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-7ba478aca5634fa101695d3f10ff14ae5b2b227d3ecd87058e2f2a69a1f097bd | There is a test among the students in each of \( n \) papers (\( n \leq 3 \)). It is known that for any paper exactly three students got the best score, and in any two papers exactly one student got the best score in both papers. Find the smallest \( n \) such that the above conditions imply that exactly one student go... | [
"8"
] | [
"There is a test among the students in each of \\( n \\) papers (\\( n \\leq 3 \\)). It is known that for any paper exactly three students got the best score, and in any two papers exactly one student got the best score in both papers. Find the smallest \\( n \\) such that the above conditions imply that exactly on... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-59050332479272cc8c38e90c15aa6a4b8aa4807e5258f9870572971ca6e5758b | There is a test with $k$ questions. Each question awards 1 point if answered correctly. There are $k$ students taking the test independently. For any pair of questions, exactly one student can solve both questions. Let $M$ be the sum of points from all the students. Find the maximum value of $M$. | [
"\\( k \\times (k-1) \\)"
] | [
"There is a test with $k$ questions. Each question awards 1 point if answered correctly. There are $k$ students taking the test independently. For any pair of questions, exactly one student can solve both questions. Let $M$ be the sum of points from all the students. Find the maximum value of $M$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2a25883509304bdf1619ab4bfff4607ff73c314634d96794ca8a02ff4588f7fe | There is a track with length $2017$ meters. There are $2018$ people in a line, with person $k$ starting to run at time $t=k-1$ s with speed $k$ m/s for $1 \le k \le 2018$. If person $A$ is the first person to finish running across the track and person $B$ is the last, compute $B-A$. | [
"1973"
] | [
"There is a track with length $2017$ meters. There are $2018$ people in a line, with person $k$ starting to run at time $t=k-1$ s with speed $k$ m/s for $1 \\le k \\le 2018$. If person $A$ is the first person to finish running across the track and person $B$ is the last, compute $B-A$."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 22
} | 43 | 0 | 0 |
omr-ws-8fd838fb5b3754f77f8d233325df54956f1779032233f3dfe1f2b3a0d398073a | There is a triangle $\triangle PQR$ on the plane $ax + by + cz = 4$ that passes through the point $(0, 3, 2)$. When the orthogonal projections of $\triangle PQR$ onto the planes $xy$, $yz$, and $xz$ have areas 2, 3, and 4 respectively, find the values of $a$, $b$, and $c$. | [
"\\frac{1}{2}"
] | [
"There is a triangle $\\triangle PQR$ on the plane $ax + by + cz = 4$ that passes through the point $(0, 3, 2)$. When the orthogonal projections of $\\triangle PQR$ onto the planes $xy$, $yz$, and $xz$ have areas 2, 3, and 4 respectively, find the values of $a$, $b$, and $c$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-948257d911a5c9d3cdc6a43d64228416204e0038b99bfea8b4ac166106d61a11 | There is a unique angle $\theta$ between $0^{\circ}$ and $90^{\circ}$ such that for nonnegative integers $n$, the value of $\tan{\left(2^{n}\theta\right)}$ is positive when $n$ is a multiple of $3$, and negative otherwise. The degree measure of $\theta$ is $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime integers... | [
"547"
] | [
"There is a unique angle $\\theta$ between $0^{\\circ}$ and $90^{\\circ}$ such that for nonnegative integers $n$, the value of $\\tan{\\left(2^{n}\\theta\\right)}$ is positive when $n$ is a multiple of $3$, and negative otherwise. The degree measure of $\\theta$ is $\\tfrac{p}{q}$, where $p$ and $q$ are relatively ... | [
