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-b8a74aa2c53ad22d86c70c09a578b89d75e831b2466e659c5cfac1473b6a0c11 | "Counting" is defined as listing positive integers, each one greater than the previous, up to (and including) an integer \( n \). In terms of \( n \), write the number of ways to count to \( n \). | [
"\\( 2^{n-1} \\)"
] | [
"\"Counting\" is defined as listing positive integers, each one greater than the previous, up to (and including) an integer \\( n \\). In terms of \\( n \\), write the number of ways to count to \\( n \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 7,
"tir": 7
} | 27 | 0 | 0 |
omr-ws-4d0b03c05cf04487195d2366969640bccd3fe23229114404cc918b28fb86a676 | "Modulo $m$ graph paper" consists of a grid of $m^2$ points, representing all pairs of integer residues $(x,y)$ where $0\le x, y <m$. To graph a congruence on modulo $m$ graph paper, we mark every point $(x,y)$ that satisfies the congruence. For example, a graph of $y\equiv x^2\pmod 5$ would consist of the points $(0,0... | [
"32"
] | [
"\"Modulo $m$ graph paper\" consists of a grid of $m^2$ points, representing all pairs of integer residues $(x,y)$ where $0\\le x, y <m$. To graph a congruence on modulo $m$ graph paper, we mark every point $(x,y)$ that satisfies the congruence. For example, a graph of $y\\equiv x^2\\pmod 5$ would consist of the po... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4b3d2f23901142ffa078114a57a75553e77184d9eba66546b2c37b6542c84900 | "Modulo $m$ graph paper" consists of a grid of $m^2$ points, representing all pairs of integer residues $(x,y)$ where $0\le x<m$. To graph a congruence on modulo $m$ graph paper, we mark every point $(x,y)$ that satisfies the congruence. For example, a graph of $y\equiv x^2\pmod 5$ would consist of the points $(0,0)$, ... | [
"10"
] | [
"\"Modulo $m$ graph paper\" consists of a grid of $m^2$ points, representing all pairs of integer residues $(x,y)$ where $0\\le x<m$. To graph a congruence on modulo $m$ graph paper, we mark every point $(x,y)$ that satisfies the congruence. For example, a graph of $y\\equiv x^2\\pmod 5$ would consist of the points... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-07c76b4dab8e6f1ca4b314a8cbb7485b66d2421462a6039da48c14b06b94e57a | $(1+\cos 4x)\sin 4x=\cos^2 2x$ | [
"$x=\\frac{\\pi n}{2}+\\frac{\\pi }{4};\\ x=\\frac{\\pi n}{4}+(-1)^n \\frac{\\pi }{24}$"
] | [
"$(1+\\cos 4x)\\sin 4x=\\cos^2 2x$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2ee078ecde8b5bc185d7f7d4c7123c31a0936ba12337de29b050bf849197bdea | $(1+\cos x)\left(\frac{1}{\sin x}-1\right)=0$. | [
"$\\frac{\\pi }{2}+2\\pi k$"
] | [
"$(1+\\cos x)\\left(\\frac{1}{\\sin x}-1\\right)=0$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e7f5ffd0affbeb415e641973541f9fa6acbdcfd3aef2e0983e9ae882049d1ab2 | $(1+\cos x)\sqrt {\tan \frac{x}{2}}-2+\sin x=2\cos x$. | [
"$\\frac{\\pi }{2}+2\\pi k$"
] | [
"$(1+\\cos x)\\sqrt {\\tan \\frac{x}{2}}-2+\\sin x=2\\cos x$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1ab8d1d762143b61af8c5b387254e8dde3407f0a5abe19f3572325c434dd0daf | $(1+\cos x)\tan\frac{x}{2}=0$. | [
"$2\\pi k$"
] | [
"$(1+\\cos x)\\tan\\frac{x}{2}=0$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-75b3b7a574962cb6997d56436c4b4ef10d136dd4eda38c90c0aadaf7c5521efc | $(1-i)\overline{z}=(1+i)z$. | [
"The straight line $y=-x$"
] | [
"$(1-i)\\overline{z}=(1+i)z$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e389794291d0cd0fe02b2055c2cd5d0b60c7971de9a9c4ca74e31366db7da653 | $(2^x+1)2^{y+1}=9$ \ $\sqrt {x+y^2}=x+y$. | [
"$x_1=0,\\ y_1=2\\log _2 3-2$; $x_2=3,\\ y_2=-1$"
] | [
"$(2^x+1)2^{y+1}=9$ \\ $\\sqrt {x+y^2}=x+y$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-94d1a12c2434bafe8824a499b3389e56ac8f872e1f2e04f79c74a75b105bb164 | $(81)^{\frac12}=3^m$. Find $m$. | [
"2"
] | [
"$(81)^{\\frac12}=3^m$. Find $m$."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2f5f86583609b5f339b7b84f8c2458e79fbcfbcf6631d86618da49efc84e52be | $(\log _a \sin x)^2+\log _a \sin x-a=0,\ a>0,\ a\ne 1$. | [
