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question stringlengths 16 5.64k	has_final_verification bool 2 classes	before_verification stringlengths 0 64.1k	after_verification stringlengths 58 76.2k	full_response stringlengths 878 76.2k	verification_info dict	verification_check bool 1 class	length_difference int64 0 0	uuid stringlengths 3 36 ⌀	content stringlengths 878 76.2k	problem stringlengths 16 5.64k	answer stringlengths 1 28	is_reasoning_complete sequencelengths 2 6	correctness_math_verify sequencelengths 2 6	template_token_length int64 398 21.2k
3. (6 points) A construction company was building a tunnel. When $\frac{1}{3}$ of the tunnel was completed at the original speed, they started using new equipment, which increased the construction speed by $20 \%$ and reduced the working hours to $80 \%$ of the original. As a result, it took a total of 185 days to comp...	true	<reasoning> Okay, let me try to figure out this problem. So, a construction company was building a tunnel. When 1/3 of the tunnel was completed at the original speed, they started using new equipment. This new equipment increased their construction speed by 20% and reduced their working hours to 80% of the original. Th...	Wait, so the answer is 180 days? Let me confirm. If original time is 180 days, then original rate is 1/180 per day. Phase 1: 1/3 of the tunnel at 1/180 per day. Time = (1/3)/(1/180) = 60 days. Phase 2: 2/3 of the tunnel at the new rate. The new rate is 0.96*(1/180) = 0.96/180 = 0.005333... per day. Time for Phase 2...	<reasoning> Okay, let me try to figure out this problem. So, a construction company was building a tunnel. When 1/3 of the tunnel was completed at the original speed, they started using new equipment. This new equipment increased their construction speed by 20% and reduced their working hours to 80% of the original. Th...	{ "explanation": "이 부분은 180일이라는 답을 찾은 후에 시작되는 최종 검증 과정입니다. 이후 원래 속도로 계산했을 때 Phase 1(60일)과 Phase 2(125일)의 합이 주어진 185일과 일치하는지 확인하는 검산 과정을 수행합니다. '확인하자(Let me confirm)'라는 표현으로 시작하여, 원래 문제의 조건들을 사용해 답을 검증하고 있습니다. 특히 기존에 구한 답(180일)을 원래 문제 조건에 대입하여 주어진 185일이 나오는지 확인함으로써 답의 정확성을 최종적으로 입증하고 있습니다.", "has_final_verification"...	true	0	1d57f4b3-c516-5522-91cc-a60dd91f6f66	<reasoning> Okay, let me try to figure out this problem. So, a construction company was building a tunnel. When 1/3 of the tunnel was completed at the original speed, they started using new equipment. This new equipment increased their construction speed by 20% and reduced their working hours to 80% of the original. Th...	3. (6 points) A construction company was building a tunnel. When $\frac{1}{3}$ of the tunnel was completed at the original speed, they started using new equipment, which increased the construction speed by $20 \%$ and reduced the working hours to $80 \%$ of the original. As a result, it took a total of 185 days to comp...	180	[ true, true ]	[ true, true ]	2,328
Prove that number $1$ can be represented as a sum of a finite number $n$ of real numbers, less than $1,$ not necessarily distinct, which contain in their decimal representation only the digits $0$ and/or $7.$ Which is the least possible number $n$?	false		<reasoning> Okay, so the problem is to prove that the number 1 can be represented as a sum of a finite number n of real numbers, each less than 1, where each number in the sum has a decimal representation consisting only of the digits 0 and/or 7. And then we need to find the least possible n. First, let me make sure I...	<reasoning> Okay, so the problem is to prove that the number 1 can be represented as a sum of a finite number n of real numbers, each less than 1, where each number in the sum has a decimal representation consisting only of the digits 0 and/or 7. And then we need to find the least possible n. First, let me make sure I...	