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[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
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"content": "During the softball season, Judy had $35$ hits. Among her hits were $1$ home run, $1$ tr... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "80\\%", "solution": "1. **Identify the total number of hits**: Judy had a total of 35 hits during the season.\n\n2. **Determine the number of hits that were not singles**: \n - Home runs: 1\n - Triples: 1\n - Doubles: 5\n - Total non-single hits = 1 (home run) + 1 (triple) + 5 (doub... | 83 | c2b369b9-093e-5969-a298-47dae2f3efba |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
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"tool_call_id": null
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{
"role": "user",
"content": "Quadrilateral $ABCD$ satisfies $\\angle ABC = \\angle ACD = 90^{\\circ}, AC=20,$ and $CD=... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "360", "solution": "1. **Assign Variables and Use Pythagorean Theorem in $\\triangle ABC$:**\n Let $AB = x$ and $BC = y$. Since $\\angle ABC = 90^\\circ$, by the Pythagorean theorem, we have:\n \\[\n x^2 + y^2 = AC^2 = 20^2 = 400.\n \\]\n\n2. **Calculate Area of $\\triangle ACD$:**\n... | 105 | 8daa180a-91f7-59c3-8c04-cca5edd5138d |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
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"role": "user",
"content": "Five positive consecutive integers starting with $a$ have average $b$. What is the averag... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "$a+4$", "solution": "1. **Define the sequence and calculate the average $b$:** \n The five consecutive integers starting with $a$ are $a, a+1, a+2, a+3, a+4$. The average of these integers, $b$, is calculated as follows:\n \\[\n b = \\frac{a + (a+1) + (a+2) + (a+3) + (a+4)}{5} = \\fr... | 62 | 65f06c47-4325-5df9-91de-7cfc6aae5daf |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
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"tool_call_id": null
},
{
"role": "user",
"content": "If $a$ and $b$ are digits for which\n$\\begin{array}{ccc}& 2 & a\\ \\times & b & 3\\ \\hl... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "7", "solution": "1. **Identify the multiplication setup**: The multiplication setup is given as:\n \\[\n \\begin{array}{ccc}\n & 2 & a\\\\ \n \\times & b & 3\\\\ \n \\hline \n & 6 & 9\\\\ \n 9 & 2\\\\ \n \\hline \n 9 & 8 & 9\n \\end{array}\n \\]\n This represents the... | 103 | ab82b110-b4ce-5300-bf38-7636239f7c0b |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "Four circles, no two of which are congruent, have centers at $A$, $B$, $C$, and $D$, and ... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "192", "solution": "1. **Understanding the Problem**: We are given four circles with centers at $A$, $B$, $C$, and $D$. Points $P$ and $Q$ lie on all four circles. The radius of circle $A$ is $\\frac{5}{8}$ times the radius of circle $B$, and similarly for circles $C$ and $D$. The distances ... | 168 | ab6e887a-ad7e-5838-b824-aa5b794f625a |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "For all non-zero numbers $x$ and $y$ such that $x = 1/y$, $\\left(x-\\frac{1}{x}\\right)\... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "x^2-y^2", "solution": "1. **Substitute $y$ in terms of $x$:** Given $x = \\frac{1}{y}$, we can rewrite $y$ as $y = \\frac{1}{x}$. \n\n2. **Substitute and simplify the expression:** We start by substituting $y = \\frac{1}{x}$ into the expression $\\left(x-\\frac{1}{x}\\right)\\left(y+\\frac{... | 76 | a40118ca-fc9e-5fcc-859a-017d1d31e27d |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "The numbers $-2, 4, 6, 9$ and $12$ are rearranged according to these rules:\n\n 1.... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "6.5", "solution": "1. **Identify the largest and smallest numbers**: The largest number in the set $\\{-2, 4, 6, 9, 12\\}$ is $12$, and the smallest is $-2$.\n\n2. **Apply rule 1**: The largest number, $12$, cannot be first but must be in one of the first three places. Thus, $12$ can be in ... | 126 | 9e938364-dab1-5f52-af76-2fc1b3ee416d |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "Quadrilateral $ABCD$ has $AB = BC = CD$, $m\\angle ABC = 70^\\circ$ and $m\\angle BCD = 1... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "85", "solution": "1. **Assign the unknown and draw necessary diagonals**: Let $\\angle BAD = x$. Draw diagonals $BD$ and $AC$. Let $I$ be the intersection of diagonals $BD$ and $AC$.\n\n2. **Analyze the isosceles triangles**: Since $AB = BC = CD$, triangles $\\triangle ABC$ and $\\triangle ... | 83 | e1a5b100-d3a5-587d-9537-3b5618d36572 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "In $\\triangle ABC$ with right angle at $C$, altitude $CH$ and median $CM$ trisect the r... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "4K", "solution": "1. **Draw the triangle and identify key components**: In $\\triangle ABC$, where $\\angle C = 90^\\circ$, draw altitude $CH$ from $C$ to hypotenuse $AB$ and median $CM$ from $C$ to the midpoint $M$ of $AB$. Since $CM$ trisects the right angle, $\\angle MCB = 30^\\circ$ and... | 82 | e96bfc6c-7a47-51c9-ade1-f38c1fe0a622 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "Sides $\\overline{AB}$ and $\\overline{AC}$ of equilateral triangle $ABC$ are tangent to ... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "\\frac{4}{3}-\\frac{4\\sqrt{3}\\pi}{27}", "solution": "1. **Identify the Geometry and Relationships**:\n - Let the radius of the circle be $r$, and its center be $O$.\n - Since $\\overline{AB}$ and $\\overline{AC}$ are tangent to the circle at points $B$ and $C$, respectively, $\\angle ... | 84 | 620242c8-29ac-5490-9b49-4de79d91ab4d |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
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{
"role": "user",
