id stringlengths 71 71 | problem stringlengths 8 3.62k | reference_answers listlengths 1 7 | original_problem_variants listlengths 1 3 | problem_sources listlengths 0 4 | source_splits listlengths 1 4 | source_split_row_counts unknown | source_row_count int64 1 364 | missing_reference_answer_rows int64 0 0 | blank_reference_answer_rows int64 0 25 |
|---|---|---|---|---|---|---|---|---|---|
omr-ws-895d7733fc07bf5e06738a5bac7c66b7656a607f7276b1c9a6c220de8ec27497 | Arrange the following expressions in increasing order: $\sin 1 + \cos 1$, $\sqrt{\sin 1} + \sqrt{\cos 1}$, $\sqrt{1 + \sin 2}$, $1$. (All angles are in radians.) | [
"\\( 1 < \\sin 1 + \\cos 1 = \\sqrt{1 + \\sin 2} < \\sqrt{\\sin 1} + \\sqrt{\\cos 1} \\)"
] | [
"Arrange the following expressions in increasing order: $\\sin 1 + \\cos 1$, $\\sqrt{\\sin 1} + \\sqrt{\\cos 1}$, $\\sqrt{1 + \\sin 2}$, $1$. (All angles are in radians.)"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 31,
"genselect": 8,
"tir": 9
} | 48 | 0 | 0 |
omr-ws-af6f508b05619e0346e429b59208b74c04844262c6f5797ab9d83123dbad2171 | Arrange the following four numbers from smallest to largest: \( a = (10^{100})^{10} \), \( b = 10^{(10^{10})} \), \( c = 1000000! \), \( d = (100!)^{10} \). | [
"\\( a, d, c, b \\)"
] | [
"Arrange the following four numbers from smallest to largest: \\( a = (10^{100})^{10} \\), \\( b = 10^{(10^{10})} \\), \\( c = 1000000! \\), \\( d = (100!)^{10} \\)."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 27
} | 41 | 0 | 0 |
omr-ws-74de0c48181cebbca8f2527502444cc08c33fbf1ffd59a9a5698ff1a290bc79f | Arrange the following in ascending order: $\tan1, \cot2, \tan3, \cot4, \tan5, \cot6, \tan7, \cot8$. | [
"\\(\\tan(5), \\tan(3), \\cot(2), \\cot(6), \\cot(8), \\tan(1), \\cot(4), \\tan(7)\\)"
] | [
"Arrange the following in ascending order: $\\tan1, \\cot2, \\tan3, \\cot4, \\tan5, \\cot6, \\tan7, \\cot8$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4e7be6d06012d1f41a705c30bc0bb407f2f712f58a5648c2eac0267627af5138 | Arrange the following integrals in increasing order: \[ I_{1} = \int_{0}^{\frac{\pi}{2}} \cos(\sin x) \, dx, \] \[ I_{2} = \int_{0}^{\frac{\pi}{2}} \sin(\cos x) \, dx, \] \[ I_{3} = \int_{0}^{\frac{\pi}{2}} \cos x \, dx. \] | [
"\\( I_1 > I_3 > I_2 \\)"
] | [
"Arrange the following integrals in increasing order:\n\\[ I_{1} = \\int_{0}^{\\frac{\\pi}{2}} \\cos(\\sin x) \\, dx, \\]\n\\[ I_{2} = \\int_{0}^{\\frac{\\pi}{2}} \\sin(\\cos x) \\, dx, \\]\n\\[ I_{3} = \\int_{0}^{\\frac{\\pi}{2}} \\cos x \\, dx. \\]"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d5e81dd236ff3f50d52d23b8bfc0d528c44bc19b8fc5847654c3b5646725e143 | Arrange the following logarithms in ascending order: \[ \log_3 108, \log_4 192, \log_6 1080, \log_5 500 \] | [
"\\(\\log_4 192, \\log_5 500, \\log_6 1080, \\log_3 108\\)"
] | [
"Arrange the following logarithms in ascending order:\n\\[\n\\log_3 108, \\log_4 192, \\log_6 1080, \\log_5 500\n\\]"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 33
} | 49 | 0 | 0 |
omr-ws-a740a7f103a2eb7ad709df5d0dba8639b319bd15405bbeb42e2e7df5bdd3441e | Arrange the following logarithms in increasing order: \(\log_2 3\), \(\log_3 4\), \(\log_4 6\), and \(\log_6 8\). | [
"\\log_6 8 < \\log_3 4 < \\log_4 6 < \\log_2 3"
] | [
"Arrange the following logarithms in increasing order: \\(\\log_2 3\\), \\(\\log_3 4\\), \\(\\log_4 6\\), and \\(\\log_6 8\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-b3befea858d35e7bc6f6015159c0c328c4436af2462e58e1a9033c5e6162c218 | Arrange the following numbers according to size: $333^{3}$, $33^{33}$, $3^{333}$, $3^{3^{33}}$, $3^{3^{3^{3}}}$. | [
"$333^3, 33^{33}, 3^{333}, 3^{3^{3^3}}, 3^{3^{33}}$"
] | [
"Arrange the following numbers according to size: $333^{3}$, $33^{33}$, $3^{333}$, $3^{3^{33}}$, $3^{3^{3^{3}}}$."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 10,
"tir": 24
} | 34 | 0 | 0 |
omr-ws-631614846c25ba6fe97f642007e5972b1cb554d4e9c78309570c7836ff856adc | Arrange the following numbers in increasing order: \(2^{5555}\), \(3^{3333}\), \(6^{2222}\). | [
"\\(3^{3333}, 2^{5555}, 6^{2222}\\)"
] | [
"Arrange the following numbers in increasing order: \\(2^{5555}\\), \\(3^{3333}\\), \\(6^{2222}\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 29
} | 43 | 0 | 0 |
omr-ws-b9cd5184b7aef67e552ae05a9d71cf913d10dd0946f903a8b8a3396f4b93f33a | Arrange the following products in increasing order from left to right: \[ 1000! \] \[ (400!)(400!)(200!) \] \[ (500!)(500!) \] \[ (600!)(300!)(100!) \] \[ (700!)(300!) \] | [
"\\( 400!400!200!, 600!300!100!, 500!500!, 700!300!, 1000! \\)"
] | [
"Arrange the following products in increasing order from left to right:\n\\[ 1000! \\]\n\\[ (400!)(400!)(200!) \\]\n\\[ (500!)(500!) \\]\n\\[ (600!)(300!)(100!) \\]\n\\[ (700!)(300!) \\]"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 11,
"tir": 26
} | 37 | 0 | 0 |
omr-ws-fbe7c637c4d8beb1dda1917a755bd6d14534816ab9bd15883f941b482d0e5dbc | Arrange the following products in increasing order from left to right: \[ 1000! \] \[ \left(400!\right)^2 \cdot \left(200!\right) \] \[ \left(500!\right)^2 \] \[ \left(600!\right) \cdot \left(300!\right) \cdot \left(1000!\right) \] \[ \left(700!\right) \cdot \left(300!\right) \] | [
"\\((400!)^2 \\cdot (200!), (500!)^2, (700!) \\cdot (300!), 1000!, (600!) \\cdot (300!) \\cdot (1000!)\\)"
] | [
"Arrange the following products in increasing order from left to right:\n\\[ 1000! \\]\n\\[ \\left(400!\\right)^2 \\cdot \\left(200!\\right) \\]\n\\[ \\left(500!\\right)^2 \\]\n\\[ \\left(600!\\right) \\cdot \\left(300!\\right) \\cdot \\left(1000!\\right) \\]\n\\[ \\left(700!\\right) \\cdot \\left(300!\\right) \\]"... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 8,
"tir": 29
} | 37 | 0 | 0 |
omr-ws-86bb51c91b54a6a6cde522d9294204832e7768d985c286ed3545503707050da1 | Arrange the following quantities in ascending order for a triangle: - Sum of the angle bisectors - Sum of the altitudes - Sum of the sides - Sum of the medians | [
"\\text{Sum of the altitudes} < \\text{Sum of the angle bisectors} < \\text{Sum of the medians} < \\text{Sum of the sides}"
] | [
"Arrange the following quantities in ascending order for a triangle:\n- Sum of the angle bisectors\n- Sum of the altitudes\n- Sum of the sides\n- Sum of the medians"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d0238c6e672ab9ea748f04cda722f04ae88243e0d6142d825e043e0b01f07dea | Arrange the fractions 15/11, 19/15, and 17/13 in ascending order. | [
"\\dfrac{19}{15} < \\dfrac{17}{13} < \\dfrac{15}{11}"
] | [
"Arrange the fractions 15/11, 19/15, and 17/13 in ascending order."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-a070f20e2be95e1bcc0e38a281e620daf27768556a128e61a4b51705c02292b8 | Arrange the functions \( f_1(x) = x^e \), \( f_2(x) = e^x \), \( f_3(x) = x^x \), \( f_4(x) = 2^x \), \( f_5(x) = (\ln x)^{2x} \) in increasing order of growth. | [
"\\( f_1(x) < f_4(x) < f_2(x) < f_5(x) < f_3(x) \\)"
] | [
"Arrange the functions \\( f_1(x) = x^e \\), \\( f_2(x) = e^x \\), \\( f_3(x) = x^x \\), \\( f_4(x) = 2^x \\), \\( f_5(x) = (\\ln x)^{2x} \\) in increasing order of growth."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 25
} | 38 | 0 | 0 |
omr-ws-e0191c5ac1d7610c0ad1cca058ab1b7c9e25b1071de8499121e6fa17d22cd3d1 | Arrange the integers from 1 to 15 in a row so that the sum of any two consecutive numbers is a perfect square. In how many ways can this be done? | [
"2"
] | [
"Arrange the integers from 1 to 15 in a row so that the sum of any two consecutive numbers is a perfect square. In how many ways can this be done?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-14839f85715fbfc0b204fc4ca2ab12d881cb0893a2fca72f9a94fc889c75169a | Arrange the letters A, F, K, N, O, T alphabetically and assign a number to each combination of these letters. a) What number does the combination ANKFOT represent? b) What combination of letters does the number 711 represent? | [
"a) 55, b) TOKFAN"
] | [
