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problem string	solution string	answer string	problem_type string	question_type string	source string	uuid string	is_reasoning_complete bool	generations string	correctness_math_verify bool	correctness_llama list	finish_reasons list	correctness_count int64	messages list	generation_id int64	extracted_box_solution string	extracted_box_solution_accuracy bool	question_prompt string	full_prompt string	new_uuid string	embedding list
Let $f(x)=x^2-2x+1.$ For some constant $k, f(x+k) = x^2+2x+1$ for all real numbers $x.$ Determine the value of $k.$	1. Given the function $ f(x) = x^2 - 2x + 1 $, we can rewrite it in a factored form: \[ f(x) = (x-1)^2 \] 2. We are given that for some constant $ k $, the function $ f(x+k) $ is equal to $ x^2 + 2x + 1 $ for all real numbers $ x $. Therefore, we need to find $ k $ such that: \[ f(x+k) = x^2...	2	Algebra	math-word-problem	aops_forum	50a92404-34c1-57b4-8c33-f28ef8abc13f_1	false	<think> Okay, so I have this problem here: the function f(x) is defined as x squared minus 2x plus 1. Then, for some constant.	false	null	null	1	[ { "content": "Let $f(x)=x^2-2x+1.$ For some constant $k, f(x+k) = x^2+2x+1$ for all real numbers $x.$ Determine the value of $k.$", "role": "user" }, { "content": "<think>\nOkay, let's see here. I need to find the value of k such that when I replace x with (x + k) in the function f(x) = x² - 2x + 1,...	1	null	false	Let $f(x)=x^2-2x+1.$ For some constant $k, f(x+k) = x^2+2x+1$ for all real numbers $x.$ Determine the value of $k.$ Let's think step by step and output the final answer within \boxed{}.	Let $f(x)=x^2-2x+1.$ For some constant $k, f(x+k) = x^2+2x+1$ for all real numbers $x.$ Determine the value of $k.$ Let's think step by step and output the final answer within \boxed{}. <think> Okay, so I have this problem here: the function f(x) is defined as x squared minus 2x plus 1. Then, for some constant.	50a92404-34c1-57b4-8c33-f28ef8abc13f_1	[ -0.2109375, -0.04345703125, 0.08642578125, -0.267578125, 0.0294189453125, -0.06591796875, 0.0220947265625, 0.1591796875, -0.3671875, 0.06884765625, 0.002838134765625, -0.201171875, 0.09228515625, 0.0732421875, -0.0140380859375, -0.062255859375, -0.474609375, -0.038330078125, -0.0...
"Example 8 Given that positive integers $p, q$ are both prime numbers, and $7p+q, pq+11$ are also pr(...TRUNCATED)	"When $p \\equiv q \\equiv 1(\\bmod 2)$,\n$$\n7 p+q \\equiv 7+1 \\equiv 0(\\bmod 2) \\text{. }\n$$\n(...TRUNCATED)	5	Number Theory	math-word-problem	cn_contest	15868f24-5226-555d-bbbf-e6be193243ef_1	true	"<think>\n\n</think>\n\n你将面对一个问题。请逐步思考，并将最终答案放在\\boxed(...TRUNCATED)	false	null	null	1	[{"content":"Example 8 Given that positive integers $p, q$ are both prime numbers, and $7p+q, pq+11$(...TRUNCATED)	1		false	"Example 8 Given that positive integers $p, q$ are both prime numbers, and $7p+q, pq+11$ are also pr(...TRUNCATED)	"Example 8 Given that positive integers $p, q$ are both prime numbers, and $7p+q, pq+11$ are also pr(...TRUNCATED)	15868f24-5226-555d-bbbf-e6be193243ef_1	[-0.212890625,-0.130859375,-0.059326171875,-0.2578125,0.0224609375,-0.1962890625,0.01226806640625,0.(...TRUNCATED)
