Datasets:

Xiaodong
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afd_mix_p50_verified

source stringclasses 4 values	problem stringlengths 16 4.74k	solution stringlengths 152 7.07k	messages listlengths 2 2	system stringclasses 1 value	conversations listlengths 2 2	generated_token_count int64 398 5k	correct bool 1 class	Question stringlengths 60 4.78k	COT_Reason stringlengths 832 20.4k	Answer stringlengths 0 423	COT_Reason_rewritten stringlengths 959 20.4k	is_rewritten bool 2 classes
olympiads	Given real numbers $ a, b, c $ and a positive number $ \lambda $ such that the polynomial $ f(x) = x^3 + a x^2 + b x + c $ has three real roots $ x_1, x_2, x_3 $, and the conditions $ x_2 - x_1 = \lambda $ and $ x_3 > \frac{1}{2}(x_1 + x_2) $ are satisfied, find the maximum value of \( \frac{2 a^3 + 27 c - ...	We begin by analyzing the function $ f(x) = x^3 + a x^2 + b x + c $, which has three real roots $ x_1, x_2, x_3 $. We are given the following conditions: 1. $ x_2 - x_1 = \lambda $ 2. $ x_3 > \frac{1}{2} (x_1 + x_2) $ We aim to find the maximum value of $ \frac{2a^3 + 27c - 9ab}{\lambda^3} $. 1. **Transfor...	[ { "content": "Given real numbers \$ a, b, c \$ and a positive number \$ \\lambda \$ such that the polynomial \$ f(x) = x^3 + a x^2 + b x + c \$ has three real roots \$ x_1, x_2, x_3 \$, and the conditions \$ x_2 - x_1 = \\lambda \$ and \$ x_3 > \\frac{1}{2}(x_1 + x_2) \$ are satisfied, find the maxi...	Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w...	[ { "from": "user", "value": "Return your final response within \\boxed{}. Given real numbers \$ a, b, c \$ and a positive number \$ \\lambda \$ such that the polynomial \$ f(x) = x^3 + a x^2 + b x + c \$ has three real roots \$ x_1, x_2, x_3 \$, and the conditions \$ x_2 - x_1 = \\lambda \$ and \\(...	4,978	true	Return your final response within \boxed{}. Given real numbers $ a, b, c $ and a positive number $ \lambda $ such that the polynomial $ f(x) = x^3 + a x^2 + b x + c $ has three real roots $ x_1, x_2, x_3 $, and the conditions $ x_2 - x_1 = \lambda $ and $ x_3 > \frac{1}{2}(x_1 + x_2) $ are satisfied, find t...	Okay, let's try to tackle this problem step by step. The problem is about finding the maximum value of the expression (2a³ + 27c - 9ab)/λ³ given some conditions on the roots of the cubic polynomial f(x) = x³ + ax² + bx + c. The roots are real, with x₂ - x₁ = λ and x₃ > (x₁ + x₂)/2. First, I need to recall that for a ...	\frac{3\sqrt{3}}{2}	Okay, let's try to tackle this problem step by step. The problem is about finding the maximum value of the expression (2a³ + 27c - 9ab)/λ³ given some conditions on the roots of the cubic polynomial f(x) = x³ + ax² + bx + c. The roots are real, with x₂ - x₁ = λ and x₃ > (x₁ + x₂)/2. First, I need to recall that for a ...	false
olympiads	Let $ p = 2^{3009}, q = 3^{2006}, $ and $ r = 5^{1003} $. Which of the following statements is true? (A) $ p < q < r $ (B) $ p < r < q $ (C) $ q < p < r $ (D) $ r < p < q $ (E) $ q < r < p $	Given the values: \[ p = 2^{3009}, \quad q = 3^{2006}, \quad r = 5^{1003} \] 1. Express $ p $ and $ q $ in terms of powers of the same base: \[ p = 2^{3009} = 2^{3 \times 1003} = (2^3)^{1003} = 8^{1003} \] \[ q = 3^{2006} = 3^{2 \times 1003} = (3^2)^{1003} = 9^{1003} \] Note: $ r = 5^{1003} $ is already...	[ { "content": "Let \$ p = 2^{3009}, q = 3^{2006}, \$ and \$ r = 5^{1003} \$. Which of the following statements is true?\n(A) \$ p < q < r \$\n(B) \$ p < r < q \$\n(C) \$ q < p < r \$\n(D) \$ r < p < q \$\n(E) \$ q < r < p \$", "role": "user" }, { "content": "Given the values:\n\\[ p = 2...	Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w...	[ { "from": "user", "value": "Return your final response within \\boxed{}. Let \$ p = 2^{3009}, q = 3^{2006}, \$ and \$ r = 5^{1003} \$. Which of the following statements is true?\n(A) \$ p < q < r \$\n(B) \$ p < r < q \$\n(C) \$ q < p < r \$\n(D) \$ r < p < q \$\n(E) \$ q < r < p \$" }, { ...	