{ "explanation": "텍스트에서 마지막 검증 부분을 찾을 수 없습니다. 텍스트는 긴 추론 과정을 통해 n=8이라는 결론에 도달하지만, 이 답을 원래 문제에 대입하거나 다른 방법으로 재확인하는 과정은 포함되어 있지 않습니다. 마지막 부분은 단순히 'Thus, after this lengthy exploration, I conclude that the minimal n is 8.'로 끝나며, 이는 검증이 아닌 결론 도출입니다.", "has_final_verification": false, "verification_start_sentence": "" ...	true	0	1cd82e3d-2e7a-5eca-9571-62f5669824c4	<reasoning> Okay, so the problem is to prove that the number 1 can be represented as a sum of a finite number n of real numbers, each less than 1, where each number in the sum has a decimal representation consisting only of the digits 0 and/or 7. And then we need to find the least possible n. First, let me make sure I...	Prove that number $1$ can be represented as a sum of a finite number $n$ of real numbers, less than $1,$ not necessarily distinct, which contain in their decimal representation only the digits $0$ and/or $7.$ Which is the least possible number $n$?	8	[ true, false ]	[ true, false ]	16,537
"A right-angled triangle has side lengths that are integers. What could be the last digit of the are(...TRUNCATED)	true	"<reasoning>\nOkay, so I need to figure out the possible last digits of the area of a right-angled t(...TRUNCATED)	"Wait, let me check another case. Take m=2, n=1. Hypotenuse=5, which is divisible by5, so invalid. T(...TRUNCATED)	"<reasoning>\nOkay, so I need to figure out the possible last digits of the area of a right-angled t(...TRUNCATED)	{"explanation":"이 부분이 마지막 검증인 이유는:\n1. 답(마지막 자릿수가 0)을 (...TRUNCATED)	true	0	79e837be-2801-5a6a-bc7a-0acee3cdd5fd	"<reasoning>\nOkay, so I need to figure out the possible last digits of the area of a right-angled t(...TRUNCATED)	"A right-angled triangle has side lengths that are integers. What could be the last digit of the are(...TRUNCATED)	0	[ true, true ]	[ true, true ]	10,452
"9.6. Find the minimum value of the expression $(\\sqrt{2(1+\\cos 2 x)}-\\sqrt{36-4 \\sqrt{5}} \\sin(...TRUNCATED)	true	"<reasoning>\nAlright, so I have this problem to find the minimum value of this big expression. Let (...TRUNCATED)	"\nBut let me check once more:\n\nCompute -36 +4 sqrt5:\n\nsqrt5≈2.2360679775\n\n4 sqrt5≈8.94427(...TRUNCATED)	"<reasoning>\nAlright, so I have this problem to find the minimum value of this big expression. Let (...TRUNCATED)	{"explanation":"이 부분이 마지막 검증인 이유는: 1) 최종 답 -27을 도출한 후에 (...TRUNCATED)	true	0	9224d27e-798f-569e-a808-122e2d76c789	"<reasoning>\nAlright, so I have this problem to find the minimum value of this big expression. Let (...TRUNCATED)	"9.6. Find the minimum value of the expression $(\\sqrt{2(1+\\cos 2 x)}-\\sqrt{36-4 \\sqrt{5}} \\sin(...TRUNCATED)	-27	[ false, true, false, true ]	[ false, true, false, true ]	8,372
"Given $0 \\leqslant x, y, z \\leqslant 1$, solve the equation:\n$$\\frac{x}{1+y+z x}+\\frac{y}{1+z+(...TRUNCATED)	true		"<reasoning>\nAlright, so I have this equation to solve: \n\n\\[\n\\frac{x}{1 + y + z x} + \\frac{y}(...TRUNCATED)	"<reasoning>\nAlright, so I have this equation to solve: \n\n\\[\n\\frac{x}{1 + y + z x} + \\frac{y}(...TRUNCATED)	{"explanation":"이 문장은 긴 추론 과정의 마지막에서 모든 경우들을 검토한 후(...TRUNCATED)	true	0	c01ee438-4f54-5585-bbf8-4d10362f295a	"<reasoning>\nAlright, so I have this equation to solve: \n\n\\[\n\\frac{x}{1 + y + z x} + \\frac{y}(...TRUNCATED)	"Given $0 \\leqslant x, y, z \\leqslant 1$, solve the equation:\n$$\\frac{x}{1+y+z x}+\\frac{y}{1+z+(...TRUNCATED)	1	[ true, false, true, true ]	[ true, false, false, false ]	7,232
