Datasets:

justus27

/

prime-rl-data-adapted

like 0

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Find the matrix of the operator $\\widehat{A}$ in the basis $\\mathbf{e}_{1}^{\\prime}, \\mathbf{e}_{2}^{\\prime}, \\mathbf{e}_{3}^{\\prime}$, where\n\n$$\n\\begin{aligned}\n& \\mathbf{e}_{1}^{\\prime}=\\mathbf{e}_{1}+\\mathbf{e}_{2}+2 \\mathbf{e}_{3}, \\\\\n& \\mathbf{e}_{2}^{\\prime}=2 \\mathbf{e}_{1}-\\mathbf{e}_{2} \\\\\n& \\mathbf{e}_{3}^{\\prime}=-\\mathbf{e}_{1}+\\mathbf{e}_{2}+\\mathbf{e}_{3},\n\\end{aligned}\n$$\n\nif in the basis $\\mathbf{e}_{1}, \\mathbf{e}_{2}, \\mathbf{e}_{3}$ its matrix is given by\n\n$$\nA_{\\mathbf{e}}=\\left(\\begin{array}{rrr}\n2 & 0 & -1 \\\\\n0 & 1 & -2 \\\\\n-1 & 2 & 0\n\\end{array}\\right)\n$$", "role": "user" } ]

{ "ground_truth": "\\begin{pmatrix}\n -7 & 6 & -8 \\\\\n 11 & -9 & 12 \\\\\n 15 & -16 & 19\n \\end{pmatrix}", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Express $\\frac{21}{40}$ as a terminating decimal.", "role": "user" } ]

{ "ground_truth": "0.525", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "What is the measure, in degrees, of one interior angle of a regular octagon?", "role": "user" } ]

{ "ground_truth": "135", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Find the units digit of the following within the indicated number base: $63_9 + 74_9$", "role": "user" } ]

{ "ground_truth": "7", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Find the equation of the circle symmetric to the circle $(x-2)^2 + (y+1)^2 = 4$ with respect to the line $y=x+1$.", "role": "user" } ]

{ "ground_truth": "(x + 2)^2 + (y - 3)^2 = 4", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Determine all possible values of $b$ if the zeros of the function $g(x) = x^2 - bx + 3b$ are integers.", "role": "user" } ]

{ "ground_truth": "3", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Given an arithmetic sequence $\\{a_n\\}$, the sum of the first $n$ terms is denoted as $S_n$. If $S_4 \\geqslant 10$ and $S_5 \\leqslant 15$, then the maximum value of $a_4$ is ______.", "role": "user" } ]

{ "ground_truth": "4", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Evaluate $\\left|\\frac{8}{7} + 3i\\right|$.", "role": "user" } ]

{ "ground_truth": "\\frac{\\sqrt{505}}{7}", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Given the universal set $U=\\{1,2,3,4,5,6,7\\}$, and the set $M=\\{x\\in\\mathbb{Z}|x^2-6x+5\\leq0\\}$, then the complement of $M$ in $U$, denoted as $\\complement_{U}M=$.", "role": "user" } ]

{ "ground_truth": "\\{6,7\\}", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "On a beautiful Sunday spring morning, the father of a family went for a walk with his sons.\n\n- Have you noticed, - he said to them, - that the age of the oldest of you is equal to the sum of the ages of your two other brothers?\n- Yes. And we also noticed - they answered in unison - that the product of our ages and your age equals the sum of the cube of the number of your sons, multiplied by a thousand, and ten times the square of this number.\n\nCan you determine from this conversation the age of the father at the time of the birth of the second son?", "role": "user" } ]

{ "ground_truth": "34", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "If $(1 + i)^2 = a + bi$ ($a, b \\in \\mathbb{R}$), calculate the value of $a + b$.", "role": "user" } ]

{ "ground_truth": "2", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "\n100 identical metal spheres with a radius of \\( R = 1 \\) mm are connected by equal conductive segments into a chain, where each segment of the connecting wire has a length of \\( l = 50 \\) cm, which is much greater than the radius of the sphere. Then, the resulting structure was placed in a uniform electric field with a known intensity \\( E = 1000 \\text{ V/m} \\). The spheres are arranged in a line parallel to the intensity vector. What are the magnitudes of the charges induced on the end spheres in the chain?", "role": "user" } ]