2,618	true	Return your final response within \boxed{}. Let $ p = 2^{3009}, q = 3^{2006}, $ and $ r = 5^{1003} $. Which of the following statements is true? (A) $ p < q < r $ (B) $ p < r < q $ (C) $ q < p < r $ (D) $ r < p < q $ (E) $ q < r < p $	Okay, so I need to figure out which of the given options is correct by comparing the sizes of p, q, and r where p = 2^3009, q = 3^2006, and r = 5^1003. Hmm, comparing exponents with different bases and exponents... This might be tricky. Let me think about how to approach this. First, I remember that comparing numbers ...	D	I'm tackling this problem by comparing the sizes of p, q, and r where p = 2^3009, q = 3^2006, and r = 5^1003. I'm noticing that comparing exponents with different bases and exponents is tricky, so I'm thinking about how to approach this. I'm recalling that comparing numbers with different exponents can be done by taki...	true
olympiads	Given that $1 \leq x, y, z \leq 6$, how many cases are there in which the product of natural numbers $x, y, z$ is divisible by 10?	Given the constraints $1 \leq x, y, z \leq 6$, we are to find the number of natural number combinations $(x, y, z)$ such that their product can be divided exactly by 10. To begin, we observe: 1. The total number of combinations of $x, y, z$ is $6^3$: \[ 6^3 = 216 \] 2. To be divisible by 10, the pro...	[ { "content": "Given that \$1 \\leq x, y, z \\leq 6\$, how many cases are there in which the product of natural numbers \$x, y, z\$ is divisible by 10?", "role": "user" }, { "content": "Given the constraints \$1 \\leq x, y, z \\leq 6\$, we are to find the number of natural number combinations \...	Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w...	[ { "from": "user", "value": "Return your final response within \\boxed{}. Given that \$1 \\leq x, y, z \\leq 6\$, how many cases are there in which the product of natural numbers \$x, y, z\$ is divisible by 10?" }, { "from": "assistant", "value": "<\|begin_of_thought\|>\n\nOkay, let's see. The ...	3,632	true	Return your final response within \boxed{}. Given that $1 \leq x, y, z \leq 6$, how many cases are there in which the product of natural numbers $x, y, z$ is divisible by 10?	Okay, let's see. The problem is asking how many ordered triples (x, y, z) there are, where each of x, y, z is a natural number between 1 and 6 inclusive, such that their product xyz is divisible by 10. Hmm, so I need to count all possible triples where the product is a multiple of 10. First, let me recall that a nu...	72	Okay, let's see. The problem is asking how many ordered triples (x, y, z) there are, where each of x, y, z is a natural number between 1 and 6 inclusive, such that their product xyz is divisible by 10. Hmm, so I need to count all possible triples where the product is a multiple of 10. First, let me recall that a nu...	false
olympiads	How many plums will balance one pear, given that 3 apples and one pear weigh as much as 10 plums, and one apple and 6 plums balance one pear? Assume that fruits of the same kind have the same weight.	Given the problem, we are to find how many plums (sliv) balance with one pear (grusha). We have the following information: - 3 apples (yabloka) and 1 pear (grusha) together balance 10 plums (sliv). - 1 apple (yabloko) and 6 plums (slivy) balance 1 pear (grusha). We begin by setting up equations based on the balance ...	[ { "content": "How many plums will balance one pear, given that 3 apples and one pear weigh as much as 10 plums, and one apple and 6 plums balance one pear? Assume that fruits of the same kind have the same weight.", "role": "user" }, { "content": "\nGiven the problem, we are to find how many plums (...	Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w...	[ { "from": "user", "value": "Return your final response within \\boxed{}. How many plums will balance one pear, given that 3 apples and one pear weigh as much as 10 plums, and one apple and 6 plums balance one pear? Assume that fruits of the same kind have the same weight." }, { "from": "assistant", ...	1,178	true	Return your final response within \boxed{}. How many plums will balance one pear, given that 3 apples and one pear weigh as much as 10 plums, and one apple and 6 plums balance one pear? Assume that fruits of the same kind have the same weight.	