"content": "Every week Roger pays for a movie ticket and a soda out of his allowance. Last week, Roge... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "23\\%", "solution": "1. **Define Variables:**\n Let $m$ represent the cost of the movie ticket and $s$ represent the cost of the soda.\n\n2. **Set Up Equations:**\n According to the problem, we have:\n \\[ m = 0.20(A - s) \\]\n \\[ s = 0.05(A - m) \\]\n\n3. **Convert to Fractional F... | 135 | 3f4dd73b-2f88-5eb9-ae37-7858d5264b34 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "A closed box with a square base is to be wrapped with a square sheet of wrapping paper. T... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "2(w+h)^2", "solution": "1. **Understanding the Problem Setup**: We have a square sheet of wrapping paper and a box with a square base of side $w$ and height $h$. The box is placed such that its base vertices lie on the midlines of the wrapping paper. The wrapping paper is folded up to meet ... | 471 | 1d9b3fed-5c18-5dc2-8042-160fa719a730 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "The figure may be folded along the lines shown to form a number cube. Three number faces ... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "14", "solution": "To solve this problem, we need to understand how a cube is formed and how the numbers on the faces are arranged. In a standard die, opposite faces sum up to 7. This means:\n- If one face shows 1, the opposite face shows 6.\n- If one face shows 2, the opposite face shows 5.... | 74 | 54a9490c-74bd-59c6-856c-85d3a9e5c41a |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "Six ants simultaneously stand on the six vertices of a regular octahedron, with each ant ... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "\\frac{5}{256}", "solution": "\nWe approach this problem by counting the number of ways ants can do their desired migration, and then multiply this number by the probability that each case occurs.\n\nLet the octahedron be labeled as $ABCDEF$, with points $B, C, D, E$ being coplanar. Then th... | 94 | 33ea7171-b89b-560d-9dc1-c0627f117604 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "If $f(x)=3x+2$ for all real $x$, then the statement:\n\"$|f(x)+4|<a$ whenever $|x+2|<b$ a... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "$b \\le a/3$", "solution": "1. **Substitute $f(x)$ into the inequality**: Given $f(x) = 3x + 2$, we substitute this into the inequality $|f(x) + 4| < a$:\n \\[\n |3x + 2 + 4| = |3x + 6|.\n \\]\n Simplifying further, we have:\n \\[\n |3x + 6| = |3(x + 2)| = 3|x + 2|.\n \\]\n\n2... | 87 | f3a2ebb4-99ef-5327-bad8-03f7191ebc92 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
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{
"role": "user",
"content": "A carton contains milk that is $2\\%$ fat, an amount that is $40\\%$ less fat than the am... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "\\frac{10}{3}", "solution": "1. **Understanding the Problem:**\n The problem states that a carton of milk contains 2% fat, which is 40% less than the fat content in a carton of whole milk. We need to find the percentage of fat in the whole milk.\n\n2. **Setting Up the Equation:**\n Let ... | 79 | 977eeab7-8707-5819-bfc7-96bf7c60fd7d |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "A long piece of paper $5$ cm wide is made into a roll for cash registers by wrapping it $... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "36\\pi", "solution": "1. **Understanding the Problem:**\n The paper is wrapped around a cardboard tube, forming concentric circles. Each wrap increases the diameter of the roll by the thickness of the paper, which is $5$ cm. The total number of wraps is $600$, and the final diameter of th... | 122 | b4daf454-d484-59aa-a6a3-f641e48d84ac |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "For each positive integer $n$, let \n\n$a_n = \\frac{(n+9)!}{(n-1)!}$.\nLet $k$ denote th... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "9", "solution": "1. **Expression Simplification**:\n Given $a_n = \\frac{(n+9)!}{(n-1)!}$, we can simplify this as:\n \\[\n a_n = n(n+1)(n+2)\\cdots(n+9)\n \\]\n This is the product of 10 consecutive integers starting from $n$.\n\n2. **Factorization**:\n We can express $a_n$ in ... | 93 | 6a7474ce-55d7-57ba-9db4-ce57c545550b |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
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{
"role": "user",
"content": "Let $S$ be the set of lattice points in the coordinate plane, both of whose coordinates a... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "85", "solution": "1. **Understanding the Problem:**\n The problem asks us to find the possible values of the slope $m$ of a line such that exactly $300$ lattice points from the set $S$ (where $S$ consists of points $(x,y)$ with $1 \\leq x, y \\leq 30$) lie on or below the line $y = mx$. T... | 124 | 418528be-8b49-570d-86a7-a235119ce265 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
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{
"role": "user",
"content": "At Euclid High School, the number of students taking the AMC 10 was $60$ in 2002, $66$ in... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "2002 and 2003", "solution": "To find the largest percentage increase in the number of students taking the AMC 10 between consecutive years, we calculate the percentage increase for each interval:\n\n1. **From 2002 to 2003:**\n \\[\n \\frac{66 - 60}{60} \\times 100\\% = \\frac{6}{60} \\t... | 128 | 2dd1c5fc-2736-5cfb-bac6-dc38c828832f |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "Four identical squares and one rectangle are placed together to form one large square as ... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "3", "solution": "1. **Identify the dimensions of the small squares and the large square**: Let the side length of each small square be $x$. Since there are four small squares placed side by side along one side of the large square, the side length of the large square is $4x$.\n\n2. **Analyze... | 199 | ab1542fa-8573-505a-9c5f-00b52b8926d9 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "In a small pond there are eleven lily pads in a row labeled 0 through 10. A frog is sitti... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "\\frac{63}{146}", "solution": "Define \\( P(N) \\) as the probability that the frog survives starting from pad \\( N \\). We need to find \\( P(1) \\), the probability that the frog escapes starting from pad 1.\n\nThe recursive relationship given in the problem is:\n\\[ P(N) = \\frac{N}{10}... | 178 | dc883d88-732c-522d-b7ce-6ded7914e9be |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "The numbers $1,2,\\dots,9$ are randomly placed into the $9$ squares of a $3 \\times 3$ gr... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "\\frac{1}{14}", "solution": "\nTo solve this problem, we need to ensure that the sum of the numbers in each row and each column is odd. We can achieve this by having either three odd numbers or one odd and two even numbers in each row and column. Let's analyze the possible configurations an... | 96 | 8f6f92b3-c888-54c5-800c-af5e040ebd8d |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "Let $n$ be a positive integer greater than 4 such that the decimal representation of $n!$... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "8", "solution": "To solve this problem, we need to find the number of trailing zeros in $n!$ and $(2n)!$ and set up the condition that the number of zeros in $(2n)!$ is exactly three times the number of zeros in $n!$. The number of trailing zeros in a factorial can be calculated using the f... | 106 | de5bbc86-1153-5c0c-8e8f-80a5a912271c |
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{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
