problem_id int64 0 3.39k | problem stringlengths 20 2.22k | label stringclasses 154
values | level stringclasses 5
values | type stringclasses 4
values | prompt listlengths 2 2 | completion listlengths 1 1 |
|---|---|---|---|---|---|---|
341 | The average of two 2-digit positive integers is equal to the decimal number obtained by writing one of the two-digit integers before the decimal point and the other two-digit integer after the decimal point. What is the smaller of the two integers? | 49 | Level 5 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are given: \"The average of two 2-digit positive integers is equal to the decimal number obtained by writing one of the two-digit integers before the decimal point and the other two-digit integer after the decimal point. What is the smaller of the two integers?\" And th... |
343 | The point with coordinates $(6,-10)$ is the midpoint of the segment with one endpoint at $(8,0)$. Find the sum of the coordinates of the other endpoint. | -16 | Level 3 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are asked: \"The point with coordinates (6,-10) is the midpoint of the segment with one endpoint at (8,0). Find the sum of the coordinates of the other endpoint.\" We need to solve using midpoint formula. The midpoint M = ((x1+x2)/2, (y1+y2)/2). Given M=(6,-10) and one ... |
343 | The point with coordinates $(6,-10)$ is the midpoint of the segment with one endpoint at $(8,0)$. Find the sum of the coordinates of the other endpoint. | -16 | Level 3 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are given: \"The point with coordinates (6,-10) is the midpoint of the segment with one endpoint at (8,0). Find the sum of the coordinates of the other endpoint.\" Solve using vector translation method.\n\nWe need to find the other endpoint. Let the other endpoint be (x... |
343 | The point with coordinates $(6,-10)$ is the midpoint of the segment with one endpoint at $(8,0)$. Find the sum of the coordinates of the other endpoint. | -16 | Level 3 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are given: \"The point with coordinates (6,-10) is the midpoint of the segment with one endpoint at (8,0). Find the sum of the coordinates of the other endpoint.\" And we are told to solve using the method: Reflection across midpoint.\n\nInterpretation: The midpoint is ... |
343 | The point with coordinates $(6,-10)$ is the midpoint of the segment with one endpoint at $(8,0)$. Find the sum of the coordinates of the other endpoint. | -16 | Level 3 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are asked: \"The point with coordinates (6,-10) is the midpoint of the segment with one endpoint at (8,0). Find the sum of the coordinates of the other endpoint.\" The user says: \"Solve the problem using the method: Direct sum shortcut.\" So we need to use that method.... |
344 | Suppose functions $g$ and $f$ have the properties that $g(x)=3f^{-1}(x)$ and $f(x)=\frac{24}{x+3}$. For what value of $x$ does $g(x)=15$? | 3 | Level 5 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are given that f(x) = 24/(x+3). And g(x) = 3 f^{-1}(x). We need to find x such that g(x)=15.\n\nWe can find f inverse explicitly. f(x)=24/(x+3). Let y = 24/(x+3). Solve for x: y(x+3)=24 => x+3=24/y => x=24/y -3. So f^{-1}(y)=24/y -3. So f^{-1}(x)=24/x -3.\n\nThen g(x)=3... |
344 | Suppose functions $g$ and $f$ have the properties that $g(x)=3f^{-1}(x)$ and $f(x)=\frac{24}{x+3}$. For what value of $x$ does $g(x)=15$? | 3 | Level 5 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are given: g(x) = 3 f^{-1}(x) and f(x) = 24/(x+3). Need to find x such that g(x)=15. So set g(x)=15 => 3 f^{-1}(x)=15 => f^{-1}(x)=5. So x = f(5). Compute f(5)=24/(5+3)=24/8=3. So x=3. But check: g(3)=3 f^{-1}(3). f^{-1}(3) means find y such that f(y)=3. f(y)=24/(y+3)=3... |
344 | Suppose functions $g$ and $f$ have the properties that $g(x)=3f^{-1}(x)$ and $f(x)=\frac{24}{x+3}$. For what value of $x$ does $g(x)=15$? | 3 | Level 5 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are given: g(x)=3 f^{-1}(x) and f(x)=24/(x+3). We need to find x such that g(x)=15. So we need to solve 3 f^{-1}(x) = 15 => f^{-1}(x) = 5. That means x = f(5). Because if f^{-1}(x)=5, then x = f(5). So compute f(5)=24/(5+3)=24/8=3. So x = 3? Wait f(5)=3, so x=3. Then g(... |
345 | Let $f(x) = (x+2)^2-5$. If the domain of $f$ is all real numbers, then $f$ does not have an inverse function, but if we restrict the domain of $f$ to an interval $[c,\infty)$, then $f$ may have an inverse function. What is the smallest value of $c$ we can use here, so that $f$ does have an inverse function? | -2 | Level 5 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We need to find the smallest c such that f(x) = (x+2)^2 - 5 is one-to-one on [c, ∞). The function is a parabola opening upward with vertex at x = -2. It is decreasing on (-∞, -2] and increasing on [-2, ∞). For it to have an inverse, it must be one-to-one (strictly monotoni... |
345 | Let $f(x) = (x+2)^2-5$. If the domain of $f$ is all real numbers, then $f$ does not have an inverse function, but if we restrict the domain of $f$ to an interval $[c,\infty)$, then $f$ may have an inverse function. What is the smallest value of $c$ we can use here, so that $f$ does have an inverse function? | -2 | Level 5 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are asked: \"Let f(x) = (x+2)^2 - 5. If the domain of f is all real numbers, then f does not have an inverse function, but if we restrict the domain of f to an interval [c, ∞), then f may have an inverse function. What is the smallest value of c we can use here, so that... |
345 | Let $f(x) = (x+2)^2-5$. If the domain of $f$ is all real numbers, then $f$ does not have an inverse function, but if we restrict the domain of $f$ to an interval $[c,\infty)$, then $f$ may have an inverse function. What is the smallest value of $c$ we can use here, so that $f$ does have an inverse function? | -2 | Level 5 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are given f(x) = (x+2)^2 - 5. This is a parabola opening upwards with vertex at (-2, -5). Domain all real numbers, not one-to-one because it fails horizontal line test (parabola). To have an inverse, we need to restrict domain to an interval where f is one-to-one. For a... |
