OpenMathReasoning

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cot

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expected_answer string	problem_type string	problem_source string	generation_model string	pass_rate_72b_tir string	problem string	generated_solution string	inference_mode string
$\frac{C_{n_1}^{a_1} \cdot C_{n_2}^{a_2} \cdot \ldots \cdot C_{n_C}^{a_C}}{C_N^A}$	has_answer_extracted	aops_c6_high_school_olympiads	DeepSeek-R1	0.65625	Given a group of $ N $ balls consisting of $ C $ colors, where the number of balls in each color is represented as $ n_1, n_2, \ldots, n_C $ (with $ n_1 + n_2 + \ldots + n_C = N $), what is the probability that when $ A $ balls are randomly picked (where $ A \leq N $), the picked balls consist of \( a_1, a_...	<think> Okay, so I need to find the probability that when I pick A balls out of N, where there are C different colors, the number of each color I pick is exactly a1, a2, ..., aC. Hmm, let's think about how to approach this. First, probability problems often involve combinations. The general formula for probability is ...	cot
\frac{n(n-1)}{2}	no_answer_extracted	aops_c4_high_school_math	DeepSeek-R1	n/a	How many lines can be drawn that are equidistant from $ n $ given points in the plane?	<think> Okay, so the problem is asking how many lines can be drawn that are equidistant from n given points in the plane. Hmm, let me try to wrap my head around this. First, I need to recall what it means for a line to be equidistant from points. I think when we say a line is equidistant from two points, it's the per...	cot
$ f(x) = f(1)x $	has_answer_extracted	aops_c6_high_school_olympiads	DeepSeek-R1	0.5625	Find all functions $ f: \mathbb{R} \to \mathbb{R} $ such that \[ (f(x^2) + f(xy) + f(y^2))(x - y) = f(x^3) - f(y^3) \] for all $ x, y \in \mathbb{R} $.	<think> Okay, let's try to solve this functional equation problem. The equation given is: \[ (f(x^2) + f(xy) + f(y^2))(x - y) = f(x^3) - f(y^3) \] We need to find all functions $ f: \mathbb{R} \to \mathbb{R} $ that satisfy this for all real numbers $ x $ and $ y $. Hmm, functional equations can be tricky, but m...	cot
2	has_answer_extracted	aops_c6_high_school_olympiads	DeepSeek-R1	0.625	Find the sum of the roots of the equation $(x+6)^{1/3} + (x-2)^{1/2} = 2$.	<think> Okay, let's see. I need to find the sum of the roots of the equation $(x+6)^{1/3} + (x-2)^{1/2} = 2$. Hmm, roots are the solutions for x that satisfy this equation. The sum of the roots would be all the solutions added together. But wait, how many roots does this equation have? Let me think. First, let's ana...	cot
32	has_answer_extracted	aops_c6_high_school_olympiads	QwQ-32B	0.0	Determine how many 1000 digit numbers $ A $ have the property that when any digit of $ A $, aside from the first, is deleted to form a 999 digit number $ B $, then $ B $ divides $ A $.	<think> Okay, so I need to figure out how many 109-digit numbers A have the property that if you delete any digit except the first one to form a 999-digit number B, then B divides A. Hmm, let's try to break this down step by step. First, let's make sure I understand the problem correctly. We have a 1000-digit number A...	cot
$\frac{2}{\pi} + \frac{32}{9\pi^2}$	has_answer_extracted	aops_c7_college_math	DeepSeek-R1	0.4375	Calculate the integral \[ \int^{\frac{3\pi}{2}}_{\frac{\pi}{2}} \left\|\left(\frac{2}{x^3}+\frac{1}{x}\right)\sin x\right\|dx \]	<think> Okay, let me try to solve this integral. So the problem is to compute the integral from π/2 to 3π/2 of the absolute value of (2/x³ + 1/x) times sin x dx. Hmm, that looks a bit intimidating at first glance, but let's break it down step by step. First, I remember that when dealing with integrals involving absolu...	cot
