*12=16.8. 40 ≤16.8? No. Disregard.Wait, that's not possible. Wait, perhaps I need to reverse the ratio. Wait, the constraints were L ≤ G ≤ (7/5)L. So G must be between L and 1.4L. So for example, if L=24, then G must be ≤ (7/5)*24=33.6. Since G must be an integer, G ≤33. But 480 divided by 24 is 20, which is less than 24. So (G=20, L=24). But in this case, G=20 < L=24, which violates L ≤ G. So perhaps the problem is that the only pairs (G, L) with G ≥ L and G * L=480 would have G ≥ sqrt(480) ≈21.9. So G must be at least 22? Let's check.Wait, 22 * 21.8≈480, but since they must be integers, the closest pairs around sqrt(480) are (24,20), (20,24). But only (24,20) where G=24, L=20. But since we need G ≥ L, (24,20) is acceptable, but check if 20 ≤24≤ (7/5)*20=28. 24≤28, yes. So (24,20) is a possible pair.Wait, let's go step by step.First, list all factor pairs of 480 where G ≥ L, G ≥12, L ≥10.Starting from L=10:- L=10, G=480/10=48. Check if 48 ≤ (7/5)*10=14. 48>14: no.- L=12, G=40. Check 40 ≤ (7/5)*12=16.8: 40>16.8: no.- L=15, G=32. Check 32 ≤ (7/5)*15=21: 32>21: no.- L=16, G=30. Check 30 ≤ (7/5)*16=22.4: 30>22.4: no.- L=20, G=24. Check 24 ≤ (7/5)*20=28: 24 ≤28: yes. So (24,20) is valid.Next, L=24, G=20. But G=20 < L=24: violates G ≥ L. So discard.Similarly, L=30, G=16: G < L, discard.So only (24,20) is a possible pair where G ≥ L, G * L=480, and G ≤ (7/5)L. Wait, but is that the only one?Wait, let's check other possible L values. Maybe I skipped some.Factors of 480:1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 32, 40, 48, 60, 80, 96, 120, 160, 240, 480.So L must be at least 10. So starting from L=10:But in factor pairs (G, L) where G ≥ L:- L=10, G=48- L=12, G=40- L=15, G=32- L=16, G=30- L=20, G=24- L=24, G=20 (but G < L here, so stop here)So those are the possible pairs. Now, checking each:1. (48,10): G=48, L=10. Check G ≤ (7/5)L: 48 ≤14? No.2. (40,12): 40 ≤ (7/5)*12=16.8? No.3. (32,15): 32 ≤21? No.4. (30,16): 30 ≤22.4? No.5. (24,20): 24 ≤28? Yes. So this is valid.So only (24,20) satisfies G ≤ (7/5)L. Therefore, this is the only possible pair.Wait, but hold on. The problem says "determine all possibilities for the number of ladies and gentlemen present that are consistent with these conditions." So maybe only (24,20)? But let's verify.But wait, perhaps there are other factor pairs where G and L are not integers? No, because the number of people must be integers. Therefore, the only possible pair is (24,20). Let me double-check.But let's confirm with the other constraints. Remember, each gentleman danced between 10 and 12 ladies. So total dances T must be between 10G and 12G. Similarly, for ladies, T between 12L and 14L.For (24,20):T must satisfy:10*24 ≤ T ≤12*24 ⇒ 240 ≤ T ≤288and12*20 ≤ T ≤14*20 ⇒240 ≤ T ≤280So the overlap is 240 ≤ T ≤280.So T must be in [240,280]. But also, T must equal the total number of dances, which is the same as the sum over gentlemen or the sum over ladies. So is there a T that is possible here? Let's check.But since each dance is a pair, T must also satisfy that each dance is counted once. So, with G=24, L=20, the maximum number of dances is 480, but the actual number T is between 240 and 280. So is there a T such that T is the total dances, and T is an integer, but how do we know if such a T is possible? Wait, but the problem doesn't specify the exact number of dances, only the constraints per person. So as long as there's an overlap in the intervals [240,288] and [240,280], which is [240,280], then any T in that range is possible? But the problem is part a) is only asking for the possible numbers of G and L. Since the constraints on T only require that the intervals overlap, which they do for (24,20), then that's the only possible pair.But wait, let's check if there's another pair where G * L=480 and L ≤ G ≤1.4L. Suppose we have L= 14, but 480/14≈34.28, not integer. Similarly, L=25, 480/25=19.2, not integer. So no.Therefore, the only possible pair is (24,20). Therefore, the answer is 24 gentlemen and 20 ladies.But let me double-check. Suppose G=24, L=20.Each gentleman danced between 10 and 12 ladies. So total dances T is between 24*10=240 and 24*12=288.Each lady danced between 12 and 14 gentlemen. So T is between 20*12=240 and 20*14=280.Thus, T must be between 240 and 280. Since 240 ≤ T ≤280, and T must be an integer. So possible. Therefore, such T exists. So the pair (24,20) is valid.Is there another pair? Let's see. Suppose someone thought of G=20, L=24. But G=20 < L=24, which violates L ≤ G. Therefore, no. So only (24,20) is valid.Wait, but let me check if there's any other factor pair where G * L=480, L ≤ G, and G ≤1.4L.Suppose, for example, G=28, L=480/28≈17.14, not integer.G=21, L=480/21≈22.857, not integer.So no. Only (24,20) works.Therefore, the answer to part a) is 24 gentlemen and 20 ladies.Moving to part b), which asks for a possible arrangement (rectangular scheme) of dance pairs meeting the conditions. So we need to create a dance matrix where each gentleman (rows) dances with 10-12 ladies (columns), each lady dances with 12-14 gentlemen, and no pair dances more than once. The total dances should be between 240-280.Since we need to provide one such arrangement, perhaps a regular graph where each gentleman dances exactly 12 ladies and each lady dances exactly 12 gentlemen. Wait, 24 gentlemen each dancing 12 ladies: total dances 24*12=288. But each lady would have to dance 288/20=14.4, which is not an integer. So that's impossible.Alternatively, if each gentleman dances 10 ladies: total dances 24*10=240, which would require each lady dances 240/20=12 gentlemen. Which is exactly the lower bound. So that's possible.Alternatively, if each gentleman dances 12 ladies: total dances 288, but each lady would have to dance 14.4, which is not possible. So perhaps a combination where some gentlemen dance 10, 11, or 12 ladies, such that the total dances T is an integer, and each lady dances between 12 and 14.But since part b) asks for one possible combination in the simplest form, maybe a regular bipartite graph where each gentleman dances exactly 12 ladies and each lady dances exactly 14.4 gentlemen? Wait, but that's not possible. So perhaps a semi-regular graph.Wait, perhaps it's easier to have each gentleman dance exactly 12 ladies, and each lady dance exactly (24*12)/20 = 14.4. But since we can't have fractions, perhaps some ladies dance 14 and some 15. But the problem states that ladies danced with at most 14 gentlemen, so 15 is over the limit. Therefore, this approach doesn't work.Alternatively, each gentleman dances 10 ladies: total dances 240, each lady dances 12. That works. So if every gentleman dances exactly 10 ladies, and every lady dances exactly 12 gentlemen. Since 24*10=240=20*12. So that's feasible.But the problem states that none of the gentlemen danced with fewer than 10 and none with more than 12 ladies. So dancing exactly 10 is allowed. Similarly, ladies danced with exactly 12, which is within their constraints (12-14). Wait, but 12 is the lower bound for ladies. Wait, the problem says none of the ladies danced with fewer than 12 and none with more than 14. So dancing exactly 12 is acceptable, as it's the minimum. So arranging a dance where each gentleman dances exactly 10 ladies, and each lady dances exactly 12 gentlemen would satisfy all constraints.So that's possible. Such a configuration is called a bi-regular bipartite graph, where on one side (gentlemen) each has degree 10, and on the other side (ladies) each has degree 12. The necessary condition for such a graph is that the total degrees are equal: 24*10=20*12 ⇒240=240, which holds. So such a graph exists.Therefore, constructing such a matrix where each gentleman is connected to 10 ladies, and each lady is connected to 12 gentlemen. How to present this in a rectangular scheme? Maybe a list or a matrix where each row represents a gentleman, and each column a lady, with a 1 indicating a dance. But the problem says "in the simplest form of a rectangular scheme", which might mean an incidence matrix. But with 24 rows and 20 columns, it's a bit large. Maybe the problem expects a description or a way to construct it.Alternatively, since 24 gentlemen and 20 ladies, and each lady dances with 12 gentlemen, which is 12/24 = 0.5, so each lady dances with half of the gentlemen. Similarly, each gentleman dances with 10/20 = 0.5 of the ladies. So perhaps a block matrix where each lady is paired with exactly 12 gentlemen, and each gentleman is paired with exactly 10 ladies.One way to construct this is using a combinatorial design. For example, a block design where each lady's "block" is a subset of 12 gentlemen, and each gentleman is in 10 ladies' blocks. Since 24*10 = 20*12, it's a matter of arranging these subsets.But constructing such a design might be non-trivial. However, since the problem asks for the simplest form, maybe a systematic way. For example, divide the gentlemen into groups or use a rotational scheme.Alternatively, since 24 and 20 have a common factor of 4, perhaps divide the gentlemen into 4 groups of 6, and the ladies into 4 groups of 5. Then, each group of 6 gentlemen can be connected to each group of 5 ladies in a certain way.But this might complicate. Alternatively, use a round-robin tournament style, but adjusted for bipartite.Alternatively, for each lady, assign 12 unique gentlemen such that each gentleman is assigned to 10 ladies. Since 20 ladies *12 dances=240, and 24 gentlemen *10 dances=240.One possible way is to arrange the gentlemen in a circle and use a systematic rotation. But with 24 gentlemen and 20 ladies, perhaps arranging each lady to dance with every 12th gentleman in some ordered list. However, ensuring that each gentleman is chosen exactly 10 times requires a more careful approach.Alternatively, use a finite field construction, but since 24 and 20 are not prime powers, this might not work. Alternatively, use a bipartite graph with a 10-12 regular structure. Since such a graph exists due to the degree conditions, we can assert that such a configuration exists, but constructing it explicitly is time-consuming.However, since the problem just asks for the simplest form of a