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"tool_call_id": null
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{
"role": "user",
"content": "Inside a right circular cone with base radius $5$ and height $12$ are three congruent sph... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "\\frac{90-40\\sqrt{3}}{11}", "solution": "1. **Identify the Geometry of the Problem:**\n The problem involves three congruent spheres inside a right circular cone, each sphere tangent to the others and to the base and side of the cone. The cone has a base radius of $5$ and a height of $12... | 85 | a3296124-a22f-5305-80b1-53694e62b30c |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "Square $EFGH$ is inside the square $ABCD$ so that each side of $EFGH$ can be extended to ... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "36", "solution": "1. **Understanding the Problem Setup:**\n - We have two squares, $ABCD$ and $EFGH$, where $EFGH$ is inside $ABCD$.\n - Each side of $EFGH$ can be extended to pass through a vertex of $ABCD$.\n - The side length of $ABCD$ is given as $\\sqrt{50}$.\n - The distance f... | 103 | 437dc599-330c-5320-9d5b-dc310a35fd7b |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "Points $A, B$ and $C$ on a circle of radius $r$ are situated so that $AB=AC, AB>r$, and t... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "\\frac{1}{2}\\csc{\\frac{1}{4}}", "solution": "1. **Identify the central angle corresponding to arc $BC$**: Given that the length of minor arc $BC$ is $r$, and knowing that the arc length of a circle is given by $r\\theta$ where $\\theta$ is the central angle in radians, we have:\n \\[\n ... | 87 | 7815b4dc-d39b-503a-a154-9c64df070442 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "A number of linked rings, each $1$ cm thick, are hanging on a peg. The top ring has an ou... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "173", "solution": "1. **Identify the sequence of inside diameters**: The outside diameter of the top ring is $20$ cm, and each subsequent ring has an outside diameter $1$ cm less than the ring above it. Since each ring is $1$ cm thick, the inside diameter of each ring is $2$ cm less than it... | 123 | 45e86ad8-bfe5-5785-8757-cb0a84d13eb1 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "How many squares whose sides are parallel to the axes and whose vertices have coordinates... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "50", "solution": "1. **Understanding the Problem:**\n We need to find the number of squares with integer coordinates for vertices, sides parallel to the axes, and lying entirely within the region bounded by:\n - $y = \\pi x$\n - $y = -0.1$\n - $x = 5.1$\n\n2. **Visualizing the Regio... | 83 | 2c4e0cda-a7d5-5fbf-a46f-9f93715e9f2c |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "Suppose $a$, $b$ and $c$ are positive integers with $a+b+c=2006$, and $a!b!c!=m\\cdot 10^... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "492", "solution": "1. **Understanding the Problem**: We need to find the smallest possible value of $n$ in the expression $a!b!c! = m \\cdot 10^n$, where $a+b+c = 2006$ and $m$ is not divisible by $10$. This means we need to minimize the number of factors of $10$ in $a!b!c!$, which is equiv... | 106 | 206f7239-e698-5983-9d6f-fb4b4bddb361 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "Liliane has $50\\%$ more soda than Jacqueline, and Alice has $25\\%$ more soda than Jacqu... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "$20\\%$", "solution": "Let's denote the amount of soda Jacqueline has as $J$. According to the problem, Liliane has $50\\%$ more soda than Jacqueline, and Alice has $25\\%$ more soda than Jacqueline. We can express the amounts of soda Liliane and Alice have in terms of $J$.\n\n1. **Calculat... | 76 | 218624f5-1792-5fbf-be5e-a8820baf9eca |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "The number $5^{867}$ is between $2^{2013}$ and $2^{2014}$. How many pairs of integers $(... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "279", "solution": "To solve this problem, we need to understand the relationship between the powers of $5$ and $2$ and how they are distributed between $5^n$ and $5^{n+1}$.\n\n1. **Understanding the relationship between $5^n$ and $2^m$:**\n We know that $5^{867}$ is between $2^{2013}$ and... | 111 | ba823412-35cf-5417-8ef4-38d7f177af99 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "What is the correct ordering of the three numbers $\\frac{5}{19}$, $\\frac{7}{21}$, and $... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "\\frac{5}{19} < \\frac{7}{21} < \\frac{9}{23}", "solution": "To find the correct ordering of the fractions $\\frac{5}{19}$, $\\frac{7}{21}$, and $\\frac{9}{23}$, we can compare each pair of fractions.\n\n1. **Simplify $\\frac{7}{21}$:**\n \\[\\frac{7}{21} = \\frac{1}{3}\\]\n\n2. **Compare... | 72 | f2d3a423-dbf6-5e59-8f21-897bf336b548 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "A triangle with vertices $(6, 5)$, $(8, -3)$, and $(9, 1)$ is reflected about the line $x... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "\\frac{32}{3}", "solution": "1. **Identify the vertices and their reflections**: \n - Let $A = (6, 5)$, $B = (8, -3)$, and $C = (9, 1)$. \n - Reflecting over the line $x=8$, we find:\n - $A' = (10, 5)$ since reflecting $(6, 5)$ over $x=8$ moves it 2 units to the left of $x=8$, so it... | 85 | 28ca87f8-7263-5674-9676-6a3179de252f |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "Let $B$ be a right rectangular prism (box) with edges lengths $1,$ $3,$ and $4$, together... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "19", "solution": "To solve the problem, we analyze the volume of $S(r)$ by decomposing it into different geometric regions as described in the problem statement. We then calculate the coefficients $a$, $b$, $c$, and $d$ in the volume formula $ar^3 + br^2 + cr + d$.\n\n1. **Region 1: The Rec... | 158 | 86ceb1bb-ce24-512d-a445-9ff5c349d4ff |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "Adam, Benin, Chiang, Deshawn, Esther, and Fiona have internet accounts. Some, but not al... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "170", "solution": "1. **Understanding the Problem:**\n - We have 6 individuals: Adam, Benin, Chiang, Deshawn, Esther, and Fiona.\n - Each individual has the same number of internet friends within this group.\n - The number of friends each person has ranges from 1 to 4, as no one can b... | 101 | 71581afe-d5f1-5f74-b4c3-0e21981371b5 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "The ratio of the areas of two concentric circles is $1: 3$. If the radius of the smaller ... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "0.73r", "solution": "1. **Identify the given information**: The ratio of the areas of two concentric circles is given as $1:3$. Let the radius of the smaller circle be $r$ and the radius of the larger circle be $R$.