1	converted_proof	aops_c6_high_school_olympiads	QwQ-32B	n/a	In $\triangle ABC$ with incenter $I$, points $P$ and $Q$ are on side $BC$ such that $AP$ and $AQ$ are isogonal conjugates with respect to $\angle BAC$. The tangent line at $B$ of $\odot (ABP)$ and the tangent line at $C$ of $\odot (ACQ)$ intersect at $D$. Let $E = AP \cap IB$ and $F = AQ \cap IC$. If $DI$ intersects $E...	<think> Alright, let me try to tackle this geometry problem step by step. It's a bit complex, but I'll break it down as much as possible. First, let me restate the problem to make sure I understand it correctly. We have triangle ABC with incenter I. Points P and Q are on side BC such that AP and AQ are isogonal conjug...	cot
$\frac{(3-\sqrt{3})(2-\sqrt{3})^{2016}+(3+\sqrt{3})(2+\sqrt{3})^{2016}}{6}$	has_answer_extracted	aops_c6_high_school_olympiads	DeepSeek-R1	0.0	Let $ x_0 = 1 $, and $ x_{n+1} = 2x_n + \sqrt{3x_n^2 - 2} $. Determine $ x_{2016} $.	"<think>\nOkay, let's see. The problem is about a recursive sequence where x₀ is 1, and each subse(...TRUNCATED)	cot
$\sqrt[2012]{2013!} > \sqrt[2013]{2012!}$	has_answer_extracted	aops_c6_high_school_olympiads	DeepSeek-R1	0.0625	Which is greater, $\sqrt[2012]{2013!}$ or $\sqrt[2013]{2012!}$?	"<think>\nOkay, so I need to figure out whether the 2012th root of 2013 factorial is greater than th(...TRUNCATED)	cot
20	has_answer_extracted	aops_c6_high_school_olympiads	DeepSeek-R1	0.9375	On average, how long will you have to flip a coin to get the sequence $HTHT$?	"<think>\nOkay, so I need to figure out the expected number of coin flips needed to get the sequence(...TRUNCATED)	cot

End of preview.

expected_answer string	problem_type string	problem_source string	generation_model string	pass_rate_72b_tir string	problem string	generated_solution string	inference_mode string
\(\frac{C_{n_1}^{a_1} \cdot C_{n_2}^{a_2} \cdot \ldots \cdot C_{n_C}^{a_C}}{C_N^A}\)	has_answer_extracted	aops_c6_high_school_olympiads	DeepSeek-R1	0.65625	Given a group of \( N \) balls consisting of \( C \) colors, where the number of balls in each color is represented as \( n_1, n_2, \ldots, n_C \) (with \( n_1 + n_2 + \ldots + n_C = N \)), what is the probability that when \( A \) balls are randomly picked (where \( A \leq N \)), the picked balls consist of \( a_1, a_...	<think> Okay, so I need to find the probability that when I pick A balls out of N, where there are C different colors, the number of each color I pick is exactly a1, a2, ..., aC. Hmm, let's think about how to approach this. First, probability problems often involve combinations. The general formula for probability is ...	cot
\frac{n(n-1)}{2}	no_answer_extracted	aops_c4_high_school_math	DeepSeek-R1	n/a	How many lines can be drawn that are equidistant from \( n \) given points in the plane?	<think> Okay, so the problem is asking how many lines can be drawn that are equidistant from n given points in the plane. Hmm, let me try to wrap my head around this. First, I need to recall what it means for a line to be equidistant from points. I think when we say a line is equidistant from two points, it's the per...	cot