"Arrange the letters A, F, K, N, O, T alphabetically and assign a number to each combination of these letters.\na) What number does the combination ANKFOT represent?\nb) What combination of letters does the number 711 represent?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-0987a14f5ca84231317ad2ab7c811205a9c9e4c6a5c8f29f5a8247c792745618 | Arrange the maximum number of bishops on an $8 \times 8$ chessboard such that no bishop controls a square occupied by another bishop. Calculate the number of bishops that can be placed under this condition if all bishops must be placed on the border of the chessboard. | [
"14"
] | [
"Arrange the maximum number of bishops on an $8 \\times 8$ chessboard such that no bishop controls a square occupied by another bishop. Calculate the number of bishops that can be placed under this condition if all bishops must be placed on the border of the chessboard."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-dd9df23f6cf2ac9e5508ffdd15dfdeadf6d479b98964a716a82f8858e1defea6 | Arrange the numbers $1$ to $10$ in a pentagon such that the sum of the three numbers on each side is the same. Determine the largest and smallest values of this sum. | [
"14 and 19"
] | [
"Arrange the numbers $1$ to $10$ in a pentagon such that the sum of the three numbers on each side is the same. Determine the largest and smallest values of this sum."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 19
} | 19 | 0 | 0 |
omr-ws-6eaf6a1a5aee6f633e026490dc6b814570acff352f9d565b09504e2ce40c9d73 | Arrange the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ in nine circles forming an equilateral triangle such that each side of the triangle contains four circles and the sum of the numbers in the circles on each side is equal. Find all possible combinations of numbers at the three vertices of the triangle. | [
"\\{7, 8, 9\\}"
] | [
"Arrange the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ in nine circles forming an equilateral triangle such that each side of the triangle contains four circles and the sum of the numbers in the circles on each side is equal. Find all possible combinations of numbers at the three vertices of the triangle."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-a2ca59e66c5ff5bf8dfb5bebcc97c02c9e2aa2aa8a37d326cd343b16ad692def | Arrange the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ into the regions labeled $a, b, c, \ldots, i$ without replacement such that the following equalities hold: \[ a + b = b + c + d = d + e + f = f + g + h = h + i. \] Find all possible solutions for these numbers. | [
"8, 6, 1, 7, 4, 3, 2, 9, 5"
] | [
"Arrange the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ into the regions labeled $a, b, c, \\ldots, i$ without replacement such that the following equalities hold:\n\\[ a + b = b + c + d = d + e + f = f + g + h = h + i. \\]\nFind all possible solutions for these numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-ce1aff6606246a2b31a2530f3be521b4a838c14526b66202b6a24013eaf13f4f | Arrange the numbers $1, 2, 3, \ldots, 10$ on a $4 \times 4$ chessboard such that the greatest common divisor (GCD) of any number and its adjacent numbers (up, down, left, right) is $1$. Ignore adjacent numbers that are out of bounds. How many ways can this be done? | [
"4821590113978"
] | [
"Arrange the numbers $1, 2, 3, \\ldots, 10$ on a $4 \\times 4$ chessboard such that the greatest common divisor (GCD) of any number and its adjacent numbers (up, down, left, right) is $1$. Ignore adjacent numbers that are out of bounds. How many ways can this be done?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9c4b129ba9b1caf7bf3ac35f7f7877e663e63b68b42f99289813191e88c5500b | Arrange the numbers $1, 2, 3, \ldots, 12$ in a circle such that for any three consecutive numbers $a, b, c$, the expression $b^2 - ac$ is divisible by $13$. If such an arrangement exists, calculate the sum of all possible values of $b$ for the triplet $(a, b, c)$ where $b$ is the number immediately following $1$ in the... | [
"26"
] | [
"Arrange the numbers $1, 2, 3, \\ldots, 12$ in a circle such that for any three consecutive numbers $a, b, c$, the expression $b^2 - ac$ is divisible by $13$. If such an arrangement exists, calculate the sum of all possible values of $b$ for the triplet $(a, b, c)$ where $b$ is the number immediately following $1$ ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5dc81e549b83ad8ef67f75c055c277fdf3df35b2bf00df89252c999b86f6edc1 | Arrange the numbers $1, 2, 3, \ldots, 9$ in a $3 \times 3$ square such that the numbers $1, 2, 3$ are each in different rows and columns. Find the total number of such arrangements. | [
"25920"
] | [
"Arrange the numbers $1, 2, 3, \\ldots, 9$ in a $3 \\times 3$ square such that the numbers $1, 2, 3$ are each in different rows and columns. Find the total number of such arrangements."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-bfc5e3f6f1d4ad608abe594036b8a41f4f00930fa3bd94430749bd8c280a113f | Arrange the numbers $1, 2, \ldots, 25$ arbitrarily on a circumference. Consider all $25$ sums obtained by adding $5$ consecutive numbers. Determine all possible values of $d$, the number of distinct residues of those sums modulo $5$. | [
"1, 3, 4, 5"
] | [
"Arrange the numbers $1, 2, \\ldots, 25$ arbitrarily on a circumference. Consider all $25$ sums obtained by adding $5$ consecutive numbers. Determine all possible values of $d$, the number of distinct residues of those sums modulo $5$."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-f92a7638c5c023b2c3bb65105aea82424682c1bc191d3c09aa1cda1004623ec2 | Arrange the numbers $1, 2, \ldots, 64$ on an 8x8 chessboard. Define a "strange" number as one that is the minimal value in its respective row and the maximal value in its respective column. Let $S$ be the sum of all "strange" numbers in all possible arrangements of the numbers on the chessboard. Calculate the value of ... | [
"no_valid_answer_found"
] | [
"Arrange the numbers $1, 2, \\ldots, 64$ on an 8x8 chessboard. Define a \"strange\" number as one that is the minimal value in its respective row and the maximal value in its respective column. Let $S$ be the sum of all \"strange\" numbers in all possible arrangements of the numbers on the chessboard. Calculate the... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0b5638130a40c5bbd77fc108008126aa491b529b3e652752b4a69b285e88f7ca | Arrange the numbers $1, 2, \ldots, n$ on a circle. Let $f(n)$ denote the number of arrangements such that there is no value $a$ ($a=1,2,\ldots,n-1$) that follows $a+1$, where $1$ can be considered $n+1$. Give a closed form for $f(n)$. | [
"(n-1)! \\sum_{k=0}^{n-1} \\frac{(-1)^k}{k!}"
] | [
"Arrange the numbers $1, 2, \\ldots, n$ on a circle. Let $f(n)$ denote the number of arrangements such that there is no value $a$ ($a=1,2,\\ldots,n-1$) that follows $a+1$, where $1$ can be considered $n+1$. Give a closed form for $f(n)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 5
} | 11 | 0 | 0 |
omr-ws-6eb588a0ce119fb04965ab07f9a209aae9d20c66e2d2bcba2f83f14ccd017ab8 | Arrange the numbers $n, n-1, n-2, \ldots, 1, 0, 1, \ldots, n-1, n$ into an ordered sequence such that there is exactly 1 number between each pair of "$1$s", 2 numbers between each pair of "$2$s", ..., and $n$ numbers between each pair of "$n$s". For example, when $n=2$, one such sequence is $1, 2, 1, 0, 2$. | [
"3, 1, 2, 1, 3, 2, 0"
] | [
"Arrange the numbers $n, n-1, n-2, \\ldots, 1, 0, 1, \\ldots, n-1, n$ into an ordered sequence such that there is exactly 1 number between each pair of \"$1$s\", 2 numbers between each pair of \"$2$s\", ..., and $n$ numbers between each pair of \"$n$s\". For example, when $n=2$, one such sequence is $1, 2, 1, 0, 2$... | [
"aops_c6_high_school_olympiads"
] | [
"genselect"
] | {
"genselect": 8
} | 8 | 0 | 0 |
omr-ws-6738123d48c8f17b26012bf6c9de0593bd91f4f3792a4a303a49c20bcd5f464d | Arrange the numbers 1 through 16 in a line such that the sum of each pair of adjacent numbers is a perfect square. | [
"8, 1, 15, 10, 6, 3, 13, 12, 4, 5, 11, 14, 2, 7, 9, 16"
] | [