"MATH_training_set",
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 60,
"tir": 11
} | 71 | 0 | 0 |
omr-ws-a998c42b4e195629762d0f6d6f603845c0516b75be72d388b391c734f6056fef | There is a unique increasing sequence \(p_1, p_2, \ldots, p_k\) of primes for \(k \ge 2\) such that \(p_k = 13\) and \(\frac{1}{p_1} + \frac{1}{p_2} + \cdots + \frac{1}{p_k} - \frac{1}{p_1 p_2 \cdots p_k}\) is an integer. Compute \(p_1 + p_2 + \cdots + p_k\). | [
"29"
] | [
"There is a unique increasing sequence \\(p_1, p_2, \\ldots, p_k\\) of primes for \\(k \\ge 2\\) such that \\(p_k = 13\\) and \\(\\frac{1}{p_1} + \\frac{1}{p_2} + \\cdots + \\frac{1}{p_k} - \\frac{1}{p_1 p_2 \\cdots p_k}\\) is an integer. Compute \\(p_1 + p_2 + \\cdots + p_k\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-062445e58d1e2c82b933f9817e227dd82298a6bc684c0f9a59fbf77fa1dc5375 | There is a unique nondecreasing sequence of positive integers \(a_1, a_2, \ldots, a_n\) such that \[ \left(a_1 + \frac{1}{a_1}\right)\left(a_2 + \frac{1}{a_2}\right) \cdots \left(a_n + \frac{1}{a_n}\right) = 2020. \] Compute \(a_1 + a_2 + \cdots + a_n\). | [
"19"
] | [
"There is a unique nondecreasing sequence of positive integers \\(a_1, a_2, \\ldots, a_n\\) such that\n\n\\[\n\\left(a_1 + \\frac{1}{a_1}\\right)\\left(a_2 + \\frac{1}{a_2}\\right) \\cdots \\left(a_n + \\frac{1}{a_n}\\right) = 2020.\n\\]\n\nCompute \\(a_1 + a_2 + \\cdots + a_n\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 34,
"tir": 5
} | 39 | 0 | 0 |
omr-ws-5e1f06e3552bb91774457cd6c8ee15a2cad58d1d108a4c4b7013295220e6bebf | There is a unique nondecreasing sequence of positive integers \(a_1, a_2, \ldots, a_n\) such that \[\left(a_1+\frac{1}{a_1}\right)\left(a_2+\frac{1}{a_2}\right)\cdots\left(a_n+\frac{1}{a_n}\right)=2020.\] Compute \(a_1 + a_2 + \cdots + a_n\). | [
"19"
] | [
"There is a unique nondecreasing sequence of positive integers \\(a_1, a_2, \\ldots, a_n\\) such that \\[\\left(a_1+\\frac{1}{a_1}\\right)\\left(a_2+\\frac{1}{a_2}\\right)\\cdots\\left(a_n+\\frac{1}{a_n}\\right)=2020.\\] Compute \\(a_1 + a_2 + \\cdots + a_n\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-cae4b8b77aee6cbdb77b689b4328c32fdb37e813d8f4f6e32ae297d8d227a55f | There is a unique polynomial $P(x)$ of degree $4$ with rational coefficients and leading coefficient $1$ which has $\sqrt{2}+\sqrt{5}$ as a root. What is $P(1)$? | [
"-4"
] | [
"There is a unique polynomial $P(x)$ of degree $4$ with rational coefficients and leading coefficient $1$ which has $\\sqrt{2}+\\sqrt{5}$ as a root. What is $P(1)$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2a54e3f8a2419ad56c0c8cff786e1364dc0c15771187a1864d666daf3bc9cd66 | There is a unique polynomial \( P \) with degree 8 such that \[ P\left(\frac{1}{n}\right) = \frac{1}{n+1} \quad \text{for } n=1,2,\dots,9. \] Given that \( P \) has no double roots, compute the sum of the reciprocals of the roots of \( P \). | [
"\\(46 - 10!\\)\n\nGiven the problem and the forum discussions, the solution provided in the posts consistently arrives at the same answer. The sum of the reciprocals of the roots of the polynomial \\(P\\) is \\(46 - 10!\\).\n\nTherefore, the final answer is:"
] | [
"There is a unique polynomial \\( P \\) with degree 8 such that\n\\[\nP\\left(\\frac{1}{n}\\right) = \\frac{1}{n+1} \\quad \\text{for } n=1,2,\\dots,9.\n\\]\nGiven that \\( P \\) has no double roots, compute the sum of the reciprocals of the roots of \\( P \\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f66c6146bc632722b0fab948727b05024b25bd720750b63c95870152eb60d410 | There is a unique positive integer \( n \) such that \(\log_2{(\log_{16}{n})} = \log_4{(\log_4{n})}\). What is the sum of the digits of \( n \)? | [
"13"
] | [
