question stringlengths 15 7.47k | solution stringlengths 0 11.9k | answer stringlengths 1 171 | problem_type stringclasses 8
values | question_type stringclasses 1
value | exit_loop_index listlengths 550 4.1k | final_initial_embedding_distance listlengths 550 4.1k | final_initial_kl_distance listlengths 550 4.1k | initial_tokens listlengths 550 4.1k | final_tokens listlengths 550 4.1k |
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"P.I. Find the largest natural number from which it is impossible to obtain a number divisible by 11(...TRUNCATED) | "If in the decimal representation of a number there is a digit 0 or two identical digits, then by er(...TRUNCATED) | 987654321 | Number Theory | math-word-problem | [3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3(...TRUNCATED) | [72.02496337890625,41.373390197753906,37.24734115600586,43.25685501098633,45.434959411621094,39.0107(...TRUNCATED) | [4.33967924118042,1.1540040969848633,0.8192936778068542,0.7709985971450806,1.7977226972579956,2.2768(...TRUNCATED) | [4525,198,198,33,44,21745,33,30,30,32,198,31,32,32,32,198,32,33,32,30,198,198,19,21745,198,19,198,30(...TRUNCATED) | [30,198,198,198,64,198,33,30,30,30,198,57,32,30,30,33,77,33,77,77,198,198,2954,198,57,19,57,30,30,33(...TRUNCATED) |
"G1.3 A six-digit number 1234xy is divisible by both 8 and 9 . Given that $x+y=c$, find the value of(...TRUNCATED) | "The number formed by last 3 digits must be divisible by 8 and the sum of digits must be divisible b(...TRUNCATED) | 8 | Number Theory | math-word-problem | [3,3,3,3,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3(...TRUNCATED) | [69.21195983886719,40.954036712646484,41.51908493041992,46.46404266357422,32.24943923950195,68.81080(...TRUNCATED) | [2.4850378036499023,0.8960867524147034,0.8444229364395142,0.6500551700592041,0.2922924757003784,4.04(...TRUNCATED) | [504,30,32,198,198,4525,4525,4525,4525,4525,4525,4525,4525,4525,4525,4525,4525,4525,4525,4525,4525,4(...TRUNCATED) | [32,30,32,2,0,1,1,1,198,198,198,198,198,198,198,198,198,198,198,198,198,198,198,198,198,198,198,198,(...TRUNCATED) |
"8. Find $\\sqrt{\\frac{x}{63}-32} \\times \\sqrt{\\frac{y}{63}-32}$, given that $\\frac{1}{x}+\\fra(...TRUNCATED) | "Solution: Let's make the substitution: $a=x / 63, b=y / 63$. Then we can rewrite the condition as: (...TRUNCATED) | 32 | Algebra | math-word-problem | [3,2,2,2,2,2,3,2,3,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,2,3,2,3,3,3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2(...TRUNCATED) | [41.22091293334961,19.061315536499023,27.966920852661133,32.29814910888672,36.763668060302734,25.905(...TRUNCATED) | [1.1097211837768555,0.0002535983803682029,0.00001152994082076475,0.29193025827407837,0.0006471779779(...TRUNCATED) | [198,198,1,520,9531,198,3,198,39122,28,1303,506,963,451,1732,1833,411,1833,30,378,1962,314,5617,327,(...TRUNCATED) | [2,198,1,520,9531,198,3,198,39122,28,1303,506,10848,623,1732,1833,411,1833,30,378,1962,314,5617,327,(...TRUNCATED) |
"10. It is required to fence a rectangular area adjacent to a wall. The fence should have a length o(...TRUNCATED) | "10. Solution. First method. Let the width of the site be \\(x \\mu\\), then its length will be ( \\(...TRUNCATED) | 450 | Algebra | math-word-problem | [3,3,3,3,3,2,3,3,3,3,3,2,2,2,3,3,3,3,2,3,3,3,3,3,3,2,3,3,3,3,2,3,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,2(...TRUNCATED) | [71.32349395751953,41.139976501464844,41.59312057495117,40.003658294677734,40.37685775756836,29.6941(...TRUNCATED) | [2.883164882659912,2.4451117515563965,2.9184393882751465,2.6244313716888428,4.778461456298828,0.0117(...TRUNCATED) | [198,30,1557,282,253,288,260,216,30,253,288,325,275,2866,351,378,1557,282,260,18906,314,216,347,3814(...TRUNCATED) | [32,30,1557,314,288,288,253,4923,30,1923,288,325,275,2866,30,378,3519,282,260,18906,314,1836,347,381(...TRUNCATED) |
"4. Median. In a set of 100 numbers. If one number is removed, the median of the remaining numbers i(...TRUNCATED) | "Solution. Arrange the numbers in ascending order. If we remove a number from the first half of the (...TRUNCATED) | 72 | Logic and Puzzles | math-word-problem | [3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,2,2,3,3,3,3,3,3,3,3,3,3,3,3,2,3,3,2,3(...TRUNCATED) | [71.0882797241211,44.81148147583008,47.83207321166992,43.04597854614258,46.050865173339844,41.263973(...TRUNCATED) | [4.583312034606934,0.24869512021541595,2.1431756019592285,0.6326180696487427,3.4701344966888428,0.42(...TRUNCATED) | [198,30,198,506,2988,282,216,2966,2966,41,2966,314,1885,30,284,260,33,1232,33,25,25,314,216,36,31,21(...TRUNCATED) | [1506,30,378,260,2798,282,260,216,216,33,2966,325,1792,30,965,260,104,1232,33,32,14428,446,216,37,44(...TRUNCATED) |