Okay, let's see. I need to figure out how many plums balance one pear. The problem gives me two equations involving apples, pears, and plums. All fruits of the same type weigh the same. Hmm, let's break it down step by step. First, let me restate the information given. The problem says: 1. 3 apples and one pear weigh...	7	Okay, let's see. I need to figure out how many plums balance one pear. The problem gives me two equations involving apples, pears, and plums. All fruits of the same type weigh the same. Hmm, let's break it down step by step. First, let me restate the information given. The problem says: 1. 3 apples and one pear weigh...	false
olympiads	Determine the value of $$ z=a \sqrt{a} \sqrt[4]{a} \sqrt[8]{a} \ldots \sqrt[2^{n}]{a} \ldots $$ if $ n $ is infinitely large.	1. We begin by rewriting the given infinite product expression in a more manageable form. The given product is: \[ z = a \sqrt{a} \sqrt[4]{a} \sqrt[8]{a} \cdots \sqrt[2^{n}]{a} \cdots \] 2. Observe that each term in the product is of the form $ a^{1/2^k} $ where $ k $ is a non-negative integer. Hen...	[ { "content": "Determine the value of\n\n$$\nz=a \\sqrt{a} \\sqrt[4]{a} \\sqrt[8]{a} \\ldots \\sqrt[2^{n}]{a} \\ldots\n$$\n\nif \$ n \$ is infinitely large.", "role": "user" }, { "content": "\n1. We begin by rewriting the given infinite product expression in a more manageable form. The given produ...	Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w...	[ { "from": "user", "value": "Return your final response within \\boxed{}. Determine the value of\n\n$$\nz=a \\sqrt{a} \\sqrt[4]{a} \\sqrt[8]{a} \\ldots \\sqrt[2^{n}]{a} \\ldots\n$$\n\nif \$ n \$ is infinitely large." }, { "from": "assistant", "value": "<\|begin_of_thought\|>\n\nOkay, so I need to...	2,847	true	Return your final response within \boxed{}. Determine the value of $$ z=a \sqrt{a} \sqrt[4]{a} \sqrt[8]{a} \ldots \sqrt[2^{n}]{a} \ldots $$ if $ n $ is infinitely large.	Okay, so I need to find the value of z, which is this infinite product of a multiplied by the square root of a, then the fourth root, then the eighth root, and so on, each time the exponent being 1 over 2 to the nth power as n goes to infinity. Hmm, let's break this down step by step. First, let's write out the expres...	a^2	I'm tackling this problem step by step. I need to find the value of z, which is this infinite product of a multiplied by the square root of a, then the fourth root, then the eighth root, and so on, each time the exponent being 1 over 2 to the nth power as n goes to infinity. I'm breaking this down step by step. I'm wr...	true
olympiads	Give the value of $0 - 1 + 2 - 3 + 4 - 5 + \ldots - 49 + 50$. Only a numerical answer is expected.	To find the value of the series $0 - 1 + 2 - 3 + 4 - 5 + \ldots - 49 + 50$, we group the terms in pairs: 1. Group terms in pairs: \[ (0 - 1) + (2 - 3) + (4 - 5) + \ldots + (48 - 49) + 50 \] 2. Each of these pairs, except for the last term 50, simplifies as follows: \[ (0 - 1) = -1, \quad (2 - 3) = -1...	[ { "content": "Give the value of \$0 - 1 + 2 - 3 + 4 - 5 + \\ldots - 49 + 50\$. Only a numerical answer is expected.", "role": "user" }, { "content": "\nTo find the value of the series \$0 - 1 + 2 - 3 + 4 - 5 + \\ldots - 49 + 50\$, we group the terms in pairs:\n1. Group terms in pairs:\n \\[\n ...	Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w...	[ { "from": "user", "value": "Return your final response within \\boxed{}. Give the value of \$0 - 1 + 2 - 3 + 4 - 5 + \\ldots - 49 + 50\$. Only a numerical answer is expected." }, { "from": "assistant", "value": "<\|begin_of_thought\|>\n\nOkay, let me try to figure out the value of this series: 0...	2,324	true	Return your final response within \boxed{}. Give the value of $0 - 1 + 2 - 3 + 4 - 5 + \ldots - 49 + 50$. Only a numerical answer is expected.	Okay, let me try to figure out the value of this series: 0 - 1 + 2 - 3 + 4 - 5 + ... - 49 + 50. Hmm, that's an alternating series where the signs alternate between negative and positive, starting with 0 minus 1. Let me write out the first few terms to see the pattern better. So the series is: 0, then -1, +2, -3, +4, -...	25	Okay, let me try to figure out the value of this series: 0 - 1 + 2 - 3 + 4 - 5 + ... - 49 + 50. Hmm, that's an alternating series where the signs alternate between negative and positive, starting with 0 minus 1. Let me write out the first few terms to see the pattern better. So the series is: 0, then -1, +2, -3, +4, -...	false