"There is a unique positive integer \\( n \\) such that \\(\\log_2{(\\log_{16}{n})} = \\log_4{(\\log_4{n})}\\). What is the sum of the digits of \\( n \\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 18,
"tir": 1
} | 19 | 0 | 0 |
omr-ws-7dbf521d533c270bac2b20456f0aa2159d266f6bc575bf8551303bae08dc299e | There is a unique positive integer \(n\) such that \(n-2000\), \(n\), and \(n + 2000\) are all prime numbers. Find \(n\). | [
"2003"
] | [
"There is a unique positive integer \\(n\\) such that \\(n-2000\\), \\(n\\), and \\(n + 2000\\) are all prime numbers. Find \\(n\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b57c2cc9ee8c05ae0b221d34af78dec8e2411328003bc359ec10b1f63793415f | There is a unique real value \( A \) such that for all \( x \) with \( 1 < x < 3 \) and \( x \neq 2 \), \[ \left| \frac{A}{x^2 - x - 2} + \frac{1}{x^2 - 6x + 8} \right| < 1999. \] Compute \( A \). | [
"\\( \\frac{3}{2} \\)"
] | [
"There is a unique real value \\( A \\) such that for all \\( x \\) with \\( 1 < x < 3 \\) and \\( x \\neq 2 \\),\n\\[ \\left| \\frac{A}{x^2 - x - 2} + \\frac{1}{x^2 - 6x + 8} \\right| < 1999. \\]\nCompute \\( A \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-914a9af819d962e2b22567ad17a202788692d12847df9ae5152341567a562485 | There is a unique real value \(A\) such that for all \(x\) with \(1 < x < 3\) and \(x \ne 2\), \[ \left| \frac{A}{x^2 - x - 2} + \frac{1}{x^2 - 6x + 8} \right| < 1999. \] Compute \(A\). | [
"\\(\\frac{3}{2}\\)"
] | [
"There is a unique real value \\(A\\) such that for all \\(x\\) with \\(1 < x < 3\\) and \\(x \\ne 2\\),\n\\[\n\\left| \\frac{A}{x^2 - x - 2} + \\frac{1}{x^2 - 6x + 8} \\right| < 1999.\n\\]\nCompute \\(A\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-bf4f8675eca542c95c1427268566cd674a756ca2bc8287070440478389226159 | There is a unique subfield \( F \) of \( K \) such that \( F \) has degree 2 over \( \mathbb{Q} \). Thus, \( F = \mathbb{Q}(\sqrt{D}) \) for some squarefree integer \( D \). What is the value of \( D \)? | [
"5"
] | [
"There is a unique subfield \\( F \\) of \\( K \\) such that \\( F \\) has degree 2 over \\( \\mathbb{Q} \\). Thus, \\( F = \\mathbb{Q}(\\sqrt{D}) \\) for some squarefree integer \\( D \\). What is the value of \\( D \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 2
} | 14 | 0 | 0 |
omr-ws-0928cfc5ea664a774e85bffd4f04aa66e5537ef6d23c85dc14caee9c4f993dd0 | There is a unique three-digit number $N= \underline{A}\;\underline{B}\;\underline{C}$, where $A,B,C$ are all digits in base ten and $500<N<600$ that satisfies the following property: - When $N$ is written in base $B$, its representation is $$\underline{2}\;\underline{C}\;\underline{C}\;\underline{A} _B.$$ What is the v... | [
"900"
] | [
"There is a unique three-digit number $N= \\underline{A}\\;\\underline{B}\\;\\underline{C}$, where $A,B,C$ are all digits in base ten and $500<N<600$ that satisfies the following property: \n- When $N$ is written in base $B$, its representation is $$\\underline{2}\\;\\underline{C}\\;\\underline{C}\\;\\underline{A} ... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-771cb1f4683b84fbfb8a4f6f012e6ff9e50c3def9bc7727f29040aff9f234376 | There is a unique time between 2 and 3 o'clock at which the minute and hour hands of a standard 12-hour clock form an angle of 180 degrees. How many hours after 2 o'clock is this time? Express your answer as a common fraction. | [
"\\(\\frac{8}{11}\\)"
] | [
"There is a unique time between 2 and 3 o'clock at which the minute and hour hands of a standard 12-hour clock form an angle of 180 degrees. How many hours after 2 o'clock is this time? Express your answer as a common fraction."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-6fd11ac5d0208f4c3b68a79b3fbb21dab70d56fb6fdc54100224300f18b2cf64 | There is a unique triangle \(ABC\) such that \(AC = 14\), \(\cos A = \frac{4}{5}\), and the radius of the inscribed circle is \(r = 4\). Find the area of this triangle. | [