{ "ground_truth": "2.75 \\times 10^{-9} \\text{ C}", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "In the modified function machine, the input is 12. Determine the output using the following instructions:\n1. Multiply the input by 3.\n2. Compare the result with 38.\n - If the result is less than or equal to 38, multiply by 2.\n - If the result is greater than 38, subtract 10.\n3. What is the output?", "role": "user" } ]

{ "ground_truth": "72", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, with $a=1$, $c= \\sqrt {2}$, and $\\cos C= \\frac {3}{4}$. \n$(1)$ Find the value of $\\sin A$; \n$(2)$ Find the area of $\\triangle ABC$.", "role": "user" } ]

{ "ground_truth": "\\frac { \\sqrt {7}}{4}", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Given $x^{m}y^{n} \\div \\frac{1}{4}x^{3}y=4x^{2}$, determine the values of $m$ and $n$ that satisfy the condition.", "role": "user" } ]

{ "ground_truth": "5,1", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Solve for $x$: $\\frac{4x + 2}{5x - 5} = \\frac{3}{4}$.", "role": "user" } ]

{ "ground_truth": "-23", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Find the coordinates of either focus for the hyperbola given by the equation \\[2x^2 - y^2 + 8x + 4y - 28 = 0.\\] Enter your answer as an ordered pair. Enter only one of the foci, not both.", "role": "user" } ]

{ "ground_truth": "(-2+4\\sqrt{3}, 2)", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "What is the value of the expression $(25 + 15)^2 - (25^2 + 15^2)$?", "role": "user" } ]

{ "ground_truth": "750", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "If the complex number $z$ satisfies $|z-2i|=1$, calculate the maximum value of $|z|$.", "role": "user" } ]

{ "ground_truth": "3", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "How many ordered pairs of positive integers $(M,N)$ satisfy the equation $\\frac{M}{4} = \\frac{12}{N}$", "role": "user" } ]

{ "ground_truth": "10", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "A ray of light emitted from point A($-3$, $3$) strikes the x-axis and is reflected. The line of the reflected ray is tangent to the circle C: $x^2 + y^2 - 4x - 4y + 7 = 0$. Determine:\n(1) The equations of the ray of light $\\ell$ and the line of the reflected ray;\n(2) The distance traveled by the ray of light from point A to the tangent point.", "role": "user" } ]

{ "ground_truth": "7", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Ms. Lin asked her online fitness group to report how many days last month they jogged for at least 20 minutes. The results are summarized in the following bar graph, where the heights of the bars indicate the number of participants:\n- 5 participants for 1 day\n- 4 participants for 3 days\n- 10 participants for 5 days\n- 7 participants for 10 days\n- 3 participants for 15 days\n- 1 participant for 20 days\n\nWhat was the mean number of days of jogging last month, rounded to the nearest hundredth, reported by the participants in Ms. Lin's fitness group?\nA) 3.40\nB) 4.50\nC) 5.56\nD) 6.73\nE) 7.25", "role": "user" } ]

{ "ground_truth": "(D) 6.73", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Find the value of the expression \\(\\frac{1}{a}+\\frac{1}{b}+\\frac{1}{c}\\), given that \\(a, b, c\\) are three distinct real numbers satisfying the conditions:\n\n\\[ a^{3} - 2022a + 1011 = 0, \\]\n\\[ b^{3} - 2022b + 1011 = 0, \\]\n\\[ c^{3} - 2022c + 1011 = 0. \\]", "role": "user" } ]

{ "ground_truth": "2", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Let the function $f(x)=ax^{3}-3x+1$ where $x \\in \\mathbb{R}$. If for any $x \\in [-1,1]$, $f(x) \\geqslant 0$ holds, then the value of the real number $a$ is \\_\\_\\_\\_\\_\\_.", "role": "user" } ]

{ "ground_truth": "4", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "What is the expression for the repayment amount $a_{n}$ in dollars for the $n$th month, given that Zhang Hua borrowed a principal of $480,000 at a monthly interest rate of $0.4\\%$, and agreed to repay the loan in equal principal installments over $20$ years?", "role": "user" } ]