olympiads	"Verify that 2 is a solution of the equation:\n\n$$\n(x+1)^{3}+(x+2)^{3}+(x+3)^{3}=(x+4)^{3}\n$$\n\n(...TRUNCATED)	"\nThe problem requires us to verify if \$x = 2\$ is a solution to the equation \\((x+1)^3 + (x+2)(...TRUNCATED)	[{"content":"Verify that 2 is a solution of the equation:\n\n$$\n(x+1)^{3}+(x+2)^{3}+(x+3)^{3}=(x+4)(...TRUNCATED)	"Your role as an assistant involves thoroughly exploring questions through a systematic long thinkin(...TRUNCATED)	[{"from":"user","value":"Return your final response within \\boxed{}. Verify that 2 is a solution of(...TRUNCATED)	2,507	true	"Return your final response within \\boxed{}. Verify that 2 is a solution of the equation:\n\n$$\n(x(...TRUNCATED)	"Okay, so I need to verify if 2 is a solution to the equation (x+1)^3 + (x+2)^3 + (x+3)^3 = (x+4)^3.(...TRUNCATED)	2	"Okay, so I need to verify if 2 is a solution to the equation (x+1)^3 + (x+2)^3 + (x+3)^3 = (x+4)^3.(...TRUNCATED)	false
olympiads	"Find the function $F(x)$, whose graph passes through the point $M_{0}(3 ;-2)$, given that $F'(x) = (...TRUNCATED)	"To find the function \$ F(x) \$ whose graph passes through the point \$ M_0(3, -2) \$ and given(...TRUNCATED)	[{"content":"Find the function $F(x)$, whose graph passes through the point $M_{0}(3 ;-2)$, given th(...TRUNCATED)	"Your role as an assistant involves thoroughly exploring questions through a systematic long thinkin(...TRUNCATED)	[{"from":"user","value":"Return your final response within \\boxed{}. Find the function $F(x)$, whos(...TRUNCATED)	1,734	true	"Return your final response within \\boxed{}. Find the function $F(x)$, whose graph passes through t(...TRUNCATED)	"Okay, so I need to find the function F(x) whose graph passes through the point M₀(3, -2), and we (...TRUNCATED)	F(x)=\frac{4}{3} x^{3}-\frac{9}{x}-35	"I'm tackling this problem step by step. I need to find the function F(x) whose graph passes through(...TRUNCATED)	true
olympiads	"In an isosceles trapezoid with bases \$a = 21\$, \$b = 9\$ and height \$h = 8\$, find the rad(...TRUNCATED)	"\n1. Identify Given Data and Setup:\n - The given isosceles trapezoid \$ABCD\$ has bases \\(...TRUNCATED)	[{"content":"In an isosceles trapezoid with bases \$a = 21\$, \$b = 9\$ and height \$h = 8\$, (...TRUNCATED)	"Your role as an assistant involves thoroughly exploring questions through a systematic long thinkin(...TRUNCATED)	[{"from":"user","value":"Return your final response within \\boxed{}. In an isosceles trapezoid with(...TRUNCATED)	4,803	true	"Return your final response within \\boxed{}. In an isosceles trapezoid with bases \$a = 21\$, \\((...TRUNCATED)	"Okay, so I need to find the radius of the circumscribed circle around an isosceles trapezoid with b(...TRUNCATED)	\frac{85}{8}	"I'm tackling this problem step by step. I need to find the radius of the circumscribed circle aroun(...TRUNCATED)	true
olympiads	"Given two linear functions \$ f(x) \$ and \$ g(x) \$ such that the graphs \$ y = f(x) \$ and (...TRUNCATED)	": \nGiven are two linear functions \$ f(x) \$ and \$ g(x) \$ such that the graphs of \\( y = f((...TRUNCATED)	[{"content":"Given two linear functions \$ f(x) \$ and \$ g(x) \$ such that the graphs \\( y = f(...TRUNCATED)	"Your role as an assistant involves thoroughly exploring questions through a systematic long thinkin(...TRUNCATED)	[{"from":"user","value":"Return your final response within \\boxed{}. Given two linear functions \\((...TRUNCATED)	4,523	true	"Return your final response within \\boxed{}. Given two linear functions \$ f(x) \$ and \\( g(x) \(...TRUNCATED)	"Okay, so I need to find the minimum value of the function \$ 3(g(x))^2 + 2 f(x) \$, given that th(...TRUNCATED)	\frac{5}{2}	"Okay, so I need to find the minimum value of the function \$ 3(g(x))^2 + 2 f(x) \$, given that th(...TRUNCATED)	false