rectangular scheme, perhaps a Latin square or something similar. But a Latin square would require that each gentleman dances with each lady exactly once, which is not the case here. Alternatively, maybe a rectangular array where each row (gentleman) has 10 entries (ladies), and each column (lady) has 12 entries (gentlemen). The exact arrangement can be represented as a binary matrix with 24 rows and 20 columns, each row having 10 ones, each column 12 ones.One standard way to construct such a matrix is to use a method called the "incidence matrix" of a bi-regular bipartite graph. However, without specific structure, it's hard to write down concisely. But perhaps a structured example can be given. For instance, divide the gentlemen into 10 groups of 2.4, which is not possible. Alternatively, note that 24 gentlemen can be divided into 12 pairs, and each lady dances with 12 gentlemen, which could be one from each pair. But 20 ladies would need to each select one from each of 12 pairs, but this may not ensure each gentleman is in exactly 10 dances. Alternatively, use a more modular approach.Alternatively, for simplicity, consider arranging the 24 gentlemen in a list and each lady dances with the first 12 gentlemen. But then each gentleman would dance with 20 ladies, which is way over the limit. So that's no good.Alternatively, use a cyclic pattern. For example, label the gentlemen from 0 to 23 and the ladies from 0 to 19. For each lady i, have her dance with gentlemen i, i+1, ..., i+11 modulo 24. But we need to check how many times each gentleman is chosen. Each gentleman j would be chosen by ladies j-11, j-10, ..., j modulo 19? Wait, not sure. Let me think.If lady i dances with gentlemen i to i+11 mod24, then each gentleman j is danced by ladies j-11 to j mod24. But since there are only 20 ladies, this would wrap around. For example, lady 0 dances with 0-11, lady 1 dances with 1-12, ..., lady 19 dances with 19-30 mod24 (which is 19-23 and 0-6). So each gentleman would be danced by 12 ladies? Wait, no. Let's compute how many ladies dance with gentleman j. Each gentleman is in the range of 12 consecutive ladies. But since there are 20 ladies, if we take 12 consecutive numbers modulo 20, overlapping with the gentlemen's indices modulo24. This seems messy.Alternatively, since 24 and 20 have a GCD of 4, divide both into 4 groups. For example, divide gentlemen into 4 groups of 6 (G1, G2, G3, G4) and ladies into 4 groups of 5 (L1, L2, L3, L4). Then, each lady in group Li dances with 3 gentlemen from each Gj where j ≠ i. Wait, let's see:Total dances per lady: 3 gentlemen * 3 groups =9. Not enough. Alternatively, each lady dances with 3 gentlemen from each of 4 groups: 3*4=12. So if each lady is in group Li, she dances with 3 gentlemen from each Gj. Then total per lady: 12. Each gentleman is in a group of 6. Each gentleman is danced by 3 ladies from each of 4 lady groups, except their own? Wait, no. If each lady in Li dances with 3 gentlemen from each Gj, then total dances per Gj is 5 ladies *3=15 dances. Since each gentleman is in Gj, which has 6 gentlemen, each gentleman in Gj is danced by 15/6=2.5 ladies, which is not an integer. Doesn't work.Alternatively, each lady dances with 6 gentlemen from two different groups. For example, lady group Li dances with gentlemen groups Gj and Gk. But this may complicate.Alternatively, use a pairwise design. Since each gentleman needs to dance 10 ladies, and there are 20 ladies, each gentleman doesn't dance with 10 ladies. But constructing a complement graph where each gentleman is not dancing with 10 ladies, but that might not help.Alternatively, since we need each gentleman to dance 10 ladies, and each lady to dance 12 gentlemen, perhaps the simplest way is to create a 24x20 matrix where each row has 10 ones and each column has 12 ones. One systematic way to do this is to have the first 12 columns filled in the first 10 rows, but ensuring column counts. However, this requires careful arrangement.For example, arrange the matrix such that each column (lady) has exactly 12 ones. Start filling the matrix row by row, each row (gentleman) placing 10 ones in different columns, making sure not to exceed 12 per column. This is similar to a progressive filling, but it's complex.Alternatively, use a round-robin scheduling where each gentleman is paired with 10 ladies in a rotational manner, ensuring that each lady is paired 12 times. Since it's a bit involved, but given the time constraints, perhaps we can state that such a matrix exists by combinatorial design, and since the problem only asks to provide one in the simplest form, maybe referencing a specific structure.But since the problem says "in the simplest form of a rectangular scheme", perhaps they expect an incidence matrix where the arrangement is systematic. For example, a matrix divided into blocks where each block represents a set of dance pairs.Given that 24 gentlemen and 20 ladies, and each gentleman dances 10 ladies, each lady dances 12 gentlemen, a possible way is:Divide the 24 gentlemen into 12 pairs. Assign each pair of gentlemen to 2 different ladies such that each lady is assigned to 12 gentlemen. Since there are 20 ladies, each lady needs 12 gentlemen. If we have 12 pairs of gentlemen, each pair can be assigned to 10 ladies (since 12 pairs *2 gentlemen=24). Wait, maybe not. Let's think.Alternatively, divide the ladies into 10 groups of 2. Each group of 2 ladies is danced by 12 gentlemen. But not sure.Alternatively, use a 24x20 grid where each row (gentleman) has 10 marks, and each column (lady) has 12 marks. To construct it, we can start by initializing all entries to 0, then for each gentleman, select 10 distinct ladies to mark as 1, ensuring that no lady exceeds 12 marks. This is essentially a bipartite graph construction problem.One possible method is the following:1. List all 24 gentlemen and 20 ladies.2. For each lady, maintain a count of how many gentlemen she has danced with (initially 0).3. For each gentleman, select 10 ladies who have not yet reached 12 dances.4. Assign each gentleman to 10 ladies, incrementing the ladies' counts.However, this requires a careful assignment to avoid conflicts. Since 24*10=240 and 20*12=240, it's possible, but the exact assignment would need to be systematic.A concrete example would be to arrange the dances such that each lady dances with the first 12 gentlemen, but this would mean each gentleman dances with 20 ladies, which is too many. So that doesn't work.Alternatively, divide the gentlemen into 2 groups of 12: Group A and Group B. Each lady dances with 6 gentlemen from Group A and 6 from Group B. Then each gentleman in Group A dances with 10 ladies: total dances from Group A: 12*10=120. Each lady dances with 6 from Group A: total dances from Group A: 20*6=120. Similarly for Group B. So this works. Therefore, each gentleman in Group A dances with 10 ladies, specifically 6 ladies, but wait: no. Wait, if there are 20 ladies, and each lady dances with 6 gentlemen from Group A (12 gentlemen), then each gentleman in Group A (12 gentlemen) must dance with (20*6)/12 =10 ladies. Yes. Similarly for Group B.Therefore, the scheme can be constructed as follows:- Split the 24 gentlemen into two groups, Group A (12 gentlemen) and Group B (12 gentlemen).- Each lady dances with 6 gentlemen from Group A and 6 from Group B.- Therefore, each gentleman in Group A dances with (20*6)/12 =10 ladies.- Similarly, each gentleman in Group B dances with 10 ladies.Thus, this satisfies the conditions: each gentleman dances 10 ladies, each lady dances 12 gentlemen (6+6).To represent this in a rectangular scheme, we can have a matrix where:- The first 12 rows (Group A) have 10 dances each, spread across the 20 columns (ladies). Each column (lady) has exactly 6 dances in the first 12 rows.- The last 12 rows (Group B) similarly have 10 dances each, with each column having exactly 6 dances in the last 12 rows.This ensures that each lady's total dances are 6+6=12, and each gentleman dances 10 ladies.To make it concrete, one way to arrange Group A is to have each gentleman in Group A dance with a unique set of 10 ladies, such that each lady is danced by exactly 6 gentlemen from Group A. Similarly for Group B.A specific example could be using a combinatorial design called a "block design". For instance, in Group A, each lady needs to be paired with 6 gentlemen. So we need a (12, 20, 6, 10) design? Wait, maybe not. Alternatively, since each of the 12 gentlemen in Group A needs to dance with 10 ladies, and each lady is danced by 6 gentlemen from Group A, we can arrange this as follows:Each lady has 6 slots for Group A. There are 20 ladies *6 slots=120 slots. Each gentleman in Group A needs to fill 10 slots. With 12 gentlemen, that's 12*10=120 slots. Perfect. So it's a matter of assigning each slot in the ladies' Group A slots to a gentleman in Group A, such that each gentleman is assigned 10 slots.This is equivalent to a bipartite graph matching problem, which is possible due to the degree constraints. A possible way to construct this is using a round-robin tournament where each gentleman is paired with different ladies.For example, use a cyclic schedule where each gentleman rotates through different ladies. However, explaining the entire matrix would be tedious, but since the problem asks for the simplest form, mentioning this grouped structure might suffice, or providing a specific submatrix.Alternatively, here's a structured way:For Group A (first 12 gentlemen) and ladies 1-20:Divide the 12 gentlemen into 6 pairs. Each pair will dance with 10 unique ladies, such that each lady is danced by 3 pairs. Wait, 6 pairs *10 ladies =60 dances, but each lady needs 6 dances from Group A. 20 ladies *6=120 dances. So 60 dances isn't enough. Wait, perhaps each pair dances with 10 ladies, but 12 gentlemen in Group A, divided into 6 pairs, each pair dances with 10 ladies. Total dances:6*10=60. But need 120. Hmm, not working.Alternatively, divide the 12 gentlemen into 12 individual dancers. Each needs to dance with 10 ladies, total 120 dances. Each lady needs 6 dances. So assign each lady 6 unique gentlemen from Group A. This is a bipartite graph with 12 nodes on one side, 20 on the other, each node on the left has degree 10, each on the right has degree 6. Such a graph exists because 12*10=120=20*6. Constructing such a graph can be done via the Havel-Hakimi algorithm or other methods, but it's complex to write out.Similarly for Group B.Therefore, the rectangular scheme can be represented as two 12x20 submatrices (Group A and Group B), each structured such that each row (gentleman) has 10 entries and each column (lady) has 6 entries. Overlaying these two submatrices gives the full 24x20 matrix where each lady has 12 entries (6 from each group).Since explicitly writing out a 24x20 matrix is impractical here, the answer can note this structure as the simplest form, mentioning the division into two groups with specified degrees.Final Answera) boxed{24} gentlemen and boxed{20} ladies.b) One possible arrangement is a 24×20 incidence matrix divided into two 12×20 blocks. Each gentleman in the first block dances with 10 ladies such that each lady dances with 6 gentlemen from the first block, and each gentleman in the second block dances with 10 ladies such that each lady dances with 6 gentlemen from the second block. This ensures every gentleman dances with 10 ladies and every lady dances with 12 gentlemen.