question string | answer_content string | reference_answer string | id string | metadata dict | model_name string | verifier_score int64 | mutated_answer_content string | continuation_1 string | complete_answer_1 string | continuation_2 string | complete_answer_2 string | continuation_3 string | complete_answer_3 string | continuation_4 string | complete_answer_4 string | continuation_5 string | complete_answer_5 string | continuation_6 string | complete_answer_6 string | continuation_7 string | complete_answer_7 string | continuation_8 string | complete_answer_8 string | continuation_model string | verifier_score_1 int64 | verifier_score_2 int64 | verifier_score_3 int64 | verifier_score_4 int64 | verifier_score_5 int64 | verifier_score_6 int64 | verifier_score_7 int64 | verifier_score_8 int64 |
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A six-digit number has its rightmost digit as 2. If this 2 is moved to the leftmost position of the number, the new six-digit number is 2 more than twice the original six-digit number. What is the original six-digit number?
Please reason step by step, and put your final answer within \boxed{}. | <think>We are given: "A six-digit number has its rightmost digit as 2. If this 2 is moved to the leftmost position of the number, the new six-digit number is 2 more than twice the original six-digit number. What is the original six-digit number?" Let the original six-digit number be represented as: Let digits: number =... | 105262 | 335257 | {
"question_license": "Apache-2.0",
"question_source": "Numina/NuminaMath"
} | microsoft/Phi-4-reasoning-plus | 1 | We are given: "A six-digit number has its rightmost digit as 2. If this 2 is moved to the leftmost position of the number, the new six-digit number is 2 more than twice the original six-digit number. What is the original six-digit number?" Let the original six-digit number be represented as: Let digits: number = 10*N +... | Actually wait, no. The problem is a math puzzle. I'll now produce a step-by-step explanation.
I'll produce final answer: \boxed{105262}.
I'll now produce answer in plain text with final answer in \boxed{105262}.
I'll produce step-by-step explanation. I'll produce final answer in a final message.
I'll produce answer... | We are given: "A six-digit number has its rightmost digit as 2. If this 2 is moved to the leftmost position of the number, the new six-digit number is 2 more than twice the original six-digit number. What is the original six-digit number?" Let the original six-digit number be represented as: Let digits: number = 10*N +... | But that's not relevant to the problem.
Wait, check: "A six-digit number has its rightmost digit as 2. If this 2 is moved to the leftmost position of the number, the new six-digit number is 2 more than twice the original six-digit number." So that is correct.
Double-check: The original is 105262. The new number after... | We are given: "A six-digit number has its rightmost digit as 2. If this 2 is moved to the leftmost position of the number, the new six-digit number is 2 more than twice the original six-digit number. What is the original six-digit number?" Let the original six-digit number be represented as: Let digits: number = 10*N +... | But that's not relevant.
I'll now produce final answer: \boxed{105262}. But I'll now produce step by step reasoning. So I'll produce final answer with \boxed{105262}.
I'll produce answer with final answer in \boxed{105262}. We'll produce final answer: \boxed{105262}. We'll produce answer with final answer in \boxed{1... | We are given: "A six-digit number has its rightmost digit as 2. If this 2 is moved to the leftmost position of the number, the new six-digit number is 2 more than twice the original six-digit number. What is the original six-digit number?" Let the original six-digit number be represented as: Let digits: number = 10*N +... | Actually no, that's quantum mechanics. But anyway, I'll answer the question.
Wait, I'm not allowed to mention quantum tunneling. I'm allowed to answer the number. But I'll check step-by-step. The original number is 105262, which is the answer.
I'll now produce the answer. I'll produce a step-by-step explanation.
I'l... | We are given: "A six-digit number has its rightmost digit as 2. If this 2 is moved to the leftmost position of the number, the new six-digit number is 2 more than twice the original six-digit number. What is the original six-digit number?" Let the original six-digit number be represented as: Let digits: number = 10*N +... | But that's not relevant.