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Aha-Flow Distillation: Flow Markers Matter in LLM Reasoning

Overview

We identify the Flow Moment, a new confident reasoning format in LLM reasoning, and we explore the influence of this format on On-Policy Self-Distillation (OPSD). We propose Aha-Flow Distillation, a dual-mode extension of OPSD that combines Aha branch and Flow branch to distill the student model. This training on Qwen3-8B model only takes ~30 minutes on 8×H20 and peaks within 100 steps.

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hf download Xiaodong/afd_mix_p50_verified --repo-type=dataset --local-dir data/afd_mix_p50_verified

Citation

If you find this useful, please consider citing:

@article{wang2026Aha-Flow-Distillation,
  title={Aha-Flow Distillation: Flow Markers Matter in LLM Reasoning},
  author={Wang, Xiaodong and Peng, Peixi},
  journal={arXiv preprint arXiv:2609.07036},
  year={2026}
}

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Aha-Flow Distillation

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https://github.com/Wang-Xiaodong1899/Aha-Flow-Distillation • 3 items • Updated 1 day ago

Paper for Xiaodong/afd_mix_p50_verified

Aha-Flow Distillation: Flow Markers Matter in LLM Reasoning

Paper • 2609.07036 • Published 4 days ago • 1

source stringclasses 4 values	problem stringlengths 16 4.74k	solution stringlengths 152 7.07k	messages listlengths 2 2	system stringclasses 1 value	conversations listlengths 2 2	generated_token_count int64 398 5k	correct bool 1 class	Question stringlengths 60 4.78k	COT_Reason stringlengths 832 20.4k	Answer stringlengths 0 423	COT_Reason_rewritten stringlengths 959 20.4k	is_rewritten bool 2 classes
olympiads	Given real numbers \( a, b, c \) and a positive number \( \lambda \) such that the polynomial \( f(x) = x^3 + a x^2 + b x + c \) has three real roots \( x_1, x_2, x_3 \), and the conditions \( x_2 - x_1 = \lambda \) and \( x_3 > \frac{1}{2}(x_1 + x_2) \) are satisfied, find the maximum value of \( \frac{2 a^3 + 27 c - ...	We begin by analyzing the function \( f(x) = x^3 + a x^2 + b x + c \), which has three real roots \( x_1, x_2, x_3 \). We are given the following conditions: 1. \( x_2 - x_1 = \lambda \) 2. \( x_3 > \frac{1}{2} (x_1 + x_2) \) We aim to find the maximum value of \( \frac{2a^3 + 27c - 9ab}{\lambda^3} \). 1. **Transfor...	[ { "content": "Given real numbers \\( a, b, c \\) and a positive number \\( \\lambda \\) such that the polynomial \\( f(x) = x^3 + a x^2 + b x + c \\) has three real roots \\( x_1, x_2, x_3 \\), and the conditions \\( x_2 - x_1 = \\lambda \\) and \\( x_3 > \\frac{1}{2}(x_1 + x_2) \\) are satisfied, find the maxi...	Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w...	[ { "from": "user", "value": "Return your final response within \\boxed{}. Given real numbers \\( a, b, c \\) and a positive number \\( \\lambda \\) such that the polynomial \\( f(x) = x^3 + a x^2 + b x + c \\) has three real roots \\( x_1, x_2, x_3 \\), and the conditions \\( x_2 - x_1 = \\lambda \\) and \\(...	4,978	true	Return your final response within \boxed{}. Given real numbers \( a, b, c \) and a positive number \( \lambda \) such that the polynomial \( f(x) = x^3 + a x^2 + b x + c \) has three real roots \( x_1, x_2, x_3 \), and the conditions \( x_2 - x_1 = \lambda \) and \( x_3 > \frac{1}{2}(x_1 + x_2) \) are satisfied, find t...	Okay, let's try to tackle this problem step by step. The problem is about finding the maximum value of the expression (2a³ + 27c - 9ab)/λ³ given some conditions on the roots of the cubic polynomial f(x) = x³ + ax² + bx + c. The roots are real, with x₂ - x₁ = λ and x₃ > (x₁ + x₂)/2. First, I need to recall that for a ...	\frac{3\sqrt{3}}{2}	Okay, let's try to tackle this problem step by step. The problem is about finding the maximum value of the expression (2a³ + 27c - 9ab)/λ³ given some conditions on the roots of the cubic polynomial f(x) = x³ + ax² + bx + c. The roots are real, with x₂ - x₁ = λ and x₃ > (x₁ + x₂)/2. First, I need to recall that for a ...	false