"2. (9th Canadian Mathematical Competition) $N$ is an integer, its representation in base $b$ is 777(...TRUNCATED)	true	"<reasoning>\nOkay, let's tackle this problem. So, N is an integer represented as 777 in base b, and(...TRUNCATED)	"Therefore, after checking all possible bases from 8 up to 17 and finding that none of them work, an(...TRUNCATED)	"<reasoning>\nOkay, let's tackle this problem. So, N is an integer represented as 777 in base b, and(...TRUNCATED)	{"explanation":"이 부분은 답으로 18을 도출한 후, 다음과 같은 최종 검증을 수(...TRUNCATED)	true	0	39a6c760-dfa3-5201-aa87-105127951e16	"<reasoning>\nOkay, let's tackle this problem. So, N is an integer represented as 777 in base b, and(...TRUNCATED)	"2. (9th Canadian Mathematical Competition) $N$ is an integer, its representation in base $b$ is 777(...TRUNCATED)	18	[ true, true ]	[ true, true ]	4,055
"Problem 6. (8 points) In the plane, there is a non-closed, non-self-intersecting broken line consis(...TRUNCATED)	true	"<reasoning>\nOkay, so I need to figure out the minimum number of different lines that can be obtain(...TRUNCATED)	"\nPossible solutions:\n\n4x +3y=31\n\nLet’s express y=(31 -4x)/3. Since y must be integer, 31-4x (...TRUNCATED)	"<reasoning>\nOkay, so I need to figure out the minimum number of different lines that can be obtain(...TRUNCATED)	{"explanation":"이 부분에서 답이 9라는 것을 발견한 후, 마지막으로 이 답이 맞(...TRUNCATED)	true	0	bdcc2598-b04d-5cb8-8634-d35d03fc6bf2	"<reasoning>\nOkay, so I need to figure out the minimum number of different lines that can be obtain(...TRUNCATED)	"Problem 6. (8 points) In the plane, there is a non-closed, non-self-intersecting broken line consis(...TRUNCATED)	9	[ true, true ]	[ true, true ]	7,044
"7.1. Solve the equation $\\frac{n!}{2}=k!+l!$ in natural numbers, where $n!=1 \\cdot 2 \\cdot \\ldo(...TRUNCATED)	true	"<reasoning>\nOkay, let's try to solve this equation: n! / 2 = k! + l! where n, k, l are natural num(...TRUNCATED)	"But let me check if maybe there's another way to get 12 with different factorials. For example, 24 (...TRUNCATED)	"<reasoning>\nOkay, let's try to solve this equation: n! / 2 = k! + l! where n, k, l are natural num(...TRUNCATED)	{"explanation":"이 부분은 텍스트의 가장 마지막에서 답을 10으로 결정한 후, 최(...TRUNCATED)	true	0	09902996-6e6f-5507-94c7-f8b8045a702b	"<reasoning>\nOkay, let's try to solve this equation: n! / 2 = k! + l! where n, k, l are natural num(...TRUNCATED)	"7.1. Solve the equation $\\frac{n!}{2}=k!+l!$ in natural numbers, where $n!=1 \\cdot 2 \\cdot \\ldo(...TRUNCATED)	10	[ true, true ]	[ true, true ]	5,382
"Four, (50 points) In an $n \\times n$ grid, fill each cell with one of the numbers 1 to $n^{2}$. If(...TRUNCATED)	true		"<reasoning>\nOkay, let's tackle this problem. So, we need to find the minimal n such that in any n (...TRUNCATED)	"<reasoning>\nOkay, let's tackle this problem. So, we need to find the minimal n such that in any n (...TRUNCATED)	{"explanation":"이 부분이 마지막 검증인 이유는 다음과 같습니다:\n1. 답으로 (...TRUNCATED)	true	0	0c6f58a5-af16-5fc6-b982-cdbffc29350d	"<reasoning>\nOkay, let's tackle this problem. So, we need to find the minimal n such that in any n (...TRUNCATED)	"Four, (50 points) In an $n \\times n$ grid, fill each cell with one of the numbers 1 to $n^{2}$. If(...TRUNCATED)	2020	[ true, true ]	[ false, true ]	6,765
"On a board, the numbers from 1 to 2009 are written. A couple of them are erased and instead of them(...TRUNCATED)	true	"<reasoning>\nOkay, so I need to figure out this problem where numbers from 1 to 2009 are written on(...TRUNCATED)	"But let me check again. Suppose that x is an original number. Then, since x was never erased, then (...TRUNCATED)	"<reasoning>\nOkay, so I need to figure out this problem where numbers from 1 to 2009 are written on(...TRUNCATED)	{"explanation":"이 부분은 답으로 8을 도출한 후에, 이 답이 문제의 모든 조건을(...TRUNCATED)	true	0	26ac9e7a-f8f3-5ef9-b051-f84dfb5129ce	"<reasoning>\nOkay, so I need to figure out this problem where numbers from 1 to 2009 are written on(...TRUNCATED)	"On a board, the numbers from 1 to 2009 are written. A couple of them are erased and instead of them(...TRUNCATED)	8	[ true, true ]	[ true, true ]	4,448

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