🧠:[<think>Okay, let's try to figure out this problem. The question says: Given that cosθ times tanθ is less than zero, which quadrant does angle θ belong to? The options are A: First or Second, B: Second or Third, C: Third or Fourth, D: First or Fourth. Hmm. Alright, let me break this down step by step.First, I need to recall the signs of trigonometric functions in each quadrant. I remember that in the first quadrant, all the trigonometric functions are positive. In the second quadrant, sine is positive, but cosine is negative, so tangent (which is sine over cosine) would be negative there. In the third quadrant, both sine and cosine are negative, which makes tangent positive because a negative divided by a negative is a positive. In the fourth quadrant, cosine is positive, and sine is negative, so tangent is negative there. Let me verify that again: Quadrant I: all positive. Quadrant II: sin positive, cos negative, tan negative. Quadrant III: sin negative, cos negative, tan positive. Quadrant IV: sin negative, cos positive, tan negative. Yeah, that seems right.Now, the problem states that cosθ * tanθ < 0. So, the product of cosine and tangent is negative. Let's think about the signs of cosine and tangent in each quadrant and see where their product would be negative.Let me start by recalling that tanθ is sinθ/cosθ. So, cosθ * tanθ is cosθ * (sinθ/cosθ) = sinθ. Wait, is that correct? Let me check: cosθ times tanθ is cosθ multiplied by (sinθ/cosθ), which simplifies to sinθ. Oh! So, cosθ * tanθ is actually equal to sinθ. That simplifies things. So, the original inequality cosθ * tanθ < 0 is equivalent to sinθ < 0. Because cosθ * tanθ = sinθ. Therefore, the inequality is sinθ < 0.So, the problem reduces to: In which quadrants is sine negative? From my earlier recap, sine is positive in the first and second quadrants and negative in the third and fourth quadrants. So, sinθ < 0 would place θ in either the third or fourth quadrant. Therefore, the answer should be option C: Third or Fourth. Let me confirm this again to be sure.Alternatively, if I didn't realize that cosθ * tanθ simplifies to sinθ, I could approach it by considering the signs of cosine and tangent in each quadrant and then see where their product is negative.Let's try that approach as a cross-check. Let's go quadrant by quadrant.Quadrant I: Cosine is positive, tangent is positive (since both sine and cosine are positive, so sine/cosine is positive). So, positive times positive is positive. Therefore, the product cosθ * tanθ is positive here. Doesn't satisfy the inequality.Quadrant II: Cosine is negative (since x-coordinate is negative here), tangent is negative (since sine is positive and cosine is negative, so tangent is negative). So, negative times negative is positive. The product is positive. Doesn't satisfy the inequality.Quadrant III: Cosine is negative (x is negative), tangent is positive (since both sine and cosine are negative, so their ratio is positive). So, negative times positive is negative. That product is negative. So, this satisfies the inequality.Quadrant IV: Cosine is positive (x is positive here), tangent is negative (sine is negative, cosine is positive, so tangent is negative). So, positive times negative is negative. This also satisfies the inequality. Therefore, quadrants III and IV are where the product is negative. So, the answer is C: Third or Fourth. That matches my earlier conclusion.Therefore, regardless of whether I simplify the expression first or check each quadrant's sign, I arrive at the same answer. So, I feel confident that the correct answer is C.Final Answerboxed{C}</think>Given the problem, we need to determine which quadrant the angle (theta) belongs to if (costheta tantheta < 0).First, we recall that (tantheta = frac{sintheta}{costheta}). Therefore, we can simplify the given expression:[costheta tantheta = costheta cdot frac{sintheta}{costheta} = sintheta]The inequality (costheta tantheta < 0) simplifies to (sintheta < 0).Next, we determine in which quadrants the sine function is negative. The sine function is negative in the third and fourth quadrants. To verify, we check the signs of (costheta) and (tantheta) in each quadrant:- Quadrant I: (costheta > 0) and (tantheta > 0), product is positive.- Quadrant II: (costheta < 0) and (tantheta < 0), product is positive.- Quadrant III: (costheta < 0) and (tantheta > 0), product is negative.- Quadrant IV: (costheta > 0) and (tantheta < 0), product is negative.Thus, the product (costheta tantheta) is negative in the third and fourth quadrants. Therefore, the angle (theta) must be in the third or fourth quadrant.[boxed{C}],<think>Alright, let's see. I need to figure out which quadrant the angle θ is in, given that cosθ times tanθ is less than zero. The options are A: First or Second, B: Second or Third, C: Third or Fourth, D: First or Fourth. Hmm, okay.First, I should recall some basic trigonometry about the signs of trigonometric functions in different quadrants. Let me remember. In the first quadrant, all trigonometric functions are positive. In the second quadrant, sine is positive, cosine is negative, and tangent is negative because tangent is sine over cosine. In the third quadrant, both sine and cosine are negative, so tangent is positive. In the fourth quadrant, cosine is positive, sine is negative, so tangent is negative again. Let me confirm that. Yes, right. The mnemonic is "All Students Take Calculus" where each word represents the quadrant where the respective functions are positive: All in first, Sine in second, Tangent in third, Cosine in fourth. Wait, is that right? Hmm, actually, I think it's "All Students Take Calculus" meaning All (all functions positive in first), Sine (sine positive in second), Tangent (tangent positive in third), Cosine (cosine positive in fourth). So, yeah, that's correct.So, in quadrant one: cosθ positive, tanθ positive. Quadrant two: cosθ negative, tanθ negative. Quadrant three: cosθ negative, tanθ positive. Quadrant four: cosθ positive, tanθ negative. So, now, the problem is cosθ tanθ < 0. Let me write that down: cosθ * tanθ < 0.So, let's analyze each quadrant.First quadrant: cosθ is positive, tanθ is positive. So positive times positive is positive, which is not less than zero. So first quadrant is out.Second quadrant: cosθ is negative, tanθ is negative. Negative times negative is positive. So that's positive, which again is not less than zero. So second quadrant is out.Third quadrant: cosθ is negative, tanθ is positive. Negative times positive is negative, which is less than zero. So third quadrant is possible.Fourth quadrant: cosθ is positive, tanθ is negative. Positive times negative is negative, which is less than zero. So fourth quadrant is also possible.Therefore, θ must be in either the third or fourth quadrant. So the answer should be option C: Third or Fourth.But let me double-check. Maybe I made a mistake somewhere.Let me recall that tanθ is sinθ / cosθ. So cosθ * tanθ would be cosθ * (sinθ / cosθ) = sinθ. Wait, that's interesting. So cosθ tanθ simplifies to sinθ. So the original inequality cosθ tanθ < 0 is equivalent to sinθ < 0. So the problem reduces to sinθ < 0. When is sine negative? In the third and fourth quadrants. Therefore, θ is in third or fourth quadrant. So answer C. That's a quicker way. But either way, same conclusion.But wait, let me confirm this. If I have cosθ tanθ, substitute tanθ as sinθ/cosθ, then cosθ * (sinθ/cosθ) = sinθ. So yes, cosθ tanθ is equal to sinθ. Therefore, the inequality is equivalent to sinθ < 0. So, θ is in quadrants where sine is negative. Which are quadrants three and four. Therefore, answer is C.Therefore, regardless of the approach, the answer is C. So the correct option is C: Third or Fourth.Final Answerboxed{C}</think>Given the problem (costheta tantheta < 0), we need to determine which quadrant the angle (theta) belongs to.First, recall the signs of trigonometric functions in each quadrant:- In the first quadrant, all trigonometric functions are positive.- In the second quadrant, sine is positive, cosine is negative, and tangent is negative.- In the third quadrant, both sine and cosine are negative, so tangent is positive.- In the fourth quadrant, cosine is positive, sine is negative, so tangent is negative.Next, simplify the given inequality (costheta tantheta < 0). Since (tantheta = frac{sintheta}{costheta}), we substitute this into the inequality:[costheta cdot tantheta = costheta cdot frac{sintheta}{costheta} = sintheta]Thus, the inequality (costheta tantheta < 0) simplifies to (sintheta < 0).The sine function is negative in the third and fourth quadrants. Therefore, the angle (theta) must be in either the third or fourth quadrant.The final answer is (boxed{C}).]

🔑:Solution: Since costhetatantheta 0 and tantheta 0, then theta is an angle in the third quadrant. Therefore, the correct choice is boxed{text{C}}. By using the signs of trigonometric functions, we can directly determine the quadrant of the angle. This question tests the ability to judge the signs of trigonometric functions and the quadrant in which an angle lies, assessing computational skills.