"$x=(-1)^k \\arcsin a^{\\left(\\sqrt {1+4a}-1\\right)/2}+k\\pi $ for $0<a<1$, $x=(-1)^k \\arcsin a^{-\\left(\\sqrt {1+4a}+1\\right)/2}+k\\pi $ for $a>1$"
] | [
"$(\\log _a \\sin x)^2+\\log _a \\sin x-a=0,\\ a>0,\\ a\\ne 1$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-240a9464f82f9253c1dd2c26379a276e201d1edce47ddba5e8ca1d45834eadd4 | $(\log _{10}\sin x)^2-2a\log _{10}\sin x-a^2+2=0$. | [
"$x_1=(-1)^k\\arcsin 10^{a+\\sqrt {2a^2-2}}+k\\pi $, $x_2=(-1)^k \\arcsin 10^{a-\\sqrt {2a^2-2}}+k\\pi $ for $-\\sqrt 2\\le 1\\le 1$; $x=x_2$ for $a<-\\sqrt 2$, $a=-1$, $b\\ge \\sqrt 2$, there are no solutions for the remaining values of $x$"
] | [
"$(\\log _{10}\\sin x)^2-2a\\log _{10}\\sin x-a^2+2=0$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8a9e564992200355d9ae09b29ed0c6d32c7abe5e8b45bec0b1fef623a451653c | $-x-y^2-\sqrt {x-y^2-1}\ge -1$. | [
"$x=1$, $y=0$"
] | [
"$-x-y^2-\\sqrt {x-y^2-1}\\ge -1$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-26b24d12ce075f55a2d94c4b285786c99a78db3e48b5a2960c715a77a3953e56 | $1+\cos 2x\cos 3x=\frac{1}{2}\sin ^2 3x$. | [
"$\\pi +2\\pi k$"
] | [
"$1+\\cos 2x\\cos 3x=\\frac{1}{2}\\sin ^2 3x$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5c0170a8a86a6bfcb3b2c957b913c2754fd4d7ea219892cfae00495ed5e378c8 | $1-2x-x^2=\tan ^2 (x+y)+\cot^2 (x+y)$. | [
"$\\left(-1;1+\\frac{\\pi }{4}(2n+1)\\right)$"
] | [
"$1-2x-x^2=\\tan ^2 (x+y)+\\cot^2 (x+y)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-123657f321258e983b8481fe968e115d869139e4f01bfeee4b7f7465869ff630 | $1-3\cos x+\cos 2x=\frac{\csc (\pi -x)}{\cot 2x-\cot x}$ | [
"$x=\\frac{\\pi }{3}(6n\\pm 1)$"
] | [
"$1-3\\cos x+\\cos 2x=\\frac{\\csc (\\pi -x)}{\\cot 2x-\\cot x}$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bf11736e7cdda98c3390db76207eec9001fcf3421bac626e84618b31c28d294d | $1-\cos(\pi -x)+\sin \frac{\pi +x}{2}=0$ | [
"$x=\\pi (2n+1);\\ x=\\frac{4\\pi }{3}(3n\\pm 1)$"
] | [
"$1-\\cos(\\pi -x)+\\sin \\frac{\\pi +x}{2}=0$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5a636cabfe944273d1af05101de98cac993ae4d850b7bbfe4eb50f5df28a5fc7 | $1-\cos^2 x=\sin 3x-\cos\left(\frac{\pi }{2}+x\right)$ | [
"$x=90^\\circ n$"
] | [
"$1-\\cos^2 x=\\sin 3x-\\cos\\left(\\frac{\\pi }{2}+x\\right)$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a6158cf946dae76a437be2d33f89697e07077a51a59a4f4c78efa2873349d4fe | $1-\frac{1}{4}\sin ^2 2\alpha -\sin ^2 \beta -\cos ^4 \alpha $ | [
"$\\sin(\\alpha +\\beta )\\sin(\\alpha -\\beta )$"
] | [
"$1-\\frac{1}{4}\\sin ^2 2\\alpha -\\sin ^2 \\beta -\\cos ^4 \\alpha $"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d3e06b3a239316aad669da37d9fc69a9217af85d44f20033c47c4c5140e1eb42 | $1-\sin^2 (\alpha +\beta )-\sin^2 (\alpha -\beta )$ | [
"$\\cos 2\\alpha \\cos 2\\beta $"
] | [
"$1-\\sin^2 (\\alpha +\\beta )-\\sin^2 (\\alpha -\\beta )$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9a63232e3116702fae881ecd8ae2137665822c13fc342cbabdc676d26d2ab42e | $1.7:\frac{\left(4.5\cdot 1\frac{2}{3}+3.75\right)\cdot \frac{7}{135}} {\frac{5}{9}}-\left(0.5+\frac{1}{3}-\frac{5}{12}\right)$ | [
"$1\\frac{17}{84}$"
] | [
"$1.7:\\frac{\\left(4.5\\cdot 1\\frac{2}{3}+3.75\\right)\\cdot \\frac{7}{135}} {\\frac{5}{9}}-\\left(0.5+\\frac{1}{3}-\\frac{5}{12}\\right)$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b6b2c8a475da8ed25028563663d43eec66ac33fbe937c0eda373b298212c0991 | $100$ integers are arranged in a circle. Each number is greater than the sum of the two subsequent numbers (in a clockwise order). Determine the maximal possible number of positive numbers in such a circle. | [
"49"
] | [
"$100$ integers are arranged in a circle. Each number is greater than the sum of the two subsequent numbers (in a clockwise order). Determine the maximal possible number of positive numbers in such a circle."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-de3bb9f295b6af3c504a03b46965aa1f397bb4ea026d726032856473b1c5cfa2 | $100$ numbers are written around a circle. The sum of the $100$ numbers is $100$ and the sum of any six consecutive numbers is always less than or equal to $6$. The first number is $6$. Find all the numbers. | [
"\\(6, -1, -4, 3, -2, 4\\) repeated over and over."