"4. A football made of leather in black and white colors is shown as follows. It is known that this (...TRUNCATED)	"4. 20 (Hint: Each black piece is a pentagon, and the total number of edges for 12 pieces is: $5 \\t(...TRUNCATED)	20	Combinatorics	math-word-problem	cn_contest	42eb8113-14d6-5003-a6da-20eb3e0a6541_0	true	"<think>\n\n</think>\n\n已知这个足球由黑色皮革制成，共有12块黑色皮革片。因(...TRUNCATED)	true	null	null	2	[{"content":"4. A football made of leather in black and white colors is shown as follows. It is know(...TRUNCATED)	0	20	true	"4. A football made of leather in black and white colors is shown as follows. It is known that this (...TRUNCATED)	"4. A football made of leather in black and white colors is shown as follows. It is known that this (...TRUNCATED)	42eb8113-14d6-5003-a6da-20eb3e0a6541_0	[-0.228515625,-0.051513671875,0.04150390625,-0.291015625,0.031494140625,-0.099609375,0.0205078125,0.(...TRUNCATED)
"3. Let [x] denote the greatest integer not exceeding $x$, then the sum of the squares of all real r(...TRUNCATED)	"Solve: The intersection points of the graphs of the functions $y=x^{2}$ and $y=5[x]-4$ (see the fig(...TRUNCATED)	34	Algebra	math-word-problem	cn_contest	3be86d9d-dbf1-50d3-968a-b586418dae24_3	true	"<think>\n\n</think>\n\n你将遇到一个问题。请逐步思考，并将最终答案放在\\boxed(...TRUNCATED)	false	null	null	2	[{"content":"3. Let [x] denote the greatest integer not exceeding $x$, then the sum of the squares o(...TRUNCATED)	3		false	"3. Let [x] denote the greatest integer not exceeding $x$, then the sum of the squares of all real r(...TRUNCATED)	"3. Let [x] denote the greatest integer not exceeding $x$, then the sum of the squares of all real r(...TRUNCATED)	3be86d9d-dbf1-50d3-968a-b586418dae24_3	[-0.27734375,-0.09765625,0.02783203125,-0.25,0.009765625,-0.0947265625,-0.032958984375,0.126953125,-(...TRUNCATED)
"3. Let [x] denote the greatest integer not exceeding $x$, then the sum of the squares of all real r(...TRUNCATED)	"Solve: The intersection points of the graphs of the functions $y=x^{2}$ and $y=5[x]-4$ (see the fig(...TRUNCATED)	34	Algebra	math-word-problem	cn_contest	3be86d9d-dbf1-50d3-968a-b586418dae24_0	true	"<think>\n\n</think>\n\n你将遇到一个问题，请逐步思考，并将最终答案放在\\boxed(...TRUNCATED)	false	null	null	2	[{"content":"3. Let [x] denote the greatest integer not exceeding $x$, then the sum of the squares o(...TRUNCATED)	0		false	"3. Let [x] denote the greatest integer not exceeding $x$, then the sum of the squares of all real r(...TRUNCATED)	"3. Let [x] denote the greatest integer not exceeding $x$, then the sum of the squares of all real r(...TRUNCATED)	3be86d9d-dbf1-50d3-968a-b586418dae24_0	[-0.263671875,-0.10302734375,0.0269775390625,-0.2216796875,0.016357421875,-0.08251953125,-0.02709960(...TRUNCATED)
Example 9. Find $\lim _{x \rightarrow 0}\left[\frac{\sin x}{x}\right]^{\frac{1}{x}}$.	"Solution. Since $\\lim _{x \\rightarrow 0} \\frac{\\sin x}{x}=1, \\lim _{x \\rightarrow 0} \\frac{1(...TRUNCATED)	1	Calculus	math-word-problem	olympiads	bf09d3e6-e44b-5970-a647-9426b699dc58_1	false	"<think>\nOkay, so I need to find the limit as x approaches 0 of [sinx/x]^(1/x). Hmm. Let me start b(...TRUNCATED)	false	null	null	1	[{"content":"Example 9. Find $\\lim _{x \\rightarrow 0}\\left[\\frac{\\sin x}{x}\\right]^{\\frac{1}{(...TRUNCATED)	1	null	false	"Example 9. Find $\\lim _{x \\rightarrow 0}\\left[\\frac{\\sin x}{x}\\right]^{\\frac{1}{x}}$. Let's (...TRUNCATED)	"Example 9. Find $\\lim _{x \\rightarrow 0}\\left[\\frac{\\sin x}{x}\\right]^{\\frac{1}{x}}$. Let's (...TRUNCATED)	bf09d3e6-e44b-5970-a647-9426b699dc58_1	[-0.095703125,-0.01495361328125,0.291015625,-0.318359375,0.08544921875,-0.01513671875,-0.04931640625(...TRUNCATED)