"Arrange the numbers 1 through 16 in a line such that the sum of each pair of adjacent numbers is a perfect square."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 6
} | 18 | 0 | 0 |
omr-ws-7e82c517ae1deff0a680f7d657e6bda438440dd2acfc8e24268196e2ae1d228e | Arrange the numbers 1 through 16 on a circle such that the sum of each pair of adjacent numbers is a perfect square. | [
"8, 1, 15, 10, 6, 3, 13, 12, 4, 5, 11, 14, 2, 7, 9, 16"
] | [
"Arrange the numbers 1 through 16 on a circle such that the sum of each pair of adjacent numbers is a perfect square."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-86ccc5c27fc7e13943140dc4193eab4d9dcf481ee780ce8edbbf6eff5ff15ece | Arrange the numbers 1 through 9 in the triangles labeled \(a\) through \(i\) such that the numbers in each of the \(2 \times 2\) triangles sum to the same value \(n\); that is, \[a + b + c + d = b + e + f + g = d + g + h + i = n.\] Find the sum of all possible distinct values of \(n\) for which such an arrangement exis... | [
"100"
] | [
"Arrange the numbers 1 through 9 in the triangles labeled \\(a\\) through \\(i\\) such that the numbers in each of the \\(2 \\times 2\\) triangles sum to the same value \\(n\\); that is,\n\\[a + b + c + d = b + e + f + g = d + g + h + i = n.\\]\nFind the sum of all possible distinct values of \\(n\\) for which such... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-321f2ee041e514f30c5969620a9be9a053748510a00443a63f010705ec661665 | Arrange the numbers 1 to 10 around a circle such that the sum of the differences between adjacent numbers is not more than 17. | [
"1, 2, 3, 4, 5, 6, 7, 8, 9, 10"
] | [
"Arrange the numbers 1 to 10 around a circle such that the sum of the differences between adjacent numbers is not more than 17."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-7221235c1ce6973ec4a1178933e55aa923e398ad068ae844c223f53f567db7cd | Arrange the numbers 1 to 11 on a wheel such that the sum of any three numbers in a straight line is 18. | [
"6"
] | [
"Arrange the numbers 1 to 11 on a wheel such that the sum of any three numbers in a straight line is 18."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 5
} | 11 | 0 | 0 |
omr-ws-2b92c07f4341b5cac27cdc095e3bd7c22d800d4b675a3e206d8ef819e1bd7b82 | Arrange the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 such that 1 and 10 cannot be at the ends, 5 cannot be in the middle, and 8 must be next to 4. How many ways can this be done? | [
"241920"
] | [
"Arrange the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 such that 1 and 10 cannot be at the ends, 5 cannot be in the middle, and 8 must be next to 4. How many ways can this be done?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 2
} | 4 | 0 | 0 |
omr-ws-31bd2b1d3e9a7235d409f60c5a0d959e528036b4aaede038cad3b93f59441dfd | Arrange the numbers 1, 2, 3, 4, 5, 6, and 7 in a row such that none of the first number, the sum of the first two numbers, the sum of the first three numbers, and so on, up to the sum of all seven numbers, is divisible by 3. In how many ways can this be done? | [
"360"
] | [
"Arrange the numbers 1, 2, 3, 4, 5, 6, and 7 in a row such that none of the first number, the sum of the first two numbers, the sum of the first three numbers, and so on, up to the sum of all seven numbers, is divisible by 3. In how many ways can this be done?"
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 28,
"genselect": 11,
"tir": 48
} | 87 | 0 | 0 |
omr-ws-8de97be6a545e23189e189b2f6dc3b0479109a4f0a7afda105e24e8ad7d987f3 | Arrange the numbers 11, 12, ..., 19 in a 3 * 3 grid such that the sum of the numbers in each row and each column is an odd number. | [
"\\(\\frac{1}{14}\\)"
] | [
"Arrange the numbers 11, 12, ..., 19 in a 3 * 3 grid such that the sum of the numbers in each row and each column is an odd number."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2263422c7f867f120fc993767249b5c37a93edc3f2baf5fde4abafcf14236f7c | Arrange the numbers \(1000000^{1000000}\), \(999999^{1000001}\), and \(1000001^{999999}\) in increasing order. | [
"\\(1000001^{999999} < 1000000^{1000000} < 999999^{1000001}\\)"
] | [
"Arrange the numbers \\(1000000^{1000000}\\), \\(999999^{1000001}\\), and \\(1000001^{999999}\\) in increasing order."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 8,
"tir": 29
} | 37 | 0 | 0 |
omr-ws-5ea6713f36c22778ce1dfc9a08a4bbadcb43bdbea8fbca7ca169156a95cbc1f2 | Arrange the numbers \(1^2, 2^2, 3^2, \ldots, 77^2\) into a \(7 \times 11\) rectangle such that each number is used exactly once and the average of each row and each column equals 2015. Calculate the sum of all the numbers in the first row. | [
"22165"
] | [
"Arrange the numbers \\(1^2, 2^2, 3^2, \\ldots, 77^2\\) into a \\(7 \\times 11\\) rectangle such that each number is used exactly once and the average of each row and each column equals 2015. Calculate the sum of all the numbers in the first row."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0787fc6bb3445ace8b285a404d0dd15e04716f73889195514f938f6c70bd25c4 | Arrange the numbers from 1 through 97 such that the absolute value of the difference between any two adjacent numbers is either 7 or 9. | [
"1, 10, 3, 12, 5, 14, 7, 16, 9, 18, 11, 20, 13, 22, 15, 24, 17, 26, 19, 28, 21, 30, 23, 32, 25, 34, 27, 36, 29, 38, 31, 40, 33, 42, 35, 44, 37, 46, 39, 48, 41, 50, 43, 52, 45, 54, 47, 56, 49, 58, 51, 60, 53, 62, 55, 64, 57, 66, 59, 68, 61, 70, 63, 72, 65, 74, 67, 76, 69, 78, 71, 80, 73, 82, 75, 84, 77, 86, 79, 88, ... | [
"Arrange the numbers from 1 through 97 such that the absolute value of the difference between any two adjacent numbers is either 7 or 9."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 5
} | 13 | 0 | 0 |
omr-ws-c4db77a7dc609e2f36763fa28cf6292828f3b1f6da59c88d2c391abb045190d3 | Arrange the numbers from 1 to 12 in a 4×4 grid, excluding the diagonal starting from the bottom left corner, such that each number appears exactly once. Additionally, ensure that for every pair of horizontally adjacent squares, the number on the left is greater than the number on the right, and for every pair of vertic... | [
"42"
] | [
"Arrange the numbers from 1 to 12 in a 4×4 grid, excluding the diagonal starting from the bottom left corner, such that each number appears exactly once. Additionally, ensure that for every pair of horizontally adjacent squares, the number on the left is greater than the number on the right, and for every pair of v... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 8
} | 16 | 0 | 0 |
omr-ws-6d7a25d59c311b26ac8e1ee0e3118a6f1b142fabf957afe8b03793f95ab54b42 | Arrange the numbers from 1 to 9 in a 3x3 grid such that the sum of the numbers in each row, each column, and both diagonals is different from the sums of all other rows, columns, and diagonals. Calculate the maximum possible difference between the largest and smallest of these sums. | [
"18"
] | [
"Arrange the numbers from 1 to 9 in a 3x3 grid such that the sum of the numbers in each row, each column, and both diagonals is different from the sums of all other rows, columns, and diagonals. Calculate the maximum possible difference between the largest and smallest of these sums."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-df0725cb950a19f77201ed36f46f3d4fdaa628c153c6fcdbd0936c14a17e954d | Arrange the numbers from the set \(\{1, 2, 3, \ldots, 12\}\) on a circle such that for any three consecutive numbers \(a\), \(b\), and \(c\) (where \(a\) is next to \(b\) and \(b\) is next to \(c\)), the condition \(13 \mid (bc - a^2)\) is satisfied. | [
"no_valid_answer_found"
] | [
"Arrange the numbers from the set \\(\\{1, 2, 3, \\ldots, 12\\}\\) on a circle such that for any three consecutive numbers \\(a\\), \\(b\\), and \\(c\\) (where \\(a\\) is next to \\(b\\) and \\(b\\) is next to \\(c\\)), the condition \\(13 \\mid (bc - a^2)\\) is satisfied."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-212f06b74bc960012d8da39786aa68c0633b4a48fa7367b29a10b3e9a5a85141 | Arrange the positive integers from $1$ to $2024$ in a circle such that each number is used exactly once and for any three consecutive numbers $a, b, c$, the number $a + c$ is divisible by $b + 1$. Calculate the sum of all possible values of $a + c$ when $b = 2023$. | [