345 | Let $f(x) = (x+2)^2-5$. If the domain of $f$ is all real numbers, then $f$ does not have an inverse function, but if we restrict the domain of $f$ to an interval $[c,\infty)$, then $f$ may have an inverse function. What is the smallest value of $c$ we can use here, so that $f$ does have an inverse function? | -2 | Level 5 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are given: f(x) = (x+2)^2 - 5. Domain is all real numbers, but then f is not one-to-one because it's a parabola opening upward with vertex at x = -2. To have an inverse, we need to restrict domain to an interval where f is one-to-one. Typically, we restrict to either (-... |
346 | Xanthia can read 100 pages per hour and Molly can read 50 pages per hour. If they each read the same book, and the book has 225 pages, how many more minutes than Xanthia would it take for Molly to finish reading the book? | 135 | Level 2 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We need to solve: Xanthia reads 100 pages per hour, Molly reads 50 pages per hour. Book has 225 pages. How many more minutes than Xanthia would it take for Molly to finish? Use method: Inverse proportion of rates.\n\nWe can compute time for each: time = pages / rate. Xanth... |
346 | Xanthia can read 100 pages per hour and Molly can read 50 pages per hour. If they each read the same book, and the book has 225 pages, how many more minutes than Xanthia would it take for Molly to finish reading the book? | 135 | Level 2 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are asked: \"Xanthia can read 100 pages per hour and Molly can read 50 pages per hour. If they each read the same book, and the book has 225 pages, how many more minutes than Xanthia would it take for Molly to finish reading the book?\" We need to solve using unit rate ... |
346 | Xanthia can read 100 pages per hour and Molly can read 50 pages per hour. If they each read the same book, and the book has 225 pages, how many more minutes than Xanthia would it take for Molly to finish reading the book? | 135 | Level 2 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are given: Xanthia reads 100 pages per hour, Molly reads 50 pages per hour. Book has 225 pages. Need to find how many more minutes than Xanthia it would take for Molly to finish. So compute time for Xanthia: 225/100 = 2.25 hours = 2.25*60 = 135 minutes. Time for Molly: ... |
347 | The operation $\&$ is defined as $a \& b = (a+b)(a-b)$. What is the value of $6 \& 3$? | 27 | Level 2 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are asked: \"The operation & is defined as a & b = (a+b)(a-b). What is the value of 6 & 3?\" So we simply compute: (6+3)*(6-3)=9*3=27. So answer is 27. Use \\boxed{27}.\nThe operation is defined as \\(a \\& b = (a+b)(a-b)\\). Substituting \\(a=6\\) and \\(b=3\\) gives \... |
347 | The operation $\&$ is defined as $a \& b = (a+b)(a-b)$. What is the value of $6 \& 3$? | 27 | Level 2 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are asked: \"The operation & is defined as a & b = (a+b)(a-b). What is the value of 6 & 3?\" And we are to solve using the method: difference of squares. So simply plug in a=6, b=3. Then (6+3)(6-3) = 9 * 3 = 27. So answer is 27. Put in \\boxed{27}.\nThe operation is def... |
347 | The operation $\&$ is defined as $a \& b = (a+b)(a-b)$. What is the value of $6 \& 3$? | 27 | Level 2 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are asked: \"The operation & is defined as a & b = (a+b)(a-b). What is the value of 6 & 3?\" Using algebraic expansion method: (a+b)(a-b) = a^2 - b^2. So 6 & 3 = 6^2 - 3^2 = 36 - 9 = 27. So answer is 27. Put in boxed.\nThe operation \\(a \\& b\\) is defined as \\((a+b)(... |
348 | If the two roots of the quadratic $7x^2+3x+k$ are $\frac{-3\pm i\sqrt{299}}{14}$, what is $k$? | 11 | Level 4 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are given: quadratic 7x^2+3x+k, roots are (-3 ± i√299)/14. Need to find k using Vieta's product formula.\n\nVieta: For quadratic ax^2+bx+c=0, sum of roots = -b/a, product = c/a. Here a=7, b=3, c=k. So product of roots = k/7.\n\nGiven roots: r1 = (-3 + i√299)/14, r2 = (-... |
348 | If the two roots of the quadratic $7x^2+3x+k$ are $\frac{-3\pm i\sqrt{299}}{14}$, what is $k$? | 11 | Level 4 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are asked: \"If the two roots of the quadratic $7x^2+3x+k$ are $\\frac{-3\\pm i\\sqrt{299}}{14}$, what is $k$?\" We need to solve using the method: Substitute root into equation. So we substitute one of the roots into the quadratic expression set equal to zero, and solv... |
348 | If the two roots of the quadratic $7x^2+3x+k$ are $\frac{-3\pm i\sqrt{299}}{14}$, what is $k$? | 11 | Level 4 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are given a quadratic equation: 7x^2 + 3x + k = 0. Its roots are given as (-3 ± i√299) / 14. We need to find k. The method: Discriminant comparison. We know that for a quadratic ax^2+bx+c=0, the roots are given by the quadratic formula: x = [-b ± √(b^2-4ac)]/(2a). Here ... |
348 | If the two roots of the quadratic $7x^2+3x+k$ are $\frac{-3\pm i\sqrt{299}}{14}$, what is $k$? | 11 | Level 4 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are given the quadratic 7x^2+3x+k and its roots are (-3 ± i√299)/14. We need to find k. The method: Expand factored form. That means we write the quadratic in factored form as a(x - r1)(x - r2) and expand to match the given quadratic. Since the leading coefficient is 7,... |
349 | For any number $x$, we are told that $x\&=7-x$ and $\&x = x -7$. What is the value of $\&(12\&)$? | -12 | Level 3 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are given two operations: \nx& = 7 - x\n&x = x - 7\n\nWe need to evaluate &(12&). The notation is a bit confusing but we must parse it. The expression is &(12&). Probably means apply the & operator on the result of 12&. But careful: The operations are defined as postfix... |
349 | For any number $x$, we are told that $x\&=7-x$ and $\&x = x -7$. What is the value of $\&(12\&)$? | -12 | Level 3 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are given two operations: For any number x, we are told that x& = 7 - x and &x = x - 7. We need to find the value of &(12&). This notation likely means: first compute 12&, then apply & to that result. So &(12&) means & applied to (12&). So compute 12& = 7 - 12 = -5. The... |
349 | For any number $x$, we are told that $x\&=7-x$ and $\&x = x -7$. What is the value of $\&(12\&)$? | -12 | Level 3 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are asked: \"For any number x, we are told that x&=7-x and &x = x-7. What is the value of &(12&)?\" The notation is a bit confusing: There is a symbol \"&\" that is used as a unary operator? Actually they define: x& = 7 - x and &x = x - 7. So \"&\" appears as a postfix ... |
