prompt listlengths 2 2 | system stringclasses 1
value | data_source stringclasses 1
value | ability stringclasses 1
value | reference_solution stringlengths 923 4.04k | reference_solution_full stringlengths 6.98k 31.1k | reward_model dict | extra_info dict | solution_tokens int64 2.43k 8.19k | n_correct int64 0 0 | mean_response_chars float64 8.82k 35.7k |
|---|---|---|---|---|---|---|---|---|---|---|
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "For how many positive integers $n$ is $n^{18/n}$ a whole number?",
"role": "user"
}
] | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To determine the number of positive integers \( n \) for which \( n^{18/n} \) is a whole number, we need to find \( n \) such that \( n^{18/n} \) is an integer. This means \( \frac{18}{n} \) must be an integer or a rational number that results in an integer when \( n \) is raised to that power.
### Step-by-Step Soluti... | <think>
Okay, let's tackle this problem: "For how many positive integers \( n \) is \( n^{18/n} \) a whole number?" Hmm, interesting. So I need to find all positive integers \( n \) such that when you take \( n \) and raise it to the power of \( 18/n \), the result is an integer. Let me think step by step.
First off, ... | {
"ground_truth": "9",
"style": "math_dapo"
} | {
"generation_model": "DeepSeek-R1",
"pass_rate_72b_tir": 0.09375,
"problem_source": "aops_c4_high_school_math"
} | 7,494 | 0 | 21,356.6875 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "In a mathematics competition with 25 problems, each problem awards 3 marks for a correct answer, deducts 1 mark for a wrong ans... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To determine the minimum number of candidates needed to ensure that at least two candidates have the same total score in a mathematics competition with 25 problems, we use the pigeonhole principle. Each problem awards 3 marks for a correct answer, deducts 1 mark for a wrong answer, and gives 0 marks for a blank answer.... | <think>
Okay, so I need to figure out the minimum number of candidates required to ensure that at least two of them have the same total score in this math competition. Let me start by understanding the problem step by step.
First, the competition has 25 problems. Each problem can either be answered correctly, incorrec... | {
"ground_truth": "99",
"style": "math_dapo"
} | {
"generation_model": "DeepSeek-R1",
"pass_rate_72b_tir": 0,
"problem_source": "aops_c6_high_school_olympiads"
} | 7,664 | 0 | 22,869.1875 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "Generalize the problem for an \\( n \\times n \\) board. If the lower left-hand corner square has the number 10, what are all t... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To solve the problem of finding all possible values of the upper right-hand square on an \( n \times n \) board where the lower left-hand corner is 10, we need to follow these steps:
1. **Determine the starting number \( k \):**
- The lower left-hand corner of the board is at position \((n, 1)\).
- The number at... | <think>
Okay, so I need to figure out all possible values of the upper right-hand square on an n x n board where the lower left-hand corner is 10. Hmm. Let's start by recalling how numbering works on such boards. Wait, the problem doesn't specify the type of numbering. Is it a magic square, a chessboard, or something e... | {
"ground_truth": "10",
"style": "math_dapo"
} | {
"generation_model": "DeepSeek-R1",
"pass_rate_72b_tir": 0.09375,
"problem_source": "aops_c6_high_school_olympiads"
} | 3,371 | 0 | 28,645.3125 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "How many finger-countable functions $ f $ satisfy $ f( f(1)) = 1 $ and $ f(3) = 2 $?",
"role": "user"
}
] | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To determine the number of finger-countable functions \( f \) that satisfy \( f(f(1)) = 1 \) and \( f(3) = 2 \), we consider the domain and codomain as the set \( S = \{1, 2, 3, 4, 5\} \).
1. **Condition \( f(3) = 2 \)**:
- This condition fixes \( f(3) = 2 \), leaving the mappings for \( f(1) \), \( f(2) \), \( f(4... | <think>
Okay, so I need to figure out how many functions f satisfy the conditions f(f(1)) = 1 and f(3) = 2. The problem mentions that these functions are "finger-countable," which I think means they're functions from the set {1, 2, 3, 4, 5} to itself, since that's a common scenario where you can count the number of fun... | {
"ground_truth": "200",
"style": "math_dapo"
} | {
"generation_model": "DeepSeek-R1",
"pass_rate_72b_tir": 0,
"problem_source": "aops_c4_high_school_math"
} | 3,326 | 0 | 12,085.75 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "Five men and nine women stand equally spaced around a circle in random order. The probability that every man stands diametrical... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To solve the problem of finding the probability that every man stands diametrically opposite a woman when five men and nine women are arranged equally spaced around a circle, we can follow these steps:
1. **Total Number of Arrangements**:
There are 14 positions arranged in 7 pairs of diametrically opposite seats. T... | <think>
Alright, so I have this probability problem here: Five men and nine women stand equally spaced around a circle in random order. The probability that every man stands diametrically opposite a woman is m/n, where m and n are coprime. I need to find m + n. Hmm, okay, let me think step by step.
First, let me try t... | {
"ground_truth": "191",
"style": "math_dapo"
} | {
"generation_model": "QwQ-32B",
"pass_rate_72b_tir": 0,
"problem_source": "MATH_training_set"
} | 6,694 | 0 | 29,559.6875 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "Find the number of ways to select 5 people out of 15 people sitting on a round table such that exactly two of the selected peop... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To find the number of ways to select 5 people out of 15 sitting on a round table such that exactly two of the selected people are consecutive and no other two selected people are consecutive, we can follow these steps:
1. **Identify Consecutive Pairs**:
- In a circular arrangement of 15 people, there are 15 possibl... | <think>
Okay, so I need to find the number of ways to select 5 people out of 15 sitting around a round table, with the condition that exactly two of the selected people are consecutive and no other two are consecutive. Hmm, let me think through this step by step.
First, since the table is round, the problem is circula... | {
"ground_truth": "1260",
"style": "math_dapo"
} | {
"generation_model": "DeepSeek-R1",
"pass_rate_72b_tir": 0.09375,
"problem_source": "aops_c4_high_school_math"
} | 5,106 | 0 | 31,421.5625 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "How many ways are there to trace out \"1, 2, 3, 4, 5, 6, 7\" by traversing the following diamond, starting at the top and alway... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To solve the problem of finding the number of ways to trace out "1, 2, 3, 4, 5, 6, 7" by traversing the given diamond structure, starting at the top and always moving to an adjacent number, we can use a dynamic programming approach. Each step must move to the next number in the sequence and to an adjacent position in t... | <think>
Okay, let's see. I need to find the number of ways to trace the sequence "1, 2, 3, 4, 5, 6, 7" on this diamond structure by moving to adjacent numbers each time. The starting point is the top 1. Hmm. So, each step I have to go from the current number to the next in the sequence by moving to an adjacent position... | {
"ground_truth": "20",
"style": "math_dapo"
} | {
"generation_model": "DeepSeek-R1",
"pass_rate_72b_tir": 0.03125,
"problem_source": "aops_c4_high_school_math"
} | 5,901 | 0 | 27,569.8125 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "A lighthouse facing East sends out a fan-shaped beam of light extending from S.-E to N.-E. An observer sailing due North contin... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To solve the problem, we need to determine the speed of a ship sailing due north that continues to see a fan-shaped beam of light extending from southeast to northeast for \(10\sqrt{2}\) minutes after meeting the beam. When the ship leaves the fan of light, it is 10 miles from the lighthouse.
1. **Define the coordinat... | <think>
Okay, so I need to find the speed of a ship that's sailing north. The problem says there's a lighthouse facing east with a fan-shaped beam from southeast to northeast. The observer on the ship continues to see the light for 10√2 minutes after passing the beam. When the ship leaves the fan of light, it's 10 mile... | {
"ground_truth": "60",
"style": "math_dapo"
} | {
"generation_model": "DeepSeek-R1",
"pass_rate_72b_tir": 0.0625,
"problem_source": "aops_c4_high_school_math"
} | 4,327 | 0 | 33,019.25 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "Find the largest possible value of $n$ such that there exist $n$ real numbers $x_1, \\ldots, x_n$ which satisfy $|x_i - x_j| > ... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To find the largest possible value of \( n \) such that there exist \( n \) real numbers \( x_1, x_2, \ldots, x_n \) which satisfy \( |x_i - x_j| > \frac{1 + x_i x_j}{100} \) for all \( i \neq j \), we proceed as follows:
1. **Substitution and Transformation:**
Let \( x_i = \tan \theta_i \) for \( \theta_i \in \lef... | <think>
Okay, let me try to work through this problem step by step. The question is asking for the largest possible value of n such that there exist n real numbers x₁, x₂, ..., xₙ satisfying |x_i - x_j| > (1 + x_i x_j)/100 for all i ≠ j. Hmm, that seems a bit complex, but maybe I can break it down.