"2024"
] | [
"Arrange the positive integers from $1$ to $2024$ in a circle such that each number is used exactly once and for any three consecutive numbers $a, b, c$, the number $a + c$ is divisible by $b + 1$. Calculate the sum of all possible values of $a + c$ when $b = 2023$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-05718554f7e9bc3ad3de3e844ba7555053333561d5ce9f5907205bd1d1e1d902 | Arrange the positive integers into two lines as follows: \[ 1 \quad 3 \qquad 6 \qquad\qquad\quad 11 \qquad\qquad\qquad\qquad\quad\ 19\qquad\qquad32\qquad\qquad 53\ldots\\ \mbox{\ \ } 2 \quad 4\ \ 5 \quad 7\ \ 8\ \ 9\ \ 10\quad\ 12\ 13\ 14\ 15\ 16\ 17\ 18\quad\ 20 \mbox{ to } 31\quad\ 33 \mbox{ to } 52\quad\ \ldots\] We... | [
"a_n = F_{n+3} - 2"
] | [
"Arrange the positive integers into two lines as follows:\n\\[ 1 \\quad 3 \\qquad 6 \\qquad\\qquad\\quad 11 \\qquad\\qquad\\qquad\\qquad\\quad\\ 19\\qquad\\qquad32\\qquad\\qquad 53\\ldots\\\\\n\\mbox{\\ \\ } 2 \\quad 4\\ \\ 5 \\quad 7\\ \\ 8\\ \\ 9\\ \\ 10\\quad\\ 12\\ 13\\ 14\\ 15\\ 16\\ 17\\ 18\\quad\\ 20 \\mbox{... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0f0ae6bdd0d0039f971b585faff03729b3b1bd9369d6f047e5ff695cb4e94252 | Arrange the positive integers into two lines as follows: \begin{align*} 1 \quad 3 \qquad 6 \qquad\qquad\quad 11 \qquad\qquad\qquad\qquad\quad\ 19\qquad\qquad32\qquad\qquad 53\ldots\\ \mbox{\ \ } 2 \quad 4\ \ 5 \quad 7\ \ 8\ \ 9\ \ 10\quad\ 12\ 13\ 14\ 15\ 16\ 17\ 18\quad\ 20 \mbox{ to } 31\quad\ 33 \mbox{ to } 52\quad\... | [
"\\( a_n = F_{n+3} - 2 \\)"
] | [
"Arrange the positive integers into two lines as follows:\n\\begin{align*}\n1 \\quad 3 \\qquad 6 \\qquad\\qquad\\quad 11 \\qquad\\qquad\\qquad\\qquad\\quad\\ 19\\qquad\\qquad32\\qquad\\qquad 53\\ldots\\\\\n\\mbox{\\ \\ } 2 \\quad 4\\ \\ 5 \\quad 7\\ \\ 8\\ \\ 9\\ \\ 10\\quad\\ 12\\ 13\\ 14\\ 15\\ 16\\ 17\\ 18\\quad... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 12
} | 59 | 0 | 0 |
omr-ws-fdac42d3682a1a80abe321cd5ed1c446bb67ca3db32355354d322e9b221b1900 | Arrange three numbers (a nine-digit number, a single-digit number, and another nine-digit number) one after another to form a nineteen-digit palindromic number. What are the three numbers? | [
"987654321"
] | [
"Arrange three numbers (a nine-digit number, a single-digit number, and another nine-digit number) one after another to form a nineteen-digit palindromic number. What are the three numbers?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-196a86c41ec3f01dbfa5ff5805e5b897c13f44b82cda5ba7d725a8e5d89e0645 | Arthur Wang is stuck in the Cartesian coordinate system and needs to move from the point $(0,0)$ to $(2n,0)$ using only the southeast $(+1,-1)$ and northeast $(+1,+1)$ directions. How many of these paths pass under the line $y=0$? | [
"\\(\\frac{n}{n+1} \\binom{2n}{n}\\)"
] | [
"Arthur Wang is stuck in the Cartesian coordinate system and needs to move from the point $(0,0)$ to $(2n,0)$ using only the southeast $(+1,-1)$ and northeast $(+1,+1)$ directions. How many of these paths pass under the line $y=0$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 5,
"tir": 3
} | 8 | 0 | 0 |
omr-ws-e354e797561eb5cddb66465f7fe6ff2c2de156d3a7ac09ce01124789fb31226a | Arthur Wang starts with 2 umbrellas at home and 2 umbrellas at his office. There is a 1/6 chance it will rain in the morning and a 1/18 chance it will rain in the afternoon. What is the probability that the first time he gets wet will be on the way home? | [
"\\dfrac{52625}{2128223}"
] | [
"Arthur Wang starts with 2 umbrellas at home and 2 umbrellas at his office. There is a 1/6 chance it will rain in the morning and a 1/18 chance it will rain in the afternoon. What is the probability that the first time he gets wet will be on the way home?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-fc50f491913ef5eefc777d767fb481ad6886fbd7c0ba1f7bd88529b838cf7083 | Arthur and Barbara are playing a game where they take turns writing down a digit to form a 2023-digit number, starting with a nonzero digit by Arthur. Arthur wins if the resulting number has a divisor of the form \(1\underbrace{77\ldots 7}_{n\text{ of }7}\) where \(n \ge 1\). Otherwise, Barbara wins. Determine which pl... | [
"Arthur"
] | [
"Arthur and Barbara are playing a game where they take turns writing down a digit to form a 2023-digit number, starting with a nonzero digit by Arthur. Arthur wins if the resulting number has a divisor of the form \\(1\\underbrace{77\\ldots 7}_{n\\text{ of }7}\\) where \\(n \\ge 1\\). Otherwise, Barbara wins. Deter... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 3
} | 12 | 0 | 0 |
omr-ws-0dee621c1c6e5462acabcb6bc2b0fb2abef1f00843f3441b1d6f611e05b84382 | Arthur and Ben play a game with a chocolate table of size \( k \times n \). They alternately break the chocolate table into two rectangular parts along the lines and eat up the smaller part. The player who first eats a single chocolate square loses. Determine all pairs \((k, n)\) for which Arthur has a winning strategy... | [
"(k, n) \\text{ where at least one of } k \\text{ or } n \\text{ is even}"
] | [
"Arthur and Ben play a game with a chocolate table of size \\( k \\times n \\). They alternately break the chocolate table into two rectangular parts along the lines and eat up the smaller part. The player who first eats a single chocolate square loses. Determine all pairs \\((k, n)\\) for which Arthur has a winnin... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 3
} | 10 | 0 | 0 |
omr-ws-94400fe6cbec02e024d6dd9e623a042ef881bd196e7ca07caea6eda7b7fca60a | Arthur and Betty play a game with the following rules. Initially there are one or more piles of stones, each pile containing one or more stones. A legal move consists either of removing one or more stones from one of the piles, or, if there are at least two piles, combining two piles into one (but not removing any ston... | [
"Betty"
] | [
"Arthur and Betty play a game with the following rules. Initially there are one or more piles of stones, each pile containing one or more stones. A legal move consists either of removing one or more stones from one of the piles, or, if there are at least two piles, combining two piles into one (but not removing any... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-9a24046075ade214ddb43ce3c5a887944eb428659a56f51f4632500fcd8d74bd | Arthur and MacArthur are playing a game with 10 boxes, each labeled with a number from 1 to 10, and each box contains exactly one ball labeled with a number from 1 to 10. MacArthur arranges the balls randomly in the boxes. Arthur's goal is to find a specific ball labeled \( n \) by making moves where he selects a box \... | [
"9"
] | [
"Arthur and MacArthur are playing a game with 10 boxes, each labeled with a number from 1 to 10, and each box contains exactly one ball labeled with a number from 1 to 10. MacArthur arranges the balls randomly in the boxes. Arthur's goal is to find a specific ball labeled \\( n \\) by making moves where he selects ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 3
} | 16 | 0 | 0 |
omr-ws-7caa8affc1ea8b0b8e73df139c838f1651c7bed853fb0812fe9aa48a8417e1fb | Arthur and Theodore each start with 100 units of every country's currency on an island with 10 countries. They exchange money at the exchange offices in any order they like, as long as they have at least 10 units of a currency. After any sequence of exchanges, let $A$ be the number of Bergengocian dollars Arthur has, a... | [
"900"
] | [
"Arthur and Theodore each start with 100 units of every country's currency on an island with 10 countries. They exchange money at the exchange offices in any order they like, as long as they have at least 10 units of a currency. After any sequence of exchanges, let $A$ be the number of Bergengocian dollars Arthur h... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 6