350 | Simplify: $$\sqrt[3]{2744000}$$ | 140 | Level 2 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We need to simplify the cube root of 2744000. The method is prime factorization. So we need to factor 2744000 into prime factors, then group them into triples (since cube root). Then take one factor from each triple. Multiply them to get the cube root.\n\nFirst, let's fact... |
350 | Simplify: $$\sqrt[3]{2744000}$$ | 140 | Level 2 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We need to simplify cube root of 2744000. The method: Factorization using perfect cubes. So find prime factorization and group into triples.\n\nFirst, note that 2744000. Possibly it's 2744 * 1000? Actually 2744 = 14^3? Let's check: 14^3 = 2744. Yes 14^3 = 2744. And 1000 = ... |
352 | There are constants $\alpha$ and $\beta$ such that $\frac{x-\alpha}{x+\beta} = \frac{x^2-80x+1551}{x^2+57x-2970}$. What is $\alpha+\beta$? | 137 | Level 5 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are given equation: (x-α)/(x+β) = (x^2 - 80x + 1551)/(x^2 + 57x - 2970). We need to find constants α and β such that this identity holds for all x (presumably where denominators nonzero). So the rational functions are equal. That implies that after factoring and canceli... |
352 | There are constants $\alpha$ and $\beta$ such that $\frac{x-\alpha}{x+\beta} = \frac{x^2-80x+1551}{x^2+57x-2970}$. What is $\alpha+\beta$? | 137 | Level 5 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are given the equation:\n\n(x - α)/(x + β) = (x^2 - 80x + 1551)/(x^2 + 57x - 2970)\n\nWe need to find α + β. There exist constants α and β such that this holds for all x (presumably as an identity, except at points where denominators are zero). So we need to find α and ... |
352 | There are constants $\alpha$ and $\beta$ such that $\frac{x-\alpha}{x+\beta} = \frac{x^2-80x+1551}{x^2+57x-2970}$. What is $\alpha+\beta$? | 137 | Level 5 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are given the equation:\n\n\\[\n\\frac{x-\\alpha}{x+\\beta} = \\frac{x^2-80x+1551}{x^2+57x-2970}\n\\]\n\nWe need to find constants α and β such that this holds for all x (presumably for all x where denominators are nonzero). Then compute α+β.\n\nThe suggested method: Su... |
354 | Working 22 hours in the second week of June, Xenia was able to earn $\$$47.60 more than during the first week of June when she worked 15 hours. If her hourly wage was constant, how many dollars did she earn during the first two weeks of June? Express your answer to the nearest hundredth. | 251.60 | Level 4 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are given: \"Working 22 hours in the second week of June, Xenia was able to earn $47.60 more than during the first week of June when she worked 15 hours. If her hourly wage was constant, how many dollars did she earn during the first two weeks of June? Express your answ... |
354 | Working 22 hours in the second week of June, Xenia was able to earn $\$$47.60 more than during the first week of June when she worked 15 hours. If her hourly wage was constant, how many dollars did she earn during the first two weeks of June? Express your answer to the nearest hundredth. | 251.60 | Level 4 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are given: \"Working 22 hours in the second week of June, Xenia was able to earn $47.60 more than during the first week of June when she worked 15 hours. If her hourly wage was constant, how many dollars did she earn during the first two weeks of June? Express your answ... |
354 | Working 22 hours in the second week of June, Xenia was able to earn $\$$47.60 more than during the first week of June when she worked 15 hours. If her hourly wage was constant, how many dollars did she earn during the first two weeks of June? Express your answer to the nearest hundredth. | 251.60 | Level 4 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are given: \"Working 22 hours in the second week of June, Xenia was able to earn $47.60 more than during the first week of June when she worked 15 hours. If her hourly wage was constant, how many dollars did she earn during the first two weeks of June? Express your answ... |
354 | Working 22 hours in the second week of June, Xenia was able to earn $\$$47.60 more than during the first week of June when she worked 15 hours. If her hourly wage was constant, how many dollars did she earn during the first two weeks of June? Express your answer to the nearest hundredth. | 251.60 | Level 4 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are asked: \"Working 22 hours in the second week of June, Xenia was able to earn $47.60 more than during the first week of June when she worked 15 hours. If her hourly wage was constant, how many dollars did she earn during the first two weeks of June? Express your answ... |
355 | A portion of the graph of $y = f(x)$ is shown in red below, where $f(x)$ is a quadratic function. The distance between grid lines is $1$ unit.
What is the sum of all distinct numbers $x$ such that $f(f(f(x)))=-3$ ?
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14... | -8 | Level 5 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are given a graph of a quadratic function y = f(x) shown in red. The grid lines are 1 unit apart. The graph appears to be a parabola opening upward? Let's analyze the graph from the Asymptote code. The code defines f(x) = x^2/4 + x - 3. So f(x) = (1/4)x^2 + x - 3. We ca... |
355 | A portion of the graph of $y = f(x)$ is shown in red below, where $f(x)$ is a quadratic function. The distance between grid lines is $1$ unit.
What is the sum of all distinct numbers $x$ such that $f(f(f(x)))=-3$ ?
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14... | -8 | Level 5 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are given a graph of a quadratic function y = f(x). The graph is shown in red, and we are told the distance between grid lines is 1 unit. The graph appears to be a parabola opening upward? Actually, from the Asymptote code: real f(real x) {return x^2/4+x-3;} So f(x) = (... |
355 | A portion of the graph of $y = f(x)$ is shown in red below, where $f(x)$ is a quadratic function. The distance between grid lines is $1$ unit.
What is the sum of all distinct numbers $x$ such that $f(f(f(x)))=-3$ ?