</think>Given the problem, we need to determine the number of gentlemen (G) and ladies (L) present at the party, and provide a possible arrangement of dance pairs. Part a)1. Constraints Analysis: - Each gentleman danced with between 10 and 12 ladies. - Each lady danced with between 12 and 14 gentlemen. - Total dances if everyone danced with everyone once: ( G times L = 480 ).2. Key Equations: - ( G times L = 480 ) - Total dances ( T ) must satisfy: - ( 10G leq T leq 12G ) - ( 12L leq T leq 14L )3. Overlap of Intervals: - The intervals for ( T ) must overlap: - ( max(10G, 12L) leq min(12G, 14L) ) - This leads to: - ( 10G leq 14L ) and ( 12L leq 12G ) - Simplifying: ( L leq G leq frac{7}{5}L )4. Factor Pairs of 480: - Possible pairs ((G, L)) where ( G geq L ) and ( G times L = 480 ): - The only valid pair satisfying ( L leq G leq frac{7}{5}L ) is ( (24, 20) ). Part b)1. Arrangement: - Construct a 24×20 incidence matrix divided into two 12×20 blocks. - Each gentleman in the first block (12 gentlemen) dances with 10 ladies such that each lady dances with 6 gentlemen from this block. - Each gentleman in the second block (12 gentlemen) dances with 10 ladies such that each lady dances with 6 gentlemen from this block. - This ensures each gentleman dances with 10 ladies and each lady dances with 12 gentlemen. Final Answera) The number of gentlemen and ladies present are (boxed{24}) and (boxed{20}) respectively.b) A possible arrangement is a 24×20 incidence matrix divided into two 12×20 blocks. Each gentleman in the first block dances with 10 ladies such that each lady dances with 6 gentlemen from the first block, and each gentleman in the second block dances with 10 ladies such that each lady dances with 6 gentlemen from the second block. This ensures every gentleman dances with 10 ladies and every lady dances with 12 gentlemen.,<think>Okay, let's tackle part a) of this problem. Hmm, so we have gentlemen and ladies at a party. The gentlemen each danced with at least 10 and at most 12 ladies. The ladies each danced with at least 12 and at most 14 gentlemen. If every gentleman had danced with every lady exactly once, there would have been 480 dances. But since they didn't, the actual number of dances is somewhere between the minimum and maximum based on the constraints. Wait, but how do we figure out the number of gentlemen and ladies?First, let me note down the key information:- Let G be the number of gentlemen and L be the number of ladies.- Each gentleman danced with between 10 and 12 ladies. So, for each gentleman, number of dances is 10 ≤ dances ≤ 12. Therefore, total dances D must satisfy 10G ≤ D ≤ 12G.- Each lady danced with between 12 and 14 gentlemen. So, for each lady, number of dances is 12 ≤ dances ≤ 14. Therefore, total dances D must also satisfy 12L ≤ D ≤ 14L.- If every gentleman danced with every lady once, there would be G*L = 480 dances.So, first off, G*L = 480. That's a key equation. So we need integers G and L such that G*L=480. Also, D must satisfy 10G ≤ D ≤12G and 12L ≤ D ≤14L. Therefore, D must lie in the intersection of [10G,12G] and [12L,14L]. So, the actual total number of dances D must satisfy both inequalities. So, combining these:10G ≤ 14L and 12L ≤ 12G. Wait, maybe not exactly. Let's think.Since D must be ≥10G and D must be ≥12L. Therefore, the maximum of 10G and 12L must be ≤ D. Similarly, D must be ≤ minimum of 12G and 14L. So, for D to exist, the lower bound must be ≤ upper bound. So, max(10G,12L) ≤ min(12G,14L). That gives:10G ≤14L and 12L ≤12G. Simplifying:10G ≤14L → 5G ≤7L12L ≤12G → L ≤GSo, we have two inequalities: 5G ≤7L and L ≤G. Also, G*L=480. So, we need integers G and L where G*L=480, L ≤G, and 5G ≤7L. Let me write that down:1. G*L = 4802. L ≤ G3. 5G ≤7L → 5G/7 ≤ LTherefore, combining 2 and 3:5G/7 ≤ L ≤ GBut since L must be an integer, L is in the interval [ceil(5G/7), G]But since G*L=480, we can express L=480/G. Therefore,5G/7 ≤480/G ≤GMultiply all parts by G (since G is positive):5G²/7 ≤480 ≤G²So, first inequality: 5G²/7 ≤480 → G² ≤ (480*7)/5 = 672 → G ≤ sqrt(672) ≈25.92. So G ≤25.Second inequality: 480 ≤G² → G ≥sqrt(480)≈21.908. So G≥22.Therefore, G is between 22 and 25 inclusive.But G must be an integer divisor of 480 since L=480/G must also be integer. Let's check the divisors of 480 in 22-25.First, factorize 480: 480=16*30=16*2*15=2^5*3*5.Possible divisors between 22 and 25:Check 24: 480/24=20. So L=20.Check 480/22=21.81… not integer.480/23≈20.86… not integer.480/24=20, which is integer.480/25=19.2, not integer.So only G=24 and L=20 is possible? Wait, but wait, G must be between 22 and25, but only 24 divides 480 in that range.Wait, but let me check 20. Wait, if G=24, L=20. Then check the inequalities:First, L=20, G=24. So L=20 ≤ G=24, which satisfies the second condition.Now, 5G/7=5*24/7≈17.14. So L must be ≥18? But L=20, which is okay. Wait, the first inequality is 5G/7 ≤ L. So 5*24/7 ≈17.14 ≤20. Yes, that's true.So, G=24, L=20 satisfies all three conditions.But are there any other possible G? Wait, the next possible divisor below 24 would be 20, but 20 is less than 22, so it's outside the range. Similarly, 16, 15, etc. So seems like only G=24, L=20.Wait, but let me check again. Wait, G must be between 22 and25 inclusive, but 480's divisors in that range are only 24. Because 22,23,25 don't divide 480.But the problem statement says "determine all possibilities for the number of ladies and gentlemen present that are consistent with these conditions." So maybe only one possibility: 24 gentlemen and 20 ladies.But let me confirm. Let's check if G=24 and L=20 satisfies the dance constraints.Total dances D must satisfy 10G ≤ D ≤12G → 240 ≤ D ≤288Also, D must satisfy 12L ≤ D ≤14L → 240 ≤ D ≤280So, the intersection is 240 ≤ D ≤280. So D must be between 240 and 280.But since D is also the total number of dances actually performed, which is equal to the sum over gentlemen of the number of dances each did, which is between 10G and12G. Similarly for ladies. So as long as 240 ≤D ≤280, and the individual constraints per gentleman and lady are met, it's possible. But the problem doesn't require us to find D, just the numbers G and L. So since G=24 and L=20 is the only pair that satisfies G*L=480 and the inequalities, then that's the only possible pair.Wait, but let me check again if maybe other factor pairs could satisfy the inequalities, even if G isn't between 22-25.Wait, the previous reasoning said G must be between 22 and25 because:From the inequalities:5G ≤7L and L=480/G.So substituting L=480/G into 5G ≤7*(480/G):Multiply both sides by G (positive):5G² ≤7*480 → G² ≤ (7*480)/5=672 → G ≤sqrt(672)≈25.92. So G≤25.From the other inequality, 480 ≤G² (since D=G*L=480 ≤12G*L? Wait, no. Wait, the second inequality was:From the dance constraints, the total number of dances D must be at least 12L. But D must also be at most 12G. Wait, but D can be up to min(12G,14L). However, the problem states that if every gentleman danced with every lady, there would be 480 dances, so G*L=480.But actually, D is the actual number of dances. Wait, no. Wait, hold on. The problem says "If every gentleman had danced with every lady exactly once, 480 dances would have taken place." That means G*L=480. So the actual number of dances is D, which is less than or equal to G*L, because they didn't all dance with each other. Wait, but no, actually, each dance is a pair. So if every gentleman dances with every lady once, that's G*L dances. So if each dance is a single pair, then D would be G*L. But in reality, each dance is a single pair, so the total number of dances is equal to the number of pairs that danced. So the actual number of dances is the number of edges in the bipartite graph between gentlemen and ladies, where each edge is a dance. So D is equal to the number of edges, which is equal to the sum of the degrees of all gentlemen (which is the same as the sum of the degrees of all ladies). So D is the same whether you sum over gentlemen or ladies.So in the problem, D must satisfy:For the gentlemen: each danced between 10 and 12 times, so total D is between 10G and 12G.For the ladies: each danced between 12 and14 times, so total D is between12L and14L.Therefore, D must satisfy both 10G ≤D ≤12G and 12L ≤D ≤14L. Hence, the overlap of these intervals must exist, i.e., the maximum of the lower bounds must be ≤ the minimum of the upper bounds.So:max(10G,12L) ≤ min(12G,14L)Which gives:10G ≤14L and 12L ≤12G.Simplify:10G ≤14L →5G ≤7L12L ≤12G → L ≤GAlso, from G*L=480.So the same as before.Therefore, G and L must satisfy G*L=480, L ≤G, and 5G ≤7L.From G*L=480, L=480/G. So substituting into inequalities:L ≤G →480/G ≤G →G² ≥480 →G ≥sqrt(480)≈21.9, so G≥22.5G ≤7L →5G ≤7*(480/G) →5G² ≤3360 →G² ≤672 →G ≤sqrt(672)≈25.92, so G≤25.Therefore, G can be 22,23,24,25. But since G must divide 480 (because L=480/G must be integer), we check which of these numbers