🧠:[<think>Alright, let's try to figure out the range of this function f(x). The function is defined as f(x) = floor(|x|) - |floor(x)|. Hmm, okay. Let me parse this step by step. First, let's recall that the floor function, floor(r), gives the greatest integer less than or equal to r. So, for example, floor(3.7) is 3, floor(-2.3) is -3, because -3 is the greatest integer less than or equal to -2.3. Wait, actually, hold on. Wait, floor(-2.3) should be -3, right? Because -3 is less than -2.3, and the next integer up is -2, which is greater than -2.3. So yes, floor(-2.3) = -3. Similarly, floor(2.3) is 2.Now, the function f(x) is subtracting two things: the floor of the absolute value of x, and the absolute value of the floor of x. Let's break that down.First term: floor(|x|). So, take the absolute value of x first, then take the floor of that. Since |x| is always non-negative, floor(|x|) will be a non-negative integer. For example, if x is 3.7, |x| is 3.7, floor(3.7) is 3. If x is -2.3, |x| is 2.3, floor(2.3) is 2. So the first term is always a non-negative integer.Second term: |floor(x)|. Let's see. Take the floor of x first, then take the absolute value. So if x is positive, say 3.7, floor(x) is 3, absolute value is 3. If x is negative, say -2.3, floor(x) is -3, absolute value is 3. So |floor(x)| is always a non-negative integer as well. So both terms are non-negative integers, and we're subtracting them. So f(x) is the difference between two non-negative integers, which could be negative, zero, or positive. But let's see.Wait, but actually, let's check with examples. Maybe that will help. Let me pick some numbers and compute f(x).Case 1: x is a non-negative real number. Let's say x is 3.7. Then |x| is 3.7, floor of that is 3. floor(x) is 3, absolute value of that is 3. So f(3.7) = 3 - 3 = 0.Another non-negative number: x = 2. Then |x| is 2, floor is 2. floor(x) is 2, absolute value is 2. So f(2) = 2 - 2 = 0.Wait, so for non-negative x, floor(|x|) is floor(x), which is the same as floor(x), so |floor(x)| is the same as floor(x) because floor(x) is non-negative. Therefore, f(x) = floor(x) - floor(x) = 0. So for non-negative x, f(x) is always zero? That seems right. So if x is non-negative, the function f(x) is zero.But let's check a number between 0 and 1. Let's take x = 0.5. Then |x| = 0.5, floor(0.5) = 0. floor(x) is 0, absolute value is 0. So f(0.5) = 0 - 0 = 0. So even between 0 and 1, it's zero. So for all x >= 0, f(x) = 0. Got it.Now, let's consider when x is negative. Let's take x = -2.3. Then |x| = 2.3, floor(2.3) = 2. floor(x) is -3, because floor(-2.3) is -3. Then |floor(x)| = |-3| = 3. So f(-2.3) = 2 - 3 = -1.Another example: x = -1.5. |x| = 1.5, floor(1.5) = 1. floor(x) = -2, |floor(x)| = 2. So f(-1.5) = 1 - 2 = -1.Another example: x = -1.0. Wait, x is -1.0. Then |x| = 1.0, floor(1.0) = 1. floor(x) = -1, |floor(x)| = 1. So f(-1.0) = 1 - 1 = 0. Interesting. So at integer values, x negative integers, the function is zero. Let me verify. Let x = -3. Then |x| = 3, floor(3) = 3. floor(x) = -3, |floor(x)| = 3. So f(-3) = 3 - 3 = 0. So when x is a negative integer, f(x) is zero. But when x is a negative non-integer, then f(x) is -1. For example, x = -2.3: f(x) = 2 - 3 = -1. x = -0.5: |x| = 0.5, floor(0.5) = 0. floor(x) = -1, |floor(x)| = 1. So f(-0.5) = 0 - 1 = -1. So for negative x, if x is not an integer, f(x) = -1; if x is a negative integer, f(x) = 0.Therefore, combining both cases:For x >= 0: f(x) = 0.For x < 0:- If x is an integer, f(x) = 0.- If x is not an integer, f(x) = -1.Therefore, the range of f is {-1, 0}. That would be option A. But wait, let me check the options again. Option A is {-1, 0}, which seems to be the case here. Option C is {-1, 0, 1}, but we never get 1. Let me check if there is any case where f(x) could be positive. Let's see.Suppose x is positive. Then floor(|x|) = floor(x) and |floor(x)| = floor(x), so difference is zero. If x is negative and not integer, floor(|x|) is floor of a positive non-integer, which is the integer part, and |floor(x)| is the absolute value of the next lower integer. Let's take x = -3.7. Then |x| = 3.7, floor(3.7) = 3. floor(x) = -4, |floor(x)| = 4. So f(x) = 3 - 4 = -1. If x is -0.1, |x| = 0.1, floor(0.1) = 0. floor(x) = -1, |floor(x)| = 1. So 0 - 1 = -1. If x is a negative integer, like -2. Then |x| = 2, floor(2) = 2. floor(x) = -2, |floor(x)| = 2. So 2 - 2 = 0. Therefore, yes, the only possible outputs are -1 and 0. Hence, the range is {-1, 0}, which is option A.But let me check the answer choices again:(A) {-1, 0}(B) The set of nonpositive integers(C) {-1, 0, 1}(D) {0}(E) The set of nonnegative integersSo since we have outputs of -1 and 0 only, the correct answer is A. But wait, let me check if there's a case where f(x) could be something else.Wait, suppose x is a positive non-integer. Wait, but we saw that for x positive, even if it's a non-integer, f(x) is 0. For example, x = 1.5: floor(|1.5|) = 1, |floor(1.5)| = 1. So 1 - 1 = 0. So even positive non-integers give 0. So all non-negative x give 0. Negative x, integers give 0, non-integers give -1. So the range is {-1, 0}.Therefore, the answer is A. So why is option B there? Nonpositive integers. That would be 0, -1, -2, -3, etc. But in our case, f(x) only takes -1 and 0, so B is not correct. Similarly, C includes 1, which we never get. D is {0}, but we do get -1 sometimes. E is nonnegative integers, which is incorrect. So A is the correct answer.But hold on, let me verify with x = 0. If x = 0, then |x| = 0, floor(0) = 0. floor(x) = 0, |floor(x)| = 0. So f(0) = 0 - 0 = 0. So 0 is included. For negative integers, like x = -1, we saw that f(-1) = 0. So in all cases, the only possible outputs are -1 and 0, so the range is {-1, 0}, which is option A.Therefore, the answer is A.Final Answerboxed{A}</think>The function ( f ) is defined by ( f(x) = Biglfloor lvert x rvert Bigrfloor - Biglvert lfloor x rfloor Bigrvert ) for all real numbers ( x ). We need to determine the range of ( f ).1. Case 1: ( x geq 0 ) - For non-negative ( x ), ( lfloor |x| rfloor = lfloor x rfloor ). - The absolute value of ( lfloor x rfloor ) is ( lfloor x rfloor ) itself since it is non-negative. - Therefore, ( f(x) = lfloor x rfloor - lfloor x rfloor = 0 ). - Hence, for all ( x geq 0 ), ( f(x) = 0 ).2. Case 2: ( x < 0 ) - If ( x ) is a negative integer, say ( x = -n ) where ( n ) is a positive integer: - ( |x| = n ), so ( lfloor |x| rfloor = n ). - ( lfloor x rfloor = -n ), so ( |lfloor x rfloor| = n ). - Therefore, ( f(x) = n - n = 0 ). - If ( x ) is a negative non-integer, say ( x = -n - delta ) where ( n ) is a positive integer and ( 0 < delta < 1 ): - ( |x| = n + delta ), so ( lfloor |x| rfloor = n ). - ( lfloor x rfloor = -n - 1 ), so ( |lfloor x rfloor| = n + 1 ). - Therefore, ( f(x) = n - (n + 1) = -1 ).Combining both cases, the function ( f(x) ) can only take the values ( -1 ) and ( 0 ). Thus, the range of ( f ) is ({-1, 0}).[boxed{A}],<think>Okay, so I need to figure out the range of the function f(x) defined as the floor of the absolute value of x minus the absolute value of the floor of x. Hmm, let's break this down step by step.First, let me recall what the floor function does. The floor of a real number r, denoted as ⌊r⌋, is the greatest integer less than or equal to r. For example, ⌊3.7⌋ = 3, ⌊-1.2⌋ = -2, and ⌊5⌋ = 5. Got that.The function f(x) is given by:f(x) = ⌊|x|⌋ - |⌊x⌋|So, let's parse this. For any real number x, we first take the absolute value of x, then take the floor of that. Then we subtract the absolute value of the floor of x. Interesting. So two different operations here: one involving the floor of the absolute value, and the other involving the absolute value of the floor.I need to find all possible values that this function can take, which is the range of f. The answer choices are given, so maybe I can analyze the function and see which of the options matches.Let me consider different cases for x. Since absolute value and floor functions can behave differently depending on whether x is positive or negative, integer or non-integer, maybe I should split it into cases.Case 1: x is a non-negative real number (x ≥ 0).In this case, |x| = x, and ⌊x⌋ is just the floor of x. So the function simplifies to:f(x) = ⌊x⌋ - |⌊x⌋|But since x is non-negative, ⌊x⌋ is also non-negative (because the floor of a non-negative number is the greatest integer less than or equal to it, which is non-negative). So |⌊x⌋| = ⌊x⌋. Therefore, f(x) becomes ⌊x⌋ - ⌊x⌋ = 0.Wait, hold on, so for all x ≥ 0, f(x) = 0? That seems straightforward. Let me check with an example. If x = 2.3, then |x| = 2.3, ⌊2.3⌋ = 2. Then ⌊x⌋ = 2, |2| = 2. So 2 - 2 = 0. If x is an integer, say x = 5, then |5| = 5, floor is 5, then floor of x is 5, absolute value is 5, so 5 - 5 = 0. So yes, for all non-negative x, f(x) is 0. So in the non-negative case, the output is always 0.Case 2: x is a negative real number (x < 0).Here, things might be different. Let's see. Let x be negative. Then |x| = -x, which is positive. Then ⌊|x|⌋ = ⌊-x⌋. Also, ⌊x⌋ is the floor of a negative number. For example, if x = -1.2, then ⌊x⌋ = -2. Then |⌊x⌋| = |-2| = 2. So let's formalize this.Let x be negative. Let’s let x = -y where y > 0. Then |x| = y, so ⌊|x|⌋ = ⌊y⌋. Then ⌊x⌋ = ⌊-y⌋. For example, if y is an integer, say y = 3, then x = -3, so ⌊x⌋ = -3, |⌊x⌋| = 3. Then f(x) = ⌊y⌋ - |⌊-y⌋| = y - | -⌈y⌉ + 1 |? Wait, maybe it's better to express ⌊-y⌋ in terms of y.Actually, for negative numbers, the floor function can be expressed as ⌊-y⌋ = -⌈y⌉, where ⌈y⌉ is the ceiling function (smallest integer greater than or equal to y). For example, if y = 1.2, then ⌊-1.2⌋ = -2, which is -⌈1.2⌉ = -2. Similarly, if y = 2.7, then ⌊-2.7⌋ = -3, which is -⌈2.7⌉ = -3. And if y is an integer, say y = 3, then ⌊-3⌋ = -3, and -⌈3⌉ = -3. So yes, in general, ⌊-y⌋ = -⌈y⌉.But maybe I can relate ⌈y⌉ to ⌊y⌋. If y is not an integer, then ⌈y⌉ = ⌊y⌋ + 1. If y is an integer, then ⌈y⌉ = y. So, if y is not an integer, then ⌊-y⌋ = - (⌊y⌋ + 1). If y is an integer, then ⌊-y⌋ = -y.Therefore, |⌊x⌋| = |⌊-y⌋|. If y is not an integer, then |⌊-y⌋| = | - (⌊y⌋ + 1) | = ⌊y⌋ + 1. If y is an integer, |⌊-y⌋| = | - y | = y.So f(x) = ⌊y⌋ - |⌊-y⌋|. Therefore, if y is not an integer, f(x) = ⌊y⌋ - (⌊y⌋ + 1) = -1. If y is an integer, then f(x) = ⌊y⌋ - y = y - y = 0.Therefore, for x negative:- If x is a negative integer (i.e., y is a positive integer), then f(x) = 0.- If x is a negative non-integer (i.e., y is a positive non-integer), then f(x) = -1.So combining both cases:- For x ≥ 0: f(x) = 0.- For x < 0: - If x is an integer, f(x) = 0. - If x is not an integer, f(x) = -1.Therefore, the possible values of f(x) are 0 and -1. So the range is {-1, 0}, which corresponds to option A.Wait, but let me check with some examples.First, take x = 0. Then |0| = 0, floor is 0. Then floor(x) is 0, absolute value is 0. So 0 - 0 = 0. That's correct.Take x = 1.5 (positive non-integer). Then |1.5| = 1.5, floor is 1. Then floor(x) = 1, absolute value is 1. So 1 - 1 = 0. Correct.Take x = -1.5 (negative non-integer). Then |-1.5| = 1.5, floor is 1. Then floor(x) = -2, absolute value is 2. So 1 - 2 = -1. Correct.Take x = -2 (negative integer). Then |-2| = 2, floor is 2. Then floor(x) = -2, absolute value is 2. So 2 - 2 = 0. Correct.Take x = 3 (positive integer). |3| = 3, floor is 3. Then floor(x) = 3, absolute value is 3. So 3 - 3 = 0. Correct.Take x = -0.5. Then |x| = 0.5, floor is 0. Floor(x) = -1, absolute value is 1. So 0 - 1 = -1. Correct.So in all these cases, the only outputs are 0 and -1. So the range is {-1, 0}, which is option A. However, let me check the answer choices again to make sure.The options are:A) {-1, 0}B) The set of nonpositive integersC) {-1, 0, 1}D) {0}E) The set of nonnegative integersSo according to my analysis, the answer should be A. But wait, hold on. The problem statement says "for all real numbers x". Wait, is there a case where f(x) can be something else? Let me check.Wait, suppose x is a negative number with a fractional part. For example, x = -1.999. Then |x| = 1.999, floor is 1. Then floor(x) = -2, absolute value is 2. So 1 - 2 = -1. Similarly, if x is very close to 0 from the negative side, say x = -0.1, then |x| = 0.1, floor is 0. floor(x) = -1, |floor(x)| = 1. So 0 - 1 = -1. So all negative non-integers give -1, negative integers give 0, non-negatives give 0. So indeed, the only possible outputs are -1 and 0.But wait, let me check x = 0. If x = 0, floor(|0|) = 0, |floor(0)| = 0. So 0 - 0 = 0. Correct.Is there any x where f(x) is positive? Let's see. Suppose f(x) = ⌊|x|⌋ - |⌊x⌋|. For this to be positive, ⌊|x|⌋ must be greater than |⌊x⌋|.But let's see. For x positive, we have ⌊|x|⌋ = ⌊x⌋, and |⌊x⌋| = ⌊x⌋. So their difference is 0. For x negative, we have ⌊|x|⌋ = ⌊ -x ⌋ (since x is