} | 10 | 0 | 0 |
omr-ws-a5baecf52fc134c16336879c6dd6c1ae522dcbde7a0e792034aabdfcd13f8e57 | Arthur has $n \ge 1$ co-authors who write books with him. Each book has a list of authors including Arthur himself, and no two books have the same set of authors. At a party with all his co-authors, each co-author writes down the number of books they remember having written with Arthur. The numbers written down are the... | [
"15"
] | [
"Arthur has $n \\ge 1$ co-authors who write books with him. Each book has a list of authors including Arthur himself, and no two books have the same set of authors. At a party with all his co-authors, each co-author writes down the number of books they remember having written with Arthur. The numbers written down a... | [
"aops_c6_high_school_olympiads"
] | [
"genselect"
] | {
"genselect": 5
} | 5 | 0 | 0 |
omr-ws-1279643ed6335a33a763a4e43ee61c6043cc80c31ef4f242b5baf8349a27b567 | Arthur has an equilateral triangle ABC with AB = 1. M is the midpoint of AB. He constructs two semicircles with bases AM and BM on the exterior of ABC. He glues this shape onto a non-square rectangular poster such that the shape shares an axis of symmetry with the poster and touches all sides. What is the area of the p... | [
"\\dfrac{4\\sqrt{3} + 4 - \\pi}{16}"
] | [
"Arthur has an equilateral triangle ABC with AB = 1. M is the midpoint of AB. He constructs two semicircles with bases AM and BM on the exterior of ABC. He glues this shape onto a non-square rectangular poster such that the shape shares an axis of symmetry with the poster and touches all sides. What is the area of ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0522482e0477857a3c5c6171a1e28481c3b563d46fec01f43c7622d7d3a85077 | Arthur is driving to David’s house. If he drives at 60 km/h, he will arrive 5 minutes late. If he drives at 90 km/h, he will arrive 5 minutes early. At what speed \( n \) km/h should he drive to arrive exactly on time? | [
"72"
] | [
"Arthur is driving to David’s house. If he drives at 60 km/h, he will arrive 5 minutes late. If he drives at 90 km/h, he will arrive 5 minutes early. At what speed \\( n \\) km/h should he drive to arrive exactly on time?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 16
} | 16 | 0 | 0 |
omr-ws-b280b63864e2fe36b5fa29224ef407934ffb4f764199c140f8f79ff3a5e2ed8a | Arthur stands on one of five equally spaced nodes on a circle. He flips a fair coin 12 times. Each head makes him walk to the next node in the clockwise direction, and each tail makes him reverse direction and walk to the next node in that new direction. After 12 flips, he ends up on the node where he started. If \( \f... | [
"287"
] | [
"Arthur stands on one of five equally spaced nodes on a circle. He flips a fair coin 12 times. Each head makes him walk to the next node in the clockwise direction, and each tail makes him reverse direction and walk to the next node in that new direction. After 12 flips, he ends up on the node where he started. If ... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 5
} | 5 | 0 | 0 |
omr-ws-1e393341590a510e7e780c31ed4264cf3ba2c2a19a5edbee3768d582e1d603e0 | Arthur stands on one of five equally spaced nodes on a circle. He flips a fair coin 12 times. Each head makes him walk to the next node in the current direction, and each tail makes him reverse direction and walk to the next node. After 12 flips, he ends up on the starting node. If \(a\) and \(b\) are relatively prime ... | [
"253"
] | [
"Arthur stands on one of five equally spaced nodes on a circle. He flips a fair coin 12 times. Each head makes him walk to the next node in the current direction, and each tail makes him reverse direction and walk to the next node. After 12 flips, he ends up on the starting node. If \\(a\\) and \\(b\\) are relative... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-eb638dfa04160f4fff3779440568aeb3db9bf3f8f028268be7cfea351a8a4810 | Arthur thought of a positive integer such that the sum of its three smallest divisors is $17$ and the sum of its three largest divisors is $3905$. Identify all possible numbers that Arthur may have thought of. | [
"3025, 2769"
] | [
"Arthur thought of a positive integer such that the sum of its three smallest divisors is $17$ and the sum of its three largest divisors is $3905$. Identify all possible numbers that Arthur may have thought of."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 25
} | 25 | 0 | 0 |
omr-ws-c6c074201de3da7a2976c6d0deaa58a804d0dbc685a9c0adfc9bfaf4bbf14754 | Arthur walks six blocks east and then 12 blocks north. If each block is one-third of a mile, how many miles did he walk? | [
"6"
] | [
"Arthur walks six blocks east and then 12 blocks north. If each block is one-third of a mile, how many miles did he walk?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ef7f03fb828d12be0b98fa27961b8f8e9dce9a49d0689a492e4617887a16a82e | Arthur's wife picks him up at the train station and drives him home every Friday. One Friday, Arthur catches the early train and arrives 90 minutes early, starting to walk home. His wife, who left home at the usual time, meets him on the way and they arrive home 20 minutes earlier than normal. How many minutes had Arth... | [
"80 minutes"
] | [
"Arthur's wife picks him up at the train station and drives him home every Friday. One Friday, Arthur catches the early train and arrives 90 minutes early, starting to walk home. His wife, who left home at the usual time, meets him on the way and they arrive home 20 minutes earlier than normal. How many minutes had... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-a95017b3c6ba5b38b035a3f1fc665d259724e26311b23aff66acfa7ccc1973e0 | Artie considers a string to be clownish if, for every three consecutive letters, there is at least one vowel. How many permutations of the letters in the word $CIRCUS$ are clownish? | [
"408"
] | [
"Artie considers a string to be clownish if, for every three consecutive letters, there is at least one vowel. How many permutations of the letters in the word $CIRCUS$ are clownish?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-372299621f8142a9460d4c25bffa0e6a12d158d90877813ea29cd574d8b5c6cb | Arutyun and Amayak show another effective trick. A spectator writes down on a board a sequence of \(N\) (decimal) digits. Amayak closes two adjacent digits by a black disc. Then Arutyun comes and says both closed digits (and their order). For which minimal \(N\) can they show such a trick? | [
"101"
] | [
"Arutyun and Amayak show another effective trick. A spectator writes down on a board a sequence of \\(N\\) (decimal) digits. Amayak closes two adjacent digits by a black disc. Then Arutyun comes and says both closed digits (and their order). For which minimal \\(N\\) can they show such a trick?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 6
} | 16 | 0 | 0 |
omr-ws-13e7a224d406760f5cae6f15fc0aca16099d83c0f8de5ef6b7ada775dc7ca669 | Arya and Bran are playing a game with 2008 coins arranged in a circle. They take turns, starting with Arya. On each turn, a player may remove any one coin, or if two adjacent coins remain, the player may remove both. If Bran follows an optimal strategy, what is the maximum number of coins Arya can remove before Bran wi... | [
"1003"
] | [
"Arya and Bran are playing a game with 2008 coins arranged in a circle. They take turns, starting with Arya. On each turn, a player may remove any one coin, or if two adjacent coins remain, the player may remove both. If Bran follows an optimal strategy, what is the maximum number of coins Arya can remove before Br... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 6
} | 10 | 0 | 0 |
omr-ws-ca677be56f133e34f6b0eeaeab0d5acc22db5727ff4f0f60b844440820dea14b | As $n$ ranges over the positive integers, what is the maximum possible value for the greatest common divisor of $11n+3$ and $6n+1$? | [
"7"
] | [