divide 480.480 divided by 22: 480/22≈21.818, not integer.480 divided by23≈20.869, not integer.480 divided by24=20, integer.480 divided by25=19.2, not integer.Therefore, only G=24 gives L=20 as integers. Hence, the only possible pair is 24 gentlemen and 20 ladies.Therefore, the answer is 24 gentlemen and 20 ladies.For part a), that's the only possible pair. Let me just double-check to make sure.Suppose G=24 and L=20.Check 5G=120, 7L=140. So 5G=120 ≤7L=140: yes.Check L=20 ≤G=24: yes.G*L=24*20=480: yes.So all conditions are satisfied.Any other G between 22-25 would not divide 480, so there are no other solutions. Therefore, the answer is G=24, L=20.Final Answera) boxed{24} gentlemen and boxed{20} ladies.For part b), we need to provide a rectangular scheme (which I assume is a matrix) showing one possible combination of dance pairs. Each gentleman must dance with between 10 and 12 ladies, and each lady must dance with between 12 and 14 gentlemen. Since G=24 and L=20, each gentleman dances with 10, 11, or 12 ladies, and each lady dances with 12, 13, or 14 gentlemen. The total dances D must be such that 10*24=240 ≤ D ≤12*24=288 and 12*20=240 ≤ D ≤14*20=280. Therefore, D must be between 240 and 280. The overlapping range is 240 to 280. However, since both gentlemen and ladies have constraints, the actual D must satisfy both. So D needs to be between 240 and 280. But given that each gentleman dances at least 10 and each lady at least 12, the minimal D is 240. However, depending on the distribution, D can go up to 280, but since gentlemen can only go up to 12 dances each (12*24=288), but ladies can only go up to 14 each (14*20=280), so D must be at most 280. Therefore, D is between 240 and 280. To create such a matrix, we need a 24x20 incidence matrix where each row (gentleman) has 10-12 ones, each column (lady) has 12-14 ones, and the total number of ones is between 240 and 280. However, the exact total number must be the same when counted from rows and columns. Since the minimal total from ladies is 240 and from gentlemen is also 240, and the maximum from ladies is 280 and from gentlemen is 288, the actual total D must be 240 ≤D ≤280. To construct such a matrix, one way is to set each gentleman to dance exactly 10 times and each lady to dance exactly 12 times. But 10*24=240 and 12*20=240. So that's possible. Alternatively, if some dance more. But to make it simple, a regular case where each gentleman dances exactly 10 times and each lady dances exactly 12 times. Wait, but 10*24=240 and 12*20=240, so that works. However, each lady would have exactly 12 dances, which is within their limit (12-14), and each gentleman exactly 10 dances, which is within their limit (10-12). So that's possible. But is that possible to arrange? Since each lady needs exactly 12 dances, and there are 20 ladies, total dances 240. Each gentleman has exactly 10 dances. So the incidence matrix would be a 24x20 matrix where each row has exactly 10 ones and each column has exactly 12 ones. This is a bipartite biregular graph. So yes, such a matrix exists. Alternatively, perhaps using a more balanced distribution. But since the problem says "in the simplest form of a rectangular scheme", maybe a biregular one where each gentleman dances exactly 10 ladies, each lady dances exactly 12 gentlemen. But how to represent that? Since it's a 24x20 matrix, writing out all 240 entries is impractical. So perhaps describe it as a biregular graph where each gentleman is connected to 10 ladies and each lady is connected to 12 gentlemen. Alternatively, arranging the dances in a systematic way, such as a block matrix.But since the problem says "in the simplest form of a rectangular scheme", maybe we can just state that such a matrix exists by combinatorial design, and perhaps mention the parameters. However, the problem might expect a more concrete answer, like a Latin square or something, but since the numbers are different (24 gentlemen and 20 ladies), a Latin square isn't directly applicable. Alternatively, using a round-robin tournament style scheduling, but adjusted for the different numbers.Alternatively, since each lady must dance with 12 gentlemen, and there are 24 gentlemen, we can divide the gentlemen into two groups of 12 for each lady. Each lady dances with 12 gentlemen, so perhaps each lady is paired with half of the gentlemen. But since there are 24 gentlemen, each lady dances with 12, which is half. So if we arrange the matrix such that the first 12 gentlemen dance with the first 10 ladies, and the next 12 dance with the next 10 ladies, but that might not balance it properly. Alternatively, use a cyclic permutation. But maybe it's complex.Alternatively, use a incidence matrix where each lady is paired with exactly 12 gentlemen, and each gentleman is paired with exactly 10 ladies. This is equivalent to a (24,20)-biregular bipartite graph with degrees 10 and 12. The existence of such a graph is guaranteed by the Gale-Ryser theorem. The Gale-Ryser conditions for bipartite graphs state that a bipartite graph with partitions of size n and m and degree sequences a1,...,an and b1,...,bm exists if and only if the sum of the degrees in each partition is equal and the sequence conditions are met. In our case, all gentlemen have degree 10, and all ladies have degree 12. The sum for gentlemen is 24*10=240, and for ladies is 20*12=240, so the sums are equal. Then, according to the Gale-Ryser theorem, since the gentlemen partition has degrees 10 each, sorted in non-increasing order (all 10s), and the ladies partition has degrees 12 each (sorted as 12s). The condition is that the sum of the degrees in each partition is equal (which it is) and that for every k, the sum of the first k degrees in the gentlemen partition is less than or equal to the sum of the degrees in the ladies partition for the first k ladies. But since all degrees are equal, it's automatically satisfied. Therefore, such a bipartite graph exists. Therefore, one possible combination is a biregular bipartite graph where each gentleman dances with exactly 10 ladies and each lady dances with exactly 12 gentlemen. Therefore, the rectangular scheme can be described as such. However, the problem asks for the "simplest form of a rectangular scheme". Since actually writing out a 24x20 matrix is impractical here, we can describe it as a biregular matrix where every gentleman has exactly 10 dances and every lady has exactly 12 dances, which satisfies all the given conditions. Therefore, this is a valid answer.Final Answerb) One possible combination is a rectangular scheme where each of the 24 gentlemen dances with exactly 10 ladies, and each of the 20 ladies dances with exactly 12 gentlemen, forming a biregular bipartite graph. This satisfies all dance constraints: gentlemen dance 10 times (within 10-12), ladies dance 12 times (within 12-14), and the total number of dances is 240, which is consistent with the conditions.However, to present it in a concrete rectangular scheme, one could arrange it such that the first 12 gentlemen dance with the first 10 ladies and the next 12 gentlemen dance with the next 10 ladies, ensuring each lady dances with exactly 12 gentlemen. But due to space constraints, a specific matrix isn't feasible here. Instead, the existence is assured by combinatorial design principles.boxed{text{A biregular matrix where each gentleman dances with 10 ladies and each lady dances with 12 gentlemen.}}</think>Given the problem, we need to determine the number of gentlemen (G) and ladies (L) present at the party, considering the dance constraints and the total number of dances if every gentleman danced with every lady exactly once (480 dances). Key Steps for Part (a):1. Equations and Inequalities: - The product of the number of gentlemen and ladies is 480: ( G times L = 480 ). - Each gentleman danced with between 10 and 12 ladies: ( 10G leq D leq 12G ). - Each lady danced with between 12 and 14 gentlemen: ( 12L leq D leq 14L ). - The total number of dances ( D ) must satisfy both inequalities.2. Deriving Constraints: - Combining the inequalities: ( max(10G, 12L) leq min(12G, 14L) ). - This leads to ( 5G leq 7L ) and ( L leq G ).3. Solving for G and L: - Substituting ( L = frac{480}{G} ) into the inequalities: - ( 5G leq 7 left( frac{480}{G} right) ) simplifies to ( 5G^2 leq 3360 ) or ( G leq sqrt{672} approx 25.92 ). - ( frac{480}{G} leq G ) simplifies to ( G geq sqrt{480} approx 21.91 ). - Possible integer values for ( G ) between 22 and 25 that divide 480 are checked. Only ( G = 24 ) and ( L = 20 ) satisfy all conditions. Final Answer for Part (a):a) boxed{24} gentlemen and boxed{20} ladies. Key Steps for Part (b):1. Constructing the Dance Matrix: - Each gentleman must dance with 10 to 12 ladies, and each lady must dance with 12 to 14 gentlemen. - The simplest form is a biregular bipartite graph where each gentleman dances with exactly 10 ladies and each lady dances with exactly 12 gentlemen. - This configuration satisfies all constraints: total dances ( D = 240 ), within the required ranges for both gentlemen and ladies. Final Answer for Part (b):b) boxed{text{A biregular matrix where each gentleman dances with 10 ladies and each lady dances with 12 gentlemen.}}]

🔑:}a) If the number of gentlemen is denoted by (h) and the number of ladies by (d), then (h cdot d = 480 quad) (1).If the numbers of gentlemen who danced with 10, 11, and 12 ladies, respectively, are denoted by (h_{1}, h_{2}, h_{3}), and the numbers of ladies who danced with 12, 13, and 14 gentlemen, respectively, are denoted by (d_{1}, d_{2}, d_{3}), then[h_{1} + h_{2} + h_{3} = h quad ; quad d_{1} + d_{2} + d_{3} = h]while the number (T) of dances performed that evening can be given in two ways:[T = 10 h_{1} + 11 h_{2} + 12 h_{3} quad ; quad T = 12 d_{1} + 13 d_{2} + 14 d_{3}]From the transformations[begin{array}{lll}T = 10 (h_{1} + h_{2} + h_{3}) + h_{2} + 2 h_{3} & ; & T = 12 (d_{1} + d_{2} + d_{3}) + d_{2} + 2 d_{3} T = 12 (h_{1} + h_{2} + h_{3}) - 2 h_{1} - h_{2} & ; & T = 14 (d_{1} + d_{2} + d_{3}) - 2 d_{1} - d_{2}end{array}]it follows that[10 h < T quad, quad 14 d > T]and thus (12 d < T < 12 h) and (10 h < T < 14 d), so (d < h < frac{7}{5} d) summarized as[d < h < frac{7}{5} d]Furthermore, due to the conditions (h geq 12, d geq 10). The factorizations of 480 that satisfy both conditions are[10 cdot 48 quad, quad 12 cdot 40 quad, quad 15 cdot 32 quad, quad 16 cdot 30 quad, quad 20 cdot 24]From this, it follows from (1) that (h = 24, d = 20) must be the case, because only in this case is (2) also satisfied.Thus, these values are the only possible numbers for the gentlemen and ladies.b) With these numbers, all conditions of the problem are indeed satisfied, as the following example of an overview of dance pairings shows (see figure):Gentleman No. (longrightarrow)section*{Adapted from ([5])}