G7.4 If $S=1+2-3-4+5+6-7-8+\ldots+1985$, find $S$.	"Reference: 1988 FG6.4, 1990 FG10.1, 1991 FSI.1, 1992 FI1.4\n$$\nS=1+(2-3-4+5)+(6-7-8+9)+\\ldots+(19(...TRUNCATED)	1	Algebra	math-word-problem	olympiads	5e1cda40-8147-5d9a-841b-97eace03cd31_1	false	"<think>\nOkay, so I need to find the sum S which is given by the series: 1 + 2 - 3 - 4 + 5 + 6 - 7 (...TRUNCATED)	false	null	null	1	[{"content":"G7.4 If $S=1+2-3-4+5+6-7-8+\\ldots+1985$, find $S$.","role":"user"},{"content":"<think>(...TRUNCATED)	1	null	false	"G7.4 If $S=1+2-3-4+5+6-7-8+\\ldots+1985$, find $S$. Let's think step by step and output the final a(...TRUNCATED)	"G7.4 If $S=1+2-3-4+5+6-7-8+\\ldots+1985$, find $S$. Let's think step by step and output the final a(...TRUNCATED)	5e1cda40-8147-5d9a-841b-97eace03cd31_1	[-0.14453125,-0.08203125,0.11083984375,-0.291015625,0.0400390625,-0.0751953125,0.0927734375,0.153320(...TRUNCATED)
"$\\underline{\\text { Folklore }}$\n\nFind the number of solutions in natural numbers of the equati(...TRUNCATED)	"Let $x=11 n+r$, where $n \\geq 0,0 \\leq r \\leq 10$. Then $\\left[{ }^{x} / 11\\right]=n, n+1=\\le(...TRUNCATED)	110	Number Theory	math-word-problem	olympiads	8fd34a89-a3f5-5316-b347-b803a43b5567_1	false	"<think>\nOkay, so I need to find the number of solutions in natural numbers for the equation [x/10](...TRUNCATED)	false	null	null	1	[{"content":"$\\underline{\\text { Folklore }}$\n\nFind the number of solutions in natural numbers o(...TRUNCATED)	1	null	false	"$\\underline{\\text { Folklore }}$\n\nFind the number of solutions in natural numbers of the equati(...TRUNCATED)	"$\\underline{\\text { Folklore }}$\n\nFind the number of solutions in natural numbers of the equati(...TRUNCATED)	8fd34a89-a3f5-5316-b347-b803a43b5567_1	[-0.185546875,-0.05615234375,0.1240234375,-0.11865234375,-0.059814453125,-0.091796875,0.006317138671(...TRUNCATED)
"202. Invariant remainder. Find a number that, when dividing the numbers $1108, 1453, 1844$, and 228(...TRUNCATED)	"202. Since the remainder does not change, the divisor must be odd. Further, $1453-1108=345,1844-145(...TRUNCATED)	23	Number Theory	math-word-problem	olympiads	489db8ed-8263-5c4f-8d7d-bcf1fdcd2c96_0	false	"<think>\nOkay, let's see. I need to find a number that, when it divides 1108, 1453, 1844, and 2281,(...TRUNCATED)	false	null	null	1	[{"content":"202. Invariant remainder. Find a number that, when dividing the numbers $1108, 1453, 18(...TRUNCATED)	0	null	false	"202. Invariant remainder. Find a number that, when dividing the numbers $1108, 1453, 1844$, and 228(...TRUNCATED)	"202. Invariant remainder. Find a number that, when dividing the numbers $1108, 1453, 1844$, and 228(...TRUNCATED)	489db8ed-8263-5c4f-8d7d-bcf1fdcd2c96_0	[-0.169921875,-0.03564453125,0.1943359375,-0.003814697265625,-0.041748046875,0.1337890625,0.19335937(...TRUNCATED)