First, let's unders... | {
"ground_truth": "315",
"style": "math_dapo"
} | {
"generation_model": "DeepSeek-R1",
"pass_rate_72b_tir": 0.0625,
"problem_source": "aops_c6_high_school_olympiads"
} | 5,667 | 0 | 24,172.625 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "Consider the system $|z^{2}+az+1|=1$ and $|z|=1$ where $a$ and $z$ are complex numbers. Let $S$ be the set of all points in the... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To solve the problem, we need to find the set \( S \) of all complex numbers \( a \) such that there exists a complex number \( z \) on the unit circle (i.e., \( |z| = 1 \)) satisfying the equation \( |z^2 + az + 1| = 1 \). We then need to compute the area \( A \) of \( S \) and find \( \lfloor 10A \rfloor \).
1. **Ex... | <think>
Okay, let's see. I need to solve this problem where we have two equations involving complex numbers: |z² + a z + 1| = 1 and |z| = 1. The goal is to find the set S of all points a in the complex plane such that if a is in S, then there's a solution z that satisfies both equations. Then, compute the area A of thi... | {
"ground_truth": "111",
"style": "math_dapo"
} | {
"generation_model": "DeepSeek-R1",
"pass_rate_72b_tir": 0,
"problem_source": "aops_c4_high_school_math"
} | 5,278 | 0 | 23,887.25 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "Find \\( f(2009) \\) where \\( f(n) \\) is the number of pairs \\( (x, y) \\in \\mathbb{Z}^2 \\) such that \\( |x^2 - y^2| = n ... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To find \( f(2009) \), where \( f(n) \) is the number of integer pairs \((x, y)\) such that \( |x^2 - y^2| = n \), we start by noting that \( |x^2 - y^2| = |(x - y)(x + y)| \). This means we need to find pairs \((d, e)\) such that \( d \cdot e = \pm 2009 \), where \( d = x - y \) and \( e = x + y \).
### Step-by-Step ... | <think>
Okay, so I need to find f(2009), where f(n) is the number of integer pairs (x, y) such that the absolute value of x² minus y² equals n. Hmm, okay. Let me think through this step by step.
First, the equation given is |x² - y²| = 2009. That simplifies to two cases: x² - y² = 2009 or y² - x² = 2009. But since we'... | {
"ground_truth": "24",
"style": "math_dapo"
} | {
"generation_model": "DeepSeek-R1",
"pass_rate_72b_tir": 0.0625,
"problem_source": "aops_c6_high_school_olympiads"
} | 8,014 | 0 | 24,724.5625 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "Arrange eight of the nine numbers \\(2, 3, 4, 7, 10, 11, 12, 13, 15\\) in the vacant squares of the \\(3 \\times 4\\) array so ... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To solve the problem, we need to arrange eight out of the nine numbers \(2, 3, 4, 7, 10, 11, 12, 13, 15\) in a \(3 \times 4\) grid such that the arithmetic average of the numbers in each row and each column is the same integer. The grid has some pre-filled numbers: \(1\) at (1,3), \(9\) at (2,2), \(14\) at (3,1), and \... | <think>
Okay, so I have this problem where I need to arrange eight out of the nine given numbers into a 3x4 grid. The numbers are 2, 3, 4, 7, 10, 11, 12, 13, 15. The goal is that the average of the numbers in each row and each column should be the same integer. Also, one of the nine numbers must be left out. The Asy co... | {
"ground_truth": "10",
"style": "math_dapo"
} | {
"generation_model": "DeepSeek-R1",
"pass_rate_72b_tir": 0,
"problem_source": "aops_c6_high_school_olympiads"
} | 6,960 | 0 | 27,606.125 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "Let \\(a, b, c > 0\\) such that \\(a + b + c = \\frac{3}{2}\\). Find the minimum value of the function \\(F = 27abc + \\frac{8}... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | Given \(a, b, c > 0\) such that \(a + b + c = \frac{3}{2}\), we need to find the minimum value of the function \(F = 27abc + \frac{8}{ab + bc + ca}\).
First, we consider the case where \(a = b = c\). In this case, each variable is \(\frac{1}{2}\). Calculating \(F\) for these values:
\[
abc = \left(\frac{1}{2}\right)^3... | <think>
Okay, so I have this problem where I need to find the minimum value of the function F = 27abc + 8/(ab + bc + ca) given that a, b, c are positive real numbers and their sum is 3/2. Hmm, let's see how to approach this.
First, since the variables a, b, c are all positive and their sum is fixed, maybe I can use so... | {
"ground_truth": "14",
"style": "math_dapo"
} | {
"generation_model": "DeepSeek-R1",
"pass_rate_72b_tir": 0.03125,
"problem_source": "aops_c6_high_school_olympiads"
} | 6,686 | 0 | 19,643.375 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "Triangle \\( T_1 \\) has sides of lengths 4, 5, and 6. A second triangle \\( T_2 \\) is formed by using the excenters of \\( T_... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To solve the problem, we need to find the ratio of the areas of triangle \( T_1 \) with sides 4, 5, and 6, and the excentral triangle \( T_2 \) formed by the excenters of \( T_1 \).
1. **Calculate the area of \( T_1 \) using Heron's formula:**
The semi-perimeter \( s \) is:
\[
s = \frac{4 + 5 + 6}{2} = 7.5
... | <think>
Okay, so I have this problem where Triangle T1 has sides of lengths 4, 5, and 6. Then, Triangle T2 is formed by using the excenters of T1 as vertices. I need to find the ratio of their areas, [T1]/[T2], and express it as m/n where m and n are coprime, then compute m + n.
First, let me recall what excenters are... | {
"ground_truth": "39",
"style": "math_dapo"
} | {
"generation_model": "DeepSeek-R1",
"pass_rate_72b_tir": 0.03125,
"problem_source": "aops_c4_high_school_math"
} | 7,485 | 0 | 20,438.875 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "Let \\(ABC\\) be a scalene triangle. A point \\(P\\) is called nice if the lines \\(AD\\), \\(BE\\), and \\(CF\\) are concurren... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To solve the problem of finding the number of nice points on the line \(OI\) in a scalene triangle \(ABC\), we need to understand the definition and properties of a nice point. A point \(P\) is called nice if the lines \(AD\), \(BE\), and \(CF\) are concurrent, where \(D\), \(E\), and \(F\) are the projections (feet of... | <think>
Okay, so I have this geometry problem here about finding the number of "nice" points on the line OI in a scalene triangle ABC. Let me try to break it down step by step.
First, the problem says a point P is called nice if the lines AD, BE, and CF are concurrent, where D, E, F are the projections (which I think ... | {
"ground_truth": "3",
"style": "math_dapo"
} | {
"generation_model": "QwQ-32B",
"pass_rate_72b_tir": 0,
"problem_source": "aops_c6_high_school_olympiads"
} | 5,873 | 0 | 26,854.25 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "Based on a city's rules, the buildings of a street may not have more than 9 stories. Moreover, if the number of stories of two ... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To solve the problem of finding the maximum number of buildings that can be built on one side of a street given the constraints (no building can have more than 9 stories, and if two buildings have the same number of stories, there must be a taller building between them), we can use the following approach:
1. **Underst... | <think>
Okay, so I need to figure out the maximum number of buildings that can be built on one side of a street given two rules: no building can have more than 9 stories, and if two buildings have the same number of stories, there must be a taller building between them. Hmm, let me think about how to approach this.