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14... | -8 | Level 5 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are given: \"A portion of the graph of y = f(x) is shown in red below, where f(x) is a quadratic function. The distance between grid lines is 1 unit.\" There is an Asymptote code that defines f(x) = x^2/4 + x - 3. So the quadratic is f(x) = (1/4)x^2 + x - 3. Let's confi... |
356 | Suppose the graph of $y=f(x)$ includes the points $(1,5),$ $(2,3),$ and $(3,1)$.
Based only on this information, there are two points that must be on the graph of $y=f(f(x))$. If we call those points $(a,b)$ and $(c,d),$ what is $ab+cd$? | 17 | Level 5 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are given that the graph of y = f(x) includes points (1,5), (2,3), (3,1). So f(1)=5, f(2)=3, f(3)=1. We need to find points that must be on the graph of y = f(f(x)). That is, for some x, compute f(f(x)) = f(f(x)). We need to find points (a,b) such that b = f(f(a)). Sinc... |
356 | Suppose the graph of $y=f(x)$ includes the points $(1,5),$ $(2,3),$ and $(3,1)$.
Based only on this information, there are two points that must be on the graph of $y=f(f(x))$. If we call those points $(a,b)$ and $(c,d),$ what is $ab+cd$? | 17 | Level 5 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are given: graph of y=f(x) includes points (1,5), (2,3), (3,1). So f(1)=5, f(2)=3, f(3)=1. We need points that must be on graph of y=f(f(x)). That is, points (x, f(f(x))) such that x is in domain and we can compute f(f(x)) from given information. We need to find all x f... |
356 | Suppose the graph of $y=f(x)$ includes the points $(1,5),$ $(2,3),$ and $(3,1)$.
Based only on this information, there are two points that must be on the graph of $y=f(f(x))$. If we call those points $(a,b)$ and $(c,d),$ what is $ab+cd$? | 17 | Level 5 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are given: \"Suppose the graph of $y=f(x)$ includes the points $(1,5),$ $(2,3),$ and $(3,1)$.\n\nBased only on this information, there are two points that must be on the graph of $y=f(f(x))$. If we call those points $(a,b)$ and $(c,d),$ what is $ab+cd$?\"\n\nInterpretat... |
356 | Suppose the graph of $y=f(x)$ includes the points $(1,5),$ $(2,3),$ and $(3,1)$.
Based only on this information, there are two points that must be on the graph of $y=f(f(x))$. If we call those points $(a,b)$ and $(c,d),$ what is $ab+cd$? | 17 | Level 5 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We need to find points that must be on the graph of y = f(f(x)) given that the graph of y = f(x) includes points (1,5), (2,3), (3,1). So we know f(1)=5, f(2)=3, f(3)=1. We need to compute f(f(x)) for x values where we can determine the output based on known f values. In pa... |
357 | If $x+\frac{1}{x}=6$, then what is the value of $x^{2}+\frac{1}{x^{2}}$? | 34 | Level 3 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are asked: If x + 1/x = 6, then find x^2 + 1/x^2. Method: expansion of square. So we can square both sides: (x + 1/x)^2 = 6^2 => x^2 + 2*x*(1/x) + (1/x)^2 = 36 => x^2 + 2 + 1/x^2 = 36 => x^2 + 1/x^2 = 36 - 2 = 34. So answer is 34. Put in \\boxed{34}.\nWe are given that ... |
357 | If $x+\frac{1}{x}=6$, then what is the value of $x^{2}+\frac{1}{x^{2}}$? | 34 | Level 3 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are given: x + 1/x = 6. Need to find x^2 + 1/x^2. There is a standard method: square both sides. (x + 1/x)^2 = x^2 + 2 + 1/x^2 = 36, so x^2 + 1/x^2 = 34. That's simple. But the user says: \"Solve the problem using the method: quadratic formula.\" So they want to solve f... |
357 | If $x+\frac{1}{x}=6$, then what is the value of $x^{2}+\frac{1}{x^{2}}$? | 34 | Level 3 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We need to find x^2 + 1/x^2 given x + 1/x = 6. Using Vieta's formulas: That is not directly about polynomials? But we can consider quadratic equation: if x + 1/x = 6, then multiply by x: x^2 + 1 = 6x => x^2 - 6x + 1 = 0. This is a quadratic with roots x and 1/x? Actually, ... |
357 | If $x+\frac{1}{x}=6$, then what is the value of $x^{2}+\frac{1}{x^{2}}$? | 34 | Level 3 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are asked: \"If $x+\\frac{1}{x}=6$, then what is the value of $x^{2}+\\frac{1}{x^{2}}$?\" and the user says \"Solve the problem using the method: Newton's sums.\" So we need to apply Newton's sums. Newton's sums are typically used for sums of powers of roots of a polyno... |
358 | If $27^8=9^q$, what is $q$? | 12 | Level 2 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are given: 27^8 = 9^q. Need to find q. Use common base 3. 27 = 3^3, 9 = 3^2. So 27^8 = (3^3)^8 = 3^(3*8)=3^24. 9^q = (3^2)^q = 3^(2q). So 3^24 = 3^(2q) => 24 = 2q => q = 12. Answer: \\boxed{12}.\nThe equation \\(27^8 = 9^q\\) can be solved by expressing both sides with ... |
358 | If $27^8=9^q$, what is $q$? | 12 | Level 2 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are given: 27^8 = 9^q. Need to find q using logarithms (any base). So take log of both sides. Use natural log or common log. Let's use natural log: ln(27^8) = ln(9^q). Then 8 ln 27 = q ln 9. So q = 8 ln 27 / ln 9. Simplify: ln 27 = ln(3^3)=3 ln3, ln 9 = ln(3^2)=2 ln3. S... |
358 | If $27^8=9^q$, what is $q$? | 12 | Level 2 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We need to solve 27^8 = 9^q for q. Using root extraction method: perhaps taking nth roots or expressing both sides as powers of same base. 27=3^3, 9=3^2. So (3^3)^8 = 3^{24} and 9^q = (3^2)^q = 3^{2q}. So 3^{24}=3^{2q} => 24=2q => q=12. So answer is 12. But the instruction... |
358 | If $27^8=9^q$, what is $q$? | 12 | Level 2 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are asked: \"If $27^8=9^q$, what is $q$?\" And we need to solve using the method: one base as power of other. So we need to express both sides with the same base. Note 27 = 3^3, 9 = 3^2. So 27^8 = (3^3)^8 = 3^{24}. And 9^q = (3^2)^q = 3^{2q}. So equation: 3^{24} = 3^{2q... |
359 | A portion of the graph of $f(x)=ax^3+bx^2+cx+d$ is shown below.