"168"
] | [
"There is a unique triangle \\(ABC\\) such that \\(AC = 14\\), \\(\\cos A = \\frac{4}{5}\\), and the radius of the inscribed circle is \\(r = 4\\). Find the area of this triangle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-0e2d6d1a6dd1e66d96172b77d40711a2b7af65a5e7935f05dd5f284270b7719f | There is a unique two-digit positive integer $t$ for which the last two digits of $11\cdot t$ are $36$. What is $t$? | [
"76"
] | [
"There is a unique two-digit positive integer $t$ for which the last two digits of $11\\cdot t$ are $36$.\n\nWhat is $t$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-edb5544fc374c1140cdc1567d16ec47a32828da4b49277f27d650ae16a4d044b | There is a water pipe that needs to be connected to two towns, Town X and Town Y. Town X is 4 miles directly above point A, and Town Y is 3 miles directly above point B. The distance from point A to B is 6 miles. A pump must be placed somewhere on the line segment AB, and pipes will be connected from the pump to each t... | [
"\\( \\frac{24}{7} \\) miles from Town X and \\( \\frac{18}{7} \\) miles from Town Y."
] | [
"There is a water pipe that needs to be connected to two towns, Town X and Town Y. Town X is 4 miles directly above point A, and Town Y is 3 miles directly above point B. The distance from point A to B is 6 miles. A pump must be placed somewhere on the line segment AB, and pipes will be connected from the pump to e... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-ca167e41150f752491871ddc6cdaf08686bc9fbb7f3b9909860793b728db68c6 | There is a website called RaTS. On RaTS, each user that joins will get a default profile picture that is a 3x3x3 cube, but some blocks may be missing. Each face view of the entire large cube is of one color, but each face view is a different color. There are 6 possible colors on RaTS, and there may not be any floating ... | [
"1440"
] | [
"There is a website called RaTS. On RaTS, each user that joins will get a default profile picture that is a 3x3x3 cube, but some blocks may be missing. Each face view of the entire large cube is of one color, but each face view is a different color. There are 6 possible colors on RaTS, and there may not be any floa... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 2
} | 8 | 0 | 0 |
omr-ws-266e53674134f9305e556cfddc38465f853ce39b1bb3f00562c1de5bdc89541a | There is a whole number \( A \) strictly positive, whose cube has 13 times more positive divisors than \( A \). How many positive divisors does the number \( A \) have? | [
"40"
] | [
"There is a whole number \\( A \\) strictly positive, whose cube has 13 times more positive divisors than \\( A \\). How many positive divisors does the number \\( A \\) have?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 27,
"genselect": 8,
"tir": 7
} | 42 | 0 | 0 |
omr-ws-17bdd1ef68198755ce65eb9e89ce6f2cbdea0d8d8b27c7257d5847346a2dc805 | There is a whole number \( A \) strictly positive, whose cube has 7 times more positive divisors than \( A \). How many positive divisors does the number \( A \) have? | [
"40"
] | [
"There is a whole number \\( A \\) strictly positive, whose cube has 7 times more positive divisors than \\( A \\). How many positive divisors does the number \\( A \\) have?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-87ce0d34c3cdc57b74e224bf2aafbd5cc9e9ca36d25cf3979c36fac8689f7a98 | There is a whole number \( B \) strictly positive, whose cube has 5 times more positive divisors than \( B \). How many positive divisors does the number \( B \) have? | [
"8"
] | [