] | [
"$100$ numbers are written around a circle. The sum of the $100$ numbers is $100$ and the sum of any six consecutive numbers is always less than or equal to $6$. The first number is $6$. Find all the numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ee0a92ceb9d422afc98d2d6b6ad340f9904c9bcff7fa790dcf8c261de469beae | $100$ pictures of BDMO math campers were painted using exactly $k$ colors in each picture. There is a common color in every $20$ pictures, but there is no common color in all $100$ pictures. Find the smallest possible value of $k$. | [
"\\( k = 3 \\)"
] | [
"$100$ pictures of BDMO math campers were painted using exactly $k$ colors in each picture. There is a common color in every $20$ pictures, but there is no common color in all $100$ pictures. Find the smallest possible value of $k$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-da08b9943f1e1b34a8184ab40d9a873474f1a95e0b66e51d83afdd606c68110a | $100$ pictures of BdMO math campers were painted using exactly $k$ colors in each picture. There is a common color in every $20$ pictures, but there is no common color in all $100$ pictures. Find the smallest possible value of $k$. | [
"20"
] | [
"$100$ pictures of BdMO math campers were painted using exactly $k$ colors in each picture. There is a common color in every $20$ pictures, but there is no common color in all $100$ pictures. Find the smallest possible value of $k$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 16,
"genselect": 2
} | 18 | 0 | 0 |
omr-ws-9107a8f9e26aed877d2fe16c068123e236c1392237a1fb163d6a1e402b9ef795 | $1000$ unit cubes are stacked to form a cube with side length $10$. A sphere of radius $10$ is centered at the corner of the large cube. How many small cubes are cut by the sphere? In other words, how many small cubes have interior points which lie on the surface of the sphere? | [
"488"
] | [
"$1000$ unit cubes are stacked to form a cube with side length $10$. A sphere of radius $10$ is centered at the corner of the large cube. How many small cubes are cut by the sphere? In other words, how many small cubes have interior points which lie on the surface of the sphere?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 3
} | 5 | 0 | 0 |
omr-ws-d2db06c4e177b3a4b1c369d8300b3a750254d7873bfab2ccd834114b48956498 | $100_{10}$ in base $b$ has exactly $5$ digits. What is the value of $b$? | [
"3"
] | [
"$100_{10}$ in base $b$ has exactly $5$ digits. What is the value of $b$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c78d22cf5f924a035dc326149577395bc628f0edb85103c47922cae61a843e1a | $1024$ Duke students, indexed $1$ to $1024$, are having a chat. For each $1 \le i \le 1023$, student $i$ claims that student $2^{\lfloor \log_2 i \rfloor + 1}$ has a girlfriend. ($\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.) Given that exactly $201$ people are lying, find the index of the $61$... | [
"115"
] | [
"$1024$ Duke students, indexed $1$ to $1024$, are having a chat. For each $1 \\le i \\le 1023$, student $i$ claims that student $2^{\\lfloor \\log_2 i \\rfloor + 1}$ has a girlfriend. ($\\lfloor x \\rfloor$ is the greatest integer less than or equal to $x$.) Given that exactly $201$ people are lying, find the index... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5e35bf07999ccfc98b59c40d970d7103739c78e5961d8437df1fd3cf5d1b2bec | $11$ theatrical groups participated in a festival. Each day, some of the groups were scheduled to perform while the remaining groups joined the general audience. At the conclusion of the festival, each group had seen, during its days off, at least $1$ performance of every other group. At least how many days did the fes... | [
"10"
] | [
"$11$ theatrical groups participated in a festival. Each day, some of the groups were scheduled to perform while the remaining groups joined the general audience. At the conclusion of the festival, each group had seen, during its days off, at least $1$ performance of every other group. At least how many days did th... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-19d172684fcf4b150328d1faef396d8536327afecf2f45323fa9a0931deaa386 | $12$ friends are seated at a round table. $4$ of them are chosen at random to leave. If the probability that none of the chosen friends were sitting next to each other can be expressed as $\frac{m}{n}$, find $m+n \mod{1000}$. | [
"40"
] | [
"$12$ friends are seated at a round table. $4$ of them are chosen at random to leave. If the probability that none of the chosen friends were sitting next to each other can be expressed as $\\frac{m}{n}$, find $m+n \\mod{1000}$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 13
} | 28 | 0 | 0 |
omr-ws-d3fe428fa2a39c3058d6759601182f5a82f97bd16806ace03b8a14455f4a69dc | $13$ people sit one at a time in a round table for $13$ people. The first person (the oldest) sits in any seat, and each subsequent person must sit next to someone who has already sat down. How many distinct seating arrangements are possible, considering that rotations of the same arrangement are not counted as distinc... | [
"4096"
] | [
"$13$ people sit one at a time in a round table for $13$ people. The first person (the oldest) sits in any seat, and each subsequent person must sit next to someone who has already sat down. How many distinct seating arrangements are possible, considering that rotations of the same arrangement are not counted as di... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-0cef4c2a0642b2935502d6caef0c9fefbb4c776d5345bcc3c23718ad3bf17271 | $137$ is a "very prime" number because $1, 3, 7, 13, 37, \text{and }137$ are all prime. Compute the remainder when the largest "very prime" number is divided by $1000$. Consider $1$ to be a prime for the purpose of this question. | [
"137"
] | [
"$137$ is a \"very prime\" number because $1, 3, 7, 13, 37, \\text{and }137$ are all prime. Compute the remainder when the largest \"very prime\" number is divided by $1000$. Consider $1$ to be a prime for the purpose of this question."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 2
} | 12 | 0 | 0 |
omr-ws-5da2446ea2342c640c1060f5f21f7d0dfac27beb9c061cd97fc6299eaa3209ac | $16$ people sit around a circular table. After some time, they all stand up and sit down either in the chair they were previously sitting on or on a chair next to it. Determine the number of ways this can be done, given that two or more people cannot sit on the same chair. | [
"2207"
] | [
"$16$ people sit around a circular table. After some time, they all stand up and sit down either in the chair they were previously sitting on or on a chair next to it. Determine the number of ways this can be done, given that two or more people cannot sit on the same chair."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 2
} | 7 | 0 | 0 |
omr-ws-ff94704faba8b298fd754c336517820cbd01bb076046635569c180b46e279ce3 | $1<|z|<4$. | [
"The interior part of the circle bounded by concentric circles of radii 1 and 4 with center at the origin"
] | [
"$1<|z|<4$."