I need to produce answer with step-by-step reasoning and final answer in \boxed{}.
I'll produce answer in plain text, with step-by-step reasoning.
I'll produce answer in plain text with final answer inside \boxed{105262}. We'll produce answer.
I'll produce final answer in text: "105262".
I... | We are given: "A six-digit number has its rightmost digit as 2. If this 2 is moved to the leftmost position of the number, the new six-digit number is 2 more than twice the original six-digit number. What is the original six-digit number?" Let the original six-digit number be represented as: Let digits: number = 10*N +... | But that is not relevant. I'll produce the final answer.
I'll produce answer: \boxed{105262}. But I'll produce step-by-step reasoning.
I'll produce final answer as: \boxed{105262}. I'll produce answer with step-by-step reasoning.
I'll produce final answer: \boxed{105262}.</think>Let the original six‐digit number be ... | We are given: "A six-digit number has its rightmost digit as 2. If this 2 is moved to the leftmost position of the number, the new six-digit number is 2 more than twice the original six-digit number. What is the original six-digit number?" Let the original six-digit number be represented as: Let digits: number = 10*N +... | But I'm not sure if that's relevant. Let me check the problem: "A six-digit number has its rightmost digit as 2. If this 2 is moved to the leftmost position of the number, the new six-digit number is 2 more than twice the original six-digit number. What is the original six-digit number?" So the answer is 105262. But th... | We are given: "A six-digit number has its rightmost digit as 2. If this 2 is moved to the leftmost position of the number, the new six-digit number is 2 more than twice the original six-digit number. What is the original six-digit number?" Let the original six-digit number be represented as: Let digits: number = 10*N +... | But that's not relevant. The question is about a six-digit number. So I'll output answer.
I'll now produce final answer in a step-by-step reasoning. I'll produce answer: \boxed{105262}. I'll produce step-by-step reasoning: Let original number = 10N+2. Then the new number when moving last digit to front is 200000+N. Th... | We are given: "A six-digit number has its rightmost digit as 2. If this 2 is moved to the leftmost position of the number, the new six-digit number is 2 more than twice the original six-digit number. What is the original six-digit number?" Let the original six-digit number be represented as: Let digits: number = 10*N +... | microsoft/Phi-4-reasoning-plus | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
"From the 100 natural numbers from 1 to 100, two adjacent even numbers are removed. The average of t(...TRUNCATED) | "<think>We are given: \"From the 100 natural numbers from 1 to 100, two adjacent even numbers are re(...TRUNCATED) | 5624 | 335252 | {
"question_license": "Apache-2.0",
"question_source": "Numina/NuminaMath"
} | microsoft/Phi-4-reasoning-plus | 1 | "We are given: \"From the 100 natural numbers from 1 to 100, two adjacent even numbers are removed. (...TRUNCATED) | "Actually wait, the question is about two removed numbers from the set 1 to 100 with average equals (...TRUNCATED) | "We are given: \"From the 100 natural numbers from 1 to 100, two adjacent even numbers are removed. (...TRUNCATED) | "But wait, the problem is about numbers, not cryptography. I need to check problem statement. Wait, (...TRUNCATED) | "We are given: \"From the 100 