"As $n$ ranges over the positive integers, what is the maximum possible value for the greatest common divisor of $11n+3$ and $6n+1$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ab1eee10d6088dd80cf6d587f5ccf9b69506053b90708de17cff6c9cbb8f1f1c | As $n$ ranges over the positive integers, what is the sum of all possible values of the greatest common divisor of $3n+4$ and $n$? | [
"7"
] | [
"As $n$ ranges over the positive integers, what is the sum of all possible values of the greatest common divisor of $3n+4$ and $n$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-99bbebb155fb80f94827288e69f77b6192299676ae2ed7a5059f94fc5e303501 | As $p$ ranges over the primes greater than $5$, how many different remainders can $p^2$ leave upon division by $120$? | [
"2"
] | [
"As $p$ ranges over the primes greater than $5$, how many different remainders can $p^2$ leave upon division by $120$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-87d61a6feda151c6857b73174c1c2d881a6761c9965618d529aac2b93d2d5b2c | As $t$ takes on all real values, the set of points $(x,y)$ defined by \begin{align*} x &= t^2 - 2, \\ y &= t^3 - 9t + 5 \end{align*}forms a curve that crosses itself. Compute the ordered pair $(x,y)$ where this crossing occurs. | [
"(7,5)"
] | [
"As $t$ takes on all real values, the set of points $(x,y)$ defined by\n\\begin{align*}\nx &= t^2 - 2, \\\\\ny &= t^3 - 9t + 5\n\\end{align*}forms a curve that crosses itself. Compute the ordered pair $(x,y)$ where this crossing occurs."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8e1e86bbe419e054dcc045187f11c64410022198d2e6494258047413d2a08660 | As $x \to (\pi)^+$, determine the limit of $\csc x$. | [
"\\(-\\infty\\)"
] | [
"As $x \\to (\\pi)^+$, determine the limit of $\\csc x$."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-366f9c3288c87f0293a598a152fba7e569990d0d269a48b0ba8665f2887e360f | As $x$ and $y$ range through the positive integers with $x > y$, find the maximum number of positive integer solutions $(m,n)$ to the equation $\frac{x^m-1}{x-1} = \frac{y^n-1}{y-1}$. | [
"2"
] | [
"As $x$ and $y$ range through the positive integers with $x > y$, find the maximum number of positive integer solutions $(m,n)$ to the equation $\\frac{x^m-1}{x-1} = \\frac{y^n-1}{y-1}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-372e7b7947dd05b20cb50ff70b2235f374b15c6d3a79575a141823564be8e8ae | As $x$ ranges over all real numbers, find the range of \[f(x) = \sin^4 x + \cos ^2 x.\]Enter your answer using interval notation. | [
"\\left[ \\frac{3}{4}, 1 \\right]"
] | [
"As $x$ ranges over all real numbers, find the range of\n\\[f(x) = \\sin^4 x + \\cos ^2 x.\\]Enter your answer using interval notation."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b860df191c33bd81cf6c3bbff47c23e85e18cb4daa16966aec22c2a871238c7d | As $x, y, z$ range over all possible real numbers, what is the minimum value of the function \[ f(x, y, z) = 3x^2 + 12y^2 + 27z^2 - 4xy - 12yz - 6zx - 8y - 24z + 100 \]? | [
"86"
] | [
"As $x, y, z$ range over all possible real numbers, what is the minimum value of the function\n\n\\[ f(x, y, z) = 3x^2 + 12y^2 + 27z^2 - 4xy - 12yz - 6zx - 8y - 24z + 100 \\]?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 24
} | 39 | 0 | 0 |
omr-ws-7f9a0c1e8033a4d34c952a163637aaaf3086b50287a781b0dd4154d84ad2913a | As \( n \) approaches \(\infty\), let the probability that Michelle will see exactly 1 bird on the line at some point in time approach \( p \). Estimate \( \lfloor 10000p \rfloor \). | [
"3678"
] | [
"As \\( n \\) approaches \\(\\infty\\), let the probability that Michelle will see exactly 1 bird on the line at some point in time approach \\( p \\). Estimate \\( \\lfloor 10000p \\rfloor \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-53629719354ae91680c6da90e92a16780a347812839ee3447886d8d2f3625142 | As \( n \) approaches infinity, what does the area of the figure approach when a rhombus with side length 1 and acute angle \( \theta = \frac{n}{360} \) (where \( n \) is an integer \(\geq 5\)) is rotated about one of its vertices \( n-1 \) times to form an \( n \)-pointed star? | [
"\\( 2\\pi \\)"
] | [
"As \\( n \\) approaches infinity, what does the area of the figure approach when a rhombus with side length 1 and acute angle \\( \\theta = \\frac{n}{360} \\) (where \\( n \\) is an integer \\(\\geq 5\\)) is rotated about one of its vertices \\( n-1 \\) times to form an \\( n \\)-pointed star?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 5,
"tir": 7
} | 22 | 0 | 0 |
omr-ws-5808635e1dc090b3859e909b65c05d901c824d3cb23ff53bdf686863acd910c1 | As \( n \) approaches infinity, what does the shaded area approach when two adjacent sides of a 1x1 square are divided into \( n \) equal parts and lines are drawn connecting the end of the \( n \)th vertical segment with the end of the first horizontal segment, the end of the \( (n-1) \)th vertical segment with the en... | [
"\\(\\frac{1}{6}\\)"
] | [
"As \\( n \\) approaches infinity, what does the shaded area approach when two adjacent sides of a 1x1 square are divided into \\( n \\) equal parts and lines are drawn connecting the end of the \\( n \\)th vertical segment with the end of the first horizontal segment, the end of the \\( (n-1) \\)th vertical segmen... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 35,
"tir": 16
} | 51 | 0 | 0 |
omr-ws-e44d82465d9c3ba0270c08a3b1d1262c405ceca4a25a94cc99eb603b7def351a | As \( n \) approaches infinity, what is the probability that Albert wins the game where \( n \) people write their names on slips, Albert redistributes the slips randomly, and those who receive their own name flip a fair coin, with the audience winning only if all flipped coins show the same side? | [
"\\( 1 - \\frac{2}{\\sqrt{e}} + \\frac{2}{e} \\)"
] | [
"As \\( n \\) approaches infinity, what is the probability that Albert wins the game where \\( n \\) people write their names on slips, Albert redistributes the slips randomly, and those who receive their own name flip a fair coin, with the audience winning only if all flipped coins show the same side?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 14
} | 29 | 0 | 0 |
omr-ws-ef5abd92160d5e5bebb93960322d86ed99dbff685d87a3d066172f605f7fd645 | As \( n \) gets large, what is the probability that there is some pair of mathematicians who were neighbors both before and after lunch, given that \( n \) mathematicians are seated randomly around a conference table and then reseat themselves randomly again? | [
"\\( e^{-2} \\)"
] | [
"As \\( n \\) gets large, what is the probability that there is some pair of mathematicians who were neighbors both before and after lunch, given that \\( n \\) mathematicians are seated randomly around a conference table and then reseat themselves randomly again?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-151fbe6a87c8f0cfe960c2034471fc766ce24027f86d02278d548db9b62b168f | As \( p \) ranges over the primes greater than 5, how many different remainders can \( p^2 \) leave upon division by 120? | [
"2"
] | [
"As \\( p \\) ranges over the primes greater than 5, how many different remainders can \\( p^2 \\) leave upon division by 120?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-645cfa0df6de38c9e5578d24b69a81ef9d1b748c630c39532f23229523704e48 | As \( x \) ranges over the interval \( (0, \infty) \), the function \[ \sqrt{9x^2 + 173x + 900} - \sqrt{9x^2 + 77x + 900} \] ranges over the interval \( (0, M) \). Find \( M \). | [
"16"
] | [
"As \\( x \\) ranges over the interval \\( (0, \\infty) \\), the function \\[ \\sqrt{9x^2 + 173x + 900} - \\sqrt{9x^2 + 77x + 900} \\] ranges over the interval \\( (0, M) \\). Find \\( M \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-7d0c654b1ec25daf2574e9d7880457b271e67cbc5b4561b17daad1b7aade1bc4 | As \( x \) ranges over the interval \((0, \infty)\), the function \(\sqrt{9x^2 + 173x + 900} - \sqrt{9x^2 + 77x + 900}\) ranges over the interval \((0, M)\). Find \( M \). | [
"16"
] | [