{ "ground_truth": "3928 - 8n", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Given the function $f(x) = x^2 - bx + 3$, and $f(0) = f(4)$. \n(1) Find the zeros of the function $y = f(x)$, and write the set of $x$ values for which $f(x) < 0$; \n(2) Find the maximum and minimum values of the function $y = f(x)$ in the interval $[0, 3]$.", "role": "user" } ]

{ "ground_truth": "3", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Solve the inequality\n\\[\\frac{x^2 - 49}{x + 7} < 0.\\]", "role": "user" } ]

{ "ground_truth": "(-\\infty, -7) \\cup (-7, 7)", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "$ABCDEFGH$ is a cube. Find $\\sin \\angle HAG$.", "role": "user" } ]

{ "ground_truth": "\\sin \\angle HAG = \\frac{\\sqrt{5}}{3}", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "In how many different ways can 4 men and 5 women be organized into two groups of three people and one group of three people, ensuring each group includes at least one man and one woman? Note that all groups of three people are indistinguishable.", "role": "user" } ]

{ "ground_truth": "180", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Corners are sliced off a cube with each edge of length 2 so that the six faces each become regular octagons. What is the total volume of the removed tetrahedra?\nA) $\\frac{48 - 32\\sqrt{2}}{3}$\nB) $\\frac{64 - 48\\sqrt{2}}{3}$\nC) $\\frac{96 - 64\\sqrt{2}}{3}$\nD) $\\frac{32(3 - 2\\sqrt{2})}{3}$", "role": "user" } ]

{ "ground_truth": "\\frac{32(3 - 2\\sqrt{2})}{3}", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Determine the smallest prime $p$ such that $2018!$ is divisible by $p^{3}$ , but not divisible by $p^{4}$ .", "role": "user" } ]

{ "ground_truth": "509", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Consider a rectangle $ABCD$ with $BC = 2 \\cdot AB$ . Let $\\omega$ be the circle that touches the sides $AB$ , $BC$ , and $AD$ . A tangent drawn from point $C$ to the circle $\\omega$ intersects the segment $AD$ at point $K$ . Determine the ratio $\\frac{AK}{KD}$ .\n\n*Proposed by Giorgi Arabidze, Georgia*", "role": "user" } ]

{ "ground_truth": "\\frac{1}{2}", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Let $A$ be a non-empty subset of the set of integers given by $A=\\{1,2,3,4,5\\}$. For $k \\in A$, if $k-1 \\notin A$ and $k+1 \\notin A$, then $k$ is called an \"isolated element\" of $A$. Calculate the number of subsets of $A$ that have exactly one isolated element.", "role": "user" } ]

{ "ground_truth": "13", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "The hyperbola given by the equation \n\\[\n\\frac{y^2}{16} - \\frac{x^2}{9} = 1\n\\]\nhas asymptotes $y = \\pm nx$, where $n$ is positive. Find $n$.", "role": "user" } ]

{ "ground_truth": "\\frac{4}{3}", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Convert the cylindrical coordinate $P\\left( 16, \\frac{\\pi}{3},5 \\right)$ into Cartesian coordinates.", "role": "user" } ]

{ "ground_truth": "(8,8\\sqrt{3},5)", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Given the function\n$$\nf(x) = \\log_a \\left(ax^2 - x + \\frac{1}{2}\\right)\n$$\nis always positive in the interval $[1,2]$, find the range of the real number $a$.", "role": "user" } ]

{ "ground_truth": "\\left(\\frac{1}{2}, \\frac{5}{8}\\right) \\cup \\left(\\frac{3}{2}, +\\infty\\right)", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Lucas distributed $y$ square stickers evenly among 12 children such that every child received the greatest possible number of whole stickers, and a few stickers were left over. What is the greatest possible number of stickers that could have been left over?", "role": "user" } ]

{ "ground_truth": "11", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Calculate: $(\\sqrt{3})^{0}+2^{-1}+\\sqrt{2}\\cos 45^{\\circ}-|-\\frac{1}{2}|$.", "role": "user" } ]