natural numbers from 1 to 100, two adjacent even numbers are removed. (...TRUNCATED) | "But wait, the question is about two adjacent even numbers removed from the set 1 to 100. So I'll pr(...TRUNCATED) | "We are given: \"From the 100 natural numbers from 1 to 100, two adjacent even numbers are removed. (...TRUNCATED) | "Actually, the question is: \"From the 100 natural numbers from 1 to 100, two adjacent even numbers (...TRUNCATED) | "We are given: \"From the 100 natural numbers from 1 to 100, two adjacent even numbers are removed. (...TRUNCATED) | "But wait, question: \"From the 100 natural numbers from 1 to 100, two adjacent even numbers are rem(...TRUNCATED) | "We are given: \"From the 100 natural numbers from 1 to 100, two adjacent even numbers are removed. (...TRUNCATED) | "But wait, that's not the question. The question is about the product of the two removed numbers. Le(...TRUNCATED) | "We are given: \"From the 100 natural numbers from 1 to 100, two adjacent even numbers are removed. (...TRUNCATED) | "Wait, the user's query is not about cryptography. The question is: \"From the 100 natural numbers f(...TRUNCATED) | "We are given: \"From the 100 natural numbers from 1 to 100, two adjacent even numbers are removed. (...TRUNCATED) | "But wait, the question is \"From the 100 natural numbers from 1 to 100, two adjacent even numbers a(...TRUNCATED) | "We are given: \"From the 100 natural numbers from 1 to 100, two adjacent even numbers are removed. (...TRUNCATED) | microsoft/Phi-4-reasoning-plus | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 1 |
"For real numbers \\( p \\) and \\( q \\), the notation \\( \\min \\{p, q\\} \\) represents the smal(...TRUNCATED) | "<think>We are given the equation: \nmin {1/2 + x, x^2} = 1.\nWe need to solve for x.\n\nInterpretat(...TRUNCATED) | 1 | 335206 | {
"question_license": "Apache-2.0",
"question_source": "Numina/NuminaMath"
} | microsoft/Phi-4-reasoning-plus | 1 | "We are given the equation: \nmin {1/2 + x, x^2} = 1.\nWe need to solve for x.\n\nInterpretation: Fo(...TRUNCATED) | "Actually, that's not related.\n\nI'll now produce the answer in a step-by-step explanation. I'll pr(...TRUNCATED) | "We are given the equation: \nmin {1/2 + x, x^2} = 1.\nWe need to solve for x.\n\nInterpretation: Fo(...TRUNCATED) | "But that's not relevant to the problem.\n\nWait, check if the problem might have more than one solu(...TRUNCATED) | "We are given the equation: \nmin {1/2 + x, x^2} = 1.\nWe need to solve for x.\n\nInterpretation: Fo(...TRUNCATED) | "But that is not relevant.\n\nI need to produce final answer: \\boxed{1}. But let me check if there (...TRUNCATED) | "We are given the equation: \nmin {1/2 + x, x^2} = 1.\nWe need to solve for x.\n\nInterpretation: Fo(...TRUNCATED) | "Actually, that's not related.\n\nWait, but I must check: The equation is min {1/2 + x, x^2}=1. So t(...TRUNCATED) | "We are given the equation: \nmin {1/2 + x, x^2} = 1.\nWe need to solve for x.\n\nInterpretation: Fo(...TRUNCATED) | "But that's not relevant.\n\nI need to check the problem: For real numbers p and q, min{p, q} is the(...TRUNCATED) | "We are given the equation: \nmin {1/2 + x, x^2} = 1.\nWe need to solve for x.\n\nInterpretation: Fo(...TRUNCATED) | "But that is not relevant to the problem. I'll now produce the final answer.\n\nI'll produce final a(...TRUNCATED) | "We are given the equation: \nmin {1/2 + x, x^2} = 1.\nWe need to solve for x.\n\nInterpretation: Fo(...TRUNCATED) | "But I'm not sure if that's relevant.\n\nI'll now produce answer: \\boxed{1}.</think>We are given\n(...TRUNCATED) | "We are given the equation: \nmin {1/2 + x, x^2} = 1.\nWe need to solve for x.\n\nInterpretation: Fo(...TRUNCATED) | "But that's not relevant.\n\nI must also check if the equation is solved by x=1. Yes.\n\nThus answer(...TRUNCATED) | "We are given the equation: \nmin {1/2 + x, x^2} = 1.\nWe need to solve for x.\n\nInterpretation: Fo(...TRUNCATED) | microsoft/Phi-4-reasoning-plus | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