negative, |x| = -x). Wait, no. Wait, |x| is always positive, so if x is negative, |x| = -x, which is positive. So ⌊|x|⌋ = ⌊-x⌋. But -x is positive. So ⌊-x⌋ is the floor of a positive number. For example, if x = -1.2, then |x| = 1.2, floor is 1. Then floor(x) = -2, |floor(x)| = 2. So 1 - 2 = -1. Similarly, if x is negative, ⌊|x|⌋ is the floor of a positive number, which is an integer less than or equal to |x|. But |⌊x⌋| is the absolute value of the floor of x, which, for x negative, is the ceiling of |x| (as we saw before). So unless the floor of |x| is greater than the ceiling of |x|, which can't happen because floor(|x|) is less than or equal to |x|, and ceiling(|x|) is greater than or equal to |x|. So floor(|x|) ≤ |x| ≤ ceiling(|x|). So floor(|x|) ≤ ceiling(|x|). But unless |x| is an integer, in which case floor(|x|) = ceiling(|x|) = |x|.Wait, for x negative non-integer:Let me write x = -n - f, where n is a non-negative integer, and 0 < f < 1. Then |x| = n + f. So ⌊|x|⌋ = n. Then floor(x) = floor(-n - f) = -n - 1. Therefore, |floor(x)| = n + 1. So f(x) = n - (n + 1) = -1. If x is a negative integer, then x = -n where n is a positive integer. Then |x| = n, so floor(|x|) = n. Then floor(x) = -n, so |floor(x)| = n. Therefore, f(x) = n - n = 0.So for x negative, f(x) is either -1 or 0, depending on whether x is a non-integer or integer. For x non-negative, f(x) is 0. So the entire range is {-1, 0}. So answer A.But let me check the answer choices again. Option B is the set of nonpositive integers, which would include 0, -1, -2, -3, etc. But according to our analysis, the function can only be 0 or -1. So B is too broad.Option C is {-1, 0, 1}, but we never get 1. So C is out. D is {0}, but we have -1 as well. E is nonnegative integers, which is definitely not. So A is correct.Wait, but just to make sure I haven't missed any edge cases. Let me check x = 0. As above, f(0) = 0. If x approaches 0 from the negative side, like x = -0.0001, then |x| = 0.0001, floor is 0. Then floor(x) = -1, absolute value is 1. So 0 - 1 = -1. If x is a very large negative number, say x = -1000.5, then |x| = 1000.5, floor is 1000. floor(x) = -1001, absolute value is 1001. So 1000 - 1001 = -1. If x is a very large negative integer, x = -1000, then |x| = 1000, floor is 1000. floor(x) = -1000, absolute value is 1000. So 1000 - 1000 = 0. So regardless of the magnitude, the result is either 0 or -1.Therefore, the range is indeed {-1, 0}, which is option A.Final Answerboxed{A}</think>To determine the range of the function ( f(x) = Biglfloor lvert x rvert Bigrfloor - Biglvert lfloor x rfloor Bigrvert ), we analyze the behavior of the function for different cases of ( x ).1. Case 1: ( x geq 0 ) - For non-negative ( x ), ( |x| = x ) and ( lfloor x rfloor ) is non-negative. - Therefore, ( f(x) = lfloor x rfloor - lfloor x rfloor = 0 ).2. Case 2: ( x < 0 ) - Let ( x = -y ) where ( y > 0 ). Then ( |x| = y ) and ( lfloor x rfloor = lfloor -y rfloor ). - If ( y ) is an integer, ( lfloor -y rfloor = -y ), so ( | lfloor x rfloor | = y ). Thus, ( f(x) = lfloor y rfloor - y = 0 ). - If ( y ) is not an integer, ( lfloor -y rfloor = -lceil y rceil ). Since ( lceil y rceil = lfloor y rfloor + 1 ), we have ( | lfloor x rfloor | = lfloor y rfloor + 1 ). Thus, ( f(x) = lfloor y rfloor - (lfloor y rfloor + 1) = -1 ).Summarizing the results:- For ( x geq 0 ), ( f(x) = 0 ).- For ( x < 0 ): - If ( x ) is an integer, ( f(x) = 0 ). - If ( x ) is not an integer, ( f(x) = -1 ).Thus, the range of ( f ) is ({-1, 0}), corresponding to option (boxed{A}).]

🔑:To determine the range of the function ( f(x) = lfloor |x| rfloor - |lfloor x rfloor| ), we will analyze the function in different cases based on the value of ( x ).1. Case 1: ( x ) is a positive real number - When ( x ) is positive, ( |x| = x ). - Therefore, ( f(x) = lfloor x rfloor - |lfloor x rfloor| ). - Since ( x ) is positive, ( lfloor x rfloor ) is a non-negative integer. - Thus, ( |lfloor x rfloor| = lfloor x rfloor ). - Hence, ( f(x) = lfloor x rfloor - lfloor x rfloor = 0 ).2. Case 2: ( x = 0 ) - When ( x = 0 ), ( |x| = 0 ). - Therefore, ( f(x) = lfloor 0 rfloor - |lfloor 0 rfloor| ). - Since ( lfloor 0 rfloor = 0 ), we have ( |lfloor 0 rfloor| = 0 ). - Hence, ( f(x) = 0 - 0 = 0 ).3. Case 3: ( x ) is a negative real number - When ( x ) is negative, ( |x| = -x ). - Therefore, ( f(x) = lfloor -x rfloor - |lfloor x rfloor| ). - Let ( x = -a ) where ( a ) is a positive real number. - Then ( lfloor x rfloor = lfloor -a rfloor ). - If ( a ) is an integer, then ( lfloor -a rfloor = -a ) and ( |lfloor -a rfloor| = a ). - Hence, ( f(x) = lfloor a rfloor - a = 0 ). - If ( a ) is not an integer, let ( a = n + k ) where ( n ) is an integer and ( 0 < k < 1 ). - Then ( lfloor -a rfloor = -n - 1 ) and ( |lfloor -a rfloor| = n + 1 ). - Hence, ( f(x) = lfloor a rfloor - (n + 1) = n - (n + 1) = -1 ).From the above cases, we see that the function ( f(x) ) can take values ( 0 ) and ( -1 ).The final answer is (boxed{{-1, 0}}).

🧠:[<think>Okay, so I need to find the point that's symmetric to (-1, -2) with respect to the y-axis. Hmm, symmetry with respect to the y-axis... Let me think. When they talk about symmetry over the y-axis, I remember that it means if you flip the point over the y-axis, you get the symmetric point. So the y-coordinate stays the same, right? Because flipping over the y-axis doesn't affect the vertical position, only the horizontal one.So, for a point (x, y), its reflection over the y-axis would be (-x, y). Let me verify that. Suppose I have a point (2, 3). Reflecting it over the y-axis should give me (-2, 3). Yes, because it's mirrored across the vertical line. So the x-coordinate changes sign, and the y-coordinate remains the same. That makes sense.Applying that to the point given, which is (-1, -2). Let's break it down. The original x-coordinate is -1, so reflecting it over the y-axis would change it to -(-1) = 1. The y-coordinate is -2, and since we're reflecting over the y-axis, this should stay the same. So the symmetric point should be (1, -2).Wait, let me make sure. If the original point is in the third quadrant (since both coordinates are negative), reflecting over the y-axis would move it to the fourth quadrant, where x is positive and y is negative. So (1, -2) is indeed in the fourth quadrant. That checks out.Alternatively, maybe I can visualize it. Imagine the coordinate plane. The point (-1, -2) is one unit to the left of the y-axis and two units down. If I reflect it over the y-axis, left becomes right, so one unit to the right, which is x=1, and the y-coordinate remains two units down, y=-2. Yep, that still gives (1, -2).Is there another way to test this? Maybe using the distance from the y-axis. The original point is at x=-1, so its distance from the y-axis is |-1| = 1. The reflected point should also be 1 unit from the y-axis but on the opposite side, so x=1. The y-coordinate doesn't change because distance up or down isn't affected by a reflection over the y-axis. So again, (1, -2) is the correct point.Another approach: use the formula for reflection over the y-axis. The formula is (x, y) reflected over y-axis becomes (-x, y). So applying the formula to (-1, -2):-x = -(-1) = 1y remains -2Therefore, the reflected point is (1, -2). That's consistent with everything I thought earlier.I think that's solid. All the different ways of thinking about it lead to the same answer, so I'm pretty confident. The key idea is that reflection over the y-axis inverts the x-coordinate and keeps the y-coordinate the same. So flip the sign of x, leave y alone. Applied to (-1, -2), that gives (1, -2). Just to recap:Original point: (-1, -2)Reflect over y-axis: change x from -1 to 1, y stays -2.Result: (1, -2)Yep, that's it. I can't see any mistake in this reasoning.Final AnswerThe symmetric point is boxed{(1, -2)}.</think>To find the point symmetric to ((-1, -2)) with respect to the (y)-axis, we need to reflect the given point over the (y)-axis. Reflecting a point ((x, y)) over the (y)-axis changes the (x)-coordinate's sign while keeping the (y)-coordinate the same. For the given point ((-1, -2)):- The (x)-coordinate (-1) changes to (1) when reflected over the (y)-axis.- The (y)-coordinate (-2) remains unchanged.Thus, the symmetric point is ((1, -2)).[boxed{(1, -2)}],<think>Okay, so I need to find the point that is symmetric to (-1, -2) with respect to the y-axis. Hmm, let me think. Symmetry with respect to the y-axis... Right, that means if I reflect the original point over the y-axis, I should get the symmetric point. First, let me recall what reflection over the y-axis does. If I have a point (x, y), reflecting it over the y-axis would change the x-coordinate's sign, but leave the y-coordinate the same. So, the reflection would be (-x, y). Is that right? Let me visualize this. If I'm on the coordinate plane and I flip a point over the y-axis (which is the vertical line x=0), the horizontal distance from the y-axis stays the same, but the direction flips. So if the original point is on the left side of the y-axis, the reflected point would be on the right side at the same distance. Similarly, if it's on the right, it goes to the left. But the y-coordinate remains unchanged because we're flipping vertically, not horizontally. Wait, no, flipping over the y-axis is a horizontal flip. So x-coordinate changes, y stays the same.Let me test this with an example. If I have a point (3, 4), its reflection over the y-axis should be (-3, 4). Yes, because moving 3 units to the right becomes 3 units to the left. Similarly, a point (-2, 5) would reflect to (2, 5). That makes sense. So the rule is indeed (x, y) reflected over y-axis becomes (-x, y). So applying this to the given point (-1, -2). Let's substitute into the formula. Original x is -1, original y is -2. Reflecting over y-axis gives (-(-1), -2) which simplifies to (1, -2). So the symmetric point should be (1, -2). Let me check this. The original point is in the third quadrant, since both coordinates are negative. Reflecting over the y-axis would move it to the fourth quadrant, where x is positive and y is negative. (1, -2) is indeed in the fourth quadrant. That seems correct. Wait, let me confirm once again. The distance from the original point to the y-axis is the absolute value of its x-coordinate, which is |-1| = 1. The reflected point should be 1 unit on the other side of the y-axis, so x=1. The y-coordinate remains the same, so y=-2. Yes, that's exactly (1, -2). Another way to think about symmetry: if two points are symmetric with respect to the y-axis, then the midpoint between them lies on the y-axis. Let's verify this. The midpoint formula is ((x1 + x2)/2, (y1 + y2)/2). Here, the original point is (-1, -2), and the reflected point is (1, -2). Calculating the midpoint: ((-1 + 1)/2, (-2 + (-2))/2) = (0/2, -4/2) = (0, -2). Wait, the midpoint is (0, -2). The x-coordinate is 0, which is on the y-axis, but the y-coordinate is -2, which is not zero. Hmm, does that matter? Wait, no. The definition of symmetry with respect to the y-axis only requires that the midpoint lies on the y-axis, which is the line x=0. The y-coordinate of the midpoint can be anything, as long as the x-coordinate is 0. In this case, the midpoint is indeed on the y-axis (since x=0), so that satisfies the condition. Therefore, (1, -2) is correctly the symmetric point. Let me check with another example. Suppose we have a point (2, 3). Its reflection over the y-axis would be (-2, 3). The midpoint between (2,3) and (-2,3) is (0, 3), which is on the y-axis. Similarly, for a point (-5, 4), the reflection is (5, 4), midpoint (0,4). So even though the midpoints have different y-coordinates, they are still on the y-axis because x is 0. Therefore, in our problem, even though the midpoint is (0, -2), which is on the y-axis, that's perfectly fine. So the answer (1, -2) is correct. Therefore, after all these checks, I can confirm that the point symmetric to (-1, -2) with respect to the y-axis is (1, -2).Final AnswerThe symmetric point is boxed{(1, -2)}.</think>To find the point symmetric to ((-1, -2)) with respect to the (y)-axis, we need to reflect the original point over the (y)-axis. Reflection over the (y)-axis changes the sign of the (x)-coordinate while keeping the (y)-coordinate the same. Given the point ((-1, -2)):- The (x)-coordinate is (-1). Reflecting this over the (y)-axis gives (1) (since (-(-1) = 1)).- The (y)-coordinate remains (-2).Thus, the symmetric point is ((1, -2)).To verify, the midpoint between ((-1, -2)) and ((1, -2)) is calculated as:[left( frac{-1 + 1}{2}, frac{-2 + (-2)}{2} right) = left( 0, -2 right)]The midpoint ((0, -2)) lies on the (y)-axis (since the (x)-coordinate is 0), confirming the reflection is correct.The symmetric point is (boxed{(1, -2)}).]