\n\n2. **Express the areas of the circles**: The area of a circle is given b... | 71 | ad8cc8d4-0750-55c4-8c56-cb35725cf2f0 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "How many ways are there to paint each of the integers $2, 3, \\cdots , 9$ either red, gre... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "432", "solution": "To solve this problem, we need to consider the constraints imposed by the requirement that each number must have a different color from each of its proper divisors. We start by identifying the divisors of each number from $2$ to $9$:\n\n- $2$: No proper divisors in the li... | 78 | e8aec90b-d151-5db2-9887-037a09723228 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "Let $D(n)$ denote the number of ways of writing the positive integer $n$ as a product\n\\... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "112", "solution": "To find $D(96)$, we need to consider all possible ways to write $96$ as a product of integers greater than $1$, where the order of factors matters. We start by noting the prime factorization of $96$:\n\\[ 96 = 2^5 \\cdot 3 \\]\n\nWe will consider different cases based on ... | 169 | 9aa8d86e-5fef-5f14-8a02-5546af37b976 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "The plane is tiled by congruent squares and congruent pentagons as indicated. The percent... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "56", "solution": "1. **Identify the basic unit of tiling**: The tiling pattern consists of a large square divided into 9 smaller squares, each with side length $a$. The large square thus has a side length of $3a$.\n\n2. **Calculate the area of the large square**: The area of the large squar... | 279 | 2050a723-bae9-58a0-bcf1-2fb348148d83 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "In the adjoining plane figure, sides $AF$ and $CD$ are parallel, as are sides $AB$ and $E... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "\\sqrt{3}", "solution": "1. **Identify the Shape and Symmetry**: The given figure can be visualized as composed of several triangles. Notably, the problem states that sides $AF$ and $CD$ are parallel, as are $AB$ and $EF$, and $BC$ and $ED$. Each side has a length of $1$, and the angles $\\... | 100 | 9f293098-1bfe-5bbb-bff2-7d2df188982c |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "The keystone arch is an ancient architectural feature. It is composed of congruent isosce... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "100", "solution": "1. **Understanding the Geometry of the Keystone Arch**:\n The keystone arch is composed of $9$ congruent isosceles trapezoids. These trapezoids are arranged such that their non-parallel sides (legs) meet at a common point when extended. The bottom sides of the end trape... | 235 | dc6a07fd-f125-5add-9e34-b427647e8b08 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "Laura added two three-digit positive integers. All six digits in these numbers are differ... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "4", "solution": "1. **Define the problem**: We need to find the smallest possible value for the sum of the digits of $S$, where $S$ is the sum of two three-digit numbers $a$ and $b$. Each digit among $a$ and $b$ is unique.\n\n2. **Set constraints on $a$ and $b$**: Since $a$ and $b$ are thre... | 76 | 4b6e64f1-ec8f-56de-9b96-13ad8e32bd62 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "Circles with centers $A$, $B$, and $C$ each have radius $r$, where $1 < r < 2$. The dista... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "1+\\sqrt{3(r^2-1)}", "solution": "1. **Setting up the problem**: We are given three circles with centers at $A$, $B$, and $C$, each having radius $r$ where $1 < r < 2$. The distance between each pair of centers is $2$. We need to find the length of the line segment $B'C'$, where $B'$ and $C... | 131 | 9101acb2-eec8-5277-a54a-9644259c5409 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "Each of the $12$ edges of a cube is labeled $0$ or $1$. Two labelings are considered diff... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "20", "solution": "To solve this problem, we need to ensure that each face of the cube has exactly two edges labeled with $1$ and two edges labeled with $0$. We will analyze the possible configurations that satisfy this condition.\n\n#### Step 1: Understanding the Cube's Faces and Edges\nA c... | 110 | f297f280-a5d7-57ac-9eb9-d0ccbc0bed39 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "Liquid $X$ does not mix with water. Unless obstructed, it spreads out on the surface of w... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "\\sqrt{\\frac{2160}{\\pi}}", "solution": "1. **Calculate the volume of liquid $X$:** \n The box has dimensions $6$ cm, $3$ cm, and $12$ cm. The volume $V$ of the box (and hence the volume of liquid $X$) is calculated by multiplying these dimensions:\n \\[\n V = 6 \\text{ cm} \\times ... | 119 | 798100ec-ef55-5535-8055-0b44fd929ffc |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "In triangle $ABC$, $AB=AC$ and $\\measuredangle A=80^\\circ$. If points $D, E$, and $F$ l... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "50^\\circ", "solution": "1. **Identify the properties of triangle $ABC$**: Given that $AB = AC$, triangle $ABC$ is isosceles. Also, $\\angle A = 80^\\circ$. Since the sum of angles in a triangle is $180^\\circ$, and $AB = AC$, the base angles $\\angle B$ and $\\angle C$ are equal. Thus, we ... | 99 | e7d7fb58-58f0-5a8d-8caa-1f9edd591bae |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "A regular octahedron has side length $1$. A plane parallel to two of its opposite faces c... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "14", "solution": "\nTo solve this problem, we need to understand the geometry of the octahedron and how the plane intersects it. An octahedron can be thought of as two square pyramids base-to-base. Each face of the octahedron is an equilateral triangle.\n\n#### Step 1: Understanding the Int... | 145 | 75b202f0-5cda-5ea6-b931-a6fc35a39210 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "Terri produces a sequence of positive integers by following three rules. She starts with... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "27", "solution": "To solve the problem, we first need to understand the sequence generated by the rules provided and then determine the $98^\\text{th}$ term of the sequence starting with 98.\n\n1. **Starting the sequence**: We begin with 98 and apply the rules:\n - **98** is even and grea... | 185 | f295533e-5783-51bc-aca2-8314705d61cd |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
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{
"role": "user",