"As \\( x \\) ranges over the interval \\((0, \\infty)\\), the function \\(\\sqrt{9x^2 + 173x + 900} - \\sqrt{9x^2 + 77x + 900}\\) ranges over the interval \\((0, M)\\). Find \\( M \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-33200fac4a24743cff9efa5b302a7224b324dcbb97fe480c31470120267e6dff | As \( z \) varies along the line \((1+2i)z - (1-2i) \overline{z} = i\), what curve does \( f(z) = \frac{1}{\overline{z}} \) trace? | [
"|w - (2 + i)| = \\sqrt{5}"
] | [
"As \\( z \\) varies along the line \\((1+2i)z - (1-2i) \\overline{z} = i\\), what curve does \\( f(z) = \\frac{1}{\\overline{z}} \\) trace?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f1b67be557fe44004340ca38e2cfa66a45e353053b2ecaa03db96143f4e2d7b2 | As \(a\), \(b\), and \(c\) range over all real numbers, let \(m\) be the smallest possible value of \[2\left(a+b+c\right)^2+\left(ab-4\right)^2+\left(bc-4\right)^2+\left(ca-4\right)^2\] and let \(n\) be the number of ordered triplets \((a, b, c)\) such that the above quantity is minimized. Compute \(m+n\). | [
"47"
] | [
"As \\(a\\), \\(b\\), and \\(c\\) range over all real numbers, let \\(m\\) be the smallest possible value of \n\\[2\\left(a+b+c\\right)^2+\\left(ab-4\\right)^2+\\left(bc-4\\right)^2+\\left(ca-4\\right)^2\\]\nand let \\(n\\) be the number of ordered triplets \\((a, b, c)\\) such that the above quantity is minimized.... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 16
} | 31 | 0 | 0 |
omr-ws-7417327d7073d729ce1b41bcc270b82f96527d95e7ffbc5fa5879185060d4d94 | As a particle moves along the line \( y = 7x + 2 \), what is its minimum distance from the origin? | [
"\\(\\frac{\\sqrt{2}}{5}\\)"
] | [
"As a particle moves along the line \\( y = 7x + 2 \\), what is its minimum distance from the origin?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-c4fb87c878927d27799824a8220fb298f2fc0159c73ed711f22f4c64490c6d30 | As a prank, Tim decides to steal Nathan's fork at dinner, but so he doesn't get caught, he convinces other people to do it for him. On Monday, he convinces Joe to do it. On Tuesday, he could get either Ambie or John to do it. On Wednesday, he can't convince any of those three people to do it, but there are five other p... | [
"40"
] | [
"As a prank, Tim decides to steal Nathan's fork at dinner, but so he doesn't get caught, he convinces other people to do it for him. On Monday, he convinces Joe to do it. On Tuesday, he could get either Ambie or John to do it. On Wednesday, he can't convince any of those three people to do it, but there are five ot... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8dc754744dd652e1001eda83df14022e9bf4c876acafa8b6c151dc154dda6520 | As a result of reconstruction of a factory, the number of the disengaged workers was within the limits from 1.7\% to 2.3\% of the total number of the personnel. Find the minimal number of workers which could be employed before the reconstruction. | [
"44 workers"
] | [
"As a result of reconstruction of a factory, the number of the disengaged workers was within the limits from 1.7\\% to 2.3\\% of the total number of the personnel. Find the minimal number of workers which could be employed before the reconstruction."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4789e65edd8c58385e67645ee9684695434af9bbaa5588c3f89d07470f57d78a | As an afternoon activity, Emilia will either play exactly two of four games (TwoWeeks, DigBuild, BelowSaga, and FlameSymbol) or work on homework for exactly one of three classes (CS61A, Math 1B, Anthro 3AC). How many choices of afternoon activities does Emilia have? | [
"9"
] | [
"As an afternoon activity, Emilia will either play exactly two of four games (TwoWeeks, DigBuild, BelowSaga, and FlameSymbol) or work on homework for exactly one of three classes (CS61A, Math 1B, Anthro 3AC). How many choices of afternoon activities does Emilia have?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-d93d4ddf54b0b9f1c3ca204b6f6fb993beca63b05f5977db85885e8919c18e1e | As my younger brother shouted out four consecutive integers, I divided each by my age (an integer) and added all four remainders to get 40. When he shouted out four other consecutive integers, I divided each by his age (a different integer) and added all four remainders, again getting 40. What is the sum of our ages? | [
"59"
] | [
"As my younger brother shouted out four consecutive integers, I divided each by my age (an integer) and added all four remainders to get 40. When he shouted out four other consecutive integers, I divided each by his age (a different integer) and added all four remainders, again getting 40. What is the sum of our ag... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-11db6b6be6c02ca9292c5d4aea5c3c18b0ff8258b1bd2bdd9a5310893e20be91 | As part of a game, a positive integer $n$ is selected and a coin is tossed until it has either landed heads $n$ times or tails $n$ times, at which point the game ends. Alex is the winner if the game takes at least $2n - 2$ coin tosses, and Bob is the winner if the game takes fewer than $2n - 2$ coin tosses. For which v... | [
"n \\leq 5"
] | [
"As part of a game, a positive integer $n$ is selected and a coin is tossed until it has either landed heads $n$ times or tails $n$ times, at which point the game ends. Alex is the winner if the game takes at least $2n - 2$ coin tosses, and Bob is the winner if the game takes fewer than $2n - 2$ coin tosses. For wh... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-8f22f833975f4e8bc2a3dd6bbb15d6bc6684ae297f5735139ccc1163aabaa791 | As sand leaks out of a hole in a container, it forms a conical pile with an altitude equal to the radius. If the height of the pile is increasing at a rate of 6 in/min, find the rate at which the sand is leaking when the pile has a height of 10 inches. | [
"\\( 600\\pi \\) in\\(^3\\)/min"
] | [
"As sand leaks out of a hole in a container, it forms a conical pile with an altitude equal to the radius. If the height of the pile is increasing at a rate of 6 in/min, find the rate at which the sand is leaking when the pile has a height of 10 inches."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 31
} | 45 | 0 | 0 |
omr-ws-aec5326381ad86045149bec0c1f5147c6319c963fcc0223a04d7992e323d9f5a | As shown in the figure, the big circle and the small circle are concentric circles, and the center of the circles is $O$. Draw the tangent $AC$ of the small circle through point $A$ on the big circle, the tangent point is $B$, point $C$ lies on the big circle, and $D$ is the midpoint of $AB$. The vertex $E$ of $\triang... | [
"\\sqrt{2}"
] | [
"As shown in the figure, the big circle and the small circle are concentric circles, and the center of the circles is $O$. Draw the tangent $AC$ of the small circle through point $A$ on the big circle, the tangent point is $B$, point $C$ lies on the big circle, and $D$ is the midpoint of $AB$. The vertex $E$ of $\\... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 5
} | 8 | 0 | 0 |
omr-ws-b2bd833b136b5dbfae76a1a0258a656848bca9447e22662b33a210f62b90fd1e | As the bears walk back from their home, they realize they left their fish by the river. The river is a circle centered at their home with radius $6$ miles. If they are currently $4$ miles away from their home, what is the shortest possible distance, in miles, that they must travel to get to the river and then return ho... | [
"6"
] | [
"As the bears walk back from their home, they realize they left their fish by the river. The river is a circle centered at their home with radius $6$ miles. If they are currently $4$ miles away from their home, what is the shortest possible distance, in miles, that they must travel to get to the river and then retu... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4d77faaa5804542f8c59d3e90cea6bdce5efc674efe7bcba3f82c14888079d52 | As the number of powers of $2$ increases to infinity, what fraction of the first digits of these powers are $1$? What fractions are $2$, and so on for each possible first digit? | [
"\\(\\log_{10}(2)\\) for the first digit being 1, \\(\\log_{10}\\left(\\frac{3}{2}\\right)\\) for the first digit being 2, and so on for each possible first digit."