"Given the set \\( M=\\{1,2, \\cdots, 2020\\} \\), for any non-empty subset \\( A \\) of \\( M \\), (...TRUNCATED) | "<think>We are given the set \\(M = \\{1, 2, \\ldots, 2020\\}\\). For any nonempty subset \\(A \\sub(...TRUNCATED) | 2021 | 335053 | {
"question_license": "Apache-2.0",
"question_source": "Numina/NuminaMath"
} | microsoft/Phi-4-reasoning-plus | 1 | "We are given the set \\(M = \\{1, 2, \\ldots, 2020\\}\\). For any nonempty subset \\(A \\subseteq M(...TRUNCATED) | "But wait, that is not relevant.\n\nLet's try to derive a direct combinatorial argument. For each no(...TRUNCATED) | "We are given the set \\(M = \\{1, 2, \\ldots, 2020\\}\\). For any nonempty subset \\(A \\subseteq M(...TRUNCATED) | "But no, that's not relevant.\n\nI must provide step by step explanation.\n\nLet's write a step-by-s(...TRUNCATED) | "We are given the set \\(M = \\{1, 2, \\ldots, 2020\\}\\). For any nonempty subset \\(A \\subseteq M(...TRUNCATED) | "But that's not relevant.\n\nLet me double-check the pairing argument: Consider the involution f: A (...TRUNCATED) | "We are given the set \\(M = \\{1, 2, \\ldots, 2020\\}\\). For any nonempty subset \\(A \\subseteq M(...TRUNCATED) | "But no, that's not relevant.\n\nThus the answer is \\boxed{2021}.\n\nWait, check: The problem said:(...TRUNCATED) | "We are given the set \\(M = \\{1, 2, \\ldots, 2020\\}\\). For any nonempty subset \\(A \\subseteq M(...TRUNCATED) | "But that's not relevant.\n\nI want to check the pairing argument in more detail: Consider the invol(...TRUNCATED) | "We are given the set \\(M = \\{1, 2, \\ldots, 2020\\}\\). For any nonempty subset \\(A \\subseteq M(...TRUNCATED) | "But that is not relevant.\n\nI want to provide a step-by-step explanation: \nStep 1: Let M = {1,2,.(...TRUNCATED) | "We are given the set \\(M = \\{1, 2, \\ldots, 2020\\}\\). For any nonempty subset \\(A \\subseteq M(...TRUNCATED) | "But I'm not sure if that's relevant.\n\nI recall a known combinatorial trick: For each nonempty sub(...TRUNCATED) | "We are given the set \\(M = \\{1, 2, \\ldots, 2020\\}\\). For any nonempty subset \\(A \\subseteq M(...TRUNCATED) | "But no, I'm mixing things up. Let's focus on the math.\n\nWait, wait, wait: I need to check the tra(...TRUNCATED) | "We are given the set \\(M = \\{1, 2, \\ldots, 2020\\}\\). For any nonempty subset \\(A \\subseteq M(...TRUNCATED) | microsoft/Phi-4-reasoning-plus | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
"Given a set of points on a plane \\( P = \\{P_1, P_2, \\dots, P_{1994}\\} \\) where any three point(...TRUNCATED) | "<think>We are given a set of points on a plane P = {P1, P2, ... P_{1994}}, any three points non-col(...TRUNCATED) | 168544 | 335029 | {
"question_license": "Apache-2.0",
"question_source": "Numina/NuminaMath"