🧠:[<think>Okay, so I need to find the sum of all possible values of ( x_0 ) where the elliptic curve ( y^2 = x^3 + 1 ) is tangent to a circle centered at ( (4, 0) ). Hmm, tangency condition usually means that at the point of contact, both the curves intersect and their tangent lines are the same. So, I need to use this condition here. Let me start by thinking about how to represent the circle and then set up the equations for the tangency.First, the circle is centered at ( (4, 0) ), so its equation should be ( (x - 4)^2 + y^2 = r^2 ), where ( r ) is the radius. Since the circle is tangent to the elliptic curve at ( (x_0, y_0) ), that point must lie on both curves. Therefore, substituting ( x_0 ) and ( y_0 ) into both equations:1. ( y_0^2 = x_0^3 + 1 ) (from the elliptic curve)2. ( (x_0 - 4)^2 + y_0^2 = r^2 ) (from the circle)Also, the tangency condition requires that the derivatives (slopes of the tangent lines) at ( (x_0, y_0) ) are equal for both curves. So I need to compute the derivatives of both curves and set them equal at that point.Let me start by differentiating the elliptic curve implicitly with respect to ( x ):For ( y^2 = x^3 + 1 ),( 2y frac{dy}{dx} = 3x^2 ),so ( frac{dy}{dx} = frac{3x^2}{2y} ).For the circle ( (x - 4)^2 + y^2 = r^2 ),differentiating implicitly with respect to ( x ):( 2(x - 4) + 2y frac{dy}{dx} = 0 ),so ( frac{dy}{dx} = frac{4 - x}{y} ).Setting the derivatives equal at ( (x_0, y_0) ):( frac{3x_0^2}{2y_0} = frac{4 - x_0}{y_0} ).Since ( y_0 neq 0 ) (if ( y_0 = 0 ), then from the elliptic curve equation, ( 0 = x_0^3 + 1 ), so ( x_0 = -1 ). Let me check if that's possible. If ( x_0 = -1 ), then the circle equation becomes ( (-1 - 4)^2 + 0^2 = r^2 ), so ( r = 5 ). Then check if the circle ( (x - 4)^2 + y^2 = 25 ) is tangent to the elliptic curve at ( (-1, 0) ). Let's see: substitute ( y^2 = x^3 + 1 ) into the circle equation: ( (x - 4)^2 + x^3 + 1 = 25 ), so ( x^3 + (x - 4)^2 + 1 - 25 = 0 ). Simplify ( (x - 4)^2 = x^2 - 8x + 16 ), so the equation becomes ( x^3 + x^2 - 8x + 16 + 1 - 25 = x^3 + x^2 - 8x - 8 = 0 ). Let's check ( x = -1 ): ( (-1)^3 + (-1)^2 - 8(-1) -8 = -1 + 1 + 8 -8 = 0. So yes, x = -1 is a root. Then factor the polynomial: since x = -1 is a root, divide by (x + 1). Using polynomial division or synthetic division:Divide ( x^3 + x^2 -8x -8 ) by ( x + 1 ):- Coefficients: 1 | 1 | -8 | -8Bring down 1.Multiply -1: 1*(-1) = -1. Add to next coefficient: 1 + (-1) = 0.Multiply -1: 0*(-1) = 0. Add to next coefficient: -8 + 0 = -8.Multiply -1: -8*(-1) = 8. Add to last coefficient: -8 +8 =0. So quotient is ( x^2 + 0x -8 ), which is ( x^2 -8 ). Therefore, the polynomial factors as ( (x +1)(x^2 -8) =0 ). So roots are x=-1, x=2√2, x=-2√2. So the circle intersects the elliptic curve at x=-1, 2√2, -2√2. But since the original point was (-1,0), to check tangency, we need multiplicity. Since the polynomial equation after substitution has a root at x=-1 and two other roots, but since the original intersection at x=-1 is a simple root (since (x +1) is a single factor), so the circle and elliptic curve intersect at (-1,0) but not tangentially. Therefore, y_0 cannot be zero, so we can assume y_0 ≠0, so we can safely divide by y_0 in the derivative equation.Therefore, returning to the equation:( frac{3x_0^2}{2y_0} = frac{4 - x_0}{y_0} ).Multiply both sides by ( y_0 ):( frac{3x_0^2}{2} = 4 - x_0 ).Multiply both sides by 2:( 3x_0^2 = 8 - 2x_0 ).Bring all terms to one side:( 3x_0^2 + 2x_0 -8 =0 ).This is a quadratic equation in x_0. Let me compute its discriminant:Discriminant D = (2)^2 -4*3*(-8) = 4 + 96 = 100.Therefore, solutions:x_0 = [ -2 ± √100 ] / (2*3) = [ -2 ±10 ] /6.So two solutions:x_0 = ( -2 +10 ) /6 = 8/6 = 4/3 ≈1.333...andx_0 = ( -2 -10 ) /6 = -12/6 = -2.Wait, so possible x_0 are 4/3 and -2. But earlier, when I checked x=-1, y=0, that didn't work as a tangent point. Here, x_0 = -2. Let me check if x_0 = -2 is a valid point.From the elliptic curve equation: y_0^2 = (-2)^3 +1 = -8 +1 = -7. Wait, that's impossible, since y_0^2 can't be negative. So x_0 = -2 would lead to y_0^2 = -7, which is not possible. Therefore, x_0 = -2 is not a valid solution. So only x_0 =4/3 is valid? But that contradicts the quadratic equation.Wait, so why did we get x_0 =-2? Because when we set the derivatives equal, we assumed y_0 ≠0, but even so, substituting x_0 =-2 into the elliptic curve equation gives y_0^2 = (-2)^3 +1 =-7, which is impossible. Therefore, x_0 =-2 is an extraneous solution introduced when we squared something? Wait, but where did we square? Wait, actually, the only step where we could introduce extraneous solutions is when we divided by y_0, but we already considered y_0 ≠0. Wait, perhaps when solving the quadratic equation, we get two roots, but need to check if they actually lie on the elliptic curve.So x_0 must satisfy y_0^2 =x_0^3 +1 ≥0, so x_0^3 +1 ≥0 ⇒x_0 ≥ -1. So x_0 must be ≥-1. But x_0=-2 is less than -1, so y_0^2 would be negative, which is impossible. Therefore, x_0=-2 is invalid. So the only valid solution is x_0=4/3.But the problem says "the sum of all possible values of x_0". If only x_0=4/3 is valid, then the sum is 4/3. But the problem might have more solutions? Wait, maybe there's another case when y_0=0. But we saw that x_0=-1, y_0=0 is on the elliptic curve, but the circle centered at (4,0) passing through (-1,0) has radius 5, but the intersection at x=-1 is not a tangency point. So, perhaps x_0=4/3 is the only possible value. So the sum would be 4/3.But wait, let me double-check. Let me verify for x_0=4/3.Compute y_0^2=(4/3)^3 +1=64/27 +1=64/27 +27/27=91/27. So y_0=±√(91/27)=±√91/(3√3). So valid.Then, the circle equation at (4/3, y_0) is (4/3 -4)^2 + y_0^2 = r^2. Compute (4/3 -12/3)^2 + y_0^2= (-8/3)^2 + y_0^2=64/9 +91/27= (64*3 +91)/27=(192 +91)/27=283/27. So radius squared is 283/27.Now, check if the system of equations only intersects at that point. Let's substitute y^2 =x^3 +1 into the circle equation:(x -4)^2 +x^3 +1 = r^2=283/27.So x^3 + (x -4)^2 +1 -283/27=0.Compute (x -4)^2= x^2 -8x +16.Therefore, equation becomes x^3 +x^2 -8x +16 +1 -283/27= x^3 +x^2 -8x +17 -283/27.Convert 17 to 459/27:459/27 -283/27=176/27.So equation is x^3 +x^2 -8x +176/27=0.Multiply through by 27 to eliminate denominators:27x^3 +27x^2 -216x +176=0.We know x=4/3 is a root. Let's verify:27*(4/3)^3 +27*(4/3)^2 -216*(4/3) +176.Calculate each term:27*(64/27)=6427*(16/9)=27*(16)/9=3*16=48-216*(4/3)= -72*4= -288So total: 64 +48 -288 +176= (64+48)=112; (112 -288)= -176; (-176 +176)=0. Correct.So x=4/3 is a root. Let's factor the polynomial.Divide 27x^3 +27x^2 -216x +176 by (x -4/3). Wait, but 4/3 is a root. Alternatively, write it as (3x -4) to make factoring easier.Let me perform polynomial division.Divide 27x^3 +27x^2 -216x +176 by (3x -4).Using polynomial long division:First term: 27x^3 /3x=9x^2. Multiply (3x -4) by 9x^2: 27x^3 -36x^2.Subtract from the polynomial:(27x^3 +27x^2 -216x +176) - (27x^3 -36x^2) = 63x^2 -216x +176.Next term: 63x^2 /3x=21x. Multiply (3x -4) by21x:63x^2 -84x.Subtract:(63x^2 -216x +176) - (63x^2 -84x)= (-132x +176).Next term: -132x /3x= -44. Multiply (3x -4) by -44: -132x +176.Subtract:(-132x +176) - (-132x +176)=0.Therefore, the polynomial factors as (3x -4)(9x^2 +21x -44).Now, set 9x^2 +21x -44=0.Solve using quadratic formula:x = [ -21 ±√(21^2 -4*9*(-44)) ]/(2*9)Compute discriminant:441 + 4*9*44=441 + 1584=2025.√2025=45.Thus,x=(-21 ±45)/18.First solution: (-21 +45)/18=24/18=4/3.Second solution: (-21 -45)/18= -66/18= -11/3≈-3.666...So the polynomial has roots at x=4/3 (twice?) Wait, no. Wait, original polynomial was 27x^3 +27x^2 -216x +176 which factors into (3x -4)(9x^2 +21x -44). Then 9x^2 +21x -44=0 gives x=4/3 and x=-11/3. Wait, no, the quadratic equation solutions are x= [ -21 ±45 ] /18= 24/18=4/3 and -66/18=-11/3. So the roots are x=4/3, x=4/3 (from the quadratic?), no, wait, no. Wait, the cubic factors into (3x -4)(9x^2 +21x -44). The quadratic factor 9x^2 +21x -44 has roots at x=4/3 and x=-11/3? Wait, no:Wait, when solving 9x^2 +21x -44=0, we found x=( -21 ±45 )/18. So 24/18=4/3 and -66/18=-11/3. So the roots are x=4/3 and x=-11/3. Therefore, the cubic equation has roots at x=4/3 (from the linear factor) and x=4/3, x=-11/3? No, wait, no. The cubic factors into (3x -4)(9x^2 +21x -44). The roots of 9x^2 +21x -44=0 are x=4/3 and x=-11/3. Therefore, the cubic equation has roots x=4/3, x=4/3, x=-11/3? Wait, no. Wait, if you factor (3x -4)(9x^2 +21x -44), and 9x^2 +21x -44=0 has roots x=4/3 and x=-11/3, then the cubic factors as (3x -4)(3x -4)(3x +11). Wait, no, because 9x^2 +21x -44 factors as (3x -4)(3x +11). Let me check:Multiply (3x -4)(3x +11)=9x^2 +33x -12x -44=9x^2 +21x -44. Yes, correct.Therefore, the cubic factors as (3x -4)(3x -4)(3x +11). So the roots are x=4/3 (double root) and x=-11/3.Wait, so that means when we substituted the circle equation into the elliptic curve, we ended up with a cubic equation that has a double root at x=4/3 and another root at x=-11/3. Therefore, the intersection points are x=4/3 (with multiplicity two, hence tangency) and x=-11/3. But x=-11/3 is approximately -3.666, which is less than -1, so when we plug into the elliptic curve equation y^2 =x^3 +1, x=-11/3:x^3 +1 = (-1331/27) +1 = (-1331 +27)/27 = -1304/27 <0, so y^2 negative, invalid. Therefore, the only real intersection point is x=4/3, which is a double root, hence the circle is tangent to the elliptic curve at x=4/3. Therefore, the only possible x_0 is 4/3, so the sum is 4/3.But the problem says "the sum of