Fi... | {
"ground_truth": "511",
"style": "math_dapo"
} | {
"generation_model": "DeepSeek-R1",
"pass_rate_72b_tir": 0,
"problem_source": "aops_c4_high_school_math"
} | 6,257 | 0 | 30,738.25 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "There is a unique angle $\\theta$ between $0^{\\circ}$ and $90^{\\circ}$ such that for nonnegative integers $n$, the value of $... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To solve the problem, we need to find an angle \(\theta\) between \(0^\circ\) and \(90^\circ\) such that \(\tan(2^n \theta)\) is positive when \(n\) is a multiple of 3 and negative otherwise. This implies the following conditions:
1. For \(n \equiv 0 \pmod{3}\), \(\tan(2^n \theta)\) must be positive, meaning \(2^n \th... | <think>
Okay, so I need to find an angle θ between 0° and 90° such that tan(2ⁿθ) is positive when n is a multiple of 3 and negative otherwise. Hmm, let's start by recalling that the tangent function is positive in the first and third quadrants (0° to 90° and 180° to 270°) and negative in the second and fourth quadrants... | {
"ground_truth": "547",
"style": "math_dapo"
} | {
"generation_model": "DeepSeek-R1",
"pass_rate_72b_tir": 0.09375,
"problem_source": "aops_c5_contests_amp_programs"
} | 8,000 | 0 | 16,748.875 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "In a wheel of fortune with 10 sectors, each giving a different prize with equal probability (0.1), how many turns, $n$, are nee... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To solve the problem of determining how many turns, \( n \), are needed to collect all 10 different items on a wheel of fortune with 10 sectors, given that 6 different items have already been collected and the probability of success should be at least 0.5, we can use the inclusion-exclusion principle.
### Problem Brea... | <think>
Okay, so I need to find out how many more turns, n, are required on a wheel of fortune with 10 sectors (each giving a different prize with equal probability 0.1) to collect all 10 items, given that I already have 6 different ones. The probability of success should be at least 0.5. Hmm, this reminds me of the co... | {
"ground_truth": "18",
"style": "math_dapo"
} | {
"generation_model": "DeepSeek-R1",
"pass_rate_72b_tir": 0,
"problem_source": "aops_c7_college_math"
} | 7,082 | 0 | 17,505.375 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "Let $P(x)=x^3+ax^2+bx+c$ be a cubic polynomial with $a$, $b$, and $c$ real. Given that two of the roots of $P(x)$ sum to the th... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To solve the problem, we start by analyzing the given conditions for the polynomial \(P(x) = x^3 + ax^2 + bx + c\). The roots \(r_1, r_2, r_3\) satisfy the condition \(r_1 + r_2 = r_3\). By Vieta's formulas, we know:
1. \(r_1 + r_2 + r_3 = -a\)
2. \(r_1r_2 + r_1r_3 + r_2r_3 = b\)
3. \(r_1r_2r_3 = -c\)
Given \(r_1 + r... | <think>
Okay, let's try to tackle this problem step by step. So, the problem is about a cubic polynomial P(x) = x³ + a x² + b x + c, with real coefficients a, b, c. The conditions given are that two of the roots sum to the third, and that the roots form a triangle in the complex plane with area 2015. We need to find th... | {
"ground_truth": "035",
"style": "math_dapo"
} | {
"generation_model": "DeepSeek-R1",
"pass_rate_72b_tir": 0.0625,
"problem_source": "aops_c5_contests_amp_programs"
} | 6,666 | 0 | 22,249.9375 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "Mia's mom puts 3 toys into the toy box every 30 seconds, but immediately after those 30 seconds, Mia takes 2 toys out of the bo... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To solve the problem of determining how long it will take for Mia and her mom to put all 30 toys into the toy box for the first time, we need to carefully consider the net effect of each cycle and the timing of the final addition.
1. **Cycle Analysis:**
- Every 30 seconds, Mia's mom adds 3 toys to the box.
- Imm... | <think>
Okay, let me try to figure out this problem. So, Mia's mom is adding toys to the toy box, but then Mia takes some out after each 30-second interval. The goal is to find out how long it will take until there are 30 toys in the box for the first time. Hmm.
First, let me restate the problem to make sure I underst... | {
"ground_truth": "14",
"style": "math_dapo"
} | {
"generation_model": "DeepSeek-R1",
"pass_rate_72b_tir": 0.03125,
"problem_source": "aops_c5_contests_amp_programs"
} | 4,442 | 0 | 27,008.5 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "Consider a point $P$ in triangle $ABC$ such that if you draw $27$ rays from $P$, their intersections with the perimeter of $ABC... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To solve the problem of finding the number of distinct points \( P \) in triangle \( ABC \) such that 27 rays from \( P \) intersect the perimeter of \( ABC \) to form 27 smaller triangles of equal area, we proceed as follows:
1. **Understanding the Problem**:
- We need to find points \( P \) inside triangle \( ABC... | <think>
Okay, so I have this problem here: In triangle ABC, there's a point P such that if you draw 27 rays from P, their intersections with the perimeter of ABC form 27 smaller triangles all of equal area. The question is asking how many total distinct points P are possible in triangle ABC.
Hmm, let me start by tryin... | {
"ground_truth": "325",
"style": "math_dapo"
} | {
"generation_model": "QwQ-32B",
"pass_rate_72b_tir": 0,
"problem_source": "aops_c6_high_school_olympiads"
} | 7,471 | 0 | 35,748.0625 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "Let \\(a\\), \\(b\\), and \\(c\\) be real numbers in the interval \\((0, 1)\\) such that \\(ab + bc + ca = 1\\). Find the large... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | Given real numbers \(a\), \(b\), and \(c\) in the interval \((0, 1)\) such that \(ab + bc + ca = 1\), we aim to find the largest value of \(a + b + c + abc\).
### Step-by-Step Solution:
1. **Constraint Analysis:**
The given constraint is \(ab + bc + ca = 1\). We need to maximize \(a + b + c + abc\).
2. **Approach... | <think>
Okay, let's try to figure out how to solve this problem. The question is: Given real numbers a, b, c in the interval (0,1) such that ab + bc + ca = 1, find the largest value of a + b + c + abc. Hmm, interesting. So we need to maximize the expression a + b + c + abc under the constraint ab + bc + ca = 1, with ea... | {
"ground_truth": "2",
"style": "math_dapo"
} | {
"generation_model": "DeepSeek-R1",
"pass_rate_72b_tir": 0,
"problem_source": "aops_c4_high_school_math"
} | 7,637 | 0 | 22,301.6875 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "Let $ABC$ be equilateral, and $D, E,$ and $F$ be the midpoints of $\\overline{BC}, \\overline{CA},$ and $\\overline{AB},$ resp... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To solve the problem, we start by setting up the coordinates for the equilateral triangle \(ABC\) with side length 2. Place \(A\) at \((0, \sqrt{3})\), \(B\) at \((-1, 0)\), and \(C\) at \((1, 0)\). The midpoints \(D\), \(E\), and \(F\) of sides \(BC\), \(CA\), and \(AB\) respectively are:
\[ D = (0, 0), \quad E = \lef... | <think>
Okay, let's see. So we have an equilateral triangle ABC, and D, E, F are midpoints of the sides BC, CA, and AB respectively. Then there are points P, Q, R on DE, EF, FD such that P is on CQ, Q is on AR, and R is on BP. We need to find the ratio of the area of ABC to the area of PQR, which is given in the form a... | {
"ground_truth": "083",
"style": "math_dapo"
} | {
"generation_model": "DeepSeek-R1",
"pass_rate_72b_tir": 0,
"problem_source": "MATH_training_set"
} | 7,720 | 0 | 18,737.125 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "A compact disk manufacturer charges $100 for each box of CDs ordered. However, for orders in excess of 30 boxes but less than 6... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To solve the problem, we need to determine the number of boxes of CDs ordered given the pricing structure and the total bill.
1. **Understanding the Pricing Structure:**
- The manufacturer charges $100 per box for orders of 30 boxes or fewer.
- For orders between 31 and 64 boxes, the price per box is reduced by ... | <think>
Okay, let's try to solve this problem step by step. So, the problem is about a customer who ordered some boxes of CDs and got a discount because they ordered more than 30 but less than 65 boxes. The total bill was $4200, and we need to find out how many boxes they ordered.