"2. (5 points) Given $a=0.5, b=\\frac{1}{3}$, then $a-b$ is $\\frac{1}{78}$ of $\\qquad$ times\n\n(...TRUNCATED)	"【Solution】Solve: $(a-b) \\div \\frac{1}{78}$\n$$\n\\begin{array}{l}\n=\\left(0.5-\\frac{1}{3}\\(...TRUNCATED)	13	Algebra	math-word-problem	olympiads	f10e154e-3936-5540-b1c8-a88a73b9d63e_0	false	"<think>\n嗯，好的，我现在要解决这个数学题。题目是说，给定a=0.5，b=1/3，(...TRUNCATED)	false	null	null	1	[{"content":"2. (5 points) Given $a=0.5, b=\\frac{1}{3}$, then $a-b$ is $\\frac{1}{78}$ of $\\qquad$(...TRUNCATED)	0	null	false	"2. (5 points) Given $a=0.5, b=\\frac{1}{3}$, then $a-b$ is $\\frac{1}{78}$ of $\\qquad$ times\n\n(...TRUNCATED)	"2. (5 points) Given $a=0.5, b=\\frac{1}{3}$, then $a-b$ is $\\frac{1}{78}$ of $\\qquad$ times\n\n(...TRUNCATED)	f10e154e-3936-5540-b1c8-a88a73b9d63e_0	[-0.1494140625,-0.060546875,-0.0233154296875,-0.0025177001953125,0.1513671875,-0.0181884765625,-0.01(...TRUNCATED)

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problem string	solution string	answer string	problem_type string	question_type string	source string	uuid string	is_reasoning_complete bool	generations string	correctness_math_verify bool	correctness_llama list	finish_reasons list	correctness_count int64	messages list	generation_id int64	extracted_box_solution string	extracted_box_solution_accuracy bool	question_prompt string	full_prompt string	new_uuid string	embedding list
Let $f(x)=x^2-2x+1.$ For some constant $k, f(x+k) = x^2+2x+1$ for all real numbers $x.$ Determine the value of $k.$	1. Given the function \( f(x) = x^2 - 2x + 1 \), we can rewrite it in a factored form: \[ f(x) = (x-1)^2 \] 2. We are given that for some constant \( k \), the function \( f(x+k) \) is equal to \( x^2 + 2x + 1 \) for all real numbers \( x \). Therefore, we need to find \( k \) such that: \[ f(x+k) = x^2...	2	Algebra	math-word-problem	aops_forum	50a92404-34c1-57b4-8c33-f28ef8abc13f_1	false	<think> Okay, so I have this problem here: the function f(x) is defined as x squared minus 2x plus 1. Then, for some constant.	false	null	null	1	[ { "content": "Let $f(x)=x^2-2x+1.$ For some constant $k, f(x+k) = x^2+2x+1$ for all real numbers $x.$ Determine the value of $k.$", "role": "user" }, { "content": "<think>\nOkay, let's see here. I need to find the value of k such that when I replace x with (x + k) in the function f(x) = x² - 2x + 1,...	1	null	false	Let $f(x)=x^2-2x+1.$ For some constant $k, f(x+k) = x^2+2x+1$ for all real numbers $x.$ Determine the value of $k.$ Let's think step by step and output the final answer within \boxed{}.	Let $f(x)=x^2-2x+1.$ For some constant $k, f(x+k) = x^2+2x+1$ for all real numbers $x.$ Determine the value of $k.$ Let's think step by step and output the final answer within \boxed{}. <think> Okay, so I have this problem here: the function f(x) is defined as x squared minus 2x plus 1. Then, for some constant.	50a92404-34c1-57b4-8c33-f28ef8abc13f_1	[ -0.2109375, -0.04345703125, 0.08642578125, -0.267578125, 0.0294189453125, -0.06591796875, 0.0220947265625, 0.1591796875, -0.3671875, 0.06884765625, 0.002838134765625, -0.201171875, 0.09228515625, 0.0732421875, -0.0140380859375, -0.062255859375, -0.474609375, -0.038330078125, -0.0...