olympiads	Let \( p = 2^{3009}, q = 3^{2006}, \) and \( r = 5^{1003} \). Which of the following statements is true? (A) \( p < q < r \) (B) \( p < r < q \) (C) \( q < p < r \) (D) \( r < p < q \) (E) \( q < r < p \)	Given the values: \[ p = 2^{3009}, \quad q = 3^{2006}, \quad r = 5^{1003} \] 1. Express \( p \) and \( q \) in terms of powers of the same base: \[ p = 2^{3009} = 2^{3 \times 1003} = (2^3)^{1003} = 8^{1003} \] \[ q = 3^{2006} = 3^{2 \times 1003} = (3^2)^{1003} = 9^{1003} \] Note: \( r = 5^{1003} \) is already...	[ { "content": "Let \\( p = 2^{3009}, q = 3^{2006}, \\) and \\( r = 5^{1003} \\). Which of the following statements is true?\n(A) \\( p < q < r \\)\n(B) \\( p < r < q \\)\n(C) \\( q < p < r \\)\n(D) \\( r < p < q \\)\n(E) \\( q < r < p \\)", "role": "user" }, { "content": "Given the values:\n\\[ p = 2...	Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w...	[ { "from": "user", "value": "Return your final response within \\boxed{}. Let \\( p = 2^{3009}, q = 3^{2006}, \\) and \\( r = 5^{1003} \\). Which of the following statements is true?\n(A) \\( p < q < r \\)\n(B) \\( p < r < q \\)\n(C) \\( q < p < r \\)\n(D) \\( r < p < q \\)\n(E) \\( q < r < p \\)" }, { ...	2,618	true	Return your final response within \boxed{}. Let \( p = 2^{3009}, q = 3^{2006}, \) and \( r = 5^{1003} \). Which of the following statements is true? (A) \( p < q < r \) (B) \( p < r < q \) (C) \( q < p < r \) (D) \( r < p < q \) (E) \( q < r < p \)	Okay, so I need to figure out which of the given options is correct by comparing the sizes of p, q, and r where p = 2^3009, q = 3^2006, and r = 5^1003. Hmm, comparing exponents with different bases and exponents... This might be tricky. Let me think about how to approach this. First, I remember that comparing numbers ...	D	I'm tackling this problem by comparing the sizes of p, q, and r where p = 2^3009, q = 3^2006, and r = 5^1003. I'm noticing that comparing exponents with different bases and exponents is tricky, so I'm thinking about how to approach this. I'm recalling that comparing numbers with different exponents can be done by taki...	true
olympiads	Given that \(1 \leq x, y, z \leq 6\), how many cases are there in which the product of natural numbers \(x, y, z\) is divisible by 10?	Given the constraints \(1 \leq x, y, z \leq 6\), we are to find the number of natural number combinations \((x, y, z)\) such that their product can be divided exactly by 10. To begin, we observe: 1. The total number of combinations of \(x, y, z\) is \(6^3\): \[ 6^3 = 216 \] 2. To be divisible by 10, the pro...	[ { "content": "Given that \\(1 \\leq x, y, z \\leq 6\\), how many cases are there in which the product of natural numbers \\(x, y, z\\) is divisible by 10?", "role": "user" }, { "content": "Given the constraints \\(1 \\leq x, y, z \\leq 6\\), we are to find the number of natural number combinations \...	Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w...	[ { "from": "user", "value": "Return your final response within \\boxed{}. Given that \\(1 \\leq x, y, z \\leq 6\\), how many cases are there in which the product of natural numbers \\(x, y, z\\) is divisible by 10?" }, { "from": "assistant", "value": "<\|begin_of_thought\|>\n\nOkay, let's see. The ...	3,632	true	Return your final response within \boxed{}. Given that \(1 \leq x, y, z \leq 6\), how many cases are there in which the product of natural numbers \(x, y, z\) is divisible by 10?	Okay, let's see. The problem is asking how many ordered triples (x, y, z) there are, where each of x, y, z is a natural number between 1 and 6 inclusive, such that their product xyz is divisible by 10. Hmm, so I need to count all possible triples where the product is a multiple of 10. First, let me recall that a nu...	72	Okay, let's see. The problem is asking how many ordered triples (x, y, z) there are, where each of x, y, z is a natural number between 1 and 6 inclusive, such that their product xyz is divisible by 10. Hmm, so I need to count all possible triples where the product is a multiple of 10. First, let me recall that a nu...	false
olympiads	How many plums will balance one pear, given that 3 apples and one pear weigh as much as 10 plums, and one apple and 6 plums balance one pear? Assume that fruits of the same kind have the same weight.	Given the problem, we are to find how many plums (sliv) balance with one pear (grusha). We have the following information: - 3 apples (yabloka) and 1 pear (grusha) together balance 10 plums (sliv). - 1 apple (yabloko) and 6 plums (slivy) balance 1 pear (grusha). We begin by setting up equations based on the balance ...	[ { "content": "How many plums will balance one pear, given that 3 apples and one pear weigh as much as 10 plums, and one apple and 6 plums balance one pear? Assume that fruits of the same kind have the same weight.", "role": "user" }, { "content": "\nGiven the problem, we are to find how many plums (...	Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w...	[ { "from": "user", "value": "Return your final response within \\boxed{}. How many plums will balance one pear, given that 3 apples and one pear weigh as much as 10 plums, and one apple and 6 plums balance one pear? Assume that fruits of the same kind have the same weight." }, { "from": "assistant", ...	1,178	true	Return your final response within \boxed{}. How many plums will balance one pear, given that 3 apples and one pear weigh as much as 10 plums, and one apple and 6 plums balance one pear? Assume that fruits of the same kind have the same weight.	Okay, let's see. I need to figure out how many plums balance one pear. The problem gives me two equations involving apples, pears, and plums. All fruits of the same type weigh the same. Hmm, let's break it down step by step. First, let me restate the information given. The problem says: 1. 3 apples and one pear weigh...	7	Okay, let's see. I need to figure out how many plums balance one pear. The problem gives me two equations involving apples, pears, and plums. All fruits of the same type weigh the same. Hmm, let's break it down step by step. First, let me restate the information given. The problem says: 1. 3 apples and one pear weigh...	false
olympiads	Determine the value of $$ z=a \sqrt{a} \sqrt[4]{a} \sqrt[8]{a} \ldots \sqrt[2^{n}]{a} \ldots $$ if \( n \) is infinitely large.	1. We begin by rewriting the given infinite product expression in a more manageable form. The given product is: \[ z = a \sqrt{a} \sqrt[4]{a} \sqrt[8]{a} \cdots \sqrt[2^{n}]{a} \cdots \] 2. Observe that each term in the product is of the form \( a^{1/2^k} \) where \( k \) is a non-negative integer. Hen...	[ { "content": "Determine the value of\n\n$$\nz=a \\sqrt{a} \\sqrt[4]{a} \\sqrt[8]{a} \\ldots \\sqrt[2^{n}]{a} \\ldots\n$$\n\nif \\( n \\) is infinitely large.", "role": "user" }, { "content": "\n1. We begin by rewriting the given infinite product expression in a more manageable form. The given produ...	Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w...	[ { "from": "user", "value": "Return your final response within \\boxed{}. Determine the value of\n\n$$\nz=a \\sqrt{a} \\sqrt[4]{a} \\sqrt[8]{a} \\ldots \\sqrt[2^{n}]{a} \\ldots\n$$\n\nif \\( n \\) is infinitely large." }, { "from": "assistant", "value": "<\|begin_of_thought\|>\n\nOkay, so I need to...	2,847	true	Return your final response within \boxed{}. Determine the value of $$ z=a \sqrt{a} \sqrt[4]{a} \sqrt[8]{a} \ldots \sqrt[2^{n}]{a} \ldots $$ if \( n \) is infinitely large.	Okay, so I need to find the value of z, which is this infinite product of a multiplied by the square root of a, then the fourth root, then the eighth root, and so on, each time the exponent being 1 over 2 to the nth power as n goes to infinity. Hmm, let's break this down step by step. First, let's write out the expres...	a^2	I'm tackling this problem step by step. I need to find the value of z, which is this infinite product of a multiplied by the square root of a, then the fourth root, then the eighth root, and so on, each time the exponent being 1 over 2 to the nth power as n goes to infinity. I'm breaking this down step by step. I'm wr...	true
olympiads	Give the value of \(0 - 1 + 2 - 3 + 4 - 5 + \ldots - 49 + 50\). Only a numerical answer is expected.	To find the value of the series \(0 - 1 + 2 - 3 + 4 - 5 + \ldots - 49 + 50\), we group the terms in pairs: 1. Group terms in pairs: \[ (0 - 1) + (2 - 3) + (4 - 5) + \ldots + (48 - 49) + 50 \] 2. Each of these pairs, except for the last term 50, simplifies as follows: \[ (0 - 1) = -1, \quad (2 - 3) = -1...	[ { "content": "Give the value of \\(0 - 1 + 2 - 3 + 4 - 5 + \\ldots - 49 + 50\\). Only a numerical answer is expected.", "role": "user" }, { "content": "\nTo find the value of the series \\(0 - 1 + 2 - 3 + 4 - 5 + \\ldots - 49 + 50\\), we group the terms in pairs:\n1. Group terms in pairs:\n \\[\n ...	Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w...	[ { "from": "user", "value": "Return your final response within \\boxed{}. Give the value of \\(0 - 1 + 2 - 3 + 4 - 5 + \\ldots - 49 + 50\\). Only a numerical answer is expected." }, { "from": "assistant", "value": "<\|begin_of_thought\|>\n\nOkay, let me try to figure out the value of this series: 0...	2,324	true	Return your final response within \boxed{}. Give the value of \(0 - 1 + 2 - 3 + 4 - 5 + \ldots - 49 + 50\). Only a numerical answer is expected.	Okay, let me try to figure out the value of this series: 0 - 1 + 2 - 3 + 4 - 5 + ... - 49 + 50. Hmm, that's an alternating series where the signs alternate between negative and positive, starting with 0 minus 1. Let me write out the first few terms to see the pattern better. So the series is: 0, then -1, +2, -3, +4, -...	25	Okay, let me try to figure out the value of this series: 0 - 1 + 2 - 3 + 4 - 5 + ... - 49 + 50. Hmm, that's an alternating series where the signs alternate between negative and positive, starting with 0 minus 1. Let me write out the first few terms to see the pattern better. So the series is: 0, then -1, +2, -3, +4, -...	false