What is the value of $8a-4b+2c-d$?
[asy]
import graph; size(7cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-3.25,xmax=4.25,ymin=-9.25,ymax=4.25;
pen cqcqcq=rgb(0.75,0.75,0.75);
/*grid*/ pen gs=linewidt... | 3 | Level 5 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are given a graph of a cubic function f(x)=ax^3+bx^2+cx+d. We need to find the value of 8a - 4b + 2c - d.\n\nWe can see from the graph that the cubic has x-intercepts at x=0, x=1, x=2? Actually the Asymptote code: real f1(real x){return x*(x-1)*(x-2)/8;} So they used th... |
359 | A portion of the graph of $f(x)=ax^3+bx^2+cx+d$ is shown below.
What is the value of $8a-4b+2c-d$?
[asy]
import graph; size(7cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-3.25,xmax=4.25,ymin=-9.25,ymax=4.25;
pen cqcqcq=rgb(0.75,0.75,0.75);
/*grid*/ pen gs=linewidt... | 3 | Level 5 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are given a graph of a cubic function f(x)=ax^3+bx^2+cx+d. The graph is shown. We need to find the value of 8a-4b+2c-d. Notice that 8a-4b+2c-d = a(8) + b(-4) + c(2) + d(-1). This looks like evaluating f(2) maybe? f(2)=a*8+b*4+c*2+d. That's not it. f(-2)=a*(-8)+b*4+c*(-2... |
360 | The value $2^8 -1$ is divisible by 3 prime numbers. What is the sum of the three prime numbers? | 25 | Level 2 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We need to find the three prime divisors of 2^8 - 1, then sum them. 2^8=256, 2^8-1=255. Factor 255. 255=3*85=3*5*17. So primes: 3,5,17. Sum=3+5+17=25. So answer is 25.\nThe value \\(2^8 - 1 = 255\\) factors as \\(3 \\times 5 \\times 17\\). The sum of these three primes is ... |
360 | The value $2^8 -1$ is divisible by 3 prime numbers. What is the sum of the three prime numbers? | 25 | Level 2 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are asked: \"The value $2^8 -1$ is divisible by 3 prime numbers. What is the sum of the three prime numbers?\" And we are instructed to solve using difference of squares factorization.\n\nSo compute 2^8 - 1 = 256 - 1 = 255. Factor 255 using difference of squares? Actual... |
360 | The value $2^8 -1$ is divisible by 3 prime numbers. What is the sum of the three prime numbers? | 25 | Level 2 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We need to find the three prime divisors of 2^8 - 1 = 255. Actually 2^8-1=255. 255 factors as 3*5*17. So the three primes are 3,5,17 sum = 25. But the user wants us to use cyclotomic polynomial factorization method. So we explain using cyclotomic polynomials: x^n - 1 = ∏_{... |
360 | The value $2^8 -1$ is divisible by 3 prime numbers. What is the sum of the three prime numbers? | 25 | Level 2 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We need to find the three prime numbers that divide 2^8 -1. 2^8 -1 = 256-1=255. Factor 255: 255 = 3*5*17. So the three prime numbers are 3,5,17. Their sum is 3+5+17=25. But the instruction says: \"Solve the problem using the method: Modular arithmetic divisibility check.\"... |
361 | Line $a$ is parallel to line $y=2x+4$ and passes through the point $(2,5)$. What is the y-intercept of line $a$? | 1 | Level 3 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are asked: \"Line a is parallel to line y=2x+4 and passes through the point (2,5). What is the y-intercept of line a?\" We need to solve using the method: substitution into slope-intercept.\n\nInterpretation: We know line a is parallel to y=2x+4, so it has the same slop... |
361 | Line $a$ is parallel to line $y=2x+4$ and passes through the point $(2,5)$. What is the y-intercept of line $a$? | 1 | Level 3 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We need to solve: Line a is parallel to line y=2x+4 and passes through (2,5). Find y-intercept. Use point-slope form. Parallel lines have same slope. So slope m = 2. Point-slope form: y - y1 = m(x - x1). Here (x1,y1)=(2,5). So y-5=2(x-2). Simplify to slope-intercept: y-5=2... |
361 | Line $a$ is parallel to line $y=2x+4$ and passes through the point $(2,5)$. What is the y-intercept of line $a$? | 1 | Level 3 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We need to find the y-intercept of line a, which is parallel to y=2x+4 and passes through (2,5). Since parallel lines have the same slope, slope of line a is 2. Using point-slope form: y - 5 = 2(x - 2) => y = 2x -4 +5 = 2x +1. So y-intercept is 1. The method: slope formula... |
361 | Line $a$ is parallel to line $y=2x+4$ and passes through the point $(2,5)$. What is the y-intercept of line $a$? | 1 | Level 3 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are asked: \"Line a is parallel to line y=2x+4 and passes through the point (2,5). What is the y-intercept of line a?\" And we need to use the \"vertical shift method\". I'm not sure what exactly the vertical shift method is, but it likely refers to using the fact that ... |
362 | Find the sum: $1+2+3+4+\dots +48+49$ | 1225 | Level 2 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We need to find sum 1+2+...+49. Gaussian pairing: pair first and last: 1+49=50, 2+48=50, ... up to 24+26=50, and then the middle number 25 remains. Number of pairs: from 1 to 49, there are 49 numbers. Excluding 25, we have 48 numbers, so 24 pairs each sum 50. So total = 24... |
362 | Find the sum: $1+2+3+4+\dots +48+49$ | 1225 | Level 2 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are asked: \"Find the sum: 1+2+3+4+...+48+49\". Solve using arithmetic sum formula. The sum of first n natural numbers is n(n+1)/2. Here n=49. So sum = 49*50/2 = 49*25 = 1225. So answer: \\boxed{1225}.\nThe sum of the first 49 natural numbers is given by the formula \\(... |
362 | Find the sum: $1+2+3+4+\dots +48+49$ | 1225 | Level 2 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We need to find sum of integers from 1 to 49. The method is combinatorial counting. Usually, sum of first n natural numbers is n(n+1)/2. Here n=49, so sum = 49*50/2 = 1225. But we need to derive using combinatorial counting. Possibly using the identity that the number of w... |