] | [
"As the number of powers of $2$ increases to infinity, what fraction of the first digits of these powers are $1$? What fractions are $2$, and so on for each possible first digit?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 11,
"tir": 31
} | 42 | 0 | 0 |
omr-ws-77ce1ce29668b6843666c36dcb04ea8bd8be759a4eb1b9889a2d1913bd39a0df | As you add more digits, the sequence \( \frac{991191}{882282}, \frac{999111991191}{888222882282}, \ldots \) converges to what number? | [
"\\(\\frac{9}{8}\\)"
] | [
"As you add more digits, the sequence \\( \\frac{991191}{882282}, \\frac{999111991191}{888222882282}, \\ldots \\) converges to what number?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 22
} | 26 | 0 | 0 |
omr-ws-63fe343b9cae6e471ba23c8eeabe78dbaab2b5bf884f13517a2435c261fed90c | Ascertain $\max_{x^m+y^m=1}\ x^py^q$ without derivatives, where $\{m,p,q\}\subset\mathbb N^*$ and $\{x,y\}\subset \mathbb R^*_+$. | [
"\\left( \\dfrac{p^p q^q}{(p+q)^{p+q}} \\right)^{1/m}"
] | [
"Ascertain $\\max_{x^m+y^m=1}\\ x^py^q$ without derivatives, where $\\{m,p,q\\}\\subset\\mathbb N^*$ and $\\{x,y\\}\\subset \\mathbb R^*_+$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ad2ca8c6b7a86e6b79c084f47769e9145e294ca7cfb7074a01d6cac68e757892 | Ascertain $m \in \mathbb{R}$ so that the irrational equation $\sqrt[3]{x+3} + \sqrt[3]{x-4} = m$ has at least one real root (without using derivatives). | [
"\\( m \\in \\mathbb{R} \\)"
] | [
"Ascertain $m \\in \\mathbb{R}$ so that the irrational equation $\\sqrt[3]{x+3} + \\sqrt[3]{x-4} = m$ has at least one real root (without using derivatives)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-f8d2c2f7235223243386dad04765d45f609acf58532dd480c6de1f8844fe3d9b | Ascertain $x > -2$ for which $2^x + 6^x + 3^x = 2x^2 + 6x + 3$. | [
"\\(x = 0\\) or \\(x = 1\\)"
] | [
"Ascertain $x > -2$ for which $2^x + 6^x + 3^x = 2x^2 + 6x + 3$."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 15
} | 15 | 0 | 0 |
omr-ws-881e1128ae4e68bc3bacec7cbc49014377f3791f94b9ccd09c4643c9acae0dc6 | Ascertain \( I_n = \int_0^{\frac{\pi}{2}} \frac{\sin nx}{\sin x} \, \mathrm{dx} \), where \( n \in \mathbb{N}^* \). | [
"\\( I_n = \\frac{\\pi}{2} \\) for odd \\( n \\), and \\( I_n = 2 + \\sum_{k=1}^{n/2-1} \\frac{2(-1)^k}{2k+1} \\) for even \\( n \\)."
] | [
"Ascertain \\( I_n = \\int_0^{\\frac{\\pi}{2}} \\frac{\\sin nx}{\\sin x} \\, \\mathrm{dx} \\), where \\( n \\in \\mathbb{N}^* \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 2,
"tir": 9
} | 15 | 0 | 0 |
omr-ws-824a3ee1fec0f37849a477fb963fcd6dd7c2fe119abbe16cc92a3758141ca0a5 | Ascertain \( L_n = \int_0^{\frac{\pi}{2}} \frac{\sin 2nx \cos x}{\sin x} \, \mathrm{dx} \), where \( n \in \mathbb{N}^* \). | [
"\\(\\frac{\\pi}{2}\\)"
] | [
"Ascertain \\( L_n = \\int_0^{\\frac{\\pi}{2}} \\frac{\\sin 2nx \\cos x}{\\sin x} \\, \\mathrm{dx} \\), where \\( n \\in \\mathbb{N}^* \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-851fabad64a9b9083ed648d02c12a9639c65841813af666aafffddea37ed62bf | Ascertain \( a \in \mathbb{R}^* \) and \( b \in \mathbb{R} \) so that \( |\sqrt{1-x} - ax - b| \le \frac{1}{8} \) for any \( x \in [0,1] \). | [
"b = \\dfrac{9}{8}"
] | [
"Ascertain \\( a \\in \\mathbb{R}^* \\) and \\( b \\in \\mathbb{R} \\) so that \\( |\\sqrt{1-x} - ax - b| \\le \\frac{1}{8} \\) for any \\( x \\in [0,1] \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-612270d71c75277e5de4bac52fa146ead5b2d365f03577b2cc5877a7c1825bce | Ascertain a primitive for the function \( f(x) = \begin{cases} \frac{|3x + 4|}{\sqrt{3x + 8 - 2\sqrt{6x + 12}}} & \text{for } -2 < x \neq -\frac{4}{3} \\ 2\sqrt{2} & \text{for } x = -\frac{4}{3} \end{cases} \). | [
"\\dfrac{2\\sqrt{3}}{3} (x + 2)^{3/2} + \\sqrt{2} \\, x"
] | [
"Ascertain a primitive for the function \\( f(x) = \\begin{cases} \\frac{|3x + 4|}{\\sqrt{3x + 8 - 2\\sqrt{6x + 12}}} & \\text{for } -2 < x \\neq -\\frac{4}{3} \\\\ 2\\sqrt{2} & \\text{for } x = -\\frac{4}{3} \\end{cases} \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-1e7b3f471fe40d9dcfbf31b03518fa36fa305ac9f386e6a4659b92099dc45882 | Ascertain a primitive for the function \( g(x) = \begin{cases} \frac{-\cot x}{\sqrt{-\ln (\sin x)}} & \text{for } 0 < x < \frac{\pi}{2} \\ -\sqrt{2} & \text{for } x = \frac{\pi}{2} \end{cases} \). | [
"2\\sqrt{-\\ln(\\sin x)}"
] | [
"Ascertain a primitive for the function \\( g(x) = \\begin{cases} \\frac{-\\cot x}{\\sqrt{-\\ln (\\sin x)}} & \\text{for } 0 < x < \\frac{\\pi}{2} \\\\ -\\sqrt{2} & \\text{for } x = \\frac{\\pi}{2} \\end{cases} \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-e067d5b6b302b80f3c231346895b9865af5f143c0dc28d80fb3b16fcb13d7564 | Ascertain the continuous function \( f: \mathbb{R} \rightarrow \mathbb{R} \) for which \( f(f(x)) - 3f(x) + 2x = 0 \) for all \( x \in \mathbb{R} \). | [
"\\( f_{m,n}(x) = \\begin{cases} \n2x - m & \\text{if } x < m \\\\\nx & \\text{if } m \\le x \\le n \\\\\n2x - n & \\text{if } x > n \n\\end{cases} \\)"
] | [
"Ascertain the continuous function \\( f: \\mathbb{R} \\rightarrow \\mathbb{R} \\) for which \\( f(f(x)) - 3f(x) + 2x = 0 \\) for all \\( x \\in \\mathbb{R} \\)."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
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