] | [] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-ca9d65f8db4ba40d6f5176f9f6096ca1c28cc9d014f5014c732b5e3e9ec48bad | $1\frac{7}{20}:2.7+2.7:1.35+\left(0.4:2\frac{1}{2}\right) \cdot \left(4.2-1\frac{3}{40}\right)$ | [
"3"
] | [
"$1\\frac{7}{20}:2.7+2.7:1.35+\\left(0.4:2\\frac{1}{2}\\right) \\cdot \\left(4.2-1\\frac{3}{40}\\right)$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a77cc840107d488b9ec9bd78c5366a71b124d7ad8ac6cf5f463ab218d9e4c237 | $1\le |z+i|\le 4$. | [
"The set of all points which lie in the interior of a circle bounded by concentric circles of radii 1 and 4 (including the circle) with center at the point $(0,-1)$"
] | [
"$1\\le |z+i|\\le 4$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7447443666a368c4ada812a4cfd3b8342b707a03574ac0c3daade7012fe5aa4d | $2(1-\sin x-\cos x)+\tan x+\cot x=0$. | [
"$-\\frac{\\pi }{4}+\\pi k;\\ \\frac{\\pi }{4}\\pm \\arccos \\frac{\\sqrt 2-\\sqrt {10}}{4}+2\\pi n$"
] | [
"$2(1-\\sin x-\\cos x)+\\tan x+\\cot x=0$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3cc298c83ace84acfb09cd3528cb82d6d77c8c004229928d1c65efd17c9d629b | $2(x^4-2x^2+3)(y^4-3y^2+4)=7$. | [
"$x_{1,2}=\\pm 1$, $y_{1,2}=\\pm \\sqrt {3/2}$; $x_{3,4}=\\pm 1$, $y_{3,4}=\\mp \\sqrt {3/2}$"
] | [
"$2(x^4-2x^2+3)(y^4-3y^2+4)=7$. "
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c6f55f4d4a9f93a87175ac6dd10c0d4c1524b5899b024d2727b779321760cd38 | $20$ points with no three collinear are given. How many obtuse triangles can be formed by these points? | [
"\\({{20}\\choose{3}}\\)"
] | [
"$20$ points with no three collinear are given. How many obtuse triangles can be formed by these points?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 2,
"tir": 1
} | 12 | 0 | 0 |
omr-ws-2354f44e08c2ec790526fcda5cb71c735e06d56fca4cacd2a51bf2f0b3c3bdaf | $2008$ different planes are chosen in space. Find the maximum number of planes that a single sphere can intersect. | [
"2008"
] | [
"$2008$ different planes are chosen in space. Find the maximum number of planes that a single sphere can intersect."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ad87f55740effde23ab4c0141050c95ae00fa2a1eb9b3579caaf9986c91661d5 | $2008$ points are located on the plane. A line does not pass through any of them. How many maximum segments with ends at given points can it cross? | [
"1008016"
] | [
"$2008$ points are located on the plane. A line does not pass through any of them. How many maximum segments with ends at given points can it cross?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-670de74e1bff93ab3559c5b2b78d46d253b2ed484a49afb919dad4a0c4d9c060 | $201$ positive integers are written on a line such that the first and the last numbers are both equal to $19999$. Each of the remaining numbers is less than the average of its neighboring numbers, and the differences between each of these numbers and the average of its neighboring numbers are all equal to a unique inte... | [
"19999"
] | [
"$201$ positive integers are written on a line such that the first and the last numbers are both equal to $19999$. Each of the remaining numbers is less than the average of its neighboring numbers, and the differences between each of these numbers and the average of its neighboring numbers are all equal to a unique... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-85c928c31dffed1711f897d47542e2e31bbb39b021caf067714a76d39740da52 | $201$ positive integers are written on a line such that the first and the last numbers are both equal to $19999$. Each of the remaining numbers is less than the average of its neighboring numbers, and the differences between each of these numbers and the average of its neighboring numbers are all equal to a unique inte... | [
"2686499"
] | [
"$201$ positive integers are written on a line such that the first and the last numbers are both equal to $19999$. Each of the remaining numbers is less than the average of its neighboring numbers, and the differences between each of these numbers and the average of its neighboring numbers are all equal to a unique... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 30,
"genselect": 8,
"tir": 28
} | 66 | 0 | 0 |
omr-ws-a0bbb398c6d392b1ca572e69f222d060c093d87e1ad167e1c7c6b38463a19232 | $2016$ bugs are sitting at different points on a 1-meter stick. Each bug runs in one direction with a constant and equal speed. If two bugs face each other, they change direction but maintain their speed. If a bug reaches either end of the stick, it flies away. What is the greatest number of contacts (collisions) that ... | [
"\\(1008^2\\)"
] | [
"$2016$ bugs are sitting at different points on a 1-meter stick. Each bug runs in one direction with a constant and equal speed. If two bugs face each other, they change direction but maintain their speed. If a bug reaches either end of the stick, it flies away. What is the greatest number of contacts (collisions) ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 45,
"tir": 29
} | 74 | 0 | 0 |
omr-ws-831baa447be15942c6d49c1cfec3594d13e60c0b6c91097b2e12d03e6ee8d160 | $2018$ little ducklings numbered $1$ through $2018$ are standing in a line, each holding a slip of paper with a nonnegative number on it. Ducklings $1$ and $2018$ have the number zero. Ducklings $2$ through $2017$ simultaneously change their number to equal the average of the numbers of the ducklings to their immediate... | [
"2000"
] | [
"$2018$ little ducklings numbered $1$ through $2018$ are standing in a line, each holding a slip of paper with a nonnegative number on it. Ducklings $1$ and $2018$ have the number zero. Ducklings $2$ through $2017$ simultaneously change their number to equal the average of the numbers of the ducklings to their imme... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-21a725e151d058e9047b2deaee06efd826abbeb9071eeab235b437a977684eaf | $2018$ people sit in a line. One by one, starting from the left, each person either stands or remains seated. The first person stands with probability $1/2$. For each $n>1$, the $n$th person stands with probability $\frac{S_n+1}{n+1}$, where $S_n$ is the number of people before the $n$th person who have stood. What is ... | [