362 | Find the sum: $1+2+3+4+\dots +48+49$ | 1225 | Level 2 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We need to find sum of 1 to 49 using geometric proof. The user wants \"Geometric proof.\" Possibly a visual proof: sum of first n numbers is n(n+1)/2. For n=49, sum = 49*50/2 = 1225. But they want geometric proof. Possibly a diagram: arrange numbers as dots in a triangle a... |
365 | Let $k, a_2, a_3$ and $k, b_2, b_3$ be nonconstant geometric sequences with different common ratios. If \[a_3-b_3=2(a_2-b_2),\]then what is the sum of the common ratios of the two sequences? | 2 | Level 5 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are given: \"Let $k, a_2, a_3$ and $k, b_2, b_3$ be nonconstant geometric sequences with different common ratios. If $a_3-b_3=2(a_2-b_2)$, then what is the sum of the common ratios of the two sequences?\" \n\nInterpretation: There are two geometric sequences. The first ... |
365 | Let $k, a_2, a_3$ and $k, b_2, b_3$ be nonconstant geometric sequences with different common ratios. If \[a_3-b_3=2(a_2-b_2),\]then what is the sum of the common ratios of the two sequences? | 2 | Level 5 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are given: \"Let $k, a_2, a_3$ and $k, b_2, b_3$ be nonconstant geometric sequences with different common ratios. If \\[a_3-b_3=2(a_2-b_2),\\] then what is the sum of the common ratios of the two sequences?\" \n\nInterpretation: We have two geometric sequences: first: ... |
365 | Let $k, a_2, a_3$ and $k, b_2, b_3$ be nonconstant geometric sequences with different common ratios. If \[a_3-b_3=2(a_2-b_2),\]then what is the sum of the common ratios of the two sequences? | 2 | Level 5 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We need to solve: Let k, a2, a3 and k, b2, b3 be nonconstant geometric sequences with different common ratios. So we have two geometric sequences that share the same first term k. They are nonconstant, so common ratios are not 1? Actually nonconstant means not constant seq... |
367 | Tiffany is constructing a fence around a rectangular tennis court. She must use exactly 300 feet of fencing. The fence must enclose all four sides of the court. Regulation states that the length of the fence enclosure must be at least 80 feet and the width must be at least 40 feet. Tiffany wants the area enclosed by th... | 5600 | Level 5 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We need to find the maximum area of a rectangle given a fixed perimeter of 300 feet, with constraints: length >= 80, width >= 40. Let length = L, width = W. Perimeter: 2L + 2W = 300 => L + W = 150 => W = 150 - L. Area A = L * W = L*(150 - L) = -L^2 + 150L. This is a quadra... |
367 | Tiffany is constructing a fence around a rectangular tennis court. She must use exactly 300 feet of fencing. The fence must enclose all four sides of the court. Regulation states that the length of the fence enclosure must be at least 80 feet and the width must be at least 40 feet. Tiffany wants the area enclosed by th... | 5600 | Level 5 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We need to solve: \"Tiffany is constructing a fence around a rectangular tennis court. She must use exactly 300 feet of fencing. The fence must enclose all four sides of the court. Regulation states that the length of the fence enclosure must be at least 80 feet and the wi... |
367 | Tiffany is constructing a fence around a rectangular tennis court. She must use exactly 300 feet of fencing. The fence must enclose all four sides of the court. Regulation states that the length of the fence enclosure must be at least 80 feet and the width must be at least 40 feet. Tiffany wants the area enclosed by th... | 5600 | Level 5 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We need to maximize area of a rectangle given perimeter fixed at 300 ft, with constraints: length >=80, width >=40. The rectangle has length L and width W. Perimeter: 2L+2W=300 => L+W=150 => W=150-L. Area A = L*W = L(150-L) = 150L - L^2. This is a quadratic opening downwar... |
367 | Tiffany is constructing a fence around a rectangular tennis court. She must use exactly 300 feet of fencing. The fence must enclose all four sides of the court. Regulation states that the length of the fence enclosure must be at least 80 feet and the width must be at least 40 feet. Tiffany wants the area enclosed by th... | 5600 | Level 5 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are asked: \"Tiffany is constructing a fence around a rectangular tennis court. She must use exactly 300 feet of fencing. The fence must enclose all four sides of the court. Regulation states that the length of the fence enclosure must be at least 80 feet and the width ... |
368 | The terms $140, a, \frac{45}{28}$ are the first, second and third terms, respectively, of a geometric sequence. If $a$ is positive, what is the value of $a$? | 15 | Level 4 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are given: first term = 140, second term = a, third term = 45/28. They form a geometric sequence. So the ratio between consecutive terms is constant: a/140 = (45/28)/a. So we have a/140 = (45/28)/a. Cross multiply: a^2 = 140 * (45/28). Simplify: 140/28 = 5, because 28*5... |
368 | The terms $140, a, \frac{45}{28}$ are the first, second and third terms, respectively, of a geometric sequence. If $a$ is positive, what is the value of $a$? | 15 | Level 4 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are asked: \"The terms $140, a, \\frac{45}{28}$ are the first, second and third terms, respectively, of a geometric sequence. If $a$ is positive, what is the value of $a$?\" Solve using geometric mean method.\n\nIn a geometric sequence, the middle term (second term) is ... |
368 | The terms $140, a, \frac{45}{28}$ are the first, second and third terms, respectively, of a geometric sequence. If $a$ is positive, what is the value of $a$? | 15 | Level 4 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are asked: \"The terms $140, a, \\frac{45}{28}$ are the first, second and third terms, respectively, of a geometric sequence. If $a$ is positive, what is the value of $a$?\" We need to solve using the \"parameter r method\". That likely means: Let the common ratio be r.... |