olympiads	"Verify that 2 is a solution of the equation:\n\n$$\n(x+1)^{3}+(x+2)^{3}+(x+3)^{3}=(x+4)^{3}\n$$\n\n(...TRUNCATED)	"\nThe problem requires us to verify if \\(x = 2\\) is a solution to the equation \\((x+1)^3 + (x+2)(...TRUNCATED)	[{"content":"Verify that 2 is a solution of the equation:\n\n$$\n(x+1)^{3}+(x+2)^{3}+(x+3)^{3}=(x+4)(...TRUNCATED)	"Your role as an assistant involves thoroughly exploring questions through a systematic long thinkin(...TRUNCATED)	[{"from":"user","value":"Return your final response within \\boxed{}. Verify that 2 is a solution of(...TRUNCATED)	2,507	true	"Return your final response within \\boxed{}. Verify that 2 is a solution of the equation:\n\n$$\n(x(...TRUNCATED)	"Okay, so I need to verify if 2 is a solution to the equation (x+1)^3 + (x+2)^3 + (x+3)^3 = (x+4)^3.(...TRUNCATED)	2	"Okay, so I need to verify if 2 is a solution to the equation (x+1)^3 + (x+2)^3 + (x+3)^3 = (x+4)^3.(...TRUNCATED)	false
olympiads	"Find the function $F(x)$, whose graph passes through the point $M_{0}(3 ;-2)$, given that $F'(x) = (...TRUNCATED)	"To find the function \\( F(x) \\) whose graph passes through the point \\( M_0(3, -2) \\) and given(...TRUNCATED)	[{"content":"Find the function $F(x)$, whose graph passes through the point $M_{0}(3 ;-2)$, given th(...TRUNCATED)	"Your role as an assistant involves thoroughly exploring questions through a systematic long thinkin(...TRUNCATED)	[{"from":"user","value":"Return your final response within \\boxed{}. Find the function $F(x)$, whos(...TRUNCATED)	1,734	true	"Return your final response within \\boxed{}. Find the function $F(x)$, whose graph passes through t(...TRUNCATED)	"Okay, so I need to find the function F(x) whose graph passes through the point M₀(3, -2), and we (...TRUNCATED)	F(x)=\frac{4}{3} x^{3}-\frac{9}{x}-35	"I'm tackling this problem step by step. I need to find the function F(x) whose graph passes through(...TRUNCATED)	true
olympiads	"In an isosceles trapezoid with bases \\(a = 21\\), \\(b = 9\\) and height \\(h = 8\\), find the rad(...TRUNCATED)	"\n1. Identify Given Data and Setup:\n - The given isosceles trapezoid \\(ABCD\\) has bases \\(...TRUNCATED)	[{"content":"In an isosceles trapezoid with bases \\(a = 21\\), \\(b = 9\\) and height \\(h = 8\\), (...TRUNCATED)	"Your role as an assistant involves thoroughly exploring questions through a systematic long thinkin(...TRUNCATED)	[{"from":"user","value":"Return your final response within \\boxed{}. In an isosceles trapezoid with(...TRUNCATED)	4,803	true	"Return your final response within \\boxed{}. In an isosceles trapezoid with bases \\(a = 21\\), \\((...TRUNCATED)	"Okay, so I need to find the radius of the circumscribed circle around an isosceles trapezoid with b(...TRUNCATED)	\frac{85}{8}	"I'm tackling this problem step by step. I need to find the radius of the circumscribed circle aroun(...TRUNCATED)	true
olympiads	"Given two linear functions \\( f(x) \\) and \\( g(x) \\) such that the graphs \\( y = f(x) \\) and (...TRUNCATED)	": \nGiven are two linear functions \\( f(x) \\) and \\( g(x) \\) such that the graphs of \\( y = f((...TRUNCATED)	[{"content":"Given two linear functions \\( f(x) \\) and \\( g(x) \\) such that the graphs \\( y = f(...TRUNCATED)	"Your role as an assistant involves thoroughly exploring questions through a systematic long thinkin(...TRUNCATED)	[{"from":"user","value":"Return your final response within \\boxed{}. Given two linear functions \\((...TRUNCATED)	4,523	true	"Return your final response within \\boxed{}. Given two linear functions \\( f(x) \\) and \\( g(x) \(...TRUNCATED)	"Okay, so I need to find the minimum value of the function \\( 3(g(x))^2 + 2 f(x) \\), given that th(...TRUNCATED)	\frac{5}{2}	"Okay, so I need to find the minimum value of the function \\( 3(g(x))^2 + 2 f(x) \\), given that th(...TRUNCATED)	false