"1009"
] | [
"$2018$ people sit in a line. One by one, starting from the left, each person either stands or remains seated. The first person stands with probability $1/2$. For each $n>1$, the $n$th person stands with probability $\\frac{S_n+1}{n+1}$, where $S_n$ is the number of people before the $n$th person who have stood. Wh... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 4
} | 19 | 0 | 0 |
omr-ws-85df65865be87db88543de79a349d2b2493b73bc8c8f56c00fc0b093f8781b47 | $2022$ lily pads are arranged in a circle, each starting with height $1$. A frog starts on one of the lily pads and jumps around clockwise as follows: if the frog is on a lily pad of height $k$, the lily pad grows by $1$ (becoming $k+1$), and then the frog jumps $k$ lily pads clockwise (i.e., jumping over $(k-1)$). The... | [
"5000"
] | [
"$2022$ lily pads are arranged in a circle, each starting with height $1$. A frog starts on one of the lily pads and jumps around clockwise as follows: if the frog is on a lily pad of height $k$, the lily pad grows by $1$ (becoming $k+1$), and then the frog jumps $k$ lily pads clockwise (i.e., jumping over $(k-1)$)... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-28317c4c528a7e725357a4fa5be1c514f12ee18c11ff4a429ba304bcc66f61d0 | $2023$ wise men are located in a circle. Each wise man either believes the earth is the center of the universe or not. Every minute, each wise man who is between two wise men with a different opinion changes his mind. Determine the smallest time required for all the wise men to have the same opinion, regardless of thei... | [
"1011"
] | [
"$2023$ wise men are located in a circle. Each wise man either believes the earth is the center of the universe or not. Every minute, each wise man who is between two wise men with a different opinion changes his mind. Determine the smallest time required for all the wise men to have the same opinion, regardless of... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 3
} | 18 | 0 | 0 |
omr-ws-9c4fd946741124b84db86ed7932c80d20abedd9b407fd51a10f217618e9b370d | $21^{105}$ is congruent to 27 modulo 666. Find $21^{970}$ modulo 666. | [
"567"
] | [
"$21^{105}$ is congruent to 27 modulo 666. Find $21^{970}$ modulo 666."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 19
} | 34 | 0 | 0 |
omr-ws-ad09913c8b248c4b474858118b391bd45d7086a37c7e7d0fe3d6aea3589b93be | $239$ wise men stand in a circle near an opaque baobab. The king places a hat of one of $16$ colors on each wise man's head. Each wise man can only see the hats of the two nearest wise men on each side. Without communicating, each wise man must guess the color of their own hat. What is the maximum $k$ such that, in any... | [
"80"
] | [
"$239$ wise men stand in a circle near an opaque baobab. The king places a hat of one of $16$ colors on each wise man's head. Each wise man can only see the hats of the two nearest wise men on each side. Without communicating, each wise man must guess the color of their own hat. What is the maximum $k$ such that, i... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ace533f653e9569d30201996a14c416b9509f876db444a42df744a3690ce8af3 | $24$ lemons are required to make $32$ gallons of lemonade. How many lemons are needed to make $4$ gallons of lemonade? | [
"3"
] | [
"$24$ lemons are required to make $32$ gallons of lemonade. How many lemons are needed to make $4$ gallons of lemonade?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-173ac2fc02af87d3ada9b10d9de41b12425fce73de14d74fef0950eb243345ae | $26:\left[\frac{3:(0.2-0.1)}{2.5\cdot (0.8+1.2)} +\frac{(34.06-33.81)\cdot 4}{6.84:(28.57-25.15)}\right] +\frac{2}{3}:\frac{4}{21}$ | [
"$7\\frac{1}{2}$"
] | [
"$26:\\left[\\frac{3:(0.2-0.1)}{2.5\\cdot (0.8+1.2)} +\\frac{(34.06-33.81)\\cdot 4}{6.84:(28.57-25.15)}\\right] +\\frac{2}{3}:\\frac{4}{21}$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d39182b9852e361db1388ca264f5b4c71f85d5facd8b0f1b23773f6521b89f27 | $2N$ people are seated in $N$ rows of $2$. If each person must move to an adjacent chair, how many different reseatings are there? | [
"\\( a_n = a_{n-1} + a_{n-2} + 2 \\left( a_{n-2} + a_{n-3} + \\cdots + a_0 \\right) \\)"
] | [
"$2N$ people are seated in $N$ rows of $2$. If each person must move to an adjacent chair, how many different reseatings are there?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dbede3ffd05d5306480931b083d4f63e7368f3bfbd49c9a67ca977125e806503 | $2\arcsin^2 x-5\arcsin x+2=0$. | [
"$\\sin \\frac{1}{2}$"
] | [
"$2\\arcsin^2 x-5\\arcsin x+2=0$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7ec656703104b719cd6fba3257c3f4fdc00c6c12089544f3e53a02fc0d139390 | $2\cos 3x+\sqrt 3\sin x+\cos x=0$. | [
"$\\frac{\\pi }{3}+\\frac{\\pi }{2}k$"
] | [
"$2\\cos 3x+\\sqrt 3\\sin x+\\cos x=0$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f9b38d0cd7efabc7c9d18451aa4238b9190a1588ff806430c34bd5a5d3b07461 | $2\cos ^2 x+4\cos x=3\sin ^2 x$ | [
"$x=2\\pi n\\pm \\arccos \\frac{\\sqrt {19}-2}{5}$"
] | [
"$2\\cos ^2 x+4\\cos x=3\\sin ^2 x$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8cc4fee7de51de628f079b2e56da9c5114749bc54de356cac11352e1ac489701 | $2\cos ^2 x+\cos 5x=1$. | [
"$\\frac{\\pi }{7}+\\frac{2\\pi }{7}k;\\ \\frac{\\pi }{3}+\\frac{2\\pi }{3}n$"
] | [
"$2\\cos ^2 x+\\cos 5x=1$."
] | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 23
} | 37 | 0 | 0 |
omr-ws-81018ec58feef5d3925c84a858dd4868697af6f837239ee1cfedd645d5b54e49 | $2\cos^2 x+5\sin x-4=0$. | [
"$(-1)^k \\frac{\\pi }{6}+\\pi k$"
] | [
"$2\\cos^2 x+5\\sin x-4=0$."
] | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 12,
"tir": 26
} | 38 | 0 | 0 |
omr-ws-f05bbd42c78af1cc25cf4f05e83d27f0c092393a8639ffd89624ad3ef6c42ce3 | $2\cos^2 x+\sin x=2$. | [
"$\\pi k;\\ (-1)^n \\frac{\\pi }{6}+\\pi n$"
] | [
"$2\\cos^2 x+\\sin x=2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-60648794969def3c95bd1d21d2cc25cf24f513d8299a965db1bd08e47f97e9f7 | $2\cos^2 x-3\sin x\cos x+5\sin^2 x=3$. | [
"$\\arctan \\frac{3\\pm \\sqrt {17}}{4}+\\pi k$"
] | [
"$2\\cos^2 x-3\\sin x\\cos x+5\\sin^2 x=3$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-23adf756e081570fd707b0bc7a485c294da58a14e540bdd1fe99e17dd8fa1e50 | $2\cot x-3\cos 3x=\tan 2x$. | [
"$\\emptyset $"
] | [
"$2\\cot x-3\\cos 3x=\\tan 2x$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-93f54b110d351fbf38eb1ae420e7d983bfd038e7c5529287f03dc4edc9e0ecbc | $2\cot(x-\pi )-(\cos x+\sin x)(\csc x-\sec x)=4$ | [
"$x=\\frac{\\pi }{2}n+(-1)^n \\frac{\\pi }{12}$"
] | [
"$2\\cot(x-\\pi )-(\\cos x+\\sin x)(\\csc x-\\sec x)=4$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b2b3c5acd106d6d7b9a86635e0f881655dbd6ff8e7c6a22d49aae9a0f2ef441c | $2\left[1-\sin\left(\frac{3\pi }{2}-x\right)\right]=\sqrt 3\tan\frac{\pi -x}{2}$ | [
"$x=\\pi (2n+1);\\ x=\\pi n+(-1)^n \\frac{\\pi }{3}$"
] | [
"$2\\left[1-\\sin\\left(\\frac{3\\pi }{2}-x\\right)\\right]=\\sqrt 3\\tan\\frac{\\pi -x}{2}$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e66e8054cc6bb7fee0b06998b4df049449a071682330c13190c0cf415aa29f09 | $2\sin 17x+\sqrt 3\cos 5x+\sin 5x=0$. | [
"$-\\frac{\\pi }{66}+\\frac{\\pi }{11}k;\\ \\frac{\\pi }{9}+\\frac{\\pi }{6}n$"
] | [
"$2\\sin 17x+\\sqrt 3\\cos 5x+\\sin 5x=0$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-051e4809d6b4d61cdcaf9144cbb08829be1a92656ae5f882a504a0f0ea87d41c | $2\sin \left(\frac{2}{3}x-\frac{\pi }{6}\right) -3\cos \left(2x+\frac{\pi }{3}\right)=5$. | [
"$\\emptyset $"
] | [
"$2\\sin \\left(\\frac{2}{3}x-\\frac{\\pi }{6}\\right) -3\\cos \\left(2x+\\frac{\\pi }{3}\\right)=5$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0919978d0150a94a71dbd857cdedf1ba3b7bb8685a12638bfbb1e8f22ade08a0 | $2\sin ^2 \left(\frac{\pi }{2}\cos ^2 x\right) =1-\cos (\pi \sin 2x)$. | [
"$\\frac{\\pi }{2}+\\pi k;\\ \\pm \\arctan \\frac{1}{2}+\\pi n$"
] | [
"$2\\sin ^2 \\left(\\frac{\\pi }{2}\\cos ^2 x\\right) =1-\\cos (\\pi \\sin 2x)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-196de0b059e97a284d5721c1ec99cd116c67a9b882b5b5aa8aadcaa31fd6382d | $2\sin x-3\cos x=3$. | [
"$\\pi +2\\pi k;\\ 2\\arctan \\frac{3}{2}+2\\pi n$"
] | [
"$2\\sin x-3\\cos x=3$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cef5830536525915d1392629c22fdd71abc8057f3ad0f6dc050b56ba54de8c59 | $2\sin^2 \alpha +\sqrt 3\sin 2\alpha -1$ | [
"$2\\sin(2\\alpha -30^\\circ)$"
] | [
"$2\\sin^2 \\alpha +\\sqrt 3\\sin 2\\alpha -1$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a5a8fb9074eef2a2919c5d1b486d2252638cd59d19e7bfa149d26f01e54f990c | $2\sin^3 x-\cos 2x-\sin x=0$. | [
"$\\frac{\\pi }{4}+\\frac{\\pi }{2}k;\\ -\\frac{\\pi }{2}+2\\pi n$"
] | [
"$2\\sin^3 x-\\cos 2x-\\sin x=0$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4ac40bcf484632fbd824efc62768dbf219c241ba4070704699f6b24bcd1303fe | $2\sqrt 2(\sin x+\cos x)\cos y=3+\cos 2y$. | [
"$x_1=(\\pi /4)+2k\\pi $, $y_1=(2n+1)\\pi $; $x_2=(5\\pi /4)+2k\\pi $, $y_2=2n\\pi $. Using the substitution $\\cos y=z$, reduce the equation to the form $z^2-2z\\sin [x+(\\pi /4)]+1=0$ or $[z-\\sin [x+(\\pi /4)]]^2+[1-\\sin ^2 [x+(\\pi /4)]]=0$"
] | [
"$2\\sqrt 2(\\sin x+\\cos x)\\cos y=3+\\cos 2y$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ed14977322407df4d5c48bfda086ea99ebeb69401353b9ea7bc4b3baa088d341 | $2\sqrt 2\sin(45^\circ+x)=\frac{1+\cos 2x}{1+\sin x}$ | [
"$x=2\\pi n;\\ x=2(\\pi n+\\arctan 2)$"
] | [
"$2\\sqrt 2\\sin(45^\\circ+x)=\\frac{1+\\cos 2x}{1+\\sin x}$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-623d3b9b09d9c613ef3afc74386afbc16a34ff534d7ad66f7e73995d542eb013 | $2\sqrt x-7x^2=2\left(\frac{x}{2}+\sqrt x\right)$ and $2\sqrt x-7x^2=2x+2\sqrt x$. | [
"Yes"
] | [
"$2\\sqrt x-7x^2=2\\left(\\frac{x}{2}+\\sqrt x\\right)$ and $2\\sqrt x-7x^2=2x+2\\sqrt x$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-065f99934f013fd3373124ff4de3c4d3bec1ffa0a3d44f7736fdb0949c738fb5 | $2\sqrt x-7x^2=2x+2\sqrt x$ and $-7x^2=2x$. | [
"No"
] | [
"$2\\sqrt x-7x^2=2x+2\\sqrt x$ and $-7x^2=2x$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fe8e1af3fe027657853913720e8fd736c2cc38052473b0329122bd6cd61348fd | $2n$ bullets with distinct speeds are fired in the same direction with an interval of 1 second in a random order. When two bullets meet, they annihilate each other. Let $P(n)$ be the probability of total annihilation. Calculate the value of $P(5)$. | [
"\\dfrac{1}{10}"
] | [
"$2n$ bullets with distinct speeds are fired in the same direction with an interval of 1 second in a random order. When two bullets meet, they annihilate each other. Let $P(n)$ be the probability of total annihilation. Calculate the value of $P(5)$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5487f4225ed80bea8b1548513173cf1299ef8a19ccb4ffda79ce94ed8c5bf67a | $2n$ points are given on the plane. Two players take turns choosing one point until they end (one point cannot be selected twice). The loser is the one who has the sum of the pairs of distances between the points chosen by him less than those of the opponent. Who wins if they both play optimally, assuming all distances... | [
"The first player wins."