368 | The terms $140, a, \frac{45}{28}$ are the first, second and third terms, respectively, of a geometric sequence. If $a$ is positive, what is the value of $a$? | 15 | Level 4 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are given: \"The terms $140, a, \\frac{45}{28}$ are the first, second and third terms, respectively, of a geometric sequence. If $a$ is positive, what is the value of $a$?\" Solve using logarithm method.\n\nIn a geometric sequence, the ratio between consecutive terms is... |
369 | Let $a_1, a_2, a_3,\dots$ be an increasing arithmetic sequence of integers. If $a_4a_5 = 13$, what is $a_3a_6$? | -275 | Level 5 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We need to solve: \"Let $a_1, a_2, a_3,\\dots$ be an increasing arithmetic sequence of integers. If $a_4a_5 = 13$, what is $a_3a_6$?\" We need to use factor pair enumeration method.\n\nInterpretation: an increasing arithmetic sequence of integers: so a_n = a_1 + (n-1)d, wi... |
369 | Let $a_1, a_2, a_3,\dots$ be an increasing arithmetic sequence of integers. If $a_4a_5 = 13$, what is $a_3a_6$? | -275 | Level 5 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are given: \"Let $a_1, a_2, a_3,\\dots$ be an increasing arithmetic sequence of integers. If $a_4a_5 = 13$, what is $a_3a_6$?\" Solve using method: Algebraic expression using d.\n\nInterpretation: We have an arithmetic sequence: a1, a2, a3, ... with common difference d ... |
369 | Let $a_1, a_2, a_3,\dots$ be an increasing arithmetic sequence of integers. If $a_4a_5 = 13$, what is $a_3a_6$? | -275 | Level 5 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are given: \"Let $a_1, a_2, a_3,\\dots$ be an increasing arithmetic sequence of integers. If $a_4a_5 = 13$, what is $a_3a_6$?\" We need to solve using the method: \"Discriminant perfect square.\" So we need to find the product a3 * a6.\n\nInterpretation: We have an arit... |
369 | Let $a_1, a_2, a_3,\dots$ be an increasing arithmetic sequence of integers. If $a_4a_5 = 13$, what is $a_3a_6$? | -275 | Level 5 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are given that a1, a2, a3, ... is an increasing arithmetic sequence of integers. So the terms are of the form a_n = a + (n-1)d, where a is the first term and d is the common difference, with d > 0 since increasing. Also all terms are integers, so a and d are integers (s... |
370 | What is the $x$-intercept of the line perpendicular to the line defined by $3x-2y = 6$ and whose $y$-intercept is 2? | 3 | Level 4 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are asked: \"What is the x-intercept of the line perpendicular to the line defined by 3x-2y = 6 and whose y-intercept is 2?\" We need to use slope-intercept method.\n\nFirst, find slope of given line. Equation: 3x - 2y = 6. Solve for y: -2y = -3x + 6 => y = (3/2)x - 3. ... |
370 | What is the $x$-intercept of the line perpendicular to the line defined by $3x-2y = 6$ and whose $y$-intercept is 2? | 3 | Level 4 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We need to find the x-intercept of a line that is perpendicular to the line 3x - 2y = 6 and has y-intercept 2. Use intercept form method.\n\nFirst, find slope of given line. 3x - 2y = 6 => -2y = -3x + 6 => y = (3/2)x - 3. So slope m1 = 3/2. Perpendicular slope m2 = -2/3 (n... |
370 | What is the $x$-intercept of the line perpendicular to the line defined by $3x-2y = 6$ and whose $y$-intercept is 2? | 3 | Level 4 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are asked: \"What is the $x$-intercept of the line perpendicular to the line defined by $3x-2y = 6$ and whose $y$-intercept is 2?\" The user says: \"Solve the problem using the method: normal form method.\" So we need to use the normal form method. But normal form metho... |
370 | What is the $x$-intercept of the line perpendicular to the line defined by $3x-2y = 6$ and whose $y$-intercept is 2? | 3 | Level 4 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are asked: \"What is the x-intercept of the line perpendicular to the line defined by 3x-2y=6 and whose y-intercept is 2?\" Solve using the parametric vector method. So we need to find the x-intercept of a line L2 that is perpendicular to line L1: 3x-2y=6, and L2 has y-... |
371 | Simplify $(3-2i)-(5-2i)$. | -2 | Level 2 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are asked to simplify (3-2i) - (5-2i). The method is component-wise subtraction. So we subtract the real parts and the imaginary parts separately.\n\nReal part: 3 - 5 = -2.\nImaginary part: (-2i) - (-2i) = -2i + 2i = 0i = 0.\nSo the result is -2 + 0i, which is simply -2... |
371 | Simplify $(3-2i)-(5-2i)$. | -2 | Level 2 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We need to simplify (3-2i)-(5-2i). The user wants us to use the \"Additive inverse method\". That likely means we treat subtraction as adding the additive inverse. So (3-2i) - (5-2i) = (3-2i) + (-(5-2i)) = (3-2i) + (-5+2i). Then combine real parts: 3 + (-5) = -2, and imagi... |
371 | Simplify $(3-2i)-(5-2i)$. | -2 | Level 2 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are asked to simplify (3-2i) - (5-2i). The method specified is \"Geometric vector subtraction.\" This likely means we treat complex numbers as vectors in the plane, subtract them component-wise. So we have (3-2i) - (5-2i) = (3-5) + (-2i - (-2i))? Wait careful: subtracti... |
373 | Given that the polynomial $x^2-5x+t$ has only positive integer roots, find the average of all distinct possible values of $t$. | 5 | Level 5 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are asked: \"Given that the polynomial x^2 - 5x + t has only positive integer roots, find the average of all distinct possible values of t.\" Use Vieta's enumeration.\n\nInterpretation: Polynomial is x^2 - 5x + t. Its roots are r and s (maybe equal). Then by Vieta: r + ... |
373 | Given that the polynomial $x^2-5x+t$ has only positive integer roots, find the average of all distinct possible values of $t$. | 5 | Level 5 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are asked: \"Given that the polynomial x^2 - 5x + t has only positive integer roots, find the average of all distinct possible values of t.\" And we need to solve using the method: Discriminant perfect square.\n\nInterpretation: The quadratic x^2 - 5x + t has two roots ... |