First, let me understand the pricing ... | {
"ground_truth": "60",
"style": "math_dapo"
} | {
"generation_model": "DeepSeek-R1",
"pass_rate_72b_tir": 0.0625,
"problem_source": "aops_c7_college_math"
} | 4,773 | 0 | 27,186.3125 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "Find the number of permutations of the 12 letters A C D G H I K N O R S T that do not contain the patterns KRISHNA, GANDHI, CHR... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To find the number of permutations of the 12 letters A, C, D, G, H, I, K, N, O, R, S, T that do not contain the patterns KRISHNA, GANDHI, CHRIST, or GOD, we use the principle of inclusion-exclusion.
1. **Total permutations**: The total number of permutations of 12 distinct letters is \(12!\).
2. **Subtract permutatio... | <think>
Okay, so I need to find the number of permutations of the 12 letters A, C, D, G, H, I, K, N, O, R, S, T. But I have to exclude any permutations that contain the substrings KRISHNA, GANDHI, CHRIST, or GOD. Hmm, this sounds like an inclusion-exclusion principle problem. Let me think step by step.
First, let me r... | {
"ground_truth": "475362144",
"style": "math_dapo"
} | {
"generation_model": "DeepSeek-R1",
"pass_rate_72b_tir": 0,
"problem_source": "aops_c4_high_school_math"
} | 6,134 | 0 | 27,252.75 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "Given the function \\( f \\) is twice differentiable such that \\( f(x) + f''(x) = -x|\\sin x| f'(x) \\) for \\( x \\geq 0 \\),... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To find the maximum value of the function \( f(x) \) given the differential equation \( f''(x) + x |\sin x| f'(x) + f(x) = 0 \) with initial conditions \( f(0) = -3 \) and \( f'(0) = 4 \), we analyze the energy of the system.
Define the energy function \( E(x) \) as:
\[
E(x) = \frac{1}{2} (f'(x)^2 + f(x)^2).
\]
Differ... | <think>
Okay, so I need to solve this differential equation: f(x) + f''(x) = -x |sin x| f'(x), with initial conditions f(0) = -3 and f'(0) = 4. The question is asking for the maximum value of f(x) for x ≥ 0. Hmm, let's see.
First, the equation is a second-order differential equation, but it's nonlinear because of the... | {
"ground_truth": "5",
"style": "math_dapo"
} | {
"generation_model": "DeepSeek-R1",
"pass_rate_72b_tir": 0,
"problem_source": "aops_c7_college_math"
} | 8,012 | 0 | 24,147.375 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "Let \\(a, b, c, d\\) be non-negative real numbers such that \\(a + b + c + d = 1\\). Find the number of ways to choose \\(a, b,... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To solve the problem where \(a, b, c, d\) are non-negative real numbers such that \(a + b + c + d = 1\) and \(a^2 + b^2 + c^2 + d^2 = \max\{a, b, c, d\}\), we will consider different cases for the maximum value among \(a, b, c, d\).
### Case 1: One variable is 1, others are 0
Assume \(a = 1\) and \(b = c = d = 0\). Th... | <think>
Okay, so I need to solve this problem where we have four non-negative real numbers a, b, c, d that add up to 1. The condition is that the sum of their squares equals the maximum of the four numbers. I have to find how many such tuples (a, b, c, d) exist. Hmm, let's break this down.
First, since a, b, c, d are ... | {
"ground_truth": "15",
"style": "math_dapo"
} | {
"generation_model": "DeepSeek-R1",
"pass_rate_72b_tir": 0,
"problem_source": "aops_c6_high_school_olympiads"
} | 6,725 | 0 | 26,638.125 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "In how many ways can we paint 16 seats in a row, each red or green, such that the number of consecutive seats painted in the sa... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To solve the problem of painting 16 seats in a row such that the number of consecutive seats painted in the same color is always odd, we can model the solution using a recurrence relation similar to the Fibonacci sequence.
### Key Steps:
1. **Identify the Problem Constraints**:
- Each run of consecutive seats of t... | <think>
Okay, let's tackle this problem. We need to find the number of ways to paint 16 seats in a row, each either red or green, such that every consecutive sequence of the same color has an odd length. Hmm, interesting. So each time the color changes, the previous run must have been odd in length. Let me think about ... | {
"ground_truth": "1974",
"style": "math_dapo"
} | {
"generation_model": "DeepSeek-R1",
"pass_rate_72b_tir": 0,
"problem_source": "aops_c6_high_school_olympiads"
} | 6,879 | 0 | 25,135.5 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "Find all natural numbers \\( n \\) such that when \\( n \\) is alternately divided by \\( 29 \\), \\( 41 \\), and \\( 59 \\), t... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To find all natural numbers \( n \) such that when \( n \) is divided by \( 29 \), \( 41 \), and \( 59 \), the sum of the three nonzero remainders equals \( n \), we proceed as follows:
1. **Define the remainders:**
Let \( r_1 \), \( r_2 \), and \( r_3 \) be the remainders when \( n \) is divided by \( 29 \), \( 41... | <think>
Okay, so I need to find all natural numbers n such that when n is divided by 29, 41, and 59, the sum of the three nonzero remainders equals n itself. Hmm, let's break this down step by step.
First, let me understand the problem. When they say "alternately divided by 29, 41, and 59," I think that just means div... | {
"ground_truth": "114",
"style": "math_dapo"
} | {
"generation_model": "DeepSeek-R1",
"pass_rate_72b_tir": 0.03125,
"problem_source": "aops_c6_high_school_olympiads"
} | 7,659 | 0 | 20,951 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "A class has ten students, and the teacher has twenty distinct stones. The teacher randomly gives each stone to one of the stude... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To find the expected value of the happiness of the class, which is defined as the product of the number of stones each student has, we start by noting that each stone is independently distributed to one of the 10 students. The happiness is the product of the counts of stones each student receives, and we need to comput... | <think>
Okay, let's see. The problem is about finding the expected value of the happiness of a class, where happiness is defined as the product of the number of stones each student has. The teacher randomly distributes 20 distinct stones to 10 students. Each stone is given to a student independently, so each stone has ... | {
"ground_truth": "788",
"style": "math_dapo"
} | {
"generation_model": "DeepSeek-R1",
"pass_rate_72b_tir": 0.0625,
"problem_source": "aops_c4_high_school_math"
} | 6,323 | 0 | 25,604.25 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "Given the functional equation \\( f(x) - 2\\left(\\frac{x}{2}\\right) + f\\left(\\frac{x}{4}\\right) = x^2 \\), find \\( f'(0) ... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To solve the functional equation \( f(x) - 2f\left(\frac{x}{2}\right) + f\left(\frac{x}{4}\right) = x^2 \) and find \( f'(0) \), we proceed as follows:
1. **Assume a Quadratic Solution**:
Let \( f(x) = ax^2 + bx + c \). Substitute this into the functional equation:
\[
f(x) - 2f\left(\frac{x}{2}\right) + f\lef... | <think>
Okay, so I have this functional equation: f(x) minus 2 times f of x over 2 plus f of x over 4 equals x squared. And I need to find f prime at 0. Hmm, functional equations can be tricky, but maybe I can figure this out step by step. Let me write it down again to make sure I have it right:
f(x) - 2f(x/2) + f(x/4... | {
"ground_truth": "0",
"style": "math_dapo"
} | {
"generation_model": "QwQ-32B",
"pass_rate_72b_tir": 0.0625,
"problem_source": "aops_c6_high_school_olympiads"
} | 7,690 | 0 | 16,863.8125 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "What is the minimum possible value of Aaron's age, to the nearest year, if he has been alive for all or part of three separate ... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To determine the minimum possible value of Aaron's age, rounded to the nearest year, given that he has been alive for all or part of three separate decades, we need to carefully consider the definition of a decade and how Aaron's lifespan can overlap these decades.
A decade is a period of ten years. For example, the 1... | <think>
Okay, let's see... The question is asking for the minimum possible age Aaron could be, rounded to the nearest year, given that he's lived through parts of three separate decades. Hmm. So, I need to figure out the youngest age someone can be while having their lifespan touch three different decades.