] | [
"$2n$ points are given on the plane. Two players take turns choosing one point until they end (one point cannot be selected twice). The loser is the one who has the sum of the pairs of distances between the points chosen by him less than those of the opponent. Who wins if they both play optimally, assuming all dist... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 2,
"tir": 5
} | 10 | 0 | 0 |
omr-ws-768490eaa8c4218a6debe37c9b599277d1d0b8a606454abdcede7a397e91351b | $2{\bf a}+3{\bf b}$ if $${\bf a}=(1,1,-1), {\bf b}=(2,0,0).$$ | [
"$6\\sqrt 2$"
] | [
"$2{\\bf a}+3{\\bf b}$ if $${\\bf a}=(1,1,-1), {\\bf b}=(2,0,0).$$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-93f5b89f1f4cda80a47bbc7f8629d897a1833c6600b2c4f91b28ebcf60290a9e | $2|\log _{10}(ax)|\cdot \log _x 10=(4\log _{10}a-3)\log _{x^2}10-\frac{1}{2}\log _{10}x$. | [
"$x=1/10$ for $a=10$; $x_1=1/10$, $x_2=10^{2-\\sqrt {1+8\\log _{10}a}}$ for $10<a<1000$; $x_1=1/10$, $x_2=1/1000$ for $a\\ge 1000$; there are no solutions for the remaining values of $a$"
] | [
"$2|\\log _{10}(ax)|\\cdot \\log _x 10=(4\\log _{10}a-3)\\log _{x^2}10-\\frac{1}{2}\\log _{10}x$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-293ee531d34c08ddd8c8d61a25ff5e8f0a0629e0a4f8433bbe1c0ccd762d9a64 | $3$ points are independently chosen at random on a circle. What is the probability that they form an acute triangle? | [
"\\dfrac{1}{4}"
] | [
"$3$ points are independently chosen at random on a circle. What is the probability that they form an acute triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-eb088f0a9365e9ff1a34178530002ff1a04cdeabda98f290b733ab8652f58f4d | $3(1-\sin x)=1+\cos 2x$. | [
"$\\frac{\\pi }{2}+2\\pi k;\\ (-1)^n \\frac{\\pi }{6}+\\pi n$"
] | [
"$3(1-\\sin x)=1+\\cos 2x$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a9a90103383f551b5a508520f68516b4b37f2ab404d828649fd7d2e95774b2ea | $300$ points are placed around a circle. Some pairs of these points are connected by segments, and among any four points there are two that are not connected by a segment. What is the largest possible number of triangles with vertices at these points and sides along the drawn segments? | [
"1000000"
] | [
"$300$ points are placed around a circle. Some pairs of these points are connected by segments, and among any four points there are two that are not connected by a segment. What is the largest possible number of triangles with vertices at these points and sides along the drawn segments?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-9ba2ad6a4853d282fdb142eca63315edb85eb40875edc2763317e82150c40fbd | $32\cos ^6 x-\cos 6x=1$. | [
"$\\frac{\\pi }{2}+\\pi k;\\ \\pm \\frac{1}{2}\\arccos \\left(-\\frac{1}{4}\\right)+\\pi n$"
] | [
"$32\\cos ^6 x-\\cos 6x=1$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-043e73dde1f149c8e833eda5d6c887aa450dbb0c30c68df1fb37952a4e7ebd70 | $361+2(19)(6)+36=x$. Solve for $x$. | [
"625"
] | [
"$361+2(19)(6)+36=x$. Solve for $x$."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-679411957bb39cd6710fbc2d9ffd58a22b26528162133f280b744da4f434c7f6 | $3\cos^2 x-\sin^2 x-\sin 2x=0$ | [
"$x=\\pi n+\\frac{\\pi }{4};\\ x=\\pi n-\\arctan 3$"
] | [
"$3\\cos^2 x-\\sin^2 x-\\sin 2x=0$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a6ecf7cad09464ff82b8b534f964e47ce4bb7f21441143a929d6e2a7c0e081b2 | $3\sin 2x+\cos 2x=2$. | [
"$\\arctan \\frac{3\\pm \\sqrt 6}{3}+\\pi n$"
] | [
"$3\\sin 2x+\\cos 2x=2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b2a849955fb414f49046bd5ea5f67dbb492655644776284521cdfdac62719c84 | $3\sin ^2 \frac{x}{3}+5\sin ^2 x=8$. | [
"$\\frac{3\\pi }{2}+3\\pi k$"
] | [
"$3\\sin ^2 \\frac{x}{3}+5\\sin ^2 x=8$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c19ffa38100d4f6a1f03fc716f62a091b03334366cb76d8507df6bce0fd45181 | $3\sin x=2\cos x$. | [
"$\\arctan \\frac{2}{3}+\\pi k$"
] | [
"$3\\sin x=2\\cos x$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8249eb39e27d603b8f0a2c3833c95b8d93f9572fe94c1914847a4bb889804e0a | $3\sin^2 x+2\sin x\cos x=2$. | [
"$\\arctan \\left(-1\\pm \\sqrt 3\\right)+\\pi k$"
] | [
"$3\\sin^2 x+2\\sin x\\cos x=2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ca624fd549b1f18a3d079a6c1b7f05c966e85d3474966cecaf51a55f5a098d8b | $3^n = 3 \cdot 9^3 \cdot 81^2$. What is the value of $n$? | [
"15"
] | [
"$3^n = 3 \\cdot 9^3 \\cdot 81^2$. What is the value of $n$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-630b7e38f62bb7ef3527215ed755ee87f6be527b77281fb796e110e375eb6d01 | $3m$ balls numbered $1, 1, 1, 2, 2, 2, 3, 3, 3, \ldots, m, m, m$ are distributed into $8$ boxes so that any two boxes contain identical balls. Find the minimal possible value of $m$. | [
"11"
] | [