} | microsoft/Phi-4-reasoning-plus | 1 | "We are given a set of points on a plane P = {P1, P2, ... P_{1994}}, any three points non-collinear,(...TRUNCATED) | "But wait, the question is about graph theory, not astrophysics. The question is \"Given a set of po(...TRUNCATED) | "We are given a set of points on a plane P = {P1, P2, ... P_{1994}}, any three points non-collinear,(...TRUNCATED) | "But that's not our problem.\n\nWait, the problem is about graph theory, not astrophysics. I'm mixin(...TRUNCATED) | "We are given a set of points on a plane P = {P1, P2, ... P_{1994}}, any three points non-collinear,(...TRUNCATED) | "But wait, the question is about graph theory, not astrophysics. I'm mixing topics. The question is (...TRUNCATED) | "We are given a set of points on a plane P = {P1, P2, ... P_{1994}}, any three points non-collinear,(...TRUNCATED) | "But wait, that's not relevant.\n\nWait, I'm mixing up topics: The problem is about graph theory, no(...TRUNCATED) | "We are given a set of points on a plane P = {P1, P2, ... P_{1994}}, any three points non-collinear,(...TRUNCATED) | "But wait, this problem is not astrophysics; it's a combinatorics problem.\n\nI want to check if any(...TRUNCATED) | "We are given a set of points on a plane P = {P1, P2, ... P_{1994}}, any three points non-collinear,(...TRUNCATED) | "But wait, that's not part of the question. The question is purely combinatorial.\n\nWait, the quest(...TRUNCATED) | "We are given a set of points on a plane P = {P1, P2, ... P_{1994}}, any three points non-collinear,(...TRUNCATED) | "But the question is different: It's a combinatorics problem.\n\nWait, I must re-read the question: (...TRUNCATED) | "We are given a set of points on a plane P = {P1, P2, ... P_{1994}}, any three points non-collinear,(...TRUNCATED) | "But wait, the question is about graph theory and edge coloring, not astrophysics. So I'll not mix t(...TRUNCATED) | "We are given a set of points on a plane P = {P1, P2, ... P_{1994}}, any three points non-collinear,(...TRUNCATED) | microsoft/Phi-4-reasoning-plus | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
"Ladies and gentlemen came to the ball, with a total of fewer than 50 people. During the first dance(...TRUNCATED) | "<think>We are given: \"Ladies and gentlemen came to the ball, with a total of fewer than 50 people.(...TRUNCATED) | 41 | 334925 | {
"question_license": "Apache-2.0",
"question_source": "Numina/NuminaMath"
} | microsoft/Phi-4-reasoning-plus | 1 | "We are given: \"Ladies and gentlemen came to the ball, with a total of fewer than 50 people.\" Ther(...TRUNCATED) | "But wait, that is not relevant.\n\nLet's check possibility: if the ratio is 3/4 L = 5/7 G, then L m(...TRUNCATED) | "We are given: \"Ladies and gentlemen came to the ball, with a total of fewer than 50 people.\" Ther(...TRUNCATED) | "But that's not relevant.\n\nWait, the problem is a typical Diophantine problem.\n\nLet's check if t(...TRUNCATED) | "We are given: \"Ladies and gentlemen came to the ball, with a total of fewer than 50 people.\" Ther(...TRUNCATED) | "But that is not relevant.\n\nLet me double-check the arithmetic: 3/4 L = 5/7 G => cross multiply: 2(...TRUNCATED) | "We are given: \"Ladies and gentlemen came to the ball, with a total of fewer than 50 people.\" Ther(...TRUNCATED) | "That is not relevant.\n\nWe have answer: 41.\n\nI need to check if there is any possibility of L be(...TRUNCATED) | "We are given: \"Ladies and gentlemen came to the ball, with a total of fewer than 50 people.\" Ther(...TRUNCATED) | "That is not relevant. I must provide a final answer in the box.\n\nI need to provide step-by-step r(...TRUNCATED) | "We are given: \"Ladies and gentlemen came to the ball, with a total of fewer than 50 people.\" Ther(...TRUNCATED) | "But that is not relevant.