"content": "Six regular hexagons surround a regular hexagon of side length $1$ as shown. What is the ... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "$3\\sqrt{3}$", "solution": "1. **Understanding the Configuration**: We are given a central regular hexagon surrounded by six regular hexagons, all of the same side length $1$. We need to find the area of $\\triangle{ABC}$, which is formed by connecting the centers of three adjacent outer he... | 61 | 6034acbd-a1e6-5b72-abf2-dffffc5da988 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "Bernardo and Silvia play the following game. An integer between $0$ and $999$ inclusive i... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "7", "solution": "\nTo solve this problem, we need to determine the smallest initial number $N$ such that Bernardo wins the game. We will analyze the sequence of operations and the conditions under which Bernardo wins.\n\n1. **Sequence of Operations**:\n - Bernardo receives $N$, doubles it... | 151 | eb450654-16df-5fbc-aa36-1fc52fff7afa |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "Convex quadrilateral $ABCD$ has $AB = 9$ and $CD = 12$. Diagonals $AC$ and $BD$ intersect... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "6", "solution": "1. **Given Information and Assumptions**:\n - Convex quadrilateral $ABCD$ with $AB = 9$, $CD = 12$, and diagonals $AC = 14$.\n - Diagonals $AC$ and $BD$ intersect at $E$.\n - $\\triangle AED$ and $\\triangle BEC$ have equal areas.\n\n2. **Using Equal Areas to Infer Si... | 98 | 8aa44f67-5751-5526-a419-e33425848260 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "Michael walks at the rate of $5$ feet per second on a long straight path. Trash pails are... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "5", "solution": "1. **Set up the problem**: Michael walks at $5$ feet per second and trash pails are located every $200$ feet. The garbage truck travels at $10$ feet per second and stops for $30$ seconds at each pail. Michael sees the truck leaving a pail $200$ feet ahead as he passes a pai... | 126 | c872d841-c9cd-50ae-839e-b25122ec3b66 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "Makarla attended two meetings during her $9$-hour work day. The first meeting took $45$ m... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "25", "solution": "1. **Convert the work day into minutes**: \n Makarla's work day is $9$ hours long. Since there are $60$ minutes in an hour, the total number of minutes in her work day is:\n \\[\n 9 \\times 60 = 540 \\text{ minutes}\n \\]\n\n2. **Calculate the total time spent in m... | 76 | 76d406e2-8d36-5a4e-ad27-cf8a8498d36a |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "A paint brush is swept along both diagonals of a square to produce the symmetric painted ... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "2\\sqrt{2}+2", "solution": "\nLet's denote the side length of the square as $s$ and the width of the brush as $w$. We are given that half the area of the square is painted, so the painted area is $\\frac{s^2}{2}$.\n\n#### Step 1: Analyze the geometry\nThe brush sweeps along both diagonals o... | 78 | e4076c9f-396a-5a82-bf8c-634a4bc2bfe6 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "A sequence of numbers is defined by $D_0=0,D_1=0,D_2=1$ and $D_n=D_{n-1}+D_{n-3}$ for $n\... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "(E,O,E)", "solution": "1. **Define the sequence and initial conditions**: The sequence $\\{D_n\\}$ is defined by $D_0 = 0$, $D_1 = 0$, $D_2 = 1$, and for $n \\geq 3$, $D_n = D_{n-1} + D_{n-3}$.\n\n2. **Compute initial terms**: We calculate the first few terms of the sequence to observe any ... | 133 | 4069bfa2-79f4-58c3-aef9-c6672f98e89e |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "For how many integers $n$ between $1$ and $50$, inclusive, is $\\frac{(n^2-1)!}{(n!)^n}$ ... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "34", "solution": "1. **Understanding the Expression**: We start by analyzing the expression \\[\\frac{(n^2-1)!}{(n!)^n}.\\] We need to determine for how many integers $n$ between $1$ and $50$, inclusive, this expression is an integer.\n\n2. **Rewriting the Expression**: We can rewrite the e... | 69 | 6cc18cfb-2f95-5df9-ad0e-7a7af546dcc5 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "If $3^{2x}+9=10\\left(3^{x}\\right)$, then the value of $(x^2+1)$ is",
"reasoning_con... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "1 or 5", "solution": "1. **Rewrite the equation**: Start by rewriting the given equation:\n \\[\n 3^{2x} + 9 = 10 \\cdot 3^x\n \\]\n Rearrange the terms to form a quadratic equation in terms of $3^x$:\n \\[\n 3^{2x} - 10 \\cdot 3^x + 9 = 0\n \\]\n\n2. **Substitute $u$ for $3^x... | 65 | f35359fe-8524-52fc-b19e-0dc37f34c3cd |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "At the beginning of a trip, the mileage odometer read $56,200$ miles. The driver filled t... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "26.9", "solution": "1. **Identify the total distance traveled**: \n The odometer readings at the start and end of the trip are given as $56,200$ miles and $57,060$ miles, respectively. Therefore, the total distance traveled during the trip is:\n \\[\n 57,060 - 56,200 = 860 \\text{ mil... | 152 | 88bccb8c-693f-51fd-baf2-06f6e7cd3b26 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "A square with side length 8 is cut in half, creating two congruent rectangles. What are t... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "4 \\times 8", "solution": "1. **Identify the original dimensions of the square**: The square has a side length of 8 units. Therefore, each side of the square is 8 units long.\n\n2. **Understand the effect of cutting the square in half**: When the square is cut in half, one dimension is halv... | 59 | 3205332a-5856-5e14-90de-61d71009ec9b |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "Professor Gamble buys a lottery ticket, which requires that he pick six different integer... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "\\frac{1}{4}", "solution": "1. **Understanding the Problem**: Professor Gamble needs to pick six different integers from 1 to 46 such that the sum of the base-ten logarithms of these numbers is an integer. This implies that the product of these numbers must be a power of 10, as the logarith... | 122 | 89fc4040-3f88-5833-886e-cdaf375bb874 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "Right triangle $ABC$ has leg lengths $AB=20$ and $BC=21$. Including $\\overline{AB}$ and ... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "12", "solution": "1. **Identify the Triangle and Hypotenuse**: \n Given a right triangle $ABC$ with legs $AB = 20$ and $BC = 21$, we first calculate the length of the hypotenuse $AC$ using the Pythagorean theorem:\n \\[\n AC = \\sqrt{AB^2 + BC^2} = \\sqrt{20^2 + 21^2} = \\sqrt{400 + 4... | 96 | 618694d8-c9d0-5dfe-b01c-4a47f4854872 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "Two distinct numbers $a$ and $b$ are chosen randomly from the set $\\{2, 2^2, 2^3, ..., 2... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "\\frac{31}{300}", "solution": "1. **Define the problem in terms of logarithms and powers of 2:**\n Let $a = 2^x$ and $b = 2^y$ where $x$ and $y$ are integers such that $1 \\leq x, y \\leq 25$. We are given that $\\log_a b$ is an integer, say $z$. Therefore, we have:\n \\[\n \\log_a b ... | 84 | 761f37e5-b456-56b0-8535-4ae4904a7e5d |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "If $n$ is a multiple of $4$, the sum $s=1+2i+3i^2+\\cdots+(n+1)i^n$, where $i=\\sqrt{-1}$... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "\\frac{1}{2}(n+2-ni)", "solution": "To solve the problem, we need to evaluate the sum $s = 1 + 2i + 3i^2 + \\cdots + (n+1)i^n$ where $i = \\sqrt{-1}$ and $n$ is a multiple of $4$. We start by analyzing the powers of $i$:\n\n1. **Powers of $i$:**\n - $i^0 = 1$\n - $i^1 = i$\n - $i^2 = ... | 77 | 7ea839a7-3cee-5265-b468-a819467f8ffe |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "Miki has a dozen oranges of the same size and a dozen pears of the same size. Miki uses h... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "40", "solution": "1. **Determine the amount of juice from pears and oranges:**\n - Miki extracts 8 ounces of pear juice from 3 pears. Therefore, the amount of pear juice per pear is:\n \\[\n \\frac{8 \\text{ ounces}}{3 \\text{ pears}} = \\frac{8}{3} \\text{ ounces per pear}\n ... | 108 | 4ac73f10-5f82-5351-a5e6-2f3968917704 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "When one ounce of water is added to a mixture of acid and water, the new mixture is $20\\... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "25\\%", "solution": "1. **Define Variables:**\n Let $a$ be the original number of ounces of acid and $w$ be the original number of ounces of water in the mixture.\n\n2. **Set Up Equations:**\n - After adding one ounce of water, the mixture becomes $20\\%$ acid. Therefore, the equation i... | 94 | 6f7e0968-fb70-5946-b495-43205f18e50c |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "Triangle $ABC$ has $AB=2 \\cdot AC$. Let $D$ and $E$ be on $\\overline{AB}$ and $\\overli... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "90^{\\circ}", "solution": "1. **Given Information and Setup**: \n - Triangle $ABC$ with $AB = 2 \\cdot AC$.\n - Points $D$ and $E$ are on $\\overline{AB}$ and $\\overline{BC}$ respectively, such that $\\angle BAE = \\angle ACD = x$.\n - $F$ is the intersection of segments $AE$ and $CD... | 118 | 06f8e996-aff1-5e05-a31c-f5587272bd17 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "A bug starts at one vertex of a cube and moves along the edges of the cube according to t... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "\\frac{1}{729}", "solution": "1. **Understanding the Problem**: A bug starts at one vertex of a cube and moves along the edges. At each vertex, it can choose any of the three connected edges with equal probability. We need to find the probability that after seven moves, the bug visits every... | 108 | d46d5917-3a98-5bb2-9adc-ea2345245230 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "Values for $A, B, C,$ and $D$ are to be selected from $\\{1, 2, 3, 4, 5, 6\\}$ without re... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "90", "solution": "1. **Understanding the Problem**: We need to find the number of ways to choose values for $A, B, C, D$ from the set $\\{1, 2, 3, 4, 5, 6\\}$ without replacement such that the curves $y = Ax^2 + B$ and $y = Cx^2 + D$ intersect.\n\n2. **Condition for Intersection**: The curv... | 177 | 50cbc4c9-e462-536a-91d2-2a345497daa1 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "For how many positive integers $n \\le 1000$ is$\\left\\lfloor \\dfrac{998}{n} \\right\\r... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "22", "solution": "We are tasked with finding how many positive integers \\( n \\leq 1000 \\) make the expression\n\\[\n\\left\\lfloor \\frac{998}{n} \\right\\rfloor + \\left\\lfloor \\frac{999}{n} \\right\\rfloor + \\left\\lfloor \\frac{1000}{n} \\right\\rfloor\n\\]\nnot divisible by 3. We ... | 115 | 0fac5f4d-ae60-52f2-bb84-36a4c8d51ead |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "Part of an \\(n\\)-pointed regular star is shown. It is a simple closed polygon in which ... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "36", "solution": "1. **Understanding the Star Polygon**: In the given problem, we have a regular star polygon with $n$ points. Each point of the star has two angles associated with it: one at $A_i$ and one at $B_i$ for $i = 1, 2, \\ldots, n$. All $A_i$ angles are congruent, and all $B_i$ an... | 134 | 6d62cd9d-7333-52de-b880-add53e8f8edf |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "Squares $ABCD$ and $EFGH$ are congruent, $AB=10$, and $G$ is the center of square $ABCD$.... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "175", "solution": "1. **Identify the given information**: We are given two congruent squares $ABCD$ and $EFGH$ with side length $AB = 10$. The point $G$ is the center of square $ABCD$.\n\n2. **Calculate the area of each square**: \n - The area of square $ABCD$ is $AB^2 = 10^2 = 100$.\n ... | 79 | b9fead02-337c-5111-b541-07841886a5d9 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "In the xy-plane, what is the length of the shortest path from $(0,0)$ to $(12,16)$ that d... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "10\\sqrt{3}+\\frac{5\\pi}{3}", "solution": "1. **Identify the Points and Circle**: \n - Let $A(0,0)$ and $D(12,16)$ be the start and end points, respectively.\n - The circle is centered at $O(6,8)$ with radius $5$, given by the equation $(x-6)^2 + (y-8)^2 = 25$.\n\n2. **Calculate Distan... | 86 | 09409fc4-e9fb-5bdb-93b0-4922a54cd788 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a ... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "\\frac{7}{36}", "solution": "1. **Understanding the Problem**: Jason rolls three fair six-sided dice and can choose to reroll any subset of them. He wins if the sum of the dice after rerolling is exactly 7. We need to find the probability that he chooses to reroll exactly two dice to optimi... | 119 | 0e68eb0f-ce99-532c-88c7-b42ca9b83b18 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "If the pattern in the diagram continues, what fraction of eighth triangle would be shaded... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "\\frac{7}{16}", "solution": "1. **Identify the pattern of shaded triangles**: Observing the given sequence of diagrams, we notice that the number of shaded triangles in each subsequent diagram follows the sequence of triangular numbers. The sequence of triangular numbers is defined by the f... | 527 | 7e05747b-ec55-504c-a13b-e2db91665fc6 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "Convex quadrilateral $ABCD$ has $AB = 9$ and $CD = 12$. Diagonals $AC$ and $BD$ intersect... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "6", "solution": "1. **Given Information and Assumptions**:\n - Convex quadrilateral $ABCD$ has $AB = 9$, $CD = 12$, $AC = 14$.\n - Diagonals $AC$ and $BD$ intersect at $E$.\n - $\\triangle AED$ and $\\triangle BEC$ have equal areas.\n\n2. **Using Equal Areas to Infer Ratios**:\n - S... | 98 | 0190ef34-ef56-5814-b870-b183741ea605 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "Three fourths of a pitcher is filled with pineapple juice. The pitcher is emptied by pour... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "15", "solution": "1. **Assume the total capacity of the pitcher**: Let's denote the total capacity of the pitcher as $C$ ounces.\n\n2. **Calculate the amount of juice in the pitcher**: Since the pitcher is filled to three-fourths of its capacity with pineapple juice, the amount of juice in ... | 76 | b10845e1-96d7-5ae2-b136-0bbc8d8d9c20 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "Marie does three equally time-consuming tasks in a row without taking breaks. She begins ... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "3:30 PM", "solution": "1. **Calculate the total time for the first two tasks**: Marie starts the first task at 1:00 PM and finishes the second task at 2:40 PM. The total time taken for these two tasks is calculated by subtracting the start time from the end time:\n \\[\n \\text{Total ti... | 79 | f4ffac57-ad01-5575-a421-bf0189eed176 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "How many squares whose sides are parallel to the axes and whose vertices have coordinates... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "50", "solution": "1. **Identify the region and lattice points**:\n The region is bounded by the lines $y = \\pi x$, $y = -0.1$, and $x = 5.1$. We consider only the lattice points (points with integer coordinates) that lie within this region. The relevant lattice points along the x-axis fr... | 83 | 5ffaea14-d265-5cbd-910e-baf2c4ee7e3e |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "Two boys start moving from the same point A on a circular track but in opposite direction... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "13", "solution": "1. **Define the problem setup**: Two boys start at the same point A on a circular track and move in opposite directions with speeds of 5 ft/s and 9 ft/s respectively. We need to find the number of times they meet excluding the start and finish.\n\n2. **Calculate relative s... | 100 | 82982459-f361-5568-b6d0-78e369d2e68d |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "A bakery owner turns on his doughnut machine at 8:30 AM. At 11:10 AM the machine has comp... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "4:30 PM", "solution": "1. **Calculate the time taken to complete one-third of the job**: \n The machine starts at $\\text{8:30 AM}$ and completes one-third of the job by $\\text{11:10 AM}$. The time duration for this is calculated as follows:\n \\[\n \\text{Time duration} = \\text{11... | 80 | cc8aee28-a169-5c85-92e8-1336b916a1b1 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "A permutation $(a_1,a_2,a_3,a_4,a_5)$ of $(1,2,3,4,5)$ is heavy-tailed if $a_1 + a_2 < a_... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "48", "solution": "We analyze the problem by considering the position of $a_3$ and how it affects the possible values of $a_1 + a_2$ and $a_4 + a_5$. We need to ensure $a_1 + a_2 < a_4 + a_5$ for the permutation to be heavy-tailed.\n\n1. **Case 1: $a_3 = 1$.**\n - The remaining numbers are... | 94 | 62bbda76-cafc-581d-b9e0-2c244bf2f177 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "Two distinct lines pass through the center of three concentric circles of radii 3, 2, and... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "\\frac{\\pi}{7}", "solution": "1. **Identify the areas of the circles**: \n - The largest circle has radius 3, so its area is $9\\pi$.\n - The middle circle has radius 2, so its area is $4\\pi$.\n - The smallest circle has radius 1, so its area is $\\pi$.\n\n2. **Set up the equations ... | 507 | 17d13ee6-f5cd-53cb-854e-481628759e66 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "A frog makes $3$ jumps, each exactly $1$ meter long. The directions of the jumps are chos... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "\\frac{1}{4}", "solution": "To solve this problem, we consider the frog's position in a coordinate system after each jump. Each jump is of length 1 meter, and the direction is chosen randomly. We need to calculate the probability that after 3 jumps, the frog is no more than 1 meter away fro... | 81 | 102a9de3-b9dc-50ac-9b12-55e2c9cac03e |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "In a small pond there are eleven lily pads in a row labeled 0 through 10. A frog is sitti... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "\\frac{63}{146}", "solution": "Define \\( P(N) \\) as the probability that the frog survives starting from pad \\( N \\). We need to find \\( P(1) \\).\n\nThe recursive relationship given in the problem is:\n\\[ P(N) = \\frac{N}{10} P(N-1) + \\left(1 - \\frac{N}{10}\\right) P(N+1) \\]\nfor ... | 178 | 32ba45de-e516-53a4-b7a5-e8f9e588ddc7 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "Five friends sat in a movie theater in a row containing $5$ seats, numbered $1$ to $5$ fr... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "2", "solution": "To solve this problem, we need to analyze the movements of each friend and determine the original seating arrangement based on the given movements.\n\n1. **Initial Setup**: There are 5 seats, and each friend occupies one seat. Ada leaves, creating one empty seat.\n\n2. **Mo... | 153 | c7be4bdd-d5d6-5739-923b-3ba437abddf6 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "Squares $ABCD$, $EFGH$, and $GHIJ$ are equal in area. Points $C$ and $D$ are the midpoint... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "\\frac{1}{3}", "solution": "1. **Assign Side Lengths**: Let the side length of each square $ABCD$, $EFGH$, and $GHIJ$ be $s = 1$ for simplicity.\n\n2. **Identify Key Points and Relationships**:\n - Points $C$ and $D$ are midpoints of sides $IH$ and $HE$ respectively, so $HC = CI = \\frac{... | 104 | 0293e784-c17d-51fa-8965-8929c70b4ff0 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "The number obtained from the last two nonzero digits of $90!$ is equal to $n$. What is $n... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "12", "solution": "1. **Count the number of factors of 10 in $90!$:**\n The number of factors of 10 in $90!$ is determined by the number of factors of 5, as there are more factors of 2 than 5. We calculate this using the formula for the number of factors of a prime $p$ in $n!