{ "ground_truth": "2", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Given the universal set U={1, 2, 3, 4, 5, 6, 7, 8, 9}, set A={3, 4, 5, 6}, and set B={5, 6, 7, 8, 9}, find $(A^c \\cap B)$, where $A^c$ is the complement of set A with respect to the universal set U.", "role": "user" } ]

{ "ground_truth": "{7, 8, 9}", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Compute $\\dbinom{10}{3}\\times \\dbinom{8}{3}$.", "role": "user" } ]

{ "ground_truth": "6720", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Let $h(x) = x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1.$ Calculate the remainder when the polynomial $h(x^{10})$ is divided by the polynomial $h(x)$.", "role": "user" } ]

{ "ground_truth": "8", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "In a right-angled triangle $PQR$ with angle $PQR = 90^{\\circ}$, suppose $\\cos Q = \\frac{5}{13}$. Given $PR = 13$, calculate the length of $PQ$.", "role": "user" } ]

{ "ground_truth": "12", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "The coordinates of the point $M(2,-3)$ with respect to the origin are ______.", "role": "user" } ]

{ "ground_truth": "(2, -3)", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Two tangents are drawn from an exterior point $A$ to a circle, touching the circle at points $B$ and $C$, respectively. A third tangent intersects segment $AB$ at $P$ and segment $AC$ at $R$, and touches the circle at $Q$. If $AB=24$, and $P$ divides $AB$ in a 3:1 ratio while $R$ divides $AC$ in a 1:3 ratio, find the perimeter of $\\triangle APR$.\nA) 40\nB) 42\nC) 48\nD) 45\nE) 50", "role": "user" } ]

{ "ground_truth": "\\textbf{(C)}\\ 48", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "What is the sum of the distinct prime factors of 924?", "role": "user" } ]

{ "ground_truth": "23", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "$\\tan 15^{\\circ}=$____.", "role": "user" } ]

{ "ground_truth": "2 - \\sqrt{3}", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "$\\tan 70^\\circ \\cdot \\cos 10^\\circ (\\tan 20^\\circ - 1)$ equals \n   A．1               B．2              C．－1           D．－2", "role": "user" } ]

{ "ground_truth": "\\text{C. } -1", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Given vectors $\\overrightarrow {a}=(6,2)$, $\\overrightarrow {b}=(-4, \\frac {1}{2})$, and a line $l$ passing through point A $(3,-1)$ whose direction vector is perpendicular to the vector $\\overrightarrow {a}+2 \\overrightarrow {b}$, then the general equation of line $l$ is __________.", "role": "user" } ]

{ "ground_truth": "2x-3y-9=0", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Let set $U = \\{ x \\mid x \\text{ is a positive integer less than } 9\\}$, set $A = \\{1, 2, 3\\}$, and set $B = \\{3, 4, 5, 6\\}$. Find the intersection of the complements of $A$ and $B$ with respect to $U$.", "role": "user" } ]

{ "ground_truth": "\\{7, 8\\}", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Point $H$ is taken on the extension of side $AD$ of parallelogram $ABCD$. $BH$ intersects diagonal $AC$ at $I$ and side $DC$ at $J$. If $IJ = 40$ and $JH = 30$, then find the length of $BI$.", "role": "user" } ]

{ "ground_truth": "20", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Let \\[g(x) = \\left\\{\n\\begin{array}{cl}\nx^2 + 2 &\\text{ if } x > 6, \\\\\n3x - 4 &\\text{ if } -6 < x \\leq 6, \\\\\n4 &\\text{ if } x \\leq -6.\n\\end{array}\n\\right.\\] Determine $g(-7) + g(0) + g(8)$.", "role": "user" } ]

{ "ground_truth": "66", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Given that $x = \\frac{4}{9}$ is a solution to the equation $81x^2 - 145x + 64 = 0$, what is the other value of $x$ that solves the equation? Express your answer as a common fraction.", "role": "user" } ]

{ "ground_truth": "-\\frac{16}{9}", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Given the geometric sequence $\\{2^{a_{n}}\\}$ with a common ratio of 2, and $a_{4}+a_{3}^{2}=21$.\n(1) Find the general term formula for $\\{a_{n}\\}$;\n(2) If $a_{1}>0$, find the sum of the first $n$ terms, $S_{n}$, for the sequence $\\{ \\frac {1}{(2a_{n}-1)(2n-1)}\\}$.", "role": "user" } ]