"$3m$ balls numbered $1, 1, 1, 2, 2, 2, 3, 3, 3, \\ldots, m, m, m$ are distributed into $8$ boxes so that any two boxes contain identical balls. Find the minimal possible value of $m$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 8,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-1264053a27d746a83b00ec510514ed3254d3961b184b00e5ff16f1ea592a573f | $4$ balls are distributed uniformly at random among $6$ bins. What is the expected number of empty bins? | [
"\\dfrac{625}{216}"
] | [
"$4$ balls are distributed uniformly at random among $6$ bins. What is the expected number of empty bins?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6e9cd79478217dffb8c4239893c4d13274f584da766827155b85e7662c1860b9 | $4$ people decide they want to enjoy the weather outside for a bit in the afternoon. Avarose decides to get outside early, and leaves his house to go to the park at $2:00$PM. Gary leaves his house at $2:30$, and Beth and Ann soon follow; they leave their house at $2:35$. All $4$ of them live a distinct distance $S$ mil... | [
"5"
] | [
"$4$ people decide they want to enjoy the weather outside for a bit in the afternoon. Avarose decides to get outside early, and leaves his house to go to the park at $2:00$PM. Gary leaves his house at $2:30$, and Beth and Ann soon follow; they leave their house at $2:35$. All $4$ of them live a distinct distance $S... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-37aa915c043bfd4f38705ba6b31e6a6ede2694ecd6e14a95b7a6a36644c1718e | $4+\sin ^2 x+\cos ^2 x=5\sin ^2 x\sin ^2 y$. | [
"$\\left(\\frac{\\pi }{2}+\\pi k;\\frac{\\pi }{2}+\\pi n\\right)$"
] | [
"$4+\\sin ^2 x+\\cos ^2 x=5\\sin ^2 x\\sin ^2 y$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dcc69e4338a7fd499ce1971cbe0b9c672cf1a3ecbca41c9572da426cdf785019 | $4-4(\cos x-\sin x)-\sin 2x=0$. | [
"$2\\pi k;\\ -\\frac{\\pi }{2}+2\\pi n$"
] | [
"$4-4(\\cos x-\\sin x)-\\sin 2x=0$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-306af2ac82a31e531d6d026401b7131320127fbae64e8c156017fdb31529cc56 | $40$ people, numbered $1$ through $40$, sit around a circular table. They begin playing a game where each person is initially considered “alive”. Starting with person $1$, the first person eliminates the closest “alive” person to their right. Then the next “alive” person, moving counterclockwise, eliminates the closest... | [
"17"
] | [
"$40$ people, numbered $1$ through $40$, sit around a circular table. They begin playing a game where each person is initially considered “alive”. Starting with person $1$, the first person eliminates the closest “alive” person to their right. Then the next “alive” person, moving counterclockwise, eliminates the cl... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-6c6cfaee9999d3ba453dab23cc05e55890034e2dd179d4b8fe436f4582f103a3 | $400$ children form a circle and, and they are numbered from $1$ to $400$. Let $1 \le k \le 400$ be a fixed integer, such that start at child $1$ and mark each $k$-th child, stopping once we mark someone twice. For example, if we choose $k = 2$, we mark child $1$, $3$, $5$, $7$, etc. For how many values of $k$ will we ... | [
"160"
] | [
"$400$ children form a circle and, and they are numbered from $1$ to $400$. Let $1 \\le k \\le 400$ be a fixed integer, such that start at child $1$ and mark each $k$-th child, stopping once we mark someone twice. For example, if we choose $k = 2$, we mark child $1$, $3$, $5$, $7$, etc. For how many values of $k$ w... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f9e148b0cdd14f578bf75b132f82e16cd29882c0d9c6e802aa52151bedf1053e | $4\arctan (x^2-3x+3)=\frac{\pi }{4}$. | [
"1; 2"
] | [
"$4\\arctan (x^2-3x+3)=\\frac{\\pi }{4}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6d7344089c916472c7ea0f7899568b6252d6e388cc10b288e41e08c099f1550e | $4\arctan x-6{\rm arccot }x=\pi $. | [
"$\\tan \\frac{2}{5}\\pi $"
] | [
"$4\\arctan x-6{\\rm arccot }x=\\pi $."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9da02cea749900529ac3c6d026715201284a5750e7c922ede38f99f7a7ed3e09 | $4\cos ^3 \frac{x}{2}+3\sqrt 2\sin x=8\cos \frac{x}{2}$. | [
"$\\pi +2\\pi k;\\ (-1)^n \\frac{\\pi }{2}+2\\pi n$"
] | [
"$4\\cos ^3 \\frac{x}{2}+3\\sqrt 2\\sin x=8\\cos \\frac{x}{2}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ab6272dfff90f34a3783264661bc6cf9bbaeab0562d9b3cc80896014c5f116d4 | $4\cos^2 2x+8\cos^2 x=7$. | [
"$\\pm \\frac{\\pi }{6}+\\pi k$"
] | [
"$4\\cos^2 2x+8\\cos^2 x=7$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d9a45e42546b829b339a77149cd2e1c1d2388f2e7697a88a99e4f508616df908 | $4\sin 2x-\tan ^2 \left(x-\frac{\pi }{4}\right)=4$. | [
"$\\frac{\\pi }{4}+\\pi k$"
] | [
"$4\\sin 2x-\\tan ^2 \\left(x-\\frac{\\pi }{4}\\right)=4$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
End of preview. Expand in Data Studio
Processed thinking block outputs of OpenMathInstruct-2 by Qwen3.8-27B
The model was instructed to solve the dataset problems with a "thinking block" CoT style, with a prompt given specifically for this format.
The output reasoning chain was then processed by Qwen3.8-27B to 3 discrete candidates.
- Downloads last month
- 45