all possible values of x_0". So, unless there are other tangent points, which would require other circles centered at (4,0) tangent to the elliptic curve at different x_0. Wait, is there another circle centered at (4,0) that is tangent to the elliptic curve at a different point? For example, maybe at the other intersection point when x=-11/3, but that point is invalid because y^2 would be negative. So perhaps only x=4/3 is valid.Wait, but perhaps there are two different tangent points corresponding to two different circles? But the problem states "a circle centered at (4,0)", so maybe there can be multiple circles centered at (4,0) tangent to the elliptic curve at different points. But in our analysis, we found only one valid x_0=4/3. The other solution x_0=-2 was invalid. But wait, in the quadratic equation for the derivative, we had x_0=4/3 and x_0=-2, but x_0=-2 is invalid. So only one value. Therefore, the sum is 4/3.But the problem says "the sum of all possible values of x_0". If there's only one value, the sum is itself. But maybe I missed something. Let's see.Alternatively, perhaps there's another tangent point when y_0=0. But as we saw earlier, at x=-1, y=0, but that point is not a tangent point. Let me check again. The circle centered at (4,0) passing through (-1,0) has radius 5, and the equation is (x-4)^2 + y^2=25. When we substitute y^2=x^3 +1 into this, we get x^3 + (x-4)^2 +1=25, which simplifies to x^3 +x^2 -8x +16 +1 -25= x^3 +x^2 -8x -8=0. As before, this factors as (x+1)(x^2 -8)=0, so roots x=-1, x=2√2≈2.828, x=-2√2≈-2.828. The point x=-1 is valid (y=0), but x=2√2 and x=-2√2. Let's check if at x=2√2, y^2=(2√2)^3 +1=16√2 +1. Wait, (2√2)^3=8*(√2)^3=8*(2√2)=16√2. So y^2=16√2 +1≈16*1.414 +1≈22.624 +1≈23.624, so real y. Similarly, x=-2√2: y^2=(-2√2)^3 +1= -16√2 +1≈-22.624 +1≈-21.624 <0, invalid. Therefore, circle intersects the elliptic curve at x=-1, y=0 and x=2√2, y=±√(16√2 +1). But since multiplicity at x=-1 is 1 (simple root), so the circle is not tangent there. So tangency only occurs at x=4/3.But wait, when we set up the derivative condition, we found x_0=4/3 is the only valid solution. So perhaps that's the only tangent point.But the problem says "the sum of all possible values of x_0". If only one value, sum is 4/3. But maybe the problem is expecting more? Wait, perhaps when setting up the equations, I should consider that the system could have higher multiplicity? Or maybe circles that are tangent at two different points?But the problem states "the elliptic curve is tangent to a circle centered at (4,0) at the point (x_0, y_0)". So it's tangent at one point (x_0,y_0). But there could be multiple such circles, each tangent at a different point, hence different x_0's. So we need to find all x_0's such that there exists a circle centered at (4,0) tangent to the elliptic curve at (x_0, y_0), and then sum those x_0's.But in our previous analysis, we found only x_0=4/3 as valid. But let me check again.Wait, maybe when we solved the quadratic equation, we got x_0=4/3 and x_0=-2. But x_0=-2 is invalid. So only one x_0. So sum is 4/3.But maybe there is another scenario where y_0=0. But as checked earlier, x_0=-1 gives y=0, but the circle through (-1,0) isn't tangent there. So maybe no. Therefore, the answer is 4/3. However, the problem says "the sum of all possible values of x_0", which implies there might be multiple x_0. But according to our analysis, only one. Hmm.Wait, but let me check again. Let me parametrize the circle. The circle is centered at (4,0) and passes through (x_0,y_0) with radius r. The tangency condition gives two equations: the point is on both curves, and the derivatives are equal. So we have:1. ( y_0^2 = x_0^3 + 1 )2. ( (x_0 -4)^2 + y_0^2 = r^2 )3. ( frac{3x_0^2}{2y_0} = frac{4 - x_0}{y_0} ) ⇒ ( 3x_0^2 = 2(4 - x_0) )From equation 3: 3x_0² + 2x_0 -8 =0, as before. Solutions x_0=( -2 ±√(4 +96 ))/6=( -2 ±10)/6 ⇒ x_0=8/6=4/3 and x_0=-12/6=-2. As x_0=-2 is invalid, only x_0=4/3.Hence, the only possible x_0 is 4/3, so sum is 4/3.But the problem is from an exam, perhaps expecting an integer or a fraction. 4/3 is a fraction. But let me check again. Maybe there is another case when the point is at y=0, but in that case, the derivative is undefined (since y=0 in the elliptic curve, the derivative would involve division by zero). However, the circle's derivative at y=0 would be slope (4 -x)/0, which is vertical or undefined. So, the tangent line would be vertical. Let me see if the elliptic curve has a vertical tangent at y=0.For the elliptic curve ( y^2 =x^3 +1 ), the derivative at y=0 is undefined (since dy/dx = 3x²/(2y)), which tends to infinity as y approaches 0, so the tangent line is vertical. Similarly, the circle at (x_0,0) would have a vertical tangent line if the derivative is undefined. The derivative of the circle at (x_0,0) is (4 -x)/y, which at y=0 would be undefined. So if both have vertical tangent lines at that point, then they could be tangent. So let's check if at (x_0,0), both curves have vertical tangent lines.From the elliptic curve, vertical tangent occurs when y=0, which implies x_0^3 +1=0 ⇒x_0=-1. So point (-1,0). For the circle centered at (4,0), the radius would be the distance from (4,0) to (-1,0), which is 5. So the circle is (x-4)^2 + y^2=25. Now, check if this circle is tangent to the elliptic curve at (-1,0). As before, substituting y^2=x³ +1 into the circle equation gives (x-4)^2 +x³ +1=25. Which simplifies to x³ +x² -8x -8=0, which factors as (x+1)(x² -8)=0. So roots at x=-1, x=2√2, x=-2√2. So the circle intersects the elliptic curve at three points: (-1,0), (2√2, ±sqrt((2√2)^3 +1)). Since there are three intersection points, the circle is not tangent at (-1,0); it just passes through it. Hence, even though both curves have vertical tangents at (-1,0), the circle cuts through the elliptic curve there, so it's not a point of tangency.Therefore, the only possible x_0 is 4/3, so the sum is 4/3. Therefore, the answer is boxed{dfrac{4}{3}}.Final Answerboxed{dfrac{4}{3}}</think>To determine the sum of all possible values of ( x_0 ) where the elliptic curve ( y^2 = x^3 + 1 ) is tangent to a circle centered at ( (4, 0) ), we start by noting that the point of tangency ((x_0, y_0)) must satisfy both the equations of the elliptic curve and the circle, as well as having equal derivatives at that point.1. Equations of the curves: - Elliptic curve: ( y_0^2 = x_0^3 + 1 ) - Circle: ( (x_0 - 4)^2 + y_0^2 = r^2 )2. Derivatives at the point of tangency: - For the elliptic curve ( y^2 = x^3 + 1 ), the derivative is ( frac{dy}{dx} = frac{3x^2}{2y} ). - For the circle ( (x - 4)^2 + y^2 = r^2 ), the derivative is ( frac{dy}{dx} = frac{4 - x}{y} ).Setting the derivatives equal at ((x_0, y_0)) gives:[ frac{3x_0^2}{2y_0} = frac{4 - x_0}{y_0} ]Multiplying both sides by ( y_0 ) (assuming ( y_0 neq 0 )):[ frac{3x_0^2}{2} = 4 - x_0 ]Multiplying through by 2:[ 3x_0^2 = 8 - 2x_0 ]Rearranging into a quadratic equation:[ 3x_0^2 + 2x_0 - 8 = 0 ]Solving this quadratic equation using the quadratic formula:[ x_0 = frac{-2 pm sqrt{4 + 96}}{6} = frac{-2 pm 10}{6} ]This gives the solutions:[ x_0 = frac{8}{6} = frac{4}{3} quad text{and} quad x_0 = frac{-12}{6} = -2 ]Checking the validity of these solutions:- For ( x_0 = -2 ), substituting into the elliptic curve equation gives ( y_0^2 = (-2)^3 + 1 = -7 ), which is not possible.- For ( x_0 = frac{4}{3} ), substituting into the elliptic curve equation gives ( y_0^2 = left(frac{4}{3}right)^3 + 1 = frac{91}{27} ), which is valid.Thus, the only valid solution is ( x_0 = frac{4}{3} ). The sum of all possible values of ( x_0 ) is therefore:[ boxed{dfrac{4}{3}} ],<think>Okay, so I need to figure out the sum of all possible x₀ values where the elliptic curve y² = x³ + 1 is tangent to a circle centered at (4, 0). Hmm, let's start by recalling that if two curves are tangent to each other at a point, they must satisfy two conditions: they both pass through that point, and their tangent lines at that point are equal. So, for the point (x₀, y₀), it must lie on both the elliptic curve and the circle, and their derivatives at that point must be the same.First, let's write the equation of the circle. Since it's centered at (4, 0), the general equation is (x - 4)² + (y - 0)² = r², where r is the radius. But since we don't know the radius, maybe we can express it in terms of x₀ and y₀. Since (x₀, y₀) is a point on the circle, substituting gives (x₀ - 4)² + y₀² = r². So, the equation of the circle is (x - 4)² + y² = (x₀ - 4)² + y₀².Now, the point (x₀, y₀) is also on the elliptic curve y² = x³ + 1. Therefore, we can substitute y₀² with x₀³ + 1. So, the equation of the circle becomes (x - 4)² + y² = (x₀ - 4)² + x₀³ + 1.Next, we need to ensure that the derivatives of both curves at (x₀, y₀) are equal. Let's compute the derivatives.For the elliptic curve y² = x³ + 1, we can take the derivative implicitly with respect to x. Differentiating both sides:2y * dy/dx = 3x²So, dy/dx = (3x²) / (2y)At the point (x₀, y₀), the slope of the tangent line is (3x₀²) / (2y₀).Now, for the circle (x - 4)² + y² = (x₀ - 4)² + x₀³ + 1, let's find the derivative. Again, differentiate implicitly with respect to x:2(x - 4) + 2y * dy/dx = 0Solving for dy/dx:dy/dx = (4 - x) / yAt the point (x₀, y₀), the slope is (4 - x₀) / y₀.Since the slopes must be equal at the point of tangency, we set the two expressions for dy/dx equal to each other:(3x₀²) / (2y₀) = (4 - x₀) / y₀Assuming y₀ ≠ 0 (if y₀ = 0, then from the elliptic curve equation, 0² = x₀³ + 1, which would require x₀³ = -1, so x₀ = -1. Let's check if that's a possible point. If x₀ = -1, then y₀ = 0. Then the circle equation would be ( -1 - 4 )² + 0² = (-5)² = 25. So the circle would be (x - 4)² + y² = 25. But we need to check if this circle is tangent to the elliptic curve at (-1, 0). Let's see. The elliptic curve at x = -1 has y = 0. Let's check if the circle intersects the elliptic curve only at (-1, 0). Plug x = -1 into the circle equation: ( -1 -4 )² + y² = 25 + y² = 25 implies y²=0, so y=0. So the circle meets the elliptic curve only at (-1, 0). But does the circle also have the same tangent there? Let's check the slopes. For the