{ "ground_truth": "\\frac {n}{2n+1}", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Evaluate the series\n\\[\n\\sum_{n=1}^\\infty \\frac{n^3 + 2n^2 - n + 1}{(n+3)!}.\n\\]", "role": "user" } ]

{ "ground_truth": "\\frac{1}{3}", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "The equation of an ellipse with foci on the x-axis is given by $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 (a > b > 0)$. A triangle is formed by connecting one endpoint of the minor axis and both foci. The inscribed circle of this triangle has a radius of $\\frac{b}{3}$. Determine the eccentricity of the ellipse.", "role": "user" } ]

{ "ground_truth": "\\frac{1}{2}", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "There are four points \\( A, B, C, D \\) on the surface of a sphere with radius \\( R \\), such that \\( AB = BC = CA = 3 \\). If the maximum volume of the tetrahedron \\( ABCD \\) is \\( \\frac{9 \\sqrt{3}}{4} \\), what is the surface area of the sphere?", "role": "user" } ]

{ "ground_truth": "16\\pi", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "In the sequence $\\{a_n\\}$, $a_1=1$, $S_{n+1}=4a_n+2$, find the value of $a_{2013}$.", "role": "user" } ]

{ "ground_truth": "3019\\times2^{2012}", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Elena earns $13.25 per hour working at a store. Calculate the amount Elena earns in 4 hours.", "role": "user" } ]

{ "ground_truth": "53.00", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Given twelve coins consisting of pennies, nickels, dimes, quarters, and half-dollars with a total value of $1.63, determine the number of dimes in this collection.", "role": "user" } ]

{ "ground_truth": "5", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "The slope angle of the line $x - y + 3 = 0$ is $\\tan^{-1}(1)$.", "role": "user" } ]

{ "ground_truth": "\\frac{\\pi}{4}", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "The operation \\(\\star\\) is defined by \\(a \\star b = ab^3 - b + 2\\). What is the value of \\(2 \\star 3\\)?", "role": "user" } ]

{ "ground_truth": "53", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "How many positive four-digit integers of the form $\\_\\_90$ are divisible by 45?", "role": "user" } ]

{ "ground_truth": "11", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Given that in \\(\\triangle DEF\\), \\(DE = EF = 31\\), and \\(DF = 46\\), calculate the area of \\(\\triangle DEF\\).", "role": "user" } ]

{ "ground_truth": "477", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "The roots of the equation \\( x^{2}+(m-2) x+5-m=0 \\) are both greater than 2. Find the range of values for \\( m \\).", "role": "user" } ]

{ "ground_truth": "(-5, -4]", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "A street has 25 houses on each side, totaling 50 houses. The house addresses on both sides of the street form arithmetic sequences. On the north side, the addresses begin at 5 and increase by 7 each time (5, 12, 19, ...), while on the south side they begin at 6 and increase by 8 (6, 14, 22, ...). A sign painter charges $\\$2$ per digit to paint house numbers. Calculate the total amount of money the painter will collect if he paints each house number once on these 50 houses.", "role": "user" } ]

{ "ground_truth": "234", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Let\n\\[\n\\mathbf{A} = \\begin{pmatrix} 0 & 0 & 1 \\\\ 1 & 0 & 0 \\\\ 0 & 1 & 0 \\end{pmatrix}, \\quad \\mathbf{B} = \\begin{pmatrix} 1 & 1 & 1 \\\\ 0 & 1 & -1 \\\\ 1 & 0 & 0 \\end{pmatrix}.\n\\]\nCompute \\(\\mathbf{A}^{99} \\mathbf{B}\\).", "role": "user" } ]

{ "ground_truth": "\\begin{pmatrix} 1 & 0 & 0 \\\\ 1 & 1 & 1 \\\\ 0 & 1 & -1 \\end{pmatrix}", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "If \\( 2 \\times 3^{8} \\) can be expressed as the sum of \\( k \\) consecutive positive integers, what is the maximum value of \\( k \\)?", "role": "user" } ]