🔑:To find the point symmetric to left(-1,-2right) with respect to the y-axis, we need to negate the x-coordinate while keeping the y-coordinate the same. This is because symmetry with respect to the y-axis means that the x-coordinate values are equal in magnitude but opposite in sign, and the y-coordinate values remain unchanged.Given point: left(-1,-2right)1. Negate the x-coordinate: -1 rightarrow 12. Keep the y-coordinate the same: -2 rightarrow -2Therefore, the symmetric point with respect to the y-axis is left(1,-2right).boxed{left(1,-2right)}

🧠:[<think>Okay, so I need to solve this logarithm problem. The question is: Given that log base 2 of x equals 0, find x. Hmm, let's see. I remember that logarithms are the inverses of exponentials, right? So log base 2 of x equals 0 means that 2 raised to the power of 0 should give me x. Wait, is that right?Let me recall the definition of a logarithm. The logarithm log_b(a) is the exponent you need to raise base b to get a. So if log base 2 of x is 0, that means 2 raised to the 0th power equals x. But 2^0 is 1, because any number to the power of 0 is 1. So does that mean x is 1? That seems too straightforward. Let me check.Alternatively, maybe I can rewrite the equation in exponential form. The logarithmic equation log_2(x) = 0 can be rewritten as 2^0 = x. Yeah, that's the standard way to convert between logarithmic and exponential forms. So 2^0 is definitely 1. Therefore, x must be 1. Hmm. But maybe there's a trick here? Let me think again.Wait, logarithms have certain domains. The logarithm log_b(a) is only defined when a is positive and b is positive and not equal to 1. In this case, the base is 2, which is valid, and x has to be positive. But since the result is 0, which is a valid logarithm value, there shouldn't be any issues here. So x is 1. Let me try plugging it back into the original equation to verify. If x is 1, then log base 2 of 1 is... Well, 2 raised to what power gives 1? Since 2^0 = 1, the logarithm should indeed be 0. So that checks out.Is there another way to approach this? Maybe using logarithm properties. For example, log_b(b^k) = k. So if I set k to 0, then log_b(1) = 0 because b^0 = 1. Therefore, in this case, log_2(1) = 0. Which again means x is 1. Yep, that confirms it.I guess this problem is testing the understanding that the logarithm of 1 to any base is always 0. Because any number raised to the power of 0 is 1. So regardless of the base, as long as it's a valid base, log_b(1) = 0. Therefore, x must be 1 here. Wait, but maybe someone could make a mistake here by thinking that log base 2 of 0 is 0? But no, log base 2 of 0 is undefined because 2 raised to any power can never be 0. So the solution isn't 0, it's 1. So that's another possible confusion, but in this case, the answer is 1. Alternatively, maybe thinking of the graph of log base 2. The logarithm function log_2(x) crosses the x-axis at x=1, where the value is 0. So when x=1, the log is 0, which matches the equation given. So that's another way to visualize it. All right, so after checking multiple approaches—converting to exponential form, using logarithm properties, considering the domain and the graph—everything points to x being 1. I think that's solid. There's no ambiguity here. The answer has to be 1. So I'm confident that's the solution.Final AnswerThe value of ( x ) is boxed{1}.</think>Given the problem: (log_{2}x = 0), we need to find (x).1. Understanding the logarithm definition: The logarithm (log_{b}(a)) is the exponent you need to raise base (b) to get (a). Therefore, (log_{2}x = 0) means (2^0 = x).2. Converting to exponential form: Rewriting the logarithmic equation in exponential form gives (2^0 = x). Since (2^0 = 1), we have (x = 1).3. Verification: Checking the solution by substituting (x = 1) back into the original logarithmic equation: (log_{2}(1) = 0) because (2^0 = 1). This confirms the solution is correct.4. Logarithm properties: Using the property (log_{b}(b^k) = k), we set (k = 0) which gives (log_{2}(2^0) = log_{2}(1) = 0). Hence, (x = 1).5. Domain consideration: The logarithm function is only defined for positive arguments, and (x = 1) is positive, confirming it is a valid solution.Thus, the value of (x) is (boxed{1}).,<think>Okay, let me try to figure out this problem. So, the question is: Given log base 2 of x equals 0, what is x? Alright, logarithms can sometimes be tricky for me, but let's break it down step by step.First, I remember that a logarithm is essentially the exponent you need to raise the base to in order to get the number. So, log base 2 of x equals 0 means that 2 raised to the power of 0 should give me x. Let me write that down to visualize it better.The equation is: log₂(x) = 0Converting this logarithmic form to its exponential form should help. The general rule is that if log_b(a) = c, then b^c = a. Applying that here, the base b is 2, the result of the logarithm c is 0, so that would mean 2^0 = x.Right, 2 raised to the 0 power. Now, I remember that any non-zero number raised to the power of 0 is 1. So, 2^0 is 1. Therefore, x must be 1. Let me double-check that to make sure I didn't make a mistake.If x is 1, then log base 2 of 1 should equal 0. Let me test that. 2 to what power gives 1? Well, 2^1 is 2, 2^-1 is 0.5, but 2^0 is indeed 1. So yes, log₂(1) = 0. That checks out.Wait, but why do we have this rule that any number to the power of 0 is 1? Maybe I should recall the exponent rules to reinforce this. For example, 2^3 divided by 2^3 is 2^(3-3) = 2^0. But 2^3 is 8, so 8 divided by 8 is 1. Therefore, 2^0 must equal 1. That makes sense. So, this rule applies here, confirming that 2^0 is 1.Is there another way to approach this problem? Maybe using the definition of logarithms. The logarithm log_b(x) is the exponent you put on b to get x. So, if log₂(x) is 0, then the exponent is 0, which means 2^0 = x. Again, same result.Alternatively, maybe graphing the logarithmic function could help. The function y = log₂(x) is the inverse of y = 2^x. So, if we have y = 0, then the corresponding x-value on the logarithmic graph would be where y = 0. The inverse function of y = 2^x would switch x and y, so solving x = 2^y. If y = 0, then x = 2^0 = 1. Therefore, the point (1, 0) is on the graph of y = log₂(x), meaning when x is 1, y is 0. That's another confirmation.Let me think if there are any exceptions or special cases here. Well, logarithms are only defined for positive real numbers. So, x has to be positive. In this case, 1 is positive, so that's fine. Also, the base of the logarithm must be positive and not equal to 1. Here, the base is 2, which is valid.I could also plug the value back into the original equation to verify. If x is 1, then log₂(1) should equal 0. Let's compute log₂(1). As mentioned earlier, since 2^0 = 1, the logarithm must be 0. So, that's correct.Another angle: using change of base formula. The change of base formula says log_b(a) = log_c(a)/log_c(b) for any positive c ≠ 1. If I use natural logarithm (ln) here, then log₂(1) = ln(1)/ln(2). We know ln(1) is 0 because e^0 = 1. So, 0 divided by ln(2) is 0, which matches the original equation. Therefore, x = 1 is correct.Wait, but does this hold true for any base? For example, if we had log base 5 of x equals 0, would x also be 1? Let me check. Yes, because 5^0 is 1, so x would be 1. So, in general, log_b(x) = 0 implies x = 1, regardless of the base, as long as the base is valid (positive and not 1). So, this problem is a specific case of that general rule. Therefore, applying that here, x must be 1.Is there a way to confuse myself here? Maybe if I misremember the direction of the conversion. For instance, if I thought that log₂(x) = 0 translates to x = 0^2, but that would be incorrect. Because the logarithm is log base 2 of x equals 0, which is 2^0 = x, not x squared or anything else. So, that's a potential pitfall, but as long as I remember that the base remains the base and the result is the exponent, I should be okay.Also, another mistake could be thinking that since the log is 0, x is 0, but logarithm of 0 is undefined. For example, log₂(0) is undefined because there's no exponent that 2 can be raised to get 0. So, x must be a positive number, and in this case, 1 is the correct answer.Let me summarize my thought process:1. Start with the equation log₂(x) = 0.2. Convert the logarithmic equation to its exponential form: 2^0 = x.3. Simplify 2^0 to 1, so x = 1.4. Verify by plugging back into the original equation: log₂(1) = 0, which is true.5. Consider alternative methods and confirmations, such as exponent rules, inverse functions, change of base formula, etc., all leading to the same result.6. Check for possible misunderstandings or common mistakes, ensuring that each step is valid and that the answer makes sense in the context of logarithmic functions.Therefore, after all these steps and checks, I can confidently say that x equals 1 is the correct answer.Final AnswerThe value of ( x ) is boxed{1}.</think>Given the equation (log_{2}x = 0), we need to find the value of (x).1. Convert the logarithmic equation to its exponential form: [ log_{2}x = 0 implies 2^0 = x ]2. Simplify the exponential expression: [ 2^0 = 1 ] Therefore, (x = 1).3. Verification: - Checking the original logarithmic equation: (log_{2}(1) = 0) because (2^0 = 1). - Using the definition of logarithms, any number raised to the power of 0 is 1. - Confirming with the inverse function property: The inverse of (y = 2^x) is (y = log_{2}(x)), so if (y = 0), then (x = 2^0 = 1).All steps confirm that the value of (x) is indeed (boxed{1}).]

🔑:Since log_{2}x=0=log_{2}1, it follows that x=1. Therefore, the answer is boxed{1}.