373 | Given that the polynomial $x^2-5x+t$ has only positive integer roots, find the average of all distinct possible values of $t$. | 5 | Level 5 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are asked: \"Given that the polynomial x^2 - 5x + t has only positive integer roots, find the average of all distinct possible values of t.\" We need to solve using \"Axis symmetry method.\" Hmm, what is axis symmetry method? Possibly related to sum of roots and product... |
373 | Given that the polynomial $x^2-5x+t$ has only positive integer roots, find the average of all distinct possible values of $t$. | 5 | Level 5 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We need to solve: Given that the polynomial x^2 - 5x + t has only positive integer roots, find the average of all distinct possible values of t. Use brute force t-testing.\n\nInterpretation: The quadratic is x^2 -5x + t = 0. Let its roots be r and s. Then r+s = 5, rs = t. ... |
374 | Define the operation $\S$ as follows: $a\,\S\, b=3a+5b$. What is the value of $7\,\S\,2$? | 31 | Level 1 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are asked: \"Define the operation S as follows: a S b = 3a + 5b. What is the value of 7 S 2?\" So we just substitute a=7, b=2 into the expression: 3*7 + 5*2 = 21 + 10 = 31. So answer is 31. Put in \\boxed{31}.\nThe operation is defined as \\(a \\S b = 3a + 5b\\). Substi... |
374 | Define the operation $\S$ as follows: $a\,\S\, b=3a+5b$. What is the value of $7\,\S\,2$? | 31 | Level 1 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are asked: \"Define the operation § as follows: a § b = 3a + 5b. What is the value of 7 § 2?\" So we need to compute 7 § 2 = 3*7 + 5*2 = 21 + 10 = 31. The problem says \"Solve the problem using the method: linearity separation.\" That might be just a hint or a specific ... |
374 | Define the operation $\S$ as follows: $a\,\S\, b=3a+5b$. What is the value of $7\,\S\,2$? | 31 | Level 1 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are given operation: a § b = 3a + 5b. Need value of 7 § 2. Simply substitute: 3*7 + 5*2 = 21 + 10 = 31. So answer is 31. But the user says \"using the method: dot product evaluation.\" That is odd because this is just a linear combination, not dot product. Possibly they... |
376 | If $a @ b$ is defined as $a @ b$ = $3a - 3b$, what is the value of $3 @ 5$? | -6 | Level 2 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are asked: \"If a @ b is defined as a @ b = 3a - 3b, what is the value of 3 @ 5?\" So we need to compute 3 @ 5 = 3*3 - 3*5 = 9 - 15 = -6. So answer is -6. Put inside \\boxed{-6}.\nThe operation \\(a @ b\\) is defined as \\(3a - 3b\\). Substituting \\(a = 3\\) and \\(b =... |
376 | If $a @ b$ is defined as $a @ b$ = $3a - 3b$, what is the value of $3 @ 5$? | -6 | Level 2 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are given a definition: a @ b = 3a - 3b. We need to compute 3 @ 5. Simply substitute a=3, b=5: 3*3 - 3*5 = 9 - 15 = -6. So answer is -6. The problem mentions \"Antisymmetry\" but that might be a red herring or a method to think about properties. But we just compute dire... |
376 | If $a @ b$ is defined as $a @ b$ = $3a - 3b$, what is the value of $3 @ 5$? | -6 | Level 2 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are asked: \"If $a @ b$ is defined as $a @ b$ = $3a - 3b$, what is the value of $3 @ 5$?\" The instruction: \"Solve the problem using the method: Dot product interpretation.\" That seems odd because the operation is defined as a simple linear expression, not a dot produ... |
378 | Compute $139+27+23+11$. | 200 | Level 1 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are asked: \"Compute $139+27+23+11$.\" And we need to solve using the method: sequential addition. That likely means adding them one after another: 139 + 27 = 166, then 166 + 23 = 189, then 189 + 11 = 200. So answer is 200. So final answer: \\boxed{200}.\nThe sum is com... |
379 | Let \[f(x) =
\begin{cases}
2x + 9 &\text{if }x<-2, \\
5-2x&\text{if }x\ge -2.
\end{cases}
\]Find $f(3).$ | -1 | Level 2 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We need to find f(3). The function is piecewise: for x < -2, f(x)=2x+9; for x >= -2, f(x)=5-2x. Since 3 >= -2, we use the second piece: f(3)=5-2*3=5-6=-1. So answer is -1.\nThe function is defined as \\( f(x) = 5 - 2x \\) for \\( x \\ge -2 \\). Since \\( 3 \\ge -2 \\), we ... |
379 | Let \[f(x) =
\begin{cases}
2x + 9 &\text{if }x<-2, \\
5-2x&\text{if }x\ge -2.
\end{cases}
\]Find $f(3).$ | -1 | Level 2 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are asked: \"Let f(x) = ... Find f(3).\" The instruction: \"Solve the problem using the method: graphical evaluation.\" However, finding f(3) is just plugging in x=3 into the piecewise function. Since 3 >= -2, we use the second piece: f(3) = 5 - 2*3 = 5 - 6 = -1. So the... |
380 | The sum of the squares of two positive integers is 90. The product of the two integers is 27. What is the sum of the two integers? | 12 | Level 1 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are given: \"The sum of the squares of two positive integers is 90. The product of the two integers is 27. What is the sum of the two integers?\" The method suggested is algebraic identity (a+b)^2. So we want to find a+b. We know a^2+b^2 = 90, and ab = 27. Then (a+b)^2 ... |
380 | The sum of the squares of two positive integers is 90. The product of the two integers is 27. What is the sum of the two integers? | 12 | Level 1 | Algebra | [
{
"role": "system",
"content": "Solve the following math problem efficiently and clearly. The last line of your response should be of the following format: 'Therefore, the final answer is: $\\boxed{{ANSWER}}$ where ANSWER is just the final number or expression that solves the problem. Think step by step bef... | [
{
"role": "assistant",
"content": "We are asked: \"The sum of the squares of two positive integers is 90. The product of the two integers is 27. What is the sum of the two integers?\" And we need to use \"sum-difference substitution\" method. Probably means let sum = s, difference = d, or something. Let the... |
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