{ "ground_truth": "108", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "The complex number $z$ satisfies $z + |z| = 3 + 12i$. What is $|z|^{2}$?", "role": "user" } ]

{ "ground_truth": "650.25", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Given the parametric equations of curve C: $$\\begin{cases} x=a\\cos\\alpha \\\\ y=b\\sin\\alpha \\end{cases}$$ (where $\\alpha$ is the parameter), the point M $(1, \\frac{\\sqrt{2}}{2})$ on curve C corresponds to the parameter $\\alpha= \\frac{\\pi}{4}$. Taking the origin O as the pole and the positive x-axis as the polar axis, we establish a polar coordinate system. The polar coordinates of point P are $(\\sqrt{2}, \\frac{\\pi}{2})$. A line $l$ passes through point P and intersects curve C at two distinct points A and B. (1) Find the standard equation of curve C; \n(2) Find the range of values for $|PA|\\cdot|PB|$.", "role": "user" } ]

{ "ground_truth": "[1,2]", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Let \\(x\\) and \\(y\\) be two distinct positive real numbers, and \\(n\\) a positive integer. We define sequences \\((P_n), (G_n)\\) as follows. First, \\(P_1\\) is \\( \\sqrt[n]{\\frac{x^n + y^n}{2}} \\) and \\( G_1 \\) is \\( \\sqrt{x \\cdot y} \\). Then for \\( n \\ge 2 \\), \\( P_n \\) and \\( G_n \\) are:\n\\[ P_{n+1} = \\sqrt[n]{\\frac{(\\sin P_n)^n + (\\sin G_n)^n}{2}} \\text{ and } G_{n+1} = \\sqrt{\\sin P_n \\cdot \\sin G_n} \\] respectively.\n\nDetermine if the sequences \\( (P_n) \\) and \\( (G_n) \\) converge and identify their limits if they exist.", "role": "user" } ]

{ "ground_truth": "0", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Given that license plates can have two types: Type A with two letters followed by four digits, and Type B with four letters followed by two digits, and they are issued in equal proportions, calculate the percentage increase in the number of possible license plates compared to the previous system consisting of three letters followed by three digits.", "role": "user" } ]

{ "ground_truth": "46.8", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "The measure of each exterior angle of a regular polygon is $45$ degrees. What is the sum of the measures of the interior angles, in degrees?", "role": "user" } ]

{ "ground_truth": "1080^\\circ", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Given the function $f\\left(x\\right)=3^{-x}-3^{x}-x$, if $f\\left(2a+3\\right)+f\\left(3-a\\right) \\gt 0$, then the range of real number $a$ is ______.", "role": "user" } ]

{ "ground_truth": "(-\\infty, -6)", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Given that $\\sin \\alpha = -\\frac{5}{13}$ and $\\alpha$ is an angle in the third quadrant, find the value of $\\tan \\alpha$.", "role": "user" } ]

{ "ground_truth": "\\frac{5}{12}", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "If $a > 0$ and $a \\neq 1$, then the graph of the function $y = \\log_a(x - 1) + 2$ always passes through a fixed point.", "role": "user" } ]

{ "ground_truth": "(2, 2)", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Find the number of ordered pairs \\((a,b)\\) of complex numbers such that\n\\[ a^4b^6 = a^9b^3 = 1. \\]", "role": "user" } ]

{ "ground_truth": "37", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "A school organizes a \"Blue Sky Project Expo Course\" for students from 6 grades to visit 6 museums, including Museum A. Each grade can choose one museum to visit. Determine the number of ways in which exactly two grades choose Museum A.", "role": "user" } ]

{ "ground_truth": "C_{6}^{2} \\times 5^{4}", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Given that the image of (x, y) under the mapping f is (x+y, xy), find the image of (3, 4) and the preimage of (1, -6).", "role": "user" } ]

{ "ground_truth": "(3, -2)", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "A circle with a radius of 5 cm is tangent to three sides of a rectangle. The area of the rectangle is three times the area of the circle. Find the length of the longer side of the rectangle, in centimeters. Express your answer in terms of $\\pi$.", "role": "user" } ]