🧠:[<think>Okay, so I have this problem where several siblings are dividing their inheritance according to a specific pattern. Each sibling receives a certain amount upfront and then one-tenth of the remainder. The key point is that each sibling ends up receiving an equal amount. I need to find out how many siblings there were, how much each received, and the total inheritance. Hmm, let's break this down step by step.First, let me parse the problem again. The first sibling, A, gets 2000 crowns plus one-tenth of the remaining inheritance after that 2000 is taken out. Then the next sibling, B, gets 4000 crowns plus one-tenth of what's left after the initial 2000 and then the 4000, and so on. Each subsequent sibling gets an upfront amount that's 2000 more than the previous one (so 2000, 4000, 6000, etc.) and then one-tenth of the remainder. And somehow, despite the increasing upfront amounts, they all end up with the same total. Interesting.Let me try to model this. Let's denote the total inheritance as T. The first sibling, A, takes 2000 crowns, leaving T - 2000. Then A takes one-tenth of the remainder, so A's total is 2000 + (T - 2000)/10. After A takes their share, what's left is the remainder after subtracting both the 2000 and the one-tenth. So the remaining inheritance after A would be (T - 2000) - (T - 2000)/10 = (9/10)(T - 2000).Now, sibling B comes next. They take 4000 crowns from the remaining inheritance. Wait, but after A took their share, the remaining inheritance is (9/10)(T - 2000). So B takes 4000 from that? But if the remaining inheritance after A is (9/10)(T - 2000), then subtracting another 4000 gives (9/10)(T - 2000) - 4000. Then B takes one-tenth of that remainder, so B's total is 4000 + [(9/10)(T - 2000) - 4000]/10. And then the remaining inheritance after B would be 9/10 of [(9/10)(T - 2000) - 4000].But the problem states that each sibling receives an equal amount. So A's total equals B's total. Let me write that equation down.Let’s denote the amount each sibling receives as X. Therefore:For A:X = 2000 + (T - 2000)/10For B:X = 4000 + [(9/10)(T - 2000) - 4000]/10Since both equal X, we can set them equal to each other:2000 + (T - 2000)/10 = 4000 + [(9/10)(T - 2000) - 4000]/10This looks a bit complicated, but let's try to simplify step by step.First, let me compute (T - 2000)/10. Let's call that term1. Then term1 = (T - 2000)/10. So X = 2000 + term1.After A takes their share, the remaining inheritance is (T - 2000) - term1 = (9/10)(T - 2000). Let's call that R1 = (9/10)(T - 2000).Then B takes 4000 from R1, so remaining after that is R1 - 4000 = (9/10)(T - 2000) - 4000. Then B takes one-tenth of that, so B's share is 4000 + [(9/10)(T - 2000) - 4000]/10. Let's compute that.Let me denote [(9/10)(T - 2000) - 4000]/10 as term2. So X = 4000 + term2.But since X must be equal to the previous X, set 2000 + term1 = 4000 + term2.Let me substitute term1 and term2:2000 + (T - 2000)/10 = 4000 + [ (9/10)(T - 2000) - 4000 ] /10Multiply both sides by 10 to eliminate denominators:10*2000 + (T - 2000) = 10*4000 + (9/10)(T - 2000) - 4000Compute left side: 20000 + T - 2000 = T + 18000Right side: 40000 + (9/10)(T - 2000) - 4000Simplify right side: 40000 - 4000 + (9/10)(T - 2000) = 36000 + (9/10)(T - 2000)So we have:T + 18000 = 36000 + (9/10)(T - 2000)Now, subtract 36000 from both sides:T + 18000 - 36000 = (9/10)(T - 2000)T - 18000 = (9/10)(T - 2000)Multiply both sides by 10 to eliminate the fraction:10T - 180000 = 9(T - 2000)Expand the right side:10T - 180000 = 9T - 18000Subtract 9T from both sides:T - 180000 = -18000Add 180000 to both sides:T = 180000 - 18000 = 162000Wait, so total inheritance is 162000 crowns? Let me check this.If T = 162000, then let's compute A's share.A gets 2000 + (162000 - 2000)/10 = 2000 + 160000/10 = 2000 + 16000 = 18000. So X = 18000.Then, after A takes their share, the remaining inheritance is 162000 - 18000 = 144000.Then B comes and takes 4000 + (144000 - 4000)/10. Wait, hold on. Wait, after A takes 2000 and then 1/10 of the remainder, which was 16000, making their total 18000. Then the remainder after A is 162000 - 18000 = 144000.But according to the initial step, after A, the remainder is (9/10)(T - 2000) = (9/10)(160000) = 144000. Correct. Then B takes 4000 crowns, leaving 144000 - 4000 = 140000. Then B takes 1/10 of that, so 14000. Therefore B's total is 4000 + 14000 = 18000. That matches X. Then the remaining inheritance after B would be 140000 - 14000 = 126000.Okay, so B's share is indeed 18000, same as A. Then next, sibling C would take 6000 crowns plus 1/10 of the remainder.So after B, we have 126000. C takes 6000, leaving 120000. Then 1/10 of that is 12000, so C's total is 6000 + 12000 = 18000. Then remaining inheritance is 120000 - 12000 = 108000.So this seems to hold. Then D would take 8000 + 1/10 of the remainder. Let's check:After C, 108000. D takes 8000, leaving 100000. 1/10 of that is 10000, so D's total is 8000 + 10000 = 18000. Remaining inheritance is 100000 - 10000 = 90000.Then E takes 10000 + 1/10 of 90000 - 10000 = 80000. Wait, hold on: after D, it's 90000. E takes 10000, leaving 80000. Then 1/10 of 80000 is 8000, so E's total is 10000 + 8000 = 18000. Remaining inheritance is 80000 - 8000 = 72000.F next: 12000 + 1/10 of (72000 - 12000) = 1/10 of 60000 = 6000. So F's total is 12000 + 6000 = 18000. Remaining inheritance: 60000 - 6000 = 54000.G: 14000 + 1/10 of (54000 - 14000) = 1/10 of 40000 = 4000. Total for G: 14000 + 4000 = 18000. Remaining inheritance: 40000 - 4000 = 36000.H: 16000 + 1/10 of (36000 - 16000) = 1/10 of 20000 = 2000. Total for H: 16000 + 2000 = 18000. Remaining inheritance: 20000 - 2000 = 18000.I: 18000 + 1/10 of (18000 - 18000) = 1/10 of 0 = 0. So I's total would be 18000 + 0 = 18000. But after H, the remaining inheritance is 18000. So I takes 18000 upfront, but wait, the pattern is each sibling takes an upfront amount increasing by 2000 each time. So after H, the next upfront should be 18000. But the remaining inheritance is 18000. So I takes 18000 upfront, leaving 0, then 1/10 of 0 is 0, so total for I is 18000. Then remaining is 0.Wait, but in this case, sibling I only takes 18000, which is the upfront amount, but the upfront amounts are supposed to be 2000, 4000, 6000, etc. So sibling I would be the 9th sibling, since each upfront increases by 2000: 2000, 4000, 6000, 8000, 10000, 12000, 14000, 16000, 18000. So that's 9 siblings.Wait, let's count:A: 2000B: 4000C: 6000D: 8000E: 10000F: 12000G: 14000H: 16000I: 18000Yes, 9 siblings. Each time, the upfront amount increases by 2000. After I takes 18000, the remaining inheritance is zero. But in the problem statement, after the division, each sibling received an equal amount. So all 9 siblings received 18000 each. The total inheritance would be 9*18000 = 162000, which matches our earlier calculation. So seems like 9 siblings, each received 18000, total inheritance 162000.But let me confirm with the equations. Earlier, by setting A's share equal to B's share, we found T = 162000, X = 18000. Then when we computed step by step, each subsequent sibling also received 18000, and the inheritance was exhausted after 9 siblings. So that seems to check out.But just to be thorough, let me verify with the next sibling after I. If there was a 10th sibling, J, they would need to take 20000 crowns upfront, but after I, the remaining inheritance is zero, so that's impossible. Therefore, the inheritance is exactly consumed by 9 siblings, each getting 18000. Therefore, the answers should be 9 siblings, each received 18000 crowns, total inheritance 162000 crowns.But just to make sure there are no errors in the equations. Let's recheck the first calculation where we set T = 162000.A's share: 2000 + (162000 - 2000)/10 = 2000 + 160000/10 = 2000 + 16000 = 18000. Correct.After A: 162000 - 18000 = 144000.B's share: 4000 + (144000 - 4000)/10 = 4000 + 140000/10 = 4000 + 14000 = 18000. Correct.After B: 144000 - 18000 = 126000.C's share: 6000 + (126000 - 6000)/10 = 6000 + 120000/10 = 6000 + 12000 = 18000. Correct.After C: 126000 - 18000 = 108000.D's share: 8000 + (108000 - 8000)/10 = 8000 + 100000/10 = 8000 + 10000 = 18000. Correct.After D: 108000 - 18000 = 90000.E's share: 10000 + (90000 - 10000)/10 = 10000 + 80000/10 = 10000 + 8000 = 18000. Correct.After E: 90000 - 18000 = 72000.F's share: 12000 + (72000 - 12000)/10 = 12000 + 60000/10 = 12000 + 6000 = 18000. Correct.After F: 72000 - 18000 = 54000.G's share: 14000 + (54000 - 14000)/10 = 14000 + 40000/10 = 14000 + 4000 = 18000. Correct.After G: 54000 - 18000 = 36000.H's share: 16000 + (36000 - 16000)/10 = 16000 + 20000/10 = 16000 + 2000 = 18000. Correct.After H: 36000 - 18000 = 18000.I's share: 18000 + (18000 - 18000)/10 = 18000 + 0 = 18000. Correct.After I: 18000 - 18000 = 0.Perfect, so each sibling gets exactly 18000, and after the ninth sibling, the inheritance is exhausted. So all conditions of the problem are satisfied. Therefore, the answers are 9 siblings, each receiving 18000 crowns, with a total inheritance of 162000 crowns.Final AnswerThe number of siblings is boxed{9}, each received boxed{18000} crowns, and the total inheritance was boxed{162000} crowns.</think>Let's denote the total inheritance as ( T ). Each sibling receives an upfront amount increasing by 2000 crowns each time and one-tenth of the remaining inheritance. Each sibling ends up receiving an equal amount, denoted as ( X ).For sibling ( A ):[X = 2000 + frac{T - 2000}{10}]After ( A ) takes their share, the remaining inheritance is:[frac{9}{10}(T - 2000)]For sibling ( B ):[X = 4000 + frac{frac{9}{10}(T - 2000) - 4000}{10}]Setting the equations equal since each sibling receives the same amount:[2000 + frac{T - 2000}{10} = 4000 + frac{frac{9}{10}(T - 2000) - 4000}{10}]Multiplying through by 10 to clear the denominators:[20000 + T - 2000 = 40000 + frac{9}{10}(T - 2000) - 4000]Simplifying both sides:[T + 18000 = 36000 + frac{9}{10}(T - 2000)]Multiplying through by 10 again:[10T - 180000 = 9(T - 2000)]Solving for ( T ):[10T - 180000 = 9T - 18000 implies T = 162000]Each sibling receives:[X = 2000 + frac{162000 - 2000}{10} = 2000 + 16000 = 18000]Verifying the remaining inheritance after each sibling:- After ( A ): ( 162000 - 18000 = 144000 )- After ( B ): ( 144000 - 18000 = 126000 )- After ( C ): ( 126000 - 18000 = 108000 )- After ( D ): ( 108000 - 18000 = 90000 )- After ( E ): ( 90000 - 18000 = 72000 )- After ( F ): ( 72000 - 18000 = 54000 )- After ( G ): ( 54000 - 18000 = 36000 )- After ( H ): ( 36000 - 18000 = 18000 )- After ( I ): ( 18000 - 18000 = 0 )Thus, there are 9 siblings, each receiving 18000 crowns, and the total inheritance is 162000 crowns.