\( f(x) = f(1)x \)	has_answer_extracted	aops_c6_high_school_olympiads	DeepSeek-R1	0.5625	Find all functions \( f: \mathbb{R} \to \mathbb{R} \) such that \[ (f(x^2) + f(xy) + f(y^2))(x - y) = f(x^3) - f(y^3) \] for all \( x, y \in \mathbb{R} \).	<think> Okay, let's try to solve this functional equation problem. The equation given is: \[ (f(x^2) + f(xy) + f(y^2))(x - y) = f(x^3) - f(y^3) \] We need to find all functions \( f: \mathbb{R} \to \mathbb{R} \) that satisfy this for all real numbers \( x \) and \( y \). Hmm, functional equations can be tricky, but m...	cot
2	has_answer_extracted	aops_c6_high_school_olympiads	DeepSeek-R1	0.625	Find the sum of the roots of the equation \((x+6)^{1/3} + (x-2)^{1/2} = 2\).	<think> Okay, let's see. I need to find the sum of the roots of the equation \((x+6)^{1/3} + (x-2)^{1/2} = 2\). Hmm, roots are the solutions for x that satisfy this equation. The sum of the roots would be all the solutions added together. But wait, how many roots does this equation have? Let me think. First, let's ana...	cot
32	has_answer_extracted	aops_c6_high_school_olympiads	QwQ-32B	0.0	Determine how many 1000 digit numbers \( A \) have the property that when any digit of \( A \), aside from the first, is deleted to form a 999 digit number \( B \), then \( B \) divides \( A \).	<think> Okay, so I need to figure out how many 109-digit numbers A have the property that if you delete any digit except the first one to form a 999-digit number B, then B divides A. Hmm, let's try to break this down step by step. First, let's make sure I understand the problem correctly. We have a 1000-digit number A...	cot
\(\frac{2}{\pi} + \frac{32}{9\pi^2}\)	has_answer_extracted	aops_c7_college_math	DeepSeek-R1	0.4375	Calculate the integral \[ \int^{\frac{3\pi}{2}}_{\frac{\pi}{2}} \left\|\left(\frac{2}{x^3}+\frac{1}{x}\right)\sin x\right\|dx \]	<think> Okay, let me try to solve this integral. So the problem is to compute the integral from π/2 to 3π/2 of the absolute value of (2/x³ + 1/x) times sin x dx. Hmm, that looks a bit intimidating at first glance, but let's break it down step by step. First, I remember that when dealing with integrals involving absolu...	cot
1	converted_proof	aops_c6_high_school_olympiads	QwQ-32B	n/a	In $\triangle ABC$ with incenter $I$, points $P$ and $Q$ are on side $BC$ such that $AP$ and $AQ$ are isogonal conjugates with respect to $\angle BAC$. The tangent line at $B$ of $\odot (ABP)$ and the tangent line at $C$ of $\odot (ACQ)$ intersect at $D$. Let $E = AP \cap IB$ and $F = AQ \cap IC$. If $DI$ intersects $E...	<think> Alright, let me try to tackle this geometry problem step by step. It's a bit complex, but I'll break it down as much as possible. First, let me restate the problem to make sure I understand it correctly. We have triangle ABC with incenter I. Points P and Q are on side BC such that AP and AQ are isogonal conjug...	cot
\(\frac{(3-\sqrt{3})(2-\sqrt{3})^{2016}+(3+\sqrt{3})(2+\sqrt{3})^{2016}}{6}\)	has_answer_extracted	aops_c6_high_school_olympiads	DeepSeek-R1	0.0	Let \( x_0 = 1 \), and \( x_{n+1} = 2x_n + \sqrt{3x_n^2 - 2} \). Determine \( x_{2016} \).	"<think>\nOkay, let's see. The problem is about a recursive sequence where x₀ is 1, and each subse(...TRUNCATED)	cot
$\sqrt[2012]{2013!} > \sqrt[2013]{2012!}$	has_answer_extracted	aops_c6_high_school_olympiads	DeepSeek-R1	0.0625	Which is greater, $\sqrt[2012]{2013!}$ or $\sqrt[2013]{2012!}$?	"<think>\nOkay, so I need to figure out whether the 2012th root of 2013 factorial is greater than th(...TRUNCATED)	cot
20	has_answer_extracted	aops_c6_high_school_olympiads	DeepSeek-R1	0.9375	On average, how long will you have to flip a coin to get the sequence $HTHT$?	"<think>\nOkay, so I need to figure out the expected number of coin flips needed to get the sequence(...TRUNCATED)	cot