{ "ground_truth": "7.5\\pi", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Given that $f(x)$ is a function defined on $\\mathbb{R}$ that satisfies $f(x) = -f(-x)$, and when $x < 0$, $f(x) = x \\cdot 3 - 1 - x$, find $f(9)$ = \\_\\_\\_\\_\\_\\_.", "role": "user" } ]

{ "ground_truth": "19", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "What is the units digit of the sum of the squares of the first 2021 odd, positive integers?", "role": "user" } ]

{ "ground_truth": "1", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Given that $A$, $B$, and $C$ are all polynomials, where $A=x^{2}+xy+3y$ and $B=x^{2}-xy$.\n$(1)$ Find the value of $3A-B$;\n$(2)$ Lily made a mistake when calculating $A+\\frac{1}{3}C$ and obtained $A-\\frac{1}{3}C$, resulting in $2x^{2}+y$. Help Lily find the correct result.", "role": "user" } ]

{ "ground_truth": "2xy + 5y", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Given sets $A=\\{x|x^{2}-2x-3 \\lt 0\\}$ and $B=\\{x|2^{x-2} \\gt 1\\}$, find the expression for $A\\cap (\\complement _{R}B)$.", "role": "user" } ]

{ "ground_truth": "(-1,2]", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "A basement has a 30-foot by 40-foot rectangular floor. The basement is flooded with water to a depth of 24 inches. Four pumps are used to pump the water out of the basement. Each pump will pump 10 gallons of water per minute. If a cubic foot of water contains 7.5 gallons, how many minutes will it take to pump all of the water out of the basement using the four pumps?", "role": "user" } ]

{ "ground_truth": "450 \\text{ minutes}", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "The line $l_1$: $x+ay+6=0$ is parallel to the line $l_2$: $(a-2)x+3y+2a=0$. Find the values of $a$ for which the two lines are parallel.", "role": "user" } ]

{ "ground_truth": "3", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Stuart has drawn a pair of concentric circles, as shown, and starts drawing chords $\\overline{AB}$, $\\overline{BC}, \\ldots$ of the large circle, each tangent to the small one. If $m\\angle ABC=60^\\circ$, then how many segments will he draw before returning to his starting point at $A$? \n[asy]\nsize(100); defaultpen(linewidth(0.8));\nreal rad1 = 1/Sin(30);\ndraw(Circle(origin,1)); draw(Circle(origin,rad1));\npair A = rad1*dir(160), B = rad1*dir(160 + 120), C = rad1*dir(160 + 2*120), D = rad1*dir(160 + 300);\ndraw(A--B--C--D,EndArrow(size=5));\nlabel(\"$A$\",A,W); label(\"$B$\",B,N); label(\"$C$\",C,E);\n[/asy]", "role": "user" } ]

{ "ground_truth": "3", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Given an arithmetic-geometric sequence ${a_n}$ with the sum of the first $n$ terms denoted as $S_n$, if $S_6 : S_3 = 3$, find the value of $S_9 : S_6$.", "role": "user" } ]

{ "ground_truth": "\\frac{7}{3}", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Let $S_{1}, S_{2}, \\ldots, S_{10}$ be the first ten terms of an arithmetic progression (A.P.) of positive integers. If $S_{1} + S_{2} + \\ldots + S_{10} = 55$ and $\\left(S_{10} - S_{8}\\right) + \\left(S_{9} - S_{7}\\right) + \\ldots + \\left(S_{3} - S_{1}\\right) = d$, find $d$.", "role": "user" } ]

{ "ground_truth": "16", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Given the ellipse $E$: $\\frac{x^{2}}{a^{2}}+ \\frac{y^{2}}{b^{2}}=1(a > b > 0)$, its minor axis length is $2$, and the eccentricity is $\\frac{\\sqrt{6}}{3}$. The line $l$ passes through the point $(-1,0)$ and intersects the ellipse $E$ at points $A$ and $B$. $O$ is the coordinate origin.\n\n(1) Find the equation of the ellipse $E$;\n\n(2) Find the maximum area of $\\triangle OAB$.", "role": "user" } ]