"31. Given that $\\alpha$ is an acute angle satisfying\n$$\n\\sqrt{369-360 \\cos \\alpha}+\\sqrt{544(...TRUNCATED) | "31. Answer: 30\nLet $X=\\sqrt{369-360 \\cos \\alpha}$ and $Y=\\sqrt{544-480 \\sin \\alpha}$. Observ(...TRUNCATED) | 30 | Algebra | math-word-problem | [3,3,3,2,2,2,2,2,3,2,3,2,2,2,2,2,2,2,2,2,3,2,2,2,3,3,2,2,3,3,3,2,3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2(...TRUNCATED) | [70.375,40.548439025878906,36.22409439086914,21.450876235961914,30.790719985961914,34.62116241455078(...TRUNCATED) | [3.88413667678833,0.8482372164726257,0.17301179468631744,0.0008400215301662683,0.0000571273521927651(...TRUNCATED) | [198,216,2,198,1,520,9531,198,2068,198,39122,28,1303,506,963,288,5482,451,1732,1833,1644,28,392,457,(...TRUNCATED) | [30,198,2,198,1,520,9531,198,3,198,39122,28,1303,506,963,288,5482,451,1732,30,1644,28,392,457,354,83(...TRUNCATED) |
"## Task 7 - V00607 = V00503\n\nTo an imagined number, 16 is added, then multiplied by 7, followed b(...TRUNCATED) | "Let $x$ be the thought number. Then the operations yield the term\n\n$$\n((x+16) \\cdot 7-8): 9\n$$(...TRUNCATED) | 14 | Algebra | math-word-problem | [3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3(...TRUNCATED) | [70.5218505859375,62.16684341430664,57.505859375,49.05683135986328,48.05531692504883,40.095489501953(...TRUNCATED) | [3.486267566680908,5.685742378234863,2.6267175674438477,1.6010714769363403,2.597393751144409,3.07512(...TRUNCATED) | [4525,198,198,19,198,21745,282,198,198,5482,451,1142,282,198,359,288,325,260,1142,282,260,198,32,198(...TRUNCATED) | [216,10283,198,198,2068,3856,216,198,19,5482,451,8345,28,392,737,288,1042,260,1685,282,260,24,1792,3(...TRUNCATED) |
"IMO 1974 Problem A1 Three players play the following game. There are three cards each with a differ(...TRUNCATED) | "The player with 9 counters. The total of the scores, 39, must equal the number of rounds times the (...TRUNCATED) | 9 | Logic and Puzzles | math-word-problem | [3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3(...TRUNCATED) | [70.68334197998047,45.014163970947266,40.906402587890625,41.017398834228516,47.373207092285156,51.78(...TRUNCATED) | [2.06929087638855,1.4323279857635498,1.9030823707580566,2.226900815963745,2.0923964977264404,2.18518(...TRUNCATED) | [198,30,30,30,30,198,198,42,198,198,3814,253,260,1695,359,260,3193,28,253,198,25,33,198,30,33,260,19(...TRUNCATED) | [216,41,41,36,198,198,5974,42,198,4239,506,25832,260,2966,335,260,7005,347,1885,81,26265,1885,82,262(...TRUNCATED) |
9.1. Solve the equation:
$$
\left(x^{2}-20\right)^{2}+\left(x^{2}-19\right)^{2}=2019
$$ | "Solution. Let $x^{2}-20=y$; then $y^{2}+y-1009=0(D=4037)$ and $y=\\frac{-1 \\pm \\sqrt{4037}}{2}$. (...TRUNCATED) | \\sqrt{\frac{39+\sqrt{4037}}{2}} | Algebra | math-word-problem | [3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,2,3(...TRUNCATED) | [69.7166519165039,45.55332946777344,44.851985931396484,45.47982406616211,42.389774322509766,49.14854(...TRUNCATED) | [2.9233405590057373,2.043656349182129,1.0544002056121826,0.7147126197814941,0.4692436456680298,1.053(...TRUNCATED) | [198,198,30,30,32,30,198,198,30,30,30,30,198,30,32,32,198,32,198,30,2064,198,32,32,30,30,198,30,32,1(...TRUNCATED) | [30,198,30,30,30,34,34,32,32,32,32,198,30,30,30,198,33,33,33,30,198,33,30,33,32,198,33,30,198,33,30,(...TRUNCATED) |
"7. Encode the letters $a \\sim z$ as $(\\mathbf{a}=1, \\mathbf{b}=2, \\ldots, \\mathbf{z}=26)$, so (...TRUNCATED) | "【Analysis】(Method one) To be a \"middle ring number\", the prime factors decomposed must includ(...TRUNCATED) | 106 | Number Theory | math-word-problem | [3,2,2,2,2,2,2,2,3,2,2,2,2,2,2,2,3,3,2,2,2,3,2,2,2,2,2,3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,2,2,2,2,2,3(...TRUNCATED) | [49.53239059448242,18.06980323791504,37.34885025024414,29.873292922973633,31.589805603027344,33.6641(...TRUNCATED) | [0.21193000674247742,0.00027948414208367467,0.0000276374976238003,0.0014538747491315007,5.2273981054(...TRUNCATED) | [2,198,1,520,9531,198,2068,198,39122,28,1303,506,1576,288,4110,578,260,1732,1296,29,23141,476,22581,(...TRUNCATED) | [2,198,1,520,9531,198,3,198,39122,28,588,506,1576,288,4110,578,260,13341,1296,29,23141,476,22581,460(...TRUNCATED) |
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