First, I s... | {
"ground_truth": "10",
"style": "math_dapo"
} | {
"generation_model": "DeepSeek-R1",
"pass_rate_72b_tir": 0.03125,
"problem_source": "aops_c4_high_school_math"
} | 4,741 | 0 | 24,368.125 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "Consider a trapezoid $ABCD$ with bases $AB$ and $CD$. Point $E$ is on side $CD$ such that $AE$ is parallel to $BC$. The areas o... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To find the area of quadrilateral \(BCEQ\) in trapezoid \(ABCD\) with bases \(AB\) and \(CD\), where point \(E\) is on \(CD\) such that \(AE \parallel BC\), and the areas of triangles \(ABQ\) and \(ADQ\) are 2 m² and 3 m² respectively, we proceed as follows:
1. **Determine the Ratio \(BQ:QD\)**:
Since the areas of ... | <think>
Okay, let me try to figure out this trapezoid problem. So we have trapezoid ABCD with bases AB and CD. Point E is on side CD such that AE is parallel to BC. The areas of triangles ABQ and ADQ are 2 m² and 3 m² respectively. Q is the intersection point of BD and AE. We need to find the area of quadrilateral BCEQ... | {
"ground_truth": "8",
"style": "math_dapo"
} | {
"generation_model": "DeepSeek-R1",
"pass_rate_72b_tir": 0.03125,
"problem_source": "aops_c4_high_school_math"
} | 6,603 | 0 | 22,794.5625 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "Find all values of \\( a \\) such that the system of equations in \\( x \\) and \\( y \\):\n\n\\[ 2^{|x|} + |x| = y + x^2 + a \... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To find all values of \( a \) such that the system of equations
\[
2^{|x|} + |x| = y + x^2 + a
\]
\[
x^2 + y^2 = 1
\]
has exactly one solution, we proceed as follows:
1. **Substitute \( y \) from the first equation into the second equation:**
From the first equation, we have:
\[
y = 2^{|x|} + |x| - x^2 - a... | <think>
Okay, let's see. I need to find all values of 'a' such that the system of equations has exactly one solution. The equations are:
First equation: 2^{|x|} + |x| = y + x² + a
Second equation: x² + y² = 1
Hmm, so we have a system of two equations with variables x and y. The problem is to find the values of 'a' w... | {
"ground_truth": "0",
"style": "math_dapo"
} | {
"generation_model": "DeepSeek-R1",
"pass_rate_72b_tir": 0.09375,
"problem_source": "aops_c6_high_school_olympiads"
} | 6,784 | 0 | 24,890.125 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "Let \\(x\\) and \\(y\\) be positive integers such that \\(\\gcd(x,y)=1\\) and \\(\\tau(x)^{\\tau(y)^2}=\\tau(y)^{\\tau(x)}\\). ... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To solve the problem, we need to find positive integers \(x\) and \(y\) such that \(\gcd(x, y) = 1\) and \(\tau(x)^{\tau(y)^2} = \tau(y)^{\tau(x)}\), and then determine the sum of all possible values of \(\tau(xy)\).
Let \(a = \tau(x)\) and \(b = \tau(y)\). The given equation becomes:
\[ a^{b^2} = b^a. \]
We need to ... | <think>
Okay, let's see. I need to solve this problem where x and y are positive integers with gcd(x, y) = 1, and the equation τ(x)^{τ(y)^2} = τ(y)^{τ(x)} holds. Then I have to find the sum of all possible values of τ(xy). Alright, first, let's recall what τ(n) means. I think τ(n) is the number of positive divisors of ... | {
"ground_truth": "114",
"style": "math_dapo"
} | {
"generation_model": "DeepSeek-R1",
"pass_rate_72b_tir": 0.03125,
"problem_source": "aops_c4_high_school_math"
} | 7,451 | 0 | 19,468.0625 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "A six-dimensional \"cube\" (a $6$-cube) has vertices at the points $(\\pm 3,\\pm 3,\\pm 3,\\pm 3,\\pm 3,\\pm 3)$. This $6$-cube... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To determine the number of 2-dimensional square faces of one or more unit 6-cubes within a larger 6-cube, we proceed as follows:
1. **Understand the Structure:**
- The larger 6-cube is divided into \(6^6\) smaller unit 6-cubes.
- Each unit 6-cube has edges of length 1.
2. **Counting 2D Faces:**
- A 2-dimensi... | <think>
Okay, so I need to find the number of 2-dimensional squares that are faces of one or more unit 6-cubes in a larger 6-cube. The big cube has vertices at (±3, ±3, ±3, ±3, ±3, ±3), which means each edge of the big cube is from -3 to +3, so that's a length of 6. Each unit cube, then, must have edges of length 1 sin... | {
"ground_truth": "1296540",
"style": "math_dapo"
} | {
"generation_model": "DeepSeek-R1",
"pass_rate_72b_tir": 0,
"problem_source": "aops_c6_high_school_olympiads"
} | 3,909 | 0 | 29,296.1875 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "Let \\( R \\) be a square region and \\( n \\ge 4 \\) an integer. A point \\( X \\) in the interior of \\( R \\) is called \\( ... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To determine the number of points that are 100-ray partitional but not 60-ray partitional in a square region \( R \), we need to analyze the coordinates of such points.
1. **100-ray partitional points**: A point \( (a, b) \) is 100-ray partitional if \( a \) and \( b \) are rational numbers with denominators dividing ... | <think>
Okay, so I need to figure out how many points inside a square region R are 100-ray partitional but not 60-ray partitional. Hmm, let me start by understanding what an n-ray partitional point is. The problem says that a point X is n-ray partitional if there are n rays emanating from X that divide R into n triangl... | {
"ground_truth": "2320",
"style": "math_dapo"
} | {
"generation_model": "DeepSeek-R1",
"pass_rate_72b_tir": 0,
"problem_source": "aops_c5_contests_amp_programs"
} | 5,412 | 0 | 33,967.75 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "For all positive integers $n$, let $f(n)$ denote the remainder when $(n+1)^{(n+2)^{(n+3)^{.^{.^{.^{(2n-1)^{2n}}}}}}}$ is divide... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To solve the problem of finding \( f(101) \), which is the remainder when \( 102^{103^{104^{\cdots^{202}}}} \) is divided by \( 101^2 \), we can use properties of modular arithmetic, specifically Euler's theorem and the Chinese Remainder Theorem.
1. **Base Reduction Using Binomial Theorem:**
\[
102 \equiv 1 \pmo... | <think>
Okay, let's try to figure out how to compute f(101). The problem says that for any positive integer n, f(n) is the remainder when this big tower of exponents from (n+1) up to 2n is divided by n². So specifically, we need to compute the remainder when (102)^{(103)^{.^{.^{.^{201^{202}}}}} is divided by 101², righ... | {
"ground_truth": "506",
"style": "math_dapo"
} | {
"generation_model": "DeepSeek-R1",
"pass_rate_72b_tir": 0,
"problem_source": "aops_c4_high_school_math"
} | 4,474 | 0 | 17,150 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "Find the sum of the series\n$$1 - 2 + 3 - 5 + 8 - 13 + \\ldots + 2584 - 4181 + 6765.$$",
"role": "user"
}
] | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To find the sum of the series \(1 - 2 + 3 - 5 + 8 - 13 + \ldots + 2584 - 4181 + 6765\), we first recognize that the terms of the series are the Fibonacci numbers with alternating signs. The Fibonacci sequence starts with \(F_1 = 1\) and \(F_2 = 2\), and each subsequent term is the sum of the previous two terms.
The se... | <think>
Okay, so I have this math problem here: I need to find the sum of the series 1 - 2 + 3 - 5 + 8 - 13 + ... + 2584 - 4181 + 6765. Hmm, let me see. First, I should try to figure out the pattern of the series. Let me write down the terms to see if I can recognize a sequence here.