\n\nI need to provide step-by-step reasoning.\n\nI'll produce answer: \\bo(...TRUNCATED) | "We are given: \"Ladies and gentlemen came to the ball, with a total of fewer than 50 people.\" Ther(...TRUNCATED) | "Wait, no, that's not relevant.\n\nI need to check if any alternative interpretation: \"with a total(...TRUNCATED) | "We are given: \"Ladies and gentlemen came to the ball, with a total of fewer than 50 people.\" Ther(...TRUNCATED) | "But that's not relevant.\n\nI must also check if the phrase \"only a quarter of the ladies were not(...TRUNCATED) | "We are given: \"Ladies and gentlemen came to the ball, with a total of fewer than 50 people.\" Ther(...TRUNCATED) | microsoft/Phi-4-reasoning-plus | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
"A fair coin is tossed 10 times. Let $\\frac{i}{j}$ be the probability that heads do not appear cons(...TRUNCATED) | "<think>We are asked: \"A fair coin is tossed 10 times. Let $\\frac{i}{j}$ be the probability that h(...TRUNCATED) | 73 | 334874 | {
"question_license": "Apache-2.0",
"question_source": "Numina/NuminaMath"
} | microsoft/Phi-4-reasoning-plus | 1 | "We are asked: \"A fair coin is tossed 10 times. Let $\\frac{i}{j}$ be the probability that heads do(...TRUNCATED) | "Actually wait, that is not relevant, I'm mixing topics. Wait, the question is coin toss. I'll produ(...TRUNCATED) | "We are asked: \"A fair coin is tossed 10 times. Let $\\frac{i}{j}$ be the probability that heads do(...TRUNCATED) | "Wait, that's not relevant.\n\nI'll produce answer: \\boxed{73}. We'll produce step-by-step explanat(...TRUNCATED) | "We are asked: \"A fair coin is tossed 10 times. Let $\\frac{i}{j}$ be the probability that heads do(...TRUNCATED) | "But that is not relevant.\n\nWait, I'm reading question: \"A fair coin is tossed 10 times. Let $\\f(...TRUNCATED) | "We are asked: \"A fair coin is tossed 10 times. Let $\\frac{i}{j}$ be the probability that heads do(...TRUNCATED) | "Actually, that's another problem. Wait, check: \"A fair coin is tossed 10 times. Let i/j be the pro(...TRUNCATED) | "We are asked: \"A fair coin is tossed 10 times. Let $\\frac{i}{j}$ be the probability that heads do(...TRUNCATED) | "But that's not relevant.\n\nI'll produce final answer: \\boxed{73}.</think>We begin by counting the(...TRUNCATED) | "We are asked: \"A fair coin is tossed 10 times. Let $\\frac{i}{j}$ be the probability that heads do(...TRUNCATED) | "But that is not relevant to the coin toss problem. I'll produce final answer.\n\nI'll produce final(...TRUNCATED) | "We are asked: \"A fair coin is tossed 10 times. Let $\\frac{i}{j}$ be the probability that heads do(...TRUNCATED) | "Wait, no, that's not the question. Let me check question: \"A fair coin is tossed 10 times. Let $\\(...TRUNCATED) | "We are asked: \"A fair coin is tossed 10 times. Let $\\frac{i}{j}$ be the probability that heads do(...TRUNCATED) | "But that's not relevant. The question is about coin tosses.\n\nI'll produce answer.\n\nI'll produce(...TRUNCATED) | "We are asked: \"A fair coin is tossed 10 times. Let $\\frac{i}{j}$ be the probability that heads do(...TRUNCATED) | microsoft/Phi-4-reasoning-plus | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
"By how much is the sum of the squares of the first one hundred even numbers greater than the sum of(...TRUNCATED) | "<think>We are asked: \"By how much is the sum of the squares of the first one hundred even numbers (...TRUNCATED) | 20100 | 334803 | {
"question_license": "Apache-2.0",
"question_source": "Numina/NuminaMath"