$:\n \\[\n ... | 58 | ff5dcbbb-b0f2-576d-9101-b68f63b71013 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "At the 2013 Winnebago County Fair a vendor is offering a \"fair special\" on sandals. If ... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "30", "solution": "1. **Calculate the total cost without any discounts:** \n The regular price for one pair of sandals is $50. Therefore, the cost for three pairs of sandals at the regular price is:\n \\[\n 3 \\times 50 = 150 \\text{ dollars}\n \\]\n\n2. **Calculate the total cost w... | 125 | a599311d-9409-54d6-9d54-4b8a52ad1d17 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "Let the bisectors of the exterior angles at $B$ and $C$ of triangle $ABC$ meet at $D$. Th... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "\\frac{1}{2}(180-A)", "solution": "1. **Identify the exterior angle bisectors**: The bisectors of the exterior angles at $B$ and $C$ of triangle $ABC$ meet at point $D$. The exterior angle at $B$ is $180^\\circ - B$ and its bisector divides it into two equal parts, each being $90^\\circ - \... | 73 | fa0e8994-d20e-56ca-a248-5b2b9769f503 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "Gilda has a bag of marbles. She gives $20\\%$ of them to her friend Pedro. Then Gilda giv... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "54", "solution": "1. **Initial Amount of Marbles**: Let's denote the initial number of marbles Gilda has as $M$.\n\n2. **Marbles Given to Pedro**: Gilda gives $20\\%$ of her marbles to Pedro. Therefore, the number of marbles she gives to Pedro is $0.20M$. The number of marbles left with Gil... | 116 | 9404a550-8829-5b2a-89ed-bbdc41a6d22d |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "Walter has exactly one penny, one nickel, one dime and one quarter in his pocket. What p... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "41\\%", "solution": "1. **Identify the coins and their values**: Walter has the following coins:\n - One penny = $1$ cent\n - One nickel = $5$ cents\n - One dime = $10$ cents\n - One quarter = $25$ cents\n\n2. **Calculate the total value of the coins**:\n \\[\n 1 \\text{ cent} +... | 61 | 2855707d-c507-586e-ba0d-c44cc05c1208 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "The circular base of a hemisphere of radius $2$ rests on the base of a square pyramid of ... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "$3\\sqrt{2}$", "solution": "1. **Identify the Geometry and Setup**: Let $ABCDE$ be the pyramid with $ABCD$ as the square base. Let $O$ be the center of square $ABCD$ and $M$ be the midpoint of side $AB$. The hemisphere with radius $2$ is tangent to the triangular face $ABE$ at point $P$.\n\... | 80 | a7f37426-971c-5faf-b8a0-1bf644a70378 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "In $\\triangle ABC$, $AB = 86$, and $AC=97$. A circle with center $A$ and radius $AB$ in... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "61", "solution": "1. **Assign Variables:**\n Let $x$ represent $CX$, and let $y$ represent $BX$. Since the circle with center $A$ and radius $AB$ intersects $\\overline{BC}$ at points $B$ and $X$, we have $AB = AX = 86$.\n\n2. **Apply Stewart's Theorem:**\n Stewart's Theorem states that... | 105 | 42654484-a7c5-5f36-a17f-ab4a4343ef7c |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "A decorative window is made up of a rectangle with semicircles at either end. The ratio o... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "6:\\pi", "solution": "1. **Set up the proportion for $AD$ and $AB$:**\n Given the ratio of $AD$ to $AB$ is $3:2$, and $AB = 30$ inches, we can write:\n \\[\n \\frac{AD}{AB} = \\frac{3}{2}\n \\]\n Substituting $AB = 30$ inches into the proportion:\n \\[\n \\frac{AD}{30} = \\fra... | 94 | 1b3ec190-cc6a-5ce5-bcdb-a1a90996d8c1 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "Sides $AB$, $BC$, and $CD$ of (simple*) quadrilateral $ABCD$ have lengths $4$, $5$, and $... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "25", "solution": "1. **Given Information and Angle Relationships**:\n - Sides $AB = 4$, $BC = 5$, and $CD = 20$.\n - Angles $B$ and $C$ are obtuse.\n - $\\sin C = -\\cos B = \\frac{3}{5}$.\n\n Since $B$ and $C$ are obtuse, we have:\n \\[\n \\sin(180^\\circ - C) = \\sin C = \\fra... | 123 | f0388a7a-2823-53d6-b021-a87b92ca5959 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "A flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "70", "solution": "1. **Identify the fraction of pink and red flowers**: \n Given that six tenths (or $\\frac{6}{10}$) of the flowers are pink, we can simplify this fraction to $\\frac{3}{5}$. Consequently, the remaining flowers must be red, which is $\\frac{2}{5}$ of the total flowers (si... | 89 | 5c086d86-a181-5afc-b81d-3062cc866278 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "Let $\\{a_k\\}_{k=1}^{2011}$ be the sequence of real numbers defined by $a_1=0.201,$ $a_2... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "1341", "solution": "To solve this problem, we need to understand the behavior of the sequence $\\{a_k\\}_{k=1}^{2011}$ and determine when $a_k = b_k$, where $\\{b_k\\}_{k=1}^{2011}$ is the sequence $\\{a_k\\}_{k=1}^{2011}$ rearranged in decreasing order.\n\n#### Step 1: Analyze the sequence... | 328 | e2db6f0f-7527-5c14-afd0-9390f89e3bb2 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "The addition below is incorrect. The display can be made correct by changing one digit $d... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "8", "solution": "1. **Identify the Incorrect Sum**: First, we add the given numbers without changing any digits:\n - $742586 + 829430 = 1572016$\n - The provided sum is $1212016$.\n\n2. **Analyze the Incorrectness**: The provided sum $1212016$ differs significantly from the actual sum $... | 149 | 0503aee2-4fa1-568e-94f1-5a44bcee4e67 |
[
{
"role": "system",
"content": "Write your final answer within \\boxed{}.",
"reasoning_content": null,
"tool_calls": null,
"name": null,
"tool_call_id": null
},
{
"role": "user",
"content": "The number obtained from the last two nonzero digits of $90!$ is equal to $n$. What is $n... | null | math | agentica-org/DeepScaleR-Preview-Dataset | mit | {"ability": "math", "data_source": "agentica-org/DeepScaleR-Preview-Dataset", "agent_ref": {"type": "responses_api_agents", "name": "math_with_judge_simple_agent"}, "reasoning": true, "language": "eng"} | {"ground_truth": {"answer": "12", "solution": "1. **Count the number of factors of 10 in $90!$:**\n The number of factors of 10 in $90!$ is determined by the number of factors of 5, as there are more factors of 2 than 5. We calculate this using the formula for the number of factors of a prime $p$ in $n!$:\n \\[\n ... | 58 | 12a05fdb-bcc7-5857-8de1-1ddbc38af904 |
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