elliptic curve at (-1, 0), the derivative is (3*(-1)^2)/(2*0) which is undefined (infinite slope), meaning a vertical tangent. For the circle, the derivative at (-1, 0) is (4 - (-1))/0, which is also undefined (infinite slope). So, yes, both have vertical tangent lines there, so they are tangent. So x₀ = -1 is a possible value. But let's see if there are others where y₀ ≠ 0.Assuming y₀ ≠ 0, we can multiply both sides of the slope equation by 2y₀ to eliminate denominators:3x₀² = 2(4 - x₀)So, 3x₀² + 2x₀ - 8 = 0.Wait, that's a quadratic equation in x₀. Let's write that again:3x₀² + 2x₀ - 8 = 0.Wait, hold on, let's double-check. If we have:(3x₀²)/(2y₀) = (4 - x₀)/y₀Multiply both sides by 2y₀:3x₀² = 2(4 - x₀)So, 3x₀² + 2x₀ - 8 = 0. Wait, actually, moving terms over:3x₀² = 8 - 2x₀So 3x₀² + 2x₀ -8 =0. Correct.Now, solving for x₀, we can use quadratic formula:x₀ = [-2 ± sqrt( (2)^2 - 4*3*(-8) )]/(2*3)Compute discriminant:4 + 96 = 100So sqrt(100)=10Therefore, x₀ = [-2 ±10]/6So two solutions:x₀ = (-2 +10)/6=8/6=4/3≈1.333...x₀=(-2 -10)/6= -12/6= -2So possible x₀ values from the quadratic are 4/3 and -2. But we need to check if these points actually lie on both curves and if the circle is indeed tangent there. Also, we need to check if y₀ exists for these x₀ on the elliptic curve.First, x₀ = 4/3. Then y₀² = (4/3)^3 +1 = 64/27 +1= 64/27 +27/27=91/27. So y₀ exists (real) as sqrt(91/27). So y₀ is ±√(91/27). So that point is valid.For x₀ = -2, y₀²= (-2)^3 +1= -8+1=-7. Wait, y₀²=-7? That's impossible in real numbers. So x₀=-2 is not a real solution. Therefore, only x₀=4/3 is a valid solution from the quadratic, but we already had x₀=-1 from the case where y₀=0.Therefore, the possible x₀ values are x₀=-1 and x₀=4/3.Wait, but wait, when we solved the quadratic, we got x₀=4/3 and x₀=-2. But x₀=-2 is invalid because y₀² would be negative. So only x₀=4/3 is valid. But we also had x₀=-1 from the y₀=0 case. So total possible x₀ values are -1 and 4/3.Hence, the sum of all possible x₀ is -1 + 4/3 = 1/3.Wait, but let me confirm once more. The problem states that the elliptic curve is tangent to a circle centered at (4,0) at (x₀, y₀). So, for each x₀, there exists such a circle. We found two x₀ values: -1 and 4/3. Let's check if both are valid.First, x₀=-1, y₀=0. As we saw earlier, the circle would be (x -4)^2 + y^2 = (-1 -4)^2 +0=25. So, the circle is (x-4)^2 + y²=25. At (-1,0), the elliptic curve has a vertical tangent, and the circle also has a vertical tangent here because the radius from (4,0) to (-1,0) is horizontal, so the tangent is vertical. So yes, tangency holds.For x₀=4/3, y₀=±sqrt((4/3)^3 +1)=±sqrt(64/27 +27/27)=±sqrt(91/27)=±sqrt(91)/3√3≈±(9.539)/5.196≈±1.837. So the circle here would have equation (x-4)^2 + y²=(4/3 -4)^2 + (91/27). Let's compute that:(4/3 -12/3)^2= (-8/3)^2=64/9. Then adding y₀²=91/27. So total radius squared is 64/9 +91/27= (192 +91)/81=283/81≈3.494. So the circle is (x-4)^2 + y²=283/81.Now, we need to check if this circle is tangent to the elliptic curve at (4/3, sqrt(91/27)). To check tangency, besides equal derivatives, the system of equations should have only one solution at that point (double root). But since we already have matching derivatives and the point is common, it's tangent. However, to be thorough, maybe we can check if substituting y² from the elliptic curve into the circle equation gives a double root.The circle equation is (x -4)^2 + y²=283/81. But y²=x³ +1. So substituting:(x -4)^2 + x³ +1 =283/81Compute left side:(x² -8x +16) +x³ +1 =x³ +x² -8x +17Set equal to 283/81:x³ +x² -8x +17 =283/81Multiply both sides by 81 to eliminate denominator:81x³ +81x² -648x +1377=283Then, 81x³ +81x² -648x +1377 -283=0Compute 1377 -283=1094So equation:81x³ +81x² -648x +1094=0Now, we know that x=4/3 is a root (since the point (4/3, y₀) is on both curves). Let's check if x=4/3 is a double root.First, divide the polynomial by (x -4/3) and see if the remainder is zero, then check if the derivative at x=4/3 is also zero.Alternatively, compute the polynomial and its derivative at x=4/3.First, compute f(4/3):81*(4/3)^3 +81*(4/3)^2 -648*(4/3) +1094First, (4/3)^3=64/27, so 81*(64/27)= (81/27)*64=3*64=192(4/3)^2=16/9, so 81*(16/9)= (81/9)*16=9*16=144-648*(4/3)= -648/3 *4= -216*4= -864So summing up:192 +144 -864 +1094= (192+144)=336; 336 -864= -528; -528 +1094=566≠0.Wait, that can't be. Wait, but we set up the equation such that when we substituted x=4/3, it should satisfy the equation. But 81x³ +81x² -648x +1094=0, but f(4/3)=566≠0. That means there is a mistake in my substitution.Wait, but this suggests that I made an error in the algebra when setting up the equation. Let's retrace.Original substitution: (x -4)^2 + y²=283/81 and y²=x³ +1. Therefore, (x -4)^2 +x³ +1=283/81.But expanding (x -4)^2 gives x² -8x +16. So, x³ +x² -8x +17=283/81. Multiply both sides by 81:81x³ +81x² -648x +1377=283. Then 81x³ +81x² -648x +1094=0. But when plugging x=4/3, we get 566 instead of 0, which is inconsistent. So, where is the mistake?Wait, wait, wait. Wait, the radius squared was computed as (4/3 -4)^2 + y₀². So (4/3 -12/3)^2= (-8/3)^2=64/9. Then y₀²= x₀³ +1=(64/27 +1)=91/27. So radius squared is 64/9 +91/27. Convert 64/9 to 192/27. Then 192/27 +91/27=283/27. Wait, 283/27 is approximately 10.481, but earlier I wrote 283/81≈3.494. Wait, that's the mistake! 64/9 is 192/27, adding 91/27 gives 283/27, not 283/81. So the radius squared is 283/27, so the equation is (x-4)^2 + y²=283/27. So substituting into the circle equation:(x -4)^2 + y²=283/27. But y²=x³ +1. Therefore, (x -4)^2 +x³ +1=283/27.Then, moving 283/27 to the left:x³ + (x -4)^2 +1 -283/27=0.But (x -4)^2= x² -8x +16. So:x³ +x² -8x +16 +1 -283/27=0.Which is x³ +x² -8x +17 -283/27=0.Convert 17 to 459/27, so 459/27 -283/27=176/27.Therefore, the equation is x³ +x² -8x +176/27=0.Multiply through by 27 to eliminate denominators:27x³ +27x² -216x +176=0.Now, check x=4/3:27*(64/27) +27*(16/9) -216*(4/3) +176=64 + (27*16)/9 - (216*4)/3 +176=64 + (432)/9 - 864/3 +176=64 + 48 - 288 +176Compute step by step:64 +48=112112 -288= -176-176 +176=0Okay, that works. So the polynomial is 27x³ +27x² -216x +176=0. Now, we need to check if x=4/3 is a double root. Let's compute the derivative of this polynomial:f(x)=27x³ +27x² -216x +176f’(x)=81x² +54x -216Evaluate f’(4/3):81*(16/9) +54*(4/3) -216= (81/9)*16 + (54/3)*4 -216=9*16 +18*4 -216=144 +72 -216=216 -216=0So yes, x=4/3 is a double root. Therefore, the equation has a double root at x=4/3, confirming that the circle is tangent to the elliptic curve there. Therefore, x₀=4/3 is valid.Similarly, for x₀=-1, we had the circle (x -4)^2 + y²=25. Substituting y²=x³ +1 into the circle equation:(x -4)^2 +x³ +1=25Expand (x -4)^2:x² -8x +16 +x³ +1=25x³ +x² -8x +17=25x³ +x² -8x -8=0We know x=-1 is a root. Let's factor:Divide polynomial x³ +x² -8x -8 by (x +1):Using synthetic division:-1 | 1 1 -8 -8 -1 0 8 1 0 -8 0So, x³ +x² -8x -8=(x +1)(x² -8)Therefore, roots are x=-1, x=±√8=±2√2. But x=2√2≈2.828 and x=-2√2≈-2.828. However, these x-values must satisfy y²=x³ +1. Let's check x=2√2: y²=(2√2)^3 +1=16√2 +1, which is positive, so real y. Similarly, x=-2√2: y²=(-2√2)^3 +1= -16√2 +1, which is negative, so invalid. Therefore, the other intersection points are (2√2, ±sqrt(16√2 +1)), but since these are distinct from x=-1, the circle intersects the elliptic curve at x=-1 (with multiplicity 1) and at x=2√2 (each with multiplicity 1). Therefore, the total intersections are three points: (-1,0) once and (2√2, ±sqrt(...)) twice. Therefore, the circle is not tangent at x=-1, but intersects transversely there. Wait, but we had earlier that at x=-1, both curves have vertical tangent lines, so they should be tangent there. This seems contradictory. Wait, but according to the polynomial x³ +x² -8x -8=0, which factors as (x +1)(x² -8)=0, so x=-1 and x=±√8. But if substituting x=-1 into the original circle and elliptic curve, we get only one point (-1,0). The other roots x=±√8 give two points each (since y²=x³ +1, so for x=√8, y²=(√8)^3 +1=8^(3/2)+1=8*2.828+1≈22.624+1≈23.624, which is positive). However, when we derived the equation for the circle tangent at (-1,0), the system of equations when substituting y²=x³ +1 into the circle's equation gives a cubic equation with x=-1 as a root. But the multiplicity of the root at x=-1 is 1, as seen from the factorization (x +1)(x² -8). Therefore, the circle intersects the elliptic curve at x=-1 once and at x=±√8 once each. Therefore, the intersection at x=-1 is a simple intersection, not a tangent. But this contradicts our previous conclusion that the derivatives are both vertical, implying tangency. Hmm, where is the mistake here?Wait, the key is that when two curves have the same tangent line at a point, the multiplicity of intersection at that point is at least two. Therefore, if the circle is tangent to the elliptic curve at (-1,0), then x=-1 should be a double root of the equation x³ +x² -8x -8=0. However, in our case, the polynomial factors as (x +1)(x² -8), so x=-1 is a single root, which suggests that the intersection multiplicity is 1, meaning they cross each other there, not tangent. But we had earlier that both derivatives are vertical, so they should have the same tangent line. How come?Wait, perhaps the problem is that even though the tangent lines are the same, the intersection multiplicity could still be higher? Wait, but in algebraic geometry, if two curves have a common tangent at a point, their intersection multiplicity at that point is at least two. However, in our case, substituting the circle and the elliptic curve gives a cubic equation with x=-1 as a simple root, which suggests that despite having the same tangent, they intersect with multiplicity 1. This seems contradictory. Let's check with specific example.Take the elliptic curve y² =x³ +1 and the circle (x -4)^2 + y²=25. At x=-1, y=0. Let's parametrize near x=-1. For the elliptic curve, near x=-1, y≈±sqrt((-1)^3 +1)=0, so it's a point where the curve has a vertical tangent. The circle at (-1,0) also has a vertical tangent. So, if we zoom in around (-1,0), both curves have a vertical tangent. However, if we look at the intersection multiplicity, let's consider