"Example 8 Given that positive integers $p, q$ are both prime numbers, and $7p+q, pq+11$ are also pr(...TRUNCATED)	"When $p \\equiv q \\equiv 1(\\bmod 2)$,\n$$\n7 p+q \\equiv 7+1 \\equiv 0(\\bmod 2) \\text{. }\n$$\n(...TRUNCATED)	5	Number Theory	math-word-problem	cn_contest	15868f24-5226-555d-bbbf-e6be193243ef_1	true	"<think>\n\n</think>\n\n你将面对一个问题。请逐步思考，并将最终答案放在\\boxed(...TRUNCATED)	false	null	null	1	[{"content":"Example 8 Given that positive integers $p, q$ are both prime numbers, and $7p+q, pq+11$(...TRUNCATED)	1		false	"Example 8 Given that positive integers $p, q$ are both prime numbers, and $7p+q, pq+11$ are also pr(...TRUNCATED)	"Example 8 Given that positive integers $p, q$ are both prime numbers, and $7p+q, pq+11$ are also pr(...TRUNCATED)	15868f24-5226-555d-bbbf-e6be193243ef_1	[-0.212890625,-0.130859375,-0.059326171875,-0.2578125,0.0224609375,-0.1962890625,0.01226806640625,0.(...TRUNCATED)
"4. A football made of leather in black and white colors is shown as follows. It is known that this (...TRUNCATED)	"4. 20 (Hint: Each black piece is a pentagon, and the total number of edges for 12 pieces is: $5 \\t(...TRUNCATED)	20	Combinatorics	math-word-problem	cn_contest	42eb8113-14d6-5003-a6da-20eb3e0a6541_0	true	"<think>\n\n</think>\n\n已知这个足球由黑色皮革制成，共有12块黑色皮革片。因(...TRUNCATED)	true	null	null	2	[{"content":"4. A football made of leather in black and white colors is shown as follows. It is know(...TRUNCATED)	0	20	true	"4. A football made of leather in black and white colors is shown as follows. It is known that this (...TRUNCATED)	"4. A football made of leather in black and white colors is shown as follows. It is known that this (...TRUNCATED)	42eb8113-14d6-5003-a6da-20eb3e0a6541_0	[-0.228515625,-0.051513671875,0.04150390625,-0.291015625,0.031494140625,-0.099609375,0.0205078125,0.(...TRUNCATED)
"3. Let [x] denote the greatest integer not exceeding $x$, then the sum of the squares of all real r(...TRUNCATED)	"Solve: The intersection points of the graphs of the functions $y=x^{2}$ and $y=5[x]-4$ (see the fig(...TRUNCATED)	34	Algebra	math-word-problem	cn_contest	3be86d9d-dbf1-50d3-968a-b586418dae24_3	true	"<think>\n\n</think>\n\n你将遇到一个问题。请逐步思考，并将最终答案放在\\boxed(...TRUNCATED)	false	null	null	2	[{"content":"3. Let [x] denote the greatest integer not exceeding $x$, then the sum of the squares o(...TRUNCATED)	3		false	"3. Let [x] denote the greatest integer not exceeding $x$, then the sum of the squares of all real r(...TRUNCATED)	"3. Let [x] denote the greatest integer not exceeding $x$, then the sum of the squares of all real r(...TRUNCATED)	3be86d9d-dbf1-50d3-968a-b586418dae24_3	[-0.27734375,-0.09765625,0.02783203125,-0.25,0.009765625,-0.0947265625,-0.032958984375,0.126953125,-(...TRUNCATED)