{ "ground_truth": "\\frac{\\sqrt{6}}{3}", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Find the coefficient of $x^4$ when $4(x^2 - 2x^4 + x^3) + 2(x^4 + 3x^3 - 2x^2 + x^5) - 3(2 + x^2 - x^4 + 2x^3)$ is simplified.", "role": "user" } ]

{ "ground_truth": "-3", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Given that $f(x)$ is an even function defined on $\\mathbb{R}$, $f(x)$ is an increasing function on $[0,+\\infty)$, and $f(\\frac{1}{3})=0$, determine the solution set for the inequality $f(\\log_{\\frac{1}{8}}x) > 0$.", "role": "user" } ]

{ "ground_truth": "(0, \\frac{1}{2})\\cup(2,+\\infty)", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Given that the three lateral faces of a triangular pyramid are mutually perpendicular to each other, and their areas are $6$, $4$, and $3$ respectively, then its volume is equal to \\_\\_\\_\\_.", "role": "user" } ]

{ "ground_truth": "4", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Given $0 < A < \\frac{\\pi}{2}$, and $\\cos A= \\frac{3}{5}$, calculate the value of $\\sin 2A$.", "role": "user" } ]

{ "ground_truth": "\\frac{24}{25}", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "From 8 female students and 4 male students, 3 students are to be selected to form an extracurricular group. If stratified sampling by gender ratio is used, the number of different methods of selection is \\_\\_\\_\\_\\_\\_ . (Answer with a number)", "role": "user" } ]

{ "ground_truth": "112", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "A function $f(x)$ is defined as an odd function on $\\mathbb{R}$. When $x > 0$, $f(x)=x-2$.\n1. Find the analytic expression of the function $f(x)$.\n2. Solve the inequality $f(x) < 2$.", "role": "user" } ]

{ "ground_truth": "(-\\infty, 4)", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Roslyn has twelve boxes. Eight of the boxes contain pencils, five of the boxes contain pens, and three of the boxes contain both pens and pencils. How many boxes contain neither pens nor pencils?", "role": "user" } ]

{ "ground_truth": "2", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Carefully observe the following pattern: $2^2 - 2 = 2(2-1) = 2$; $2^3 - 2^2 = 2^2(2-1) = 2^2$; $2^4 - 2^3 = 2^3(2-1) = 2^3$... (Now you must have noticed a certain pattern, continue with the following questions; the result can be kept in exponential form) \n(1) $2^{100} - 2^{99} =$     \n(2) $2^n - 2^{n-1} =$     \n(3) Calculate: $2 - 2^2 - 2^3 - 2^4 - … - 2^{2013} + 2^{2014}$ (Don't forget to write the complete calculation process)", "role": "user" } ]

{ "ground_truth": "6", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Let $F_n$ be the Fibonacci sequence, defined as $F_0 = 0$, $F_1 = 1$, and for $n \\geq 0$, $F_{n+2} = F_{n+1} + F_n$. Compute the sum \\[ \\sum_{n=0}^\\infty \\frac{F_{2n+1}}{5^n}. \\]", "role": "user" } ]

{ "ground_truth": "\\frac{35}{3}", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "Given the fraction $\\frac{xy}{x+y}$, where $x$ and $y$ are positive numbers, determine the new value of the fraction when the values of $x$ and $y$ are both doubled.", "role": "user" } ]

{ "ground_truth": "\\frac{2xy}{x + y}", "style": "rule" }

{ "index": 0, "split": "dummy" }

[ { "content": "Solve the following math problem efficiently and clearly. Think carefully and step by step about your response and reason before providing a final response. Conclude your response with: \n\nTherefore, the final answer is: $\\boxed{answer}$. I hope it is correct.\n\nWhere [answer] is just the final number or expression that solves the problem.", "role": "system" }, { "content": "A factory produces a batch of components (unit: $cm)$, and the size $\\xi$ follows a normal distribution $N(\\mu, \\sigma ^{2})$. Given that $P(\\xi \\leq 14) = 0.1$ and $P(\\xi < 18) = 0.9$, find the value of $\\mu$.", "role": "user" } ]

{ "ground_truth": "16", "style": "rule" }

{ "index": 0, "split": "dummy" }