solving the equations together. The circle equation is (x -4)^2 + y²=25. Substitute y²=x³ +1:(x -4)^2 +x³ +1=25 ⇒x³ +x² -8x +17 -25=0⇒x³ +x² -8x -8=0. Which factors as (x +1)(x² -8)=0. So, only single root at x=-1. Therefore, multiplicity is 1. But how is that possible if they have the same tangent?Wait, perhaps because the other intersections are complex? But no, x=±√8 are real. Wait, but in the case of tangency, we expect a double root. So maybe in this case, even though the tangent lines are the same, the intersection multiplicity is still 1 because one of the branches is real and the other is complex? Wait, no. Wait, intersection multiplicity is a local concept. At the point (-1,0), even though the tangent lines are the same, the curves might intersect with multiplicity higher than 1. But according to the equation, it's a simple root, so multiplicity 1. This is confusing. Let me recall that in algebraic geometry, the intersection multiplicity at a point is the sum of the multiplicities of the corresponding factor in the resultant equation. Here, x=-1 is a simple root, so multiplicity 1. However, in the case of two curves having a common tangent, the intersection multiplicity should be at least 2. So perhaps there is a miscalculation here. Wait, but in this case, the elliptic curve has a node or a cusp at (-1,0)? Wait, no. The elliptic curve y² =x³ +1. At (-1,0), the derivative is undefined (infinite), which means it's a vertical tangent. Similarly, the circle also has a vertical tangent there. So, if both curves have a vertical tangent at (-1,0), then the intersection multiplicity should be at least 2. But according to the algebra, it's only 1. This is a contradiction. Therefore, there must be an error in my reasoning.Wait, let's consider parametrizing the curves near (-1,0). For the elliptic curve, near x=-1, y can be approximated as follows. Let’s set x = -1 + t, where t is small. Then y² = (-1 + t)^3 +1 = (-1 + 3t - 3t² + t³) +1 = 3t - 3t² + t³. So, y ≈ ±sqrt(3t) as t approaches 0. Therefore, near t=0, the curve has two branches: y ≈ sqrt(3)t^{1/2} and y ≈ -sqrt(3)t^{1/2}. So, it's like a cusp where the two branches meet at a vertical tangent.For the circle centered at (4,0) passing through (-1,0), the equation is (x -4)^2 + y²=25. Near (-1,0), let's parametrize the circle. Let’s set x = -1 + t, then the equation becomes (-1 + t -4)^2 + y² =25 ⇒(t -5)^2 + y²=25 ⇒y²=25 - (t -5)^2 =25 - (t² -10t +25)=10t -t². So, y ≈ ±sqrt(10t) as t approaches 0. Comparing to the elliptic curve's y ≈ ±sqrt(3t). So, near t=0, both curves have y ~ sqrt(t), but with different coefficients. Therefore, they intersect at t=0 with multiplicity 1? Wait, how does that work?If two curves have parametrizations y = sqrt(a t) and y = sqrt(b t), then setting sqrt(a t) = sqrt(b t) gives t=0 as the only solution for a ≠ b. Therefore, the intersection at t=0 is simple (multiplicity 1). However, if a = b, then they coincide for all t, which would be a multiplicity infinity. But in our case, a=3 and b=10, so different. Therefore, the intersection multiplicity at (-1,0) is 1, even though they share the same vertical tangent. That seems to contradict the usual statement that common tangent implies intersection multiplicity at least 2.Wait, maybe because the tangent line is the same, but the curves don't actually share a common component in that tangent line. For example, if two curves cross each other at a point with the same tangent line, but in different branches, maybe the intersection multiplicity is still 1. Wait, but according to the formal definition, intersection multiplicity is the number of intersections near the point after perturbation, counting with multiplicity. If two curves have a common tangent, a small perturbation would make them intersect in two points near the original, hence multiplicity at least 2. But in this case, the perturbation might not separate the intersection into two points because one curve has a cusp?Wait, the elliptic curve at (-1,0) has a cusp? Wait, y² =x³ +1. The derivative dy/dx is infinite there, but actually, the curve has two branches meeting at (-1,0) with vertical tangents. Wait, no, if you solve for y, you get y=±sqrt(x³ +1). At x=-1, x³ +1=0, so y=0. For x near -1, x³ +1 is near 0. If x approaches -1 from the right (x > -1), x³ +1 is positive, so real y. If x approaches from the left (x < -1), x³ +1 is negative, so y imaginary. Therefore, the point (-1,0) is an endpoint of the real curve; the curve doesn't extend to x < -1 in real numbers. Therefore, near (-1,0), the elliptic curve only exists for x ≥ -1, approaching from the right. Therefore, the curve has a vertical tangent at (-1,0), but only on one side. The circle, on the other hand, is a closed curve. So, near (-1,0), the circle passes through (-1,0) and continues on both sides. Wait, but since the circle is centered at (4,0) with radius 5, it extends from x=4 -5=-1 to x=4 +5=9. So, at x=-1, which is the leftmost point of the circle, the circle has a vertical tangent. However, the elliptic curve at x=-1 is only the point (-1,0), as for x < -1, y²=x³ +1 would be negative. Therefore, the circle and the elliptic curve meet at (-1,0), but the elliptic curve doesn't extend beyond that point. Therefore, in this case, the intersection at (-1,0) is a point where the circle meets the endpoint of the elliptic curve. In such a case, the concept of tangency might still apply if the direction of the curve at the endpoint is tangent to the circle. Since the elliptic curve has a vertical tangent at (-1,0), and the circle also has a vertical tangent there, they are tangent at that point, even though the elliptic curve doesn't extend beyond it. Therefore, even though algebraically the intersection multiplicity is 1, geometrically they are tangent at that point because their tangent lines coincide and the curves don't cross each other there (since the elliptic curve ends at that point). Therefore, x₀=-1 should be considered a valid tangent point.Therefore, the possible x₀ values are -1 and 4/3, and their sum is -1 + 4/3 = ( -3/3 +4/3 )=1/3.So, the answer is boxed{dfrac{1}{3}}.Final Answerboxed{dfrac{1}{3}}</think>To determine the sum of all possible values of ( x_0 ) where the elliptic curve ( y^2 = x^3 + 1 ) is tangent to a circle centered at ( (4, 0) ), we need to ensure that both the curves pass through the point ( (x_0, y_0) ) and have the same tangent line at that point.1. Equation of the Circle: The circle centered at ( (4, 0) ) with radius ( r ) has the equation: [ (x - 4)^2 + y^2 = r^2 ] Substituting the point ( (x_0, y_0) ) into the circle's equation, we get: [ (x_0 - 4)^2 + y_0^2 = r^2 ] Since ( (x_0, y_0) ) lies on the elliptic curve ( y^2 = x^3 + 1 ), we substitute ( y_0^2 = x_0^3 + 1 ) into the circle's equation: [ (x_0 - 4)^2 + x_0^3 + 1 = r^2 ]2. Derivatives: - For the elliptic curve ( y^2 = x^3 + 1 ), the derivative is: [ frac{dy}{dx} = frac{3x^2}{2y} ] - For the circle ( (x - 4)^2 + y^2 = r^2 ), the derivative is: [ frac{dy}{dx} = frac{4 - x}{y} ] Setting the derivatives equal at ( (x_0, y_0) ): [ frac{3x_0^2}{2y_0} = frac{4 - x_0}{y_0} ] Simplifying, we get: [ 3x_0^2 = 2(4 - x_0) implies 3x_0^2 + 2x_0 - 8 = 0 ]3. Solving the Quadratic Equation: Solving ( 3x_0^2 + 2x_0 - 8 = 0 ) using the quadratic formula: [ x_0 = frac{-2 pm sqrt{4 + 96}}{6} = frac{-2 pm 10}{6} ] This gives solutions: [ x_0 = frac{8}{6} = frac{4}{3} quad text{and} quad x_0 = frac{-12}{6} = -2 ] However, ( x_0 = -2 ) is invalid as it leads to ( y_0^2 = -7 ), which is not real.4. Checking ( y_0 = 0 ): If ( y_0 = 0 ), then ( x_0^3 + 1 = 0 implies x_0 = -1 ). The circle centered at ( (4, 0) ) with radius 5 (since ( (-1 - 4)^2 + 0^2 = 25 )) is tangent to the elliptic curve at ( (-1, 0) ).5. Valid ( x_0 ) Values: The valid ( x_0 ) values are ( -1 ) and ( frac{4}{3} ).6. Sum of Values: The sum of all possible ( x_0 ) values is: [ -1 + frac{4}{3} = frac{-3}{3} + frac{4}{3} = frac{1}{3} ]Thus, the sum of all possible values of ( x_0 ) is (boxed{dfrac{1}{3}}).]

🔑:1. Equation of the Circle: The equation of a circle centered at ((4,0)) with radius (r) is: [ (x-4)^2 + y^2 = r^2 ] Rearranging this equation, we get: [ y^2 = r^2 - (x-4)^2 = r^2 - (x^2 - 8x + 16) = r^2 - x^2 + 8x - 16 ]2. Implicit Differentiation of the Circle: Differentiating both sides of (y^2 = r^2 - x^2 + 8x - 16) with respect to (x), we get: [ 2y frac{dy}{dx} = -2x + 8 ] Solving for (frac{dy}{dx}), we have: [ frac{dy}{dx} = frac{-2x + 8}{2y} ]3. Implicit Differentiation of the Elliptic Curve: The equation of the elliptic curve is (y^2 = x^3 + 1). Differentiating both sides with respect to (x), we get: [ 2y frac{dy}{dx} = 3x^2 ] Solving for (frac{dy}{dx}), we have: [ frac{dy}{dx} = frac{3x^2}{2y} ]4. Setting the Derivatives Equal: Since the curves are tangent at ((x_0, y_0)), their derivatives must be equal at this point: [ frac{-2x_0 + 8}{2y_0} = frac{3x_0^2}{2y_0} ] Simplifying, we get: [ -2x_0 + 8 = 3x_0^2 ] Rearranging, we obtain: [ 3x_0^2 + 2x_0 - 8 = 0 ]5. Solving the Quadratic Equation: Solving the quadratic equation (3x_0^2 + 2x_0 - 8 = 0) using the quadratic formula (x = frac{-b pm sqrt{b^2 - 4ac}}{2a}), where (a = 3), (b = 2), and (c = -8), we get: [ x_0 = frac{-2 pm sqrt{2^2 - 4 cdot 3 cdot (-8)}}{2 cdot 3} = frac{-2 pm sqrt{4 + 96}}{6} = frac{-2 pm sqrt{100}}{6} = frac{-2 pm 10}{6} ] Thus, the solutions are: [ x_0 = frac{8}{6} = frac{4}{3} quad text{and} quad x_0 = frac{-12}{6} = -2 ]6. Checking Validity of Solutions: - For (x_0 = frac{4}{3}): [ y_0^2 = left(frac{4}{3}right)^3 + 1 = frac{64}{27} + 1 = frac{64}{27} + frac{27}{27} = frac{91}{27} ] Since (y_0) is real, (x_0 = frac{4}{3}) is valid. - For (x_0 = -2): [ y_0^2 = (-2)^3 + 1 = -8 + 1 = -7 ] Since (y_0^2) is negative, (y_0) is imaginary, so (x_0 = -2) is not valid.7. Considering (y_0 = 0): If (y_0 = 0), then from the circle equation: [ (x_0 - 4)^2 = r^2 ] and from the elliptic curve: [ 0 = x_0^3 + 1 implies x_0^3 = -1 implies x_0 = -1 ] This is a valid solution.8. Sum of Valid (x_0) Values: The valid (x_0) values are (frac{4}{3}) and (-1). Their sum is: [ frac{4}{3} - 1 = frac{4}{3} - frac{3}{3} = frac{1}{3} ]The final answer is (boxed{frac{1}{3}})