"3. Let [x] denote the greatest integer not exceeding $x$, then the sum of the squares of all real r(...TRUNCATED)	"Solve: The intersection points of the graphs of the functions $y=x^{2}$ and $y=5[x]-4$ (see the fig(...TRUNCATED)	34	Algebra	math-word-problem	cn_contest	3be86d9d-dbf1-50d3-968a-b586418dae24_0	true	"<think>\n\n</think>\n\n你将遇到一个问题，请逐步思考，并将最终答案放在\\boxed(...TRUNCATED)	false	null	null	2	[{"content":"3. Let [x] denote the greatest integer not exceeding $x$, then the sum of the squares o(...TRUNCATED)	0		false	"3. Let [x] denote the greatest integer not exceeding $x$, then the sum of the squares of all real r(...TRUNCATED)	"3. Let [x] denote the greatest integer not exceeding $x$, then the sum of the squares of all real r(...TRUNCATED)	3be86d9d-dbf1-50d3-968a-b586418dae24_0	[-0.263671875,-0.10302734375,0.0269775390625,-0.2216796875,0.016357421875,-0.08251953125,-0.02709960(...TRUNCATED)
Example 9. Find $\lim _{x \rightarrow 0}\left[\frac{\sin x}{x}\right]^{\frac{1}{x}}$.	"Solution. Since $\\lim _{x \\rightarrow 0} \\frac{\\sin x}{x}=1, \\lim _{x \\rightarrow 0} \\frac{1(...TRUNCATED)	1	Calculus	math-word-problem	olympiads	bf09d3e6-e44b-5970-a647-9426b699dc58_1	false	"<think>\nOkay, so I need to find the limit as x approaches 0 of [sinx/x]^(1/x). Hmm. Let me start b(...TRUNCATED)	false	null	null	1	[{"content":"Example 9. Find $\\lim _{x \\rightarrow 0}\\left[\\frac{\\sin x}{x}\\right]^{\\frac{1}{(...TRUNCATED)	1	null	false	"Example 9. Find $\\lim _{x \\rightarrow 0}\\left[\\frac{\\sin x}{x}\\right]^{\\frac{1}{x}}$. Let's (...TRUNCATED)	"Example 9. Find $\\lim _{x \\rightarrow 0}\\left[\\frac{\\sin x}{x}\\right]^{\\frac{1}{x}}$. Let's (...TRUNCATED)	bf09d3e6-e44b-5970-a647-9426b699dc58_1	[-0.095703125,-0.01495361328125,0.291015625,-0.318359375,0.08544921875,-0.01513671875,-0.04931640625(...TRUNCATED)
G7.4 If $S=1+2-3-4+5+6-7-8+\ldots+1985$, find $S$.	"Reference: 1988 FG6.4, 1990 FG10.1, 1991 FSI.1, 1992 FI1.4\n$$\nS=1+(2-3-4+5)+(6-7-8+9)+\\ldots+(19(...TRUNCATED)	1	Algebra	math-word-problem	olympiads	5e1cda40-8147-5d9a-841b-97eace03cd31_1	false	"<think>\nOkay, so I need to find the sum S which is given by the series: 1 + 2 - 3 - 4 + 5 + 6 - 7 (...TRUNCATED)	false	null	null	1	[{"content":"G7.4 If $S=1+2-3-4+5+6-7-8+\\ldots+1985$, find $S$.","role":"user"},{"content":"<think>(...TRUNCATED)	1	null	false	"G7.4 If $S=1+2-3-4+5+6-7-8+\\ldots+1985$, find $S$. Let's think step by step and output the final a(...TRUNCATED)	"G7.4 If $S=1+2-3-4+5+6-7-8+\\ldots+1985$, find $S$. Let's think step by step and output the final a(...TRUNCATED)	5e1cda40-8147-5d9a-841b-97eace03cd31_1	[-0.14453125,-0.08203125,0.11083984375,-0.291015625,0.0400390625,-0.0751953125,0.0927734375,0.153320(...TRUNCATED)