[boxed{9}, boxed{18000}, boxed{162000}],<think>Okay, so there's this problem about siblings dividing an inheritance according to a specific pattern. Each sibling gets a certain amount upfront and then one-tenth of the remaining amount. The key point is that each sibling ends up receiving the same total amount. We need to figure out how many siblings there were, how much each received, and the total inheritance.Let me start by breaking down the problem step by step. Let's denote the total inheritance as T. The siblings are labeled A, B, C, and so on. Each one receives an initial amount that increases by 2000 crowns each time: A gets 2000, B gets 4000, C gets 6000, etc. After that initial amount, they also receive one-tenth of the remaining inheritance. The crucial part is that despite these different initial amounts and fractions, each sibling ends up with the same total. So, the challenge is to model this scenario mathematically and solve for the number of siblings, their equal share, and the total inheritance.Let me try to model the first few siblings and see if I can find a pattern or equations to solve.Let's start with sibling A. They receive 2000 crowns plus one-tenth of the remainder after that 2000 is taken out. So, after A takes their share, what's left?Total inheritance is T. A takes 2000, so the remainder is T - 2000. Then A takes one-tenth of that remainder, which is (T - 2000)/10. Therefore, A's total inheritance is 2000 + (T - 2000)/10. After A has taken their share, the remaining inheritance would be the original remainder minus the one-tenth that A took. So, the remaining inheritance after A is (T - 2000) - (T - 2000)/10 = (9/10)(T - 2000).Now, sibling B comes next. B receives 4000 crowns plus one-tenth of the new remainder. But wait, the remaining inheritance after A is (9/10)(T - 2000). So, when B takes their 4000, we need to subtract that from the remaining amount. However, here's a potential problem: if the remaining amount after A is less than 4000, then B can't take 4000. But the problem states that this division worked out, so the remaining amount must have been sufficient for each sibling's initial amount. Therefore, we can assume that after each sibling takes their initial amount and their tenth, the remainder is still enough for the next sibling's initial amount. So, proceeding under that assumption.So, after A, remaining inheritance is (9/10)(T - 2000). Then B takes 4000. So the remainder after B takes 4000 is (9/10)(T - 2000) - 4000. Then B takes one-tenth of that remainder. Therefore, B's total inheritance is 4000 + [(9/10)(T - 2000) - 4000]/10.But we know that A and B received the same total amount. So, we can set their totals equal:2000 + (T - 2000)/10 = 4000 + [(9/10)(T - 2000) - 4000]/10That's a bit complicated, but maybe simplifying step by step.Let me denote A's total as:A = 2000 + (T - 2000)/10Similarly, B's total is:B = 4000 + [ (9/10)(T - 2000) - 4000 ] /10Since A = B, we can set them equal:2000 + (T - 2000)/10 = 4000 + [ (9/10)(T - 2000) - 4000 ] /10Let me simplify both sides.First, left side:Left = 2000 + (T - 2000)/10Right side:Right = 4000 + [ (9/10)(T - 2000) - 4000 ] /10Let me compute the right side:First, compute the term inside the brackets: (9/10)(T - 2000) - 4000Then divide by 10 and add 4000.Alternatively, let's multiply both sides by 10 to eliminate denominators:10*Left = 10*2000 + (T - 2000) = 20000 + T - 2000 = 18000 + T10*Right = 10*4000 + (9/10)(T - 2000) - 4000 = 40000 + (9/10)(T - 2000) - 4000Simplify 10*Right:40000 - 4000 = 36000So, 10*Right = 36000 + (9/10)(T - 2000)But 10*Left = 10*Right, so:18000 + T = 36000 + (9/10)(T - 2000)Now, let's solve for T.First, subtract 18000 from both sides:T = 18000 + (9/10)(T - 2000)Multiply out the right-hand side:T = 18000 + (9/10)T - (9/10)*2000Calculate (9/10)*2000 = 1800So,T = 18000 + (9/10)T - 1800Simplify constants:18000 - 1800 = 16200Therefore,T = 16200 + (9/10)TSubtract (9/10)T from both sides:T - (9/10)T = 16200(1/10)T = 16200Multiply both sides by 10:T = 162000So, the total inheritance is 162000 crowns.Now, check if this makes sense.Compute A's share:A = 2000 + (162000 - 2000)/10 = 2000 + 160000/10 = 2000 + 16000 = 18000So, A receives 18000 crowns.Then, after A takes their share, the remaining inheritance is 162000 - 18000 = 144000 crowns.Wait, let's confirm:Original total: 162000A takes 2000, leaving 160000, then takes 1/10 of that, which is 16000. So total taken by A is 2000 + 16000 = 18000. Remaining is 160000 - 16000 = 144000. Correct.Now, sibling B takes 4000, so remaining is 144000 - 4000 = 140000. Then B takes 1/10 of that, which is 14000. So B's total is 4000 + 14000 = 18000. Which matches A's share. Then remaining inheritance is 140000 - 14000 = 126000.So, moving on to sibling C. They take 6000, so remaining is 126000 - 6000 = 120000. Then C takes 1/10 of that, which is 12000. So total for C is 6000 + 12000 = 18000. Remaining inheritance is 120000 - 12000 = 108000.Next, sibling D would take 8000, leaving 108000 - 8000 = 100000. Then they take 1/10 of that, 10000. Total for D: 8000 + 10000 = 18000. Remaining: 100000 - 10000 = 90000.E takes 10000, remaining 90000 - 10000 = 80000. Then 1/10 is 8000. Total: 10000 + 8000 = 18000. Remaining: 80000 - 8000 = 72000.F takes 12000, remaining 72000 - 12000 = 60000. 1/10 is 6000. Total: 12000 + 6000 = 18000. Remaining: 60000 - 6000 = 54000.G takes 14000, remaining 54000 - 14000 = 40000. 1/10 is 4000. Total: 14000 + 4000 = 18000. Remaining: 40000 - 4000 = 36000.H takes 16000, remaining 36000 - 16000 = 20000. 1/10 is 2000. Total: 16000 + 2000 = 18000. Remaining: 20000 - 2000 = 18000.I takes 18000, remaining 18000 - 18000 = 0. Then 1/10 of 0 is 0. So total for I is 18000 + 0 = 18000. Remaining: 0 - 0 = 0.Wait, hold on. Wait, after H, the remaining inheritance is 18000. Then the next sibling, I, is supposed to take 18000 (since each initial amount increases by 2000: A 2000, B 4000, C 6000,... H 16000, I 18000). But the remaining inheritance is 18000. So I takes 18000, but then they also take 1/10 of the remainder after that 18000. But after taking 18000, the remainder is 0, so 1/10 of 0 is 0. So I's total is 18000 + 0 = 18000. That works. Then the inheritance is fully distributed.So, how many siblings are there? Let's count:A, B, C, D, E, F, G, H, I. That's 9 siblings. Each received 18000, and total inheritance is 162000.Wait, but let's verify the count step by step:After A: 144000After B: 126000After C: 108000After D: 90000After E: 72000After F: 54000After G: 36000After H: 18000After I: 0Yes, that's 9 siblings. Each time, the remaining inheritance reduces by 18000. Let's check:From 162000, each sibling takes 18000. 162000 divided by 18000 is 9. So, 9 siblings. That makes sense. So each sibling receives 18000, total inheritance is 162000, and there are 9 siblings.But let me double-check the calculations step by step to ensure there's no mistake.Starting with T = 162000.A takes 2000 + (162000 - 2000)/10 = 2000 + 16000 = 18000. Remaining: 162000 - 18000 = 144000. Correct.B takes 4000 + (144000 - 4000)/10 = 4000 + 14000 = 18000. Remaining: 144000 - 18000 = 126000. Correct.C takes 6000 + (126000 - 6000)/10 = 6000 + 12000 = 18000. Remaining: 126000 - 18000 = 108000. Correct.D takes 8000 + (108000 - 8000)/10 = 8000 + 10000 = 18000. Remaining: 108000 - 18000 = 90000. Correct.E takes 10000 + (90000 - 10000)/10 = 10000 + 8000 = 18000. Remaining: 90000 - 18000 = 72000. Correct.F takes 12000 + (72000 - 12000)/10 = 12000 + 6000 = 18000. Remaining: 72000 - 18000 = 54000. Correct.G takes 14000 + (54000 - 14000)/10 = 14000 + 4000 = 18000. Remaining: 54000 - 18000 = 36000. Correct.H takes 16000 + (36000 - 16000)/10 = 16000 + 2000 = 18000. Remaining: 36000 - 18000 = 18000. Correct.I takes 18000 + (18000 - 18000)/10 = 18000 + 0 = 18000. Remaining: 18000 - 18000 = 0. Correct.So, all 9 siblings received 18000 each, totaling 9*18000 = 162000, which matches the total inheritance. Therefore, the answer is 9 siblings, each received 18000 crowns, and the total inheritance was 162000 crowns.I think that's solid. But let me just check if there's another way to approach this problem, maybe through forming equations for each sibling's share and seeing a pattern.Suppose there are n siblings, each receiving x crowns. Total inheritance T = n*x.For the first sibling:x = 2000 + (T - 2000)/10Similarly, after the first sibling, remaining inheritance is T - x = T - [2000 + (T - 2000)/10] = T - 2000 - (T - 2000)/10 = (9/10)(T - 2000)Second sibling's share:x = 4000 + (Remaining after A - 4000)/10 = 4000 + [(9/10)(T - 2000) - 4000]/10But since x is the same, set equal to first equation:2000 + (T - 2000)/10 = 4000 + [(9/10)(T - 2000) - 4000]/10Which is the same equation as before, leading to T = 162000. Then x = 18000, and n = 162000 / 18000 = 9.Alternatively, maybe modeling the problem as a recurrence relation. Each time, the remaining inheritance after each sibling is 9/10 of (previous remaining - the initial amount taken by the next sibling). However, since each sibling's share is the same, perhaps we can find a pattern in the remaining inheritance.After the first sibling, remaining R1 = 9/10*(T - 2000)After the second sibling, remaining R2 = 9/10*(R1 - 4000)After the third sibling, remaining R3 = 9/10*(R2 - 6000)And so on, until Rn = 0.Since each sibling's share is x = 18000, as calculated, and the remaining inheritance reduces by x each time, which is 18000. But perhaps connecting this to the recurrence.Alternatively, since after each sibling, the remaining inheritance is 9/10*(previous remaining - next initial amount). Let's denote the initial amount for sibling k as 2000k. So for sibling 1, 2000*1; sibling 2, 2000*2; etc.So, R_{k} = (9/10)*(R_{k-1} - 2000k)But with R_0 = T.But since after the last sibling, R_n = 0.But this might be more complex. However, since we already found the answer through the first method, and verified step by step, perhaps this is sufficient.Another way to think about this: since each sibling takes an initial amount increasing by 2000 and then 1/10 of the remainder, and all end up with the same amount, perhaps the difference in their initial amounts is exactly offset by the difference in the 1/10 they receive. For example, B takes 4000 instead of 2000, which is 2000 more, but receives 1/10 of a remainder that is less by some amount. The difference in the 1/10 portions must compensate for the extra 2000.So, setting up the equation for the difference:A's 1/10 portion - B's 1/10 portion = 2000Wait, since B gets 2000 more initially but less in the 1/10 portion.Wait, A's 1/10 portion is (T - 2000)/10B's 1/10 portion is [ (9/10)(T - 2000) - 4000 ] /10So the difference between A's 1/10 and B's 1/10 is:(T - 2000)/10 - [ (9/10)(T - 2000) - 4000 ] /10Factor out 1/10:[ (T - 2000) - (9/10)(T - 2000) + 4000 ] /10Simplify inside:[ (1 - 9/10)(T - 2000) + 4000 ] /10= [ (1/10)(T - 2000) + 4000 ] /10This difference should equal the extra 2000 that B took initially. Wait, no. Wait, actually, since B receives 4000 instead of 2000, which is 2000 more, but the 1/10 portion that B receives is less than what A received. Therefore, the total for B is 4000 + (smaller 1/10) = 2000 + (larger 1/10). Therefore, the difference in the 1/10 portions must be 2000.So:A's 1/10 - B's 1/10 = 2000Which is:(T - 2000)/10 - [ (9/10)(T - 2000) - 4000 ] /10 = 2000Which simplifies to:[ (T - 2000) - (9/10)(T - 2000) + 4000 ] /10 = 2000Multiply both sides by 10:[ (1/10)(T - 2000) + 4000 ] = 20000Then:(1/10)(T - 2000) = 20000 - 4000 = 16000Multiply both sides by 10:T - 2000 = 160000Therefore:T = 162000Same result as before. So this confirms the total inheritance. Then, since each sibling receives 18000, the number of siblings is 162000 / 18000 = 9. So, 9 siblings. This method directly relates the difference in their 1/10 portions to the difference in their initial amounts, which is an elegant way to approach it.Therefore, regardless of the method used, the answer consistently comes out to 9 siblings, each receiving 18000 crowns, with a total inheritance of 162000 crowns. The step-by-step verification of each sibling's share also confirms that the distribution works perfectly without any remainder, which solidifies the conclusion.Final AnswerThe total inheritance was boxed{162000} crowns, each of the boxed{9} siblings received boxed{18000} crowns.</think>Given the problem of several siblings dividing an inheritance where each sibling receives an initial amount and one-tenth of the remainder, and each sibling ends up receiving an equal amount, we need to determine the number of siblings, the amount each received, and the total inheritance.1. Setting up the equations for the first two siblings: - Let ( T ) be the total inheritance. - Sibling A receives ( 2000 ) crowns plus one-tenth of the remainder: [ A = 2000 + frac{T - 2000}{10} ] - After A's share, the remaining inheritance is: [ frac{9}{10}(T - 2000) ] - Sibling B receives ( 4000 ) crowns plus one-tenth of the new remainder: [ B = 4000 + frac{frac{9}{10}(T - 2000) - 4000}{10} ]2. Equating the amounts received by A and B: [ 2000 + frac{T - 2000}{10} = 4000 + frac{frac{9}{10}(T - 2000) - 4000}{10} ] - Simplifying both sides and solving for ( T ): [ 10 left( 2000 + frac{T - 2000}{10} right) = 10 left( 4000 + frac{frac{9}{10}(T - 2000) - 4000}{10} right) ] [ 18000 + T = 36000 + frac{9}{10}(T - 2000) ] [ T - frac{9}{10}T = 16200 ] [ frac{1}{10}T = 16200 ] [ T = 162000 ]3. Calculating each sibling's share: - For sibling A: [ A = 2000 + frac{162000 - 2000}{10} = 2000 + 16000 = 18000 ] - Each subsequent sibling's share is calculated similarly, confirming they all receive 18000 crowns.4. Determining the number of siblings: - The total inheritance is ( 162000 ) crowns, and each sibling receives ( 18000 ) crowns. - Number of siblings: [ frac{162000}{18000} = 9 ]5. Verification: - Each sibling's share and the remaining inheritance after each step confirm the calculations are correct.Final Answer:The total inheritance was (boxed{162000}) crowns, each of the (boxed{9}) siblings received (boxed{18000}) crowns.]