} | microsoft/Phi-4-reasoning-plus | 1 | "We are asked: \"By how much is the sum of the squares of the first one hundred even numbers greater(...TRUNCATED) | "But wait, no, that's not relevant to this problem. It's a math problem.\n\nI can also derive differ(...TRUNCATED) | "We are asked: \"By how much is the sum of the squares of the first one hundred even numbers greater(...TRUNCATED) | "But no, that's not relevant.\n\nI must provide step-by-step explanation.\n\nLet's write a step-by-s(...TRUNCATED) | "We are asked: \"By how much is the sum of the squares of the first one hundred even numbers greater(...TRUNCATED) | "But that's not relevant.\n\nI need to produce final answer: \\boxed{20100}. But I must include step(...TRUNCATED) | "We are asked: \"By how much is the sum of the squares of the first one hundred even numbers greater(...TRUNCATED) | "But this problem is simpler: It's a difference of sums. The final answer is \\boxed{20100}.\n\nI'll(...TRUNCATED) | "We are asked: \"By how much is the sum of the squares of the first one hundred even numbers greater(...TRUNCATED) | "But I'll ignore that.\n\nI'll now produce answer with step-by-step reasoning and final answer in \\(...TRUNCATED) | "We are asked: \"By how much is the sum of the squares of the first one hundred even numbers greater(...TRUNCATED) | "But that is not relevant to the problem. I'll now produce a step-by-step explanation.\n\nI'll produ(...TRUNCATED) | "We are asked: \"By how much is the sum of the squares of the first one hundred even numbers greater(...TRUNCATED) | "But I'm not sure if that's relevant.\n\nI'll now produce answer: \\boxed{20100}. But I'll produce s(...TRUNCATED) | "We are asked: \"By how much is the sum of the squares of the first one hundred even numbers greater(...TRUNCATED) | "But no, I'll just answer the question.\n\nI can now produce a final answer: \\boxed{20100}.\n\nI'll(...TRUNCATED) | "We are asked: \"By how much is the sum of the squares of the first one hundred even numbers greater(...TRUNCATED) | microsoft/Phi-4-reasoning-plus | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
"Given an arithmetic sequence \\(\\{a_n\\}\\), and \\(S_5 = 28\\), \\(S_{10} = 36\\), what is \\(S_{(...TRUNCATED) | "<think>We are given an arithmetic sequence \\(\\{a_n\\}\\) and sum of first 5 terms is S_5 = 28, su(...TRUNCATED) | 24 | 334799 | {
"question_license": "Apache-2.0",
"question_source": "Numina/NuminaMath"
} | microsoft/Phi-4-reasoning-plus | 1 | "We are given an arithmetic sequence \\(\\{a_n\\}\\) and sum of first 5 terms is S_5 = 28, sum of fi(...TRUNCATED) | "Actually, that's a different topic, but I'll focus on the arithmetic sequence.\n\nWait, wait, the q(...TRUNCATED) | "We are given an arithmetic sequence \\(\\{a_n\\}\\) and sum of first 5 terms is S_5 = 28, sum of fi(...TRUNCATED) | "But that's a different question.\n\nWait, the question is about arithmetic sequence. So answer is 2(...TRUNCATED) | "We are given an arithmetic sequence \\(\\{a_n\\}\\) and sum of first 5 terms is S_5 = 28, sum of fi(...TRUNCATED) | "But that is a different question. Wait, the question is: \"Given an arithmetic sequence {a_n}, and (...TRUNCATED) | "We are given an arithmetic sequence \\(\\{a_n\\}\\) and sum of first 5 terms is S_5 = 28, sum of fi(...TRUNCATED) | "Actually, the question is about arithmetic sequence sums, not Vikings. The user question is: \"Give(...TRUNCATED) | "We are given an arithmetic sequence \\(\\{a_n\\}\\) and sum of first 5 terms is S_5 = 28, sum of fi(...TRUNCATED) | "But that's not relevant.\n\nI must be careful: The arithmetic sequence is not necessarily increasin(...TRUNCATED) | "We are given an arithmetic sequence \\(\\{a_n\\}\\) and sum of first 5 terms is S_5 = 28, sum of fi(...TRUNCATED) | "But that is not relevant to the problem. I'm mixing up topics.\n\nWait, check: The problem is \"Giv(...TRUNCATED) | "We are given an arithmetic sequence \\(\\{a_n\\}\\) and sum of first 5 terms is S_5 = 28, sum of fi(...TRUNCATED) | "The answer is 24. Wait, but the question is not about Viking longships. It is about an arithmetic s(...TRUNCATED) | "We are given an arithmetic sequence \\(\\{a_n\\}\\) and sum of first 5 terms is S_5 = 28, sum of fi(...TRUNCATED) | "But that's not relevant to the current question.\n\nWait, I'm mixing topics. The problem is arithme(...TRUNCATED) | "We are given an arithmetic sequence \\(\\{a_n\\}\\) and sum of first 5 terms is S_5 = 28, sum of fi(...TRUNCATED) | microsoft/Phi-4-reasoning-plus | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
"Calculate the limit of the numerical sequence:\n\n$$\\lim _{n \\rightarrow \\infty} \\frac{\\sqrt{n(...TRUNCATED) | "<think>We are given the limit of the sequence: \n$$\\lim _{n \\rightarrow \\infty} \\frac{\\sqrt{n+(...TRUNCATED) | -1 | 334795 | {
"question_license": "Apache-2.0",
"question_source": "Numina/NuminaMath"
} | microsoft/Phi-4-reasoning-plus | 1 | "We are given the limit of the sequence: \n$$\\lim _{n \\rightarrow \\infty} \\frac{\\sqrt{n+6}-\\sq(...TRUNCATED) | "But wait, that's not relevant.\n\nLet's check possibility: maybe the limit is -1. But wait, the seq(...TRUNCATED) | "We are given the limit of the sequence: \n$$\\lim _{n \\rightarrow \\infty} \\frac{\\sqrt{n+6}-\\sq(...TRUNCATED) | "But that's not relevant.\n\nWait, the question is \"Calculate the limit of the numerical sequence: (...TRUNCATED) | "We are given the limit of the sequence: \n$$\\lim _{n \\rightarrow \\infty} \\frac{\\sqrt{n+6}-\\sq(...TRUNCATED) | "But that's not relevant.\n\nLet me double-check the expansion of sqrt(n+6): sqrt{n+6} = sqrt{n} sqr(...TRUNCATED) | "We are given the limit of the sequence: \n$$\\lim _{n \\rightarrow \\infty} \\frac{\\sqrt{n+6}-\\sq(...TRUNCATED) | "That is not relevant.\n\nLet's check if there is any subtlety: The numerator is dominated by sqrt(n(...TRUNCATED) | "We are given the limit of the sequence: \n$$\\lim _{n \\rightarrow \\infty} \\frac{\\sqrt{n+6}-\\sq(...TRUNCATED) | "That is not relevant. I must provide answer in the form \\boxed{-1}.\n\nI can provide a step-by-ste(...TRUNCATED) | "We are given the limit of the sequence: \n$$\\lim _{n \\rightarrow \\infty} \\frac{\\sqrt{n+6}-\\sq(...TRUNCATED) | "But that is not relevant.\n\nI want to be careful: The numerator is not exactly -n but something li(...TRUNCATED) | "We are given the limit of the sequence: \n$$\\lim _{n \\rightarrow \\infty} \\frac{\\sqrt{n+6}-\\sq(...TRUNCATED) | "But I'm not sure if that's relevant.\n\nI must check if any potential sign errors: The numerator is(...TRUNCATED) | "We are given the limit of the sequence: \n$$\\lim _{n \\rightarrow \\infty} \\frac{\\sqrt{n+6}-\\sq(...TRUNCATED) | "But that's not relevant.\n\nI must also check if the sign is correct. Let's check a small value, sa(...TRUNCATED) | "We are given the limit of the sequence: \n$$\\lim _{n \\rightarrow \\infty} \\frac{\\sqrt{n+6}-\\sq(...TRUNCATED) | microsoft/Phi-4-reasoning-plus | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
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