The given series is: 1, -2, +3, -5... | {
"ground_truth": "4181",
"style": "math_dapo"
} | {
"generation_model": "QwQ-32B",
"pass_rate_72b_tir": 0.03125,
"problem_source": "aops_c6_high_school_olympiads"
} | 6,574 | 0 | 19,109 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "Determine the greatest possible jersey number of the reserve player who substituted the goal-keeper with jersey number 1, such ... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To determine the greatest possible jersey number of the reserve player who substituted the goalkeeper with jersey number 1 such that the sums of the jersey numbers of each team are exactly equal after the substitution, we can follow these steps:
1. **Assumptions and Total Sum Calculation**:
- Assume there are 22 pl... | <think>
Okay, let's try to figure out this problem. So, the question is about substituting a goalkeeper with jersey number 1 with a reserve player. We need to determine the greatest possible jersey number of this reserve player such that the sums of the jersey numbers of each team are exactly equal after the substituti... | {
"ground_truth": "122",
"style": "math_dapo"
} | {
"generation_model": "DeepSeek-R1",
"pass_rate_72b_tir": 0,
"problem_source": "aops_c6_high_school_olympiads"
} | 5,546 | 0 | 31,857.875 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "Convex pentagon \\(ABCDE\\) is inscribed in circle \\(\\omega\\) such that \\(\\frac{AC}{DE} = \\frac{2}{3}\\), \\(AE = CD\\), ... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To solve the problem, we need to find the radius \( R \) of the circle \(\omega\) in which the convex pentagon \(ABCDE\) is inscribed. The given conditions are:
- \(\frac{AC}{DE} = \frac{2}{3}\)
- \(AE = CD\)
- \(AB = BC\)
- The distance from \(B\) to line \(AC\) is 144
- The distance from \(B\) to line \(DE\) is 864
... | <think>
Okay, so I have this geometry problem here about a convex pentagon inscribed in a circle. The pentagon is named ABCDE, and there are a few conditions given: the ratio of AC to DE is 2/3, AE equals CD, and AB equals BC. Also, the distance from point B to the line AC is 144, and the distance from B to line DE is ... | {
"ground_truth": "648",
"style": "math_dapo"
} | {
"generation_model": "DeepSeek-R1",
"pass_rate_72b_tir": 0.09375,
"problem_source": "aops_c4_high_school_math"
} | 7,729 | 0 | 25,054.125 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "For every integer $n \\ge 1$, the probability that symptoms of cosineavirus show on the $n$th day is $\\frac{1}{n(n + 1)}$. Sup... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To solve the problem, we need to find the expected number of days \( r \) until someone first shows symptoms of cosineavirus, given that the probability of symptoms showing on the \( n \)-th day is \( \frac{1}{n(n + 1)} \).
### Step-by-Step Solution:
1. **Understanding the Probability Distribution:**
Each person i... | <think>
Okay, let's tackle this problem step by step. So, the question is about finding the expected number of days until someone first shows symptoms of cosineavirus. The setup is that each day starting from Day 0, one new person gets infected. On the nth day (for n ≥ 1), the probability that the person shows symptoms... | {
"ground_truth": "171",
"style": "math_dapo"
} | {
"generation_model": "DeepSeek-R1",
"pass_rate_72b_tir": 0,
"problem_source": "aops_c6_high_school_olympiads"
} | 6,157 | 0 | 22,987.3125 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "In a class of 440 students, each student votes for 2 candidates to elect 3 representatives. What is the minimum number of votes... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To determine the minimum number of votes required to guarantee a student will be chosen as a representative, we start by noting that there are 440 students, each voting for 2 candidates, resulting in a total of \(440 \times 2 = 880\) votes. We need to ensure that a candidate is among the top three vote-getters.
Let's ... | <think>
Okay, so there's this problem where there are 440 students in a class, and each student votes for 2 candidates to elect 3 representatives. The question is asking for the minimum number of votes that guarantees a student will be chosen as a representative. Hmm, let's break this down step by step.
First, let me ... | {
"ground_truth": "221",
"style": "math_dapo"
} | {
"generation_model": "DeepSeek-R1",
"pass_rate_72b_tir": 0.0625,
"problem_source": "aops_c6_high_school_olympiads"
} | 4,089 | 0 | 29,726.4375 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "Let \\( k = a_1 + a_2 + a_3 + \\cdots + a_n \\), where \\( a_i \\) are distinct positive integers and \\( n \\) is a positive i... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To find the maximum value of \( k \) given that \( k = a_1 + a_2 + a_3 + \cdots + a_n \) and \( \prod_{i=1}^n a_i = 2012 \), where \( a_i \) are distinct positive integers, we start by factorizing 2012:
\[ 2012 = 2^2 \times 503 \]
The divisors of 2012 are:
\[ 1, 2, 4, 503, 1006, 2012 \]
We need to find the combinati... | <think>
Okay, so I need to find the maximum value of k, which is the sum of distinct positive integers a1, a2, ..., an, such that their product is 2012. Hmm. Let's see. The problem says the product is 2012, and we need the sum to be as large as possible.
First, I remember that when you want to maximize the sum of num... | {
"ground_truth": "2013",
"style": "math_dapo"
} | {
"generation_model": "DeepSeek-R1",
"pass_rate_72b_tir": 0.09375,
"problem_source": "aops_c6_high_school_olympiads"
} | 5,497 | 0 | 20,986.3125 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "There are stones with a total mass of 9 tons that need to be transported by trucks. None of the stones is heavier than 1 ton, a... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To determine the minimum number of trucks necessary to transport 9 tons of stones, where each stone is no heavier than 1 ton and each truck has a capacity of 3 tons, we need to consider the worst-case scenario for stone distribution.
### Steps to Determine the Minimum Number of Trucks:
1. **Naive Approach**:
- If ... | <think>
Okay, so I have this problem here: there are stones with a total mass of 9 tons, and each stone is no heavier than 1 ton. I need to figure out the minimum number of trucks required to transport all of them at once, with each truck having a 3-ton capacity. Hmm, let me think through this step by step.
First off,... | {
"ground_truth": "4",
"style": "math_dapo"
} | {
"generation_model": "QwQ-32B",
"pass_rate_72b_tir": 0,
"problem_source": "aops_c6_high_school_olympiads"
} | 8,188 | 0 | 27,011.0625 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "In a 90-minute football match, each player must play the same integer number of minutes, but each player has to play less than ... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To solve the problem of determining the minimum number of players required for a 90-minute football match where each player must play the same integer number of minutes and each player plays less than 60 minutes, follow these steps:
1. **Total Playing Time**: In a football match, there are 11 players on the field at a... | <think>
Okay, so I need to figure out the minimum number of players required for a 90-minute football match where each player plays the same integer number of minutes, but each one has to play less than 60 minutes. Hmm, let me break this down.
First, the total playing time of the team is 90 minutes, right? Because tha... | {
"ground_truth": "18",
"style": "math_dapo"
} | {
"generation_model": "QwQ-32B",
"pass_rate_72b_tir": 0,
"problem_source": "aops_c6_high_school_olympiads"
} | 5,463 | 0 | 26,513.125 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "Each of 27 bricks with dimensions \\(a \\times b \\times c\\) (where \\(a\\), \\(b\\), and \\(c\\) are pairwise relatively prim... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To solve the problem, we need to determine the dimensions \(a\), \(b\), and \(c\) of the bricks such that they satisfy the given conditions. Let's break down the solution step-by-step.
1. **Initial Setup**:
- The original block is a \(3 \times 3 \times 3\) block formed by 27 bricks.
- Each brick has dimensions \... | <think>
Alright, let's try to solve this problem. Hmm, okay, we have bricks with dimensions a, b, c which are pairwise relatively prime positive integers. So first, there are 27 bricks arranged in a 3x3x3 block. Then, when adding a 28th brick, they form a 2x2x7 block. The new block is 1 unit taller, 1 unit wider, and 1... | {
"ground_truth": "92",
"style": "math_dapo"
} | {
"generation_model": "DeepSeek-R1",
"pass_rate_72b_tir": 0,
"problem_source": "aops_c5_contests_amp_programs"
} | 5,720 | 0 | 27,555.0625 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "An angle of depression from the top of a 75-foot tall lighthouse to a ship in the water is 31 degrees. What is the length of th... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To solve the problem, we need to determine the length of the rope attached directly from the ship to the top of the lighthouse. We are given that the height of the lighthouse is 75 feet and the angle of depression from the top of the lighthouse to the ship is 31 degrees.