"$\\underline{\\text { Folklore }}$\n\nFind the number of solutions in natural numbers of the equati(...TRUNCATED)	"Let $x=11 n+r$, where $n \\geq 0,0 \\leq r \\leq 10$. Then $\\left[{ }^{x} / 11\\right]=n, n+1=\\le(...TRUNCATED)	110	Number Theory	math-word-problem	olympiads	8fd34a89-a3f5-5316-b347-b803a43b5567_1	false	"<think>\nOkay, so I need to find the number of solutions in natural numbers for the equation [x/10](...TRUNCATED)	false	null	null	1	[{"content":"$\\underline{\\text { Folklore }}$\n\nFind the number of solutions in natural numbers o(...TRUNCATED)	1	null	false	"$\\underline{\\text { Folklore }}$\n\nFind the number of solutions in natural numbers of the equati(...TRUNCATED)	"$\\underline{\\text { Folklore }}$\n\nFind the number of solutions in natural numbers of the equati(...TRUNCATED)	8fd34a89-a3f5-5316-b347-b803a43b5567_1	[-0.185546875,-0.05615234375,0.1240234375,-0.11865234375,-0.059814453125,-0.091796875,0.006317138671(...TRUNCATED)
"202. Invariant remainder. Find a number that, when dividing the numbers $1108, 1453, 1844$, and 228(...TRUNCATED)	"202. Since the remainder does not change, the divisor must be odd. Further, $1453-1108=345,1844-145(...TRUNCATED)	23	Number Theory	math-word-problem	olympiads	489db8ed-8263-5c4f-8d7d-bcf1fdcd2c96_0	false	"<think>\nOkay, let's see. I need to find a number that, when it divides 1108, 1453, 1844, and 2281,(...TRUNCATED)	false	null	null	1	[{"content":"202. Invariant remainder. Find a number that, when dividing the numbers $1108, 1453, 18(...TRUNCATED)	0	null	false	"202. Invariant remainder. Find a number that, when dividing the numbers $1108, 1453, 1844$, and 228(...TRUNCATED)	"202. Invariant remainder. Find a number that, when dividing the numbers $1108, 1453, 1844$, and 228(...TRUNCATED)	489db8ed-8263-5c4f-8d7d-bcf1fdcd2c96_0	[-0.169921875,-0.03564453125,0.1943359375,-0.003814697265625,-0.041748046875,0.1337890625,0.19335937(...TRUNCATED)
"2. (5 points) Given $a=0.5, b=\\frac{1}{3}$, then $a-b$ is $\\frac{1}{78}$ of $\\qquad$ times\n\n(...TRUNCATED)	"【Solution】Solve: $(a-b) \\div \\frac{1}{78}$\n$$\n\\begin{array}{l}\n=\\left(0.5-\\frac{1}{3}\\(...TRUNCATED)	13	Algebra	math-word-problem	olympiads	f10e154e-3936-5540-b1c8-a88a73b9d63e_0	false	"<think>\n嗯，好的，我现在要解决这个数学题。题目是说，给定a=0.5，b=1/3，(...TRUNCATED)	false	null	null	1	[{"content":"2. (5 points) Given $a=0.5, b=\\frac{1}{3}$, then $a-b$ is $\\frac{1}{78}$ of $\\qquad$(...TRUNCATED)	0	null	false	"2. (5 points) Given $a=0.5, b=\\frac{1}{3}$, then $a-b$ is $\\frac{1}{78}$ of $\\qquad$ times\n\n(...TRUNCATED)	"2. (5 points) Given $a=0.5, b=\\frac{1}{3}$, then $a-b$ is $\\frac{1}{78}$ of $\\qquad$ times\n\n(...TRUNCATED)	f10e154e-3936-5540-b1c8-a88a73b9d63e_0	[-0.1494140625,-0.060546875,-0.0233154296875,-0.0025177001953125,0.1513671875,-0.0181884765625,-0.01(...TRUNCATED)