🔑:If the entire inheritance is x, then A's share is 2000+frac{x-2000}{10}=1800+frac{x}{10}; B's share is 4000+frac{1}{10}left[x-1800-frac{x}{10}-4000right]= 3420-frac{9 x}{100}.Thus,1800+frac{x}{10}=3420+frac{9 x}{100}from whichx=162000The number of siblings is 9, and each one's inheritance is 18000 mathrm{~K}.(Kürti Imre, Eger.)Number of solutions: 26.

🧠:[<think>Okay, let me try to figure out this problem. So, we have a point P that's at a distance of √3 from plane α. Then there's a point Q on the plane α such that the angle between the line PQ and the plane α is between 30° and 60°. We need to find the area of the region formed by all such points Q.Hmm, first, I need to visualize this. Let me imagine the plane α and point P somewhere above it. The distance from P to α is √3, which is like the perpendicular distance. Now, when they say the angle between PQ and the plane α is between 30° and 60°, I think that angle is measured between the line PQ and the plane. Wait, how do you measure the angle between a line and a plane? I remember that the angle between a line and a plane is the complement of the angle between the line and the normal to the plane. So, if θ is the angle between PQ and the plane, then the angle φ between PQ and the normal would be 90° - θ. So, θ = 30° to 60°, which means φ = 60° to 30°. Wait, that seems right. Let me confirm. If the line is parallel to the plane, the angle with the plane would be 0°, and with the normal it would be 90°. So yes, the angle with the plane plus the angle with the normal equals 90°. So θ (with the plane) = 30°-60°, so φ (with the normal) = 60°-30°.Okay, so the angle between PQ and the normal is between 30° and 60°. Now, PQ is a line from P to Q on the plane. Let me consider the projection of P onto the plane α. Let's call that point O. Then, the distance from P to O is √3, since that's the perpendicular distance. So, in the plane α, point O is the foot of the perpendicular from P.Now, for any point Q on α, PQ is a line from P to Q, and the angle φ between PQ and the normal PO is between 30° and 60°. Let's see. If we consider triangle PQO, which is a right triangle, right? Because PO is perpendicular to the plane, so OQ is on the plane, so PO is perpendicular to OQ. Therefore, triangle PQO is a right triangle with right angle at O.In triangle PQO, PO is the leg of length √3, OQ is the other leg, and PQ is the hypotenuse. The angle φ at P between PQ and PO can be found using trigonometry. The cosine of φ is adjacent over hypotenuse, which is PO / PQ. So cos φ = PO / PQ. Since PO = √3, then cos φ = √3 / PQ. Therefore, PQ = √3 / cos φ. But we also have that OQ is the other leg, so by Pythagoras, OQ = sqrt( PQ² - PO² ) = sqrt( (3 / cos² φ ) - 3 ) = sqrt( 3 (1 / cos² φ - 1 ) ) = sqrt( 3 tan² φ ) = √3 tan φ. Therefore, OQ = √3 tan φ.So, the length of OQ depends on φ, which is the angle between PQ and the normal. Since φ is between 30° and 60°, then OQ is between √3 tan 30° and √3 tan 60°. Let's compute those.tan 30° = 1/√3, so √3 * (1/√3) = 1. tan 60° = √3, so √3 * √3 = 3. Therefore, OQ is between 1 and 3. So, the set of points Q in the plane α such that their distance from O is between 1 and 3. That is, Q lies in an annulus (a ring-shaped region) centered at O with inner radius 1 and outer radius 3.Therefore, the area of this region would be the area of the outer circle minus the area of the inner circle. That is, π*(3)^2 - π*(1)^2 = 9π - 1π = 8π.Wait, but let me check if there are any restrictions. The angle is "no less than 30° and no more than 60°", so φ is between 30° and 60°, which corresponds to OQ between 1 and 3, so yes, that's an annulus. Therefore, area is 8π.But hold on, let me make sure I didn't make a mistake with the angle. So, θ is the angle between PQ and the plane. θ is between 30° and 60°, so φ, the angle between PQ and the normal, is between 30° and 60° as well? Wait, no. Wait, if θ is the angle between PQ and the plane, then φ = 90° - θ. So if θ is between 30° and 60°, then φ is between 90° - 60° = 30° and 90° - 30° = 60°. Wait, so φ is also between 30° and 60°? That seems contradictory to my initial thought, but according to the definition, θ is the angle between the line and the plane, which is the complement of φ, the angle between the line and the normal. So, θ + φ = 90°. Therefore, θ = 30°-60°, so φ = 60°-30°. Wait, that's the same as swapping the angles? So if θ is 30°, φ is 60°, and if θ is 60°, φ is 30°. So, actually, φ is between 30° and 60°, but in reverse order. So, φ ranges from 30° to 60°, but θ is from 30° to 60°, so φ is from 60° down to 30°? Wait, maybe I confused the angle ranges.Wait, let me clarify. If θ is the angle between PQ and the plane, and φ is the angle between PQ and the normal, then θ + φ = 90°. So if θ is 30°, then φ is 60°, and if θ is 60°, then φ is 30°. Therefore, φ actually ranges from 30° to 60°, but in reverse order. However, since φ is an angle, the trigonometric functions tan φ would still take values from tan 30° to tan 60°, which is increasing. Wait, tan 30° is about 0.577, tan 60° is √3 ≈ 1.732. So even though φ decreases from 60° to 30°, the tan φ decreases from tan 60° to tan 30°, but in our calculation, OQ = √3 tan φ. Wait, hold on, if φ is the angle between PQ and the normal, then when φ is 30°, OQ is √3 tan 30° = 1, and when φ is 60°, OQ is √3 tan 60° = 3. So as φ increases from 30° to 60°, OQ increases from 1 to 3. But in reality, when θ increases (angle with the plane), φ decreases. So θ between 30° and 60° corresponds to φ between 30° and 60° but in reverse. Wait, maybe I need to think differently.Wait, let's take an example. Suppose θ = 30°, which is the minimum angle with the plane. Then φ = 90° - 30° = 60°, which is the maximum angle with the normal. So when θ is smallest (30°), φ is largest (60°), leading to the largest OQ (3). When θ is largest (60°), φ is smallest (30°), leading to the smallest OQ (1). Therefore, the region of Q is between circles of radius 1 and 3. Therefore, the annulus between 1 and 3. Thus, area is π(3² - 1²) = 8π.Therefore, the answer should be 8π.Wait, but just to double-check, maybe there's a cone involved? Because when you fix the angle between a line and a plane, the set of lines from a point making a fixed angle with a plane forms a cone. Then, the intersection of that cone with the plane would be a circle.But in this case, we have two angles, 30° and 60°, so two cones. The region between them would be the area between two circles. So yes, that would result in an annulus.So the first cone (angle 60° with the plane, which is φ = 30° with the normal) would have a smaller radius, and the second cone (angle 30° with the plane, φ = 60° with the normal) would have a larger radius. Therefore, the area between them is the annulus with radii 1 and 3, area 8π.Therefore, the answer is 8π.Final Answerboxed{8pi}</think>Given a point ( P ) at a distance ( sqrt{3} ) from plane ( alpha ), we need to find the area of the region formed by points ( Q ) on ( alpha ) such that the angle between line ( PQ ) and plane ( alpha ) is between ( 30^circ ) and ( 60^circ ).1. Understanding the Angle: - The angle between line ( PQ ) and plane ( alpha ) is ( theta ), which ranges from ( 30^circ ) to ( 60^circ ). - The angle ( phi ) between line ( PQ ) and the normal to the plane is ( 90^circ - theta ). Thus, ( phi ) ranges from ( 30^circ ) to ( 60^circ ).2. Trigonometric Relationships: - The distance from ( P ) to the plane ( alpha ) is ( sqrt{3} ), which is the length of the perpendicular segment ( PO ) (where ( O ) is the projection of ( P ) onto ( alpha )). - In the right triangle ( triangle PQO ), we use the tangent function: ( tan phi = frac{OQ}{PO} ). - Solving for ( OQ ), we get ( OQ = PO cdot tan phi = sqrt{3} cdot tan phi ).3. Range of ( OQ ): - When ( phi = 30^circ ), ( OQ = sqrt{3} cdot tan 30^circ = sqrt{3} cdot frac{1}{sqrt{3}} = 1 ). - When ( phi = 60^circ ), ( OQ = sqrt{3} cdot tan 60^circ = sqrt{3} cdot sqrt{3} = 3 ).4. Area Calculation: - The region of points ( Q ) forms an annulus with inner radius 1 and outer radius 3. - The area of this annulus is ( pi cdot 3^2 - pi cdot 1^2 = 9pi - pi = 8pi ).Thus, the area of the region formed by such points ( Q ) is (boxed{8pi}).,<think>Okay, let's see. The problem says that we have a point P at a distance of √3 from plane α. Then there's a point Q on plane α such that the angle between line PQ and the plane α is between 30° and 60°. We need to find the area of the region formed by all such points Q on the plane. Hmm, okay, let me break this down step by step.First, I need to visualize the situation. There's a plane α, and point P is somewhere above it at a height of √3. The distance from P to the plane is √3, which is the perpendicular distance. Now, Q is any point on the plane α such that when we connect P to Q, the angle between PQ and the plane is between 30° and 60°. The set of all such Q points forms a region on the plane, and we need to find the area of that region.Let me recall that the angle between a line and a plane is the complement of the angle between the line and the normal to the plane. Wait, is that right? So if θ is the angle between PQ and the plane, then the angle φ between PQ and the normal to the plane would be 90° - θ. Because if the line is almost parallel to the plane, the angle with the normal would be almost 90°, so θ would be small. Yeah, that makes sense. So θ + φ = 90°, where θ is the angle with the plane and φ is the angle with the normal.Given that the angle θ is between 30° and 60°, then the angle φ between PQ and the normal would be between 30° and 60° as well. Wait, no. If θ is 30°, then φ is 60°, and if θ is 60°, then φ is 30°. So actually, φ is between 30° and 60°. Let me confirm that.Suppose θ is the angle between PQ and the plane. Then the angle between PQ and the normal to the plane is indeed 90° - θ. So if θ is 30°, then 90° - 30° = 60°, and if θ is 60°, then 90° - 60° = 30°. Therefore, the angle φ between PQ and the normal is between 30° and 60°.Now, the distance from P to the plane is √3, which is the length of the perpendicular from P to α. Let's denote the foot of the perpendicular as point O. So, O is the projection of P onto plane α, and PO = √3.Given that, for any point Q on plane α, the line PQ forms an angle φ with the normal PO. So, the angle between PQ and PO is φ, which is between 30° and 60°. To find the locus of points Q on plane α such that the angle between PQ and PO is between 30° and 60°, we can think in terms of right triangles. Let's consider triangle POQ, which is a right triangle with right angle at O. The length of PO is √3, and the angle at Q is φ, which is between 30° and 60°. Wait, no. Actually, angle at P would be φ, the angle between PQ and PO. Wait, no, the angle between PQ and the normal is φ, which is at point P. Let me clarify.In triangle POQ, we have PO = √3 (the height), OQ is the horizontal distance from O to Q on the plane, and PQ is the hypotenuse. The angle between PQ and PO is φ (30° to 60°). So in triangle POQ, angle at P is φ. Therefore, in triangle POQ, cosφ = adjacent/hypotenuse = PO / PQ. Therefore, PQ = PO / cosφ = √3 / cosφ. Similarly, sinφ = opposite/hypotenuse = OQ / PQ, so OQ = PQ sinφ = √3 / cosφ * sinφ = √3 tanφ.Therefore, the horizontal distance from O to Q on the plane α is OQ = √3 tanφ. Since φ is between 30° and 60°, tanφ ranges from tan30° = 1/√3 to tan60° = √3. Therefore, OQ ranges from √3 * (1/√3) = 1 to √3 * √3 = 3. So the points Q lie in an annular region (a ring) on plane α centered at O, with inner radius 1 and outer radius 3. Therefore, the area would be the area of the outer circle minus the area of the inner circle: π(3²) - π(1²) = π(9 - 1) = 8π.Wait, but hold on. Let me check again. The angle φ is between 30° and 60°, which corresponds to OQ = √3 tanφ. So when φ is 30°, tan30° = 1/√3, so OQ = √3*(1/√3)=1. When φ is 60°, tan60°=√3, so OQ=√3*√3=3. Therefore, the region of Q is the set of all points on plane α that are at least 1 unit away from O and at most 3 units away. So that's an annulus with inner radius 1 and outer radius 3. The area is π*(3² -1²)= π*(9 -1)=8π. So the answer should be 8π.But let me verify if there's any mistake here. Let me think again about the angle. The angle between PQ and the normal is φ, which relates to the angle θ between PQ and the plane. So θ = 90° - φ. The problem states that θ is between 30° and 60°, so φ is between 30° and 60° as we thought. Therefore, φ ranges from 30° to 60°, leading to OQ from 1 to 3, and hence the annulus area is 8π. That seems correct.Alternatively, let's approach it using trigonometry. The angle θ between the line PQ and the plane is given. The sine of θ is equal to the opposite side over the hypotenuse in the right triangle formed by P, Q, and the foot O. Wait, but θ is the angle between PQ and the plane. How is this angle defined?Actually, the angle between a line and a plane is the smallest angle between the line and its projection onto the plane. So, if θ is the angle between PQ and the plane, then sinθ = PO / PQ. Because PO is the height, which is the opposite side to angle θ, and PQ is the hypotenuse. So sinθ = PO / PQ => PQ = PO / sinθ. Then, the horizontal distance OQ can be found via Pythagoras: OQ = sqrt(PQ² - PO²) = sqrt( ( (√3 / sinθ )² - (√3)^2 ) ) = sqrt( 3 / sin²θ - 3 ) = sqrt(3 (1/sin²θ - 1 )) = sqrt(3 (cot²θ )) = sqrt(3) cotθ. Therefore, OQ = sqrt(3) cotθ.Given that θ is between 30° and 60°, so cotθ would be between cot60° and cot30°, which is between 1/√3 and √3. Therefore, OQ is between sqrt(3)*(1/√3) = 1 and sqrt(3)*sqrt(3) = 3. So again, the radius ranges from 1 to 3, leading to an annulus with area 8π.Wait a second, so using this method, we also get the same result. So, yes, whether we use the angle with the normal or the angle with the plane, we end up with the same radii. Therefore, the area is indeed 8π.Therefore, the answer should be 8π. Let me just check if there's another way this could be interpreted. Maybe the region isn't an annulus? For example, if the angle is measured differently. But no, since the problem says "the angle between line PQ and α is no less than 30° and no more than 60°", which is the angle between the line and the plane. As per definition, that angle is the complement of the angle between the line and the normal. So, the two approaches are equivalent, and both lead to OQ between 1 and 3.Another way to think about it is to parameterize Q. Let’s assume that the plane α is the xy-plane for simplicity, and point P is at (0, 0, √3). Then point Q is (x, y, 0). The vector PQ is (x, y, -√3). The angle between PQ and the plane α is the angle between PQ and its projection onto the plane α. The projection of PQ onto α is (x, y, 0). The angle θ between PQ and its projection can be found using the dot product:cosθ = (PQ · projection) / (|PQ| |projection|)PQ · projection = x^2 + y^2 + 0 = x^2 + y^2|PQ| = sqrt(x^2 + y^2 + 3)|projection| = sqrt(x^2 + y^2)Therefore, cosθ = (x^2 + y^2) / (sqrt(x^2 + y^2 + 3) * sqrt(x^2 + y^2)) ) = sqrt(x^2 + y^2) / sqrt(x^2 + y^2 + 3)But θ is the angle between PQ and the plane, so we can relate this to sinθ. Wait, actually, the angle between the line and the plane is θ, which is equal to the angle between the line and its projection. Alternatively, sometimes the angle between the line and the normal is considered. But in the problem statement, it's specified as the angle between PQ and the plane α, which should be the angle between the line and the plane, defined as the complement of the angle between the line and the normal.Wait, let me make sure. The angle between a line and a plane is the angle between the line and its orthogonal projection onto the plane. This angle is measured between 0° and 90°. So if θ is this angle, then sinθ = opposite / hypotenuse = PO / PQ. Which matches the previous approach.Therefore, sinθ = PO / PQ = √3 / |PQ|. Therefore, sinθ = √3 / sqrt(x^2 + y^2 + 3). So, given that θ is between 30° and 60°, then sinθ is between sin30°=1/2 and sin60°=√3/2.So,1/2 ≤ √3 / sqrt(x^2 + y^2 + 3) ≤ √3 / 2Wait, let's write that inequality:1/2 ≤ √3 / sqrt(x^2 + y^2 + 3) ≤ √3 / 2Wait, but let's check. If θ is between 30° and 60°, then sinθ is between 1/2 and √3/2. So:1/2 ≤ √3 / sqrt(x^2 + y^2 + 3) ≤ √3 / 2Wait, but solving these inequalities.First, left inequality:√3 / sqrt(x^2 + y^2 + 3) ≥ 1/2Multiply both sides by sqrt(x^2 + y^2 + 3):√3 ≥ (1/2) sqrt(x^2 + y^2 + 3)Multiply both sides by 2:2√3 ≥ sqrt(x^2 + y^2 + 3)Square both sides:(2√3)^2 ≥ x^2 + y^2 + 312 ≥ x^2 + y^2 + 3Therefore:x^2 + y^2 ≤ 12 - 3 = 9Similarly, the right inequality:√3 / sqrt(x^2 + y^2 + 3) ≤ √3 / 2Divide both sides by √3 (positive, so inequality direction remains):1 / sqrt(x^2 + y^2 + 3) ≤ 1 / 2Take reciprocal (inequality reverses):sqrt(x^2 + y^2 + 3) ≥ 2Square both sides:x^2 + y^2 + 3 ≥ 4Thus:x^2 + y^2 ≥ 1Therefore, combining both inequalities, we have:1 ≤ x^2 + y^2 ≤ 9Which represents an annulus with inner radius 1 and outer radius 3 (since x^2 + y^2 is r^2 in polar coordinates). Therefore, the area is π*(3^2 -1^2)=π*(9 -1)=8π. So, yes, this confirms the earlier result.Therefore, regardless of the method used—whether geometric reasoning with triangles, trigonometric relations, or coordinate algebra—we arrive at the same conclusion: the area is 8π. So, the answer should be 8π.Final AnswerThe area of the region formed by such points ( Q ) is (boxed{8pi}).</think>Given a point ( P ) at a distance of ( sqrt{3} ) from plane ( alpha ), we need to find the area of the region formed by points ( Q ) on plane ( alpha ) such that the angle between line ( PQ ) and plane ( alpha ) is between ( 30^circ ) and ( 60^circ ).1. Understanding the Angle: - The angle between line ( PQ ) and plane ( alpha ) is the complement of the angle between ( PQ ) and the normal to the plane. - If ( theta ) is the angle between ( PQ ) and the plane, then the angle ( phi ) between ( PQ ) and the normal is ( 90^circ - theta ). - Given ( 30^circ leq theta leq 60^circ ), we have ( 30^circ leq phi leq 60^circ ).2. Trigonometric Relations: - The distance from ( P ) to the plane is ( sqrt{3} ), which is the length of the perpendicular from ( P ) to ( alpha ) (denoted as ( PO )). - Using trigonometry in the right triangle ( POQ ), we find the horizontal distance ( OQ ) from ( O ) to ( Q ): - ( cosphi = frac{PO}{PQ} ) implies ( PQ = frac{sqrt{3}}{cosphi} ). - ( sinphi = frac{OQ}{PQ} ) implies ( OQ = sqrt{3} tanphi ).3. Range of Horizontal Distance: - For ( phi = 30^circ ), ( OQ = sqrt{3} tan 30^circ = 1 ). - For ( phi = 60^circ ), ( OQ = sqrt{3} tan 60^circ = 3 ).4. Area Calculation: - The region of points ( Q ) forms an annulus with inner radius 1 and outer radius 3. - The area of this annulus is ( pi times (3^2 - 1^2) = pi times (9 - 1) = 8pi ).Thus, the area of the region formed by such points ( Q ) is (boxed{8pi}).]

🔑:Answer: 8 pi.Solution: Let the projection of point P on plane alpha be O. From the given conditions, frac{O P}{O Q}=tan angle O Q P inleft[frac{sqrt{3}}{3}, sqrt{3}right], i.e., O Q in[1,3]. Therefore, the area of the required region is pi cdot 3^{2}-pi cdot 1^{2}=8 pi.