1. **Understand the Geometry:**
- The height... | <think>
Okay, so I need to solve this problem where there's a lighthouse that's 75 feet tall, and the angle of depression from the top of the lighthouse to a ship in the water is 31 degrees. The question is asking for the length of the rope that's attached directly from the ship to the top of the lighthouse. Hmm, let m... | {
"ground_truth": "146",
"style": "math_dapo"
} | {
"generation_model": "DeepSeek-R1",
"pass_rate_72b_tir": 0,
"problem_source": "aops_c4_high_school_math"
} | 4,684 | 0 | 21,980.9375 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "Let \\( P(x) \\) be a quadratic polynomial with complex coefficients whose \\( x^2 \\) coefficient is \\( 1 \\). Suppose the eq... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To solve the problem, we need to find the sum of all possible values of \((a + b)^2\) given that \(P(x)\) is a quadratic polynomial with complex coefficients and the equation \(P(P(x)) = 0\) has four distinct solutions \(x = 3, 4, a, b\).
### Step-by-Step Solution:
1. **Form of \(P(x)\)**:
Since \(P(x)\) is a quad... | <think>
Okay, let's see. I need to solve this problem where there's a quadratic polynomial P(x) with complex coefficients, leading coefficient 1. The equation P(P(x)) = 0 has four distinct solutions: 3, 4, a, b. I need to find the sum of all possible values of (a + b)^2. Hmm, okay.
First, let me recall that since P(x)... | {
"ground_truth": "85",
"style": "math_dapo"
} | {
"generation_model": "QwQ-32B",
"pass_rate_72b_tir": 0,
"problem_source": "aops_c5_contests_amp_programs"
} | 6,931 | 0 | 22,397.1875 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "Find the minimum number of colors necessary to color the integers from 1 to 2007 such that if distinct integers \\(a\\), \\(b\\... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To solve the problem of finding the minimum number of colors necessary to color the integers from 1 to 2007 such that no three distinct integers \(a\), \(b\), and \(c\) of the same color form a divisibility chain \(a \mid b \mid c\), we need to consider the structure of the poset (partially ordered set) under divisibil... | <think>
Okay, so I need to figure out the minimum number of colors required to color the integers from 1 to 2007 such that in any group of three distinct numbers of the same color, if a divides b and b divides c, then those three can't all be the same color. Hmm, wait, let me parse the problem again. The condition is t... | {
"ground_truth": "6",
"style": "math_dapo"
} | {
"generation_model": "DeepSeek-R1",
"pass_rate_72b_tir": 0.03125,
"problem_source": "aops_c4_high_school_math"
} | 8,190 | 0 | 32,334.75 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "Let \\(R\\) be a square region and \\(n \\ge 4\\) an integer. A point \\(X\\) in the interior of \\(R\\) is called \\(n\\)-ray ... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To solve the problem of finding the number of points that are 100-ray partitional but not 60-ray partitional in a square region, we need to analyze the conditions under which a point \(X\) inside the square is \(n\)-ray partitional.
A point \(X\) is \(n\)-ray partitional if there are \(n\) rays emanating from \(X\) th... | <think>
Okay, so I have this problem here about n-ray partitional points in a square region. The question is asking for the number of points that are 100-ray partitional but not 60-ray partitional. Hmm, let me try to break this down.
First, let me recall what an n-ray partitional point is. The definition says a point ... | {
"ground_truth": "2320",
"style": "math_dapo"
} | {
"generation_model": "DeepSeek-R1",
"pass_rate_72b_tir": 0,
"problem_source": "aops_c5_contests_amp_programs"
} | 6,141 | 0 | 32,549.5 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "Determine the composite numbers \\( n \\) for which there exist positive integers \\( a \\) and \\( b \\) such that the set \\(... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To determine the composite numbers \( n \) for which there exist positive integers \( a \) and \( b \) such that the set \( A = \{(a+b)^2, (a+2b)^2, \ldots, (a+nb)^2\} \) is isolated, we need to ensure that the sum of any proper subset of \( A \) is coprime with the total sum \( S \) of all elements in \( A \).
### St... | <think>
Okay, let's try to tackle this problem. Hmm, we need to find all composite numbers n such that there exist positive integers a and b where the set A = {(a+b)^2, (a+2b)^2, ..., (a+nb)^2} is isolated. Isolated means that the sum of any proper subset is coprime with the sum of the entire set.
First, let me make ... | {
"ground_truth": "6",
"style": "math_dapo"
} | {
"generation_model": "DeepSeek-R1",
"pass_rate_72b_tir": 0.03125,
"problem_source": "aops_c6_high_school_olympiads"
} | 7,840 | 0 | 22,741.0625 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "Find the least \\( M \\) such that the following inequality holds for positive reals \\( a, b, c \\):\n\\[ |(a-b)(b-c)|^{3} + |... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To find the least \( M \) such that the inequality
\[ |(a-b)(b-c)|^{3} + |(b-c)(c-a)|^{3} + |(c-a)(a-b)|^{3} \le M(a^{3} + b^{3} + c^{3})^{2} \]
holds for all positive reals \( a, b, c \), we proceed as follows:
1. **Case Analysis:**
- **Case 1: \( a = b = c \)**
- Here, the left-hand side (LHS) becomes \( 0 \... | <think>
Okay, so I need to find the smallest M such that the inequality holds for all positive real numbers a, b, c. The left side is the sum of the cubes of the absolute values of the products of differences, and the right side is M times the square of the sum of the cubes of a, b, and c. Hmm, this looks a bit complex... | {
"ground_truth": "1",
"style": "math_dapo"
} | {
"generation_model": "DeepSeek-R1",
"pass_rate_72b_tir": 0.03125,
"problem_source": "aops_c6_high_school_olympiads"
} | 7,783 | 0 | 21,208.375 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "Given the polynomial \\(x^2 - 25x - c\\) where \\(c\\) is an integer, find \\(c\\) for which the roots of the equation are both... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To solve the problem of finding the integer \( c \) for which the roots of the polynomial \( x^2 - 25x - c \) are both positive prime numbers, we can proceed as follows:
1. **Identify the roots using Vieta's formulas:**
- According to Vieta's formulas, for a quadratic equation \( ax^2 + bx + c = 0 \), the sum of th... | <think>
Okay, let's see. I need to find the value of c such that the quadratic equation x² -25x -c has both roots as positive prime numbers. Hmm, primes are numbers greater than 1 that have no divisors other than 1 and themselves. So the roots p and q must be primes, and both positive. Let's start by recalling Vieta's ... | {
"ground_truth": "46",
"style": "math_dapo"
} | {
"generation_model": "DeepSeek-R1",
"pass_rate_72b_tir": 0,
"problem_source": "aops_c4_high_school_math"
} | 2,431 | 0 | 8,816.25 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "In a regular 10 × 10 grid of students, how many students can the student in the bottom-left corner see, given that any three co... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To determine how many students the student in the bottom-left corner of a 10×10 grid can see, we need to consider the condition that any three collinear students obstruct the line-of-sight between the other two students. This means a student at position \((0,0)\) can see another student at \((x,y)\) if there are no oth... | <think>
Okay, let me try to figure out this problem. So, the question is about a regular 10x10 grid of students, and we need to determine how many students the student in the bottom-left corner can see. The catch is that if three students are collinear, they block each other's line of sight. Hmm. Let me start by visual... | {
"ground_truth": "57",
"style": "math_dapo"
} | {
"generation_model": "DeepSeek-R1",
"pass_rate_72b_tir": 0.09375,
"problem_source": "aops_c4_high_school_math"
} | 6,422 | 0 | 23,958.6875 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "Two rectangles of different sizes have the same side ratio. They are positioned such that each vertex of the smaller rectangle ... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To determine the possible side ratios of two rectangles that allow the smaller rectangle to be positioned such that each vertex lies on the sides of the larger rectangle, we start by noting that both rectangles must have the same aspect ratio. Let the larger rectangle have sides \(a\) and \(b\) with aspect ratio \(r = ... | <think>
Okay, let's see. So the problem is about two rectangles with the same aspect ratio, meaning their sides are proportional. The smaller rectangle is positioned inside the larger one such that each of its vertices lies on the sides of the larger rectangle. We need to find the possible side ratios that allow this s... | {
"ground_truth": "1",
"style": "math_dapo"
} | {
"generation_model": "DeepSeek-R1",
"pass_rate_72b_tir": 0.0625,
"problem_source": "aops_c4_high_school_math"
} | 7,132 | 0 | 28,200.9375 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "Eleven points \\( A_1, A_2, \\dots, A_{11} \\) are arranged on a circle in that order and connected by chords as follows: \\( A... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To solve the problem of finding the minimum possible value of the sum of the angles \( A_1 + A_2 + \dots + A_{11} \) formed by the chords connecting the points \( A_1, A_2, \dots, A_{11} \) on a circle, we proceed as follows:
1. **Identify the Chord Connections**:
- The points are connected such that \( A_1 \) is c... | <think>
Okay, let's see. So, there are eleven points arranged on a circle in order from A₁ to A₁₁. The problem describes connecting these points with specific chords: A₁ to A₃, A₃ to A₅, and so on, skipping one each time. Then, after reaching A₁₁, it connects back to A₂, and continues this pattern. The question is aski... | {
"ground_truth": "1260",
"style": "math_dapo"
} | {
"generation_model": "DeepSeek-R1",
"pass_rate_72b_tir": 0.03125,
"problem_source": "aops_c6_high_school_olympiads"
} | 7,883 | 0 | 26,575.3125 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "Twelve congruent disks are placed on a circle \\( C \\) of radius 1 in such a way that the twelve disks cover \\( C \\), no two... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To solve the problem, we need to determine the radius of the twelve congruent disks arranged around a circle \( C \) of radius 1, such that the disks cover \( C \), do not overlap, and each is tangent to its neighbors. The sum of the areas of the disks is expressed in the form \( \pi(a - b\sqrt{c}) \), and we need to f... | <think>
Okay, so we have this problem where twelve congruent disks are placed on a circle C of radius 1. The conditions are that the disks cover the circle C, they don't overlap, and each disk is tangent to its two neighbors. We need to find the sum of the areas of the twelve disks expressed as π(a - b√c) and then find... | {
"ground_truth": "135",
"style": "math_dapo"
} | {
"generation_model": "DeepSeek-R1",
"pass_rate_72b_tir": 0,
"problem_source": "aops_c4_high_school_math"
} | 5,533 | 0 | 26,117 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "Given \\( n \\), let \\( k = k(n) \\) be the minimal degree of any monic integral polynomial\n\\[ f(x) = x^k + a_{k-1}x^{k-1} +... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To solve the problem of finding the minimal degree \( k(n) \) of a monic integral polynomial \( f(x) \) such that \( f(x) \) is divisible by \( n \) for every integer \( x \), and specifically for \( n = 10^6 \), we follow these steps:
1. **Understanding the Problem**:
- The polynomial must be monic with integer co... | <think>
Okay, so I need to figure out the minimal degree k(n) of a monic polynomial with integer coefficients such that for every integer x, the polynomial f(x) is divisible by n. Then, specifically find k(10^6). Let's start breaking this down.
First, let's understand what the problem is asking. We need a monic polyno... | {
"ground_truth": "25",
"style": "math_dapo"
} | {
"generation_model": "DeepSeek-R1",
"pass_rate_72b_tir": 0,
"problem_source": "aops_c7_college_math"
} | 5,936 | 0 | 23,554.3125 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "In a senate of 30 people, each pair of senators are either friends or enemies, and each senator has exactly 6 enemies. Every th... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To solve the problem of finding the total number of committees of three senators that are either all friends or all enemies in a senate of 30 senators where each senator has exactly 6 enemies and 23 friends, we can use graph theory.
1. **Model the Problem:**
- Represent the senate as a graph \( G \) where each sena... | <think>
Okay, let's tackle this problem step by step. So, we have a senate with 30 senators. Each pair is either friends or enemies, and each senator has exactly 6 enemies. We need to find the number of committees (which are groups of three senators) where all three members are either all friends or all enemies with ea... | {
"ground_truth": "1990",
"style": "math_dapo"
} | {
"generation_model": "QwQ-32B",
"pass_rate_72b_tir": 0.03125,
"problem_source": "aops_c6_high_school_olympiads"
} | 6,716 | 0 | 31,668.375 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "Positive integers \\( x_1, \\ldots, x_m \\) (not necessarily distinct) are written on a blackboard. It is known that each of th... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To solve the problem of finding the smallest number \( m \) such that each of the first 2018 Fibonacci numbers can be represented as a sum of one or more of the numbers on the blackboard, we need to leverage the properties of Fibonacci numbers and their representations.
### Step-by-Step Solution:
1. **Understanding t... | <think>
Okay, so I need to figure out the smallest number of positive integers \( x_1, \ldots, x_m \) on a blackboard such that every one of the first 2018 Fibonacci numbers can be expressed as a sum of one or more of these numbers. The Fibonacci sequence starts with \( F_1 = 1 \), \( F_2 = 1 \), and each subsequent te... | {
"ground_truth": "1009",
"style": "math_dapo"
} | {
"generation_model": "QwQ-32B",
"pass_rate_72b_tir": 0,
"problem_source": "aops_c6_high_school_olympiads"
} | 7,757 | 0 | 30,617 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "We wish to find the sum of 40 given numbers using 40 processors. Initially, each processor has the number 0 on its screen. Each... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To find the sum of 40 given numbers using 40 processors, we need to follow a systematic approach. Each processor starts with the number 0 on its screen and can add a number either directly or transferred from another processor in a unit time. When a number is transferred, the sender resets to 0.
### Step-by-Step Solut... | <think>
Okay, so I need to figure out the least time required to sum 40 numbers using 40 processors. Each processor starts with 0. They can add numbers either by receiving a direct input (one of the 40 numbers) or by getting a number transferred from another processor. But when a number is transferred, the sending proc... | {
"ground_truth": "7",
"style": "math_dapo"
} | {
"generation_model": "DeepSeek-R1",
"pass_rate_72b_tir": 0.09375,
"problem_source": "aops_c6_high_school_olympiads"
} | 4,094 | 0 | 34,609.25 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "How many ordered triples $(a, b, c)$ of not necessarily distinct positive integer divisors of $216,000$ satisfy $$\\text{lcm(a,... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To solve the problem, we start by factorizing \(216,000\) into its prime factors:
\[
216,000 = 2^6 \times 3^3 \times 5^3
\]
We need to find the number of ordered triples \((a, b, c)\) of divisors of \(216,000\) such that:
\[
\text{lcm}(a, \text{gcd}(b, c)) = \text{lcm}(b, \text{gcd}(a, c)) = \text{lcm}(c, \text{gcd}(a,... | <think>
Okay, let's see. The problem is asking for the number of ordered triples (a, b, c) of positive integer divisors of 216,000 such that the least common multiple (LCM) conditions hold: LCM(a, GCD(b, c)) equals LCM(b, GCD(a, c)), and that equals LCM(c, GCD(a, b)). All three LCM expressions must be equal. The diviso... | {
"ground_truth": "33880",
"style": "math_dapo"
} | {
"generation_model": "DeepSeek-R1",
"pass_rate_72b_tir": 0.0625,
"problem_source": "aops_c4_high_school_math"
} | 7,175 | 0 | 26,742.0625 |
[
{
"content": "You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \\boxed{}.",
"role": "system"
},
{
"content": "Let \\( X = \\{1, 2, \\ldots, 2018\\} \\) and \\( A \\) be a subset of \\( X \\) such that for any \\( x, y, z \\in A \\) with ... | You are a careful mathematician. Solve the following problem. Show your work, then give the final answer inside \boxed{}. | math_dapo | math | To solve the problem of finding the maximum size of a subset \( A \) of \( X = \{1, 2, \ldots, 2018\} \) such that any three elements \( x, y, z \in A \) with \( x < y < z \) form a triangle, we need to ensure that the sum of the two smaller numbers is greater than the largest number. This translates to the condition \... | <think>
Okay, so I need to find the maximum size of a subset A of the set X = {1, 2, ..., 2018} such that any three elements x, y, z in A (with x < y < z) can form a triangle. The triangle inequality says that the sum of any two sides must be greater than the third side. For three numbers to form a triangle, the sum of... | {
"ground_truth": "1010",
"style": "math_dapo"
} | {
"generation_model": "DeepSeek-R1",
"pass_rate_72b_tir": 0.09375,
"problem_source": "aops_c6_high_school_olympiads"
} | 5,528 | 0 | 25,414.6875 |
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