"There is a whole number \\( B \\) strictly positive, whose cube has 5 times more positive divisors than \\( B \\). How many positive divisors does the number \\( B \\) have?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f758341beac6d240f93f5b7918218d7d5402307a48adc61f281664bd64759205 | There is an angle $\theta$ in the range $0^\circ < \theta < 45^\circ$ which satisfies \[\tan \theta + \tan 2 \theta + \tan 3 \theta = 0.\]Calculate $\tan \theta$ for this angle. | [
"\\frac{1}{\\sqrt{2}}"
] | [
"There is an angle $\\theta$ in the range $0^\\circ < \\theta < 45^\\circ$ which satisfies\n\\[\\tan \\theta + \\tan 2 \\theta + \\tan 3 \\theta = 0.\\]Calculate $\\tan \\theta$ for this angle."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bb14e0ae88c27b8ffbaf040d8c7bda552be9fdf7a238fa3c1c8dd1666d51e970 | There is an equiangular hexagon $OUTDIA$ such that $OU=DI$, $UT=IA$, $TD=AO$ are all integer side lengths. The incircles of triangles $OUT$ and $OTD$ are tangent to each other. Given that $OU, UT, TD$ are all distinct integers, what is the smallest possible value of the sum of the sides of the hexagon? | [
"22"
] | [
"There is an equiangular hexagon $OUTDIA$ such that $OU=DI$, $UT=IA$, $TD=AO$ are all integer side lengths. The incircles of triangles $OUT$ and $OTD$ are tangent to each other. Given that $OU, UT, TD$ are all distinct integers, what is the smallest possible value of the sum of the sides of the hexagon?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 9,
"tir": 4
} | 13 | 0 | 0 |
omr-ws-86d4c49503c05fd26ff435bcf73e89f404830bb1b08036add5442692d5c87cee | There is an equilateral triangle \(ABC\) on the plane. Three straight lines pass through \(A\), \(B\), and \(C\) respectively, such that the intersections of these lines form an equilateral triangle inside \(ABC\). On each turn, Ming chooses a two-line intersection inside \(ABC\) and draws the straight line determined ... | [
"7651"
] | [
"There is an equilateral triangle \\(ABC\\) on the plane. Three straight lines pass through \\(A\\), \\(B\\), and \\(C\\) respectively, such that the intersections of these lines form an equilateral triangle inside \\(ABC\\). On each turn, Ming chooses a two-line intersection inside \\(ABC\\) and draws the straight... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-469c47576653929d6a34ac3dae4e21501e0f7b9d6ef65edff20a759efc351b21 | There is an equilateral triangle on the plane with three circles centered at its vertices. Each circle's radius is less than the triangle’s height. Points on the plane are colored as follows: yellow if inside exactly one circle, green if inside exactly two circles, and blue if inside all three circles. Given that the y... | [
"300"
] | [
"There is an equilateral triangle on the plane with three circles centered at its vertices. Each circle's radius is less than the triangle’s height. Points on the plane are colored as follows: yellow if inside exactly one circle, green if inside exactly two circles, and blue if inside all three circles. Given that ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 5
} | 13 | 0 | 0 |
omr-ws-585373aebd94cbd6d7d55c5a5344093e1bfcac0e782b3b552434050ca4724347 | There is an infinite grid of lattice points with one point containing a leak. Water flows out from the leak, and at second $s$, water will be present at a lattice point $(a,b)$ if the taxicab distance from the leak $(x,y)$ to $(a,b)$ is less than or equal to $s$ (i.e., $|x-a|+|y-b|\leq s$). Determine the coordinates of... | [
"(x, y)"
] | [
"There is an infinite grid of lattice points with one point containing a leak. Water flows out from the leak, and at second $s$, water will be present at a lattice point $(a,b)$ if the taxicab distance from the leak $(x,y)$ to $(a,b)$ is less than or equal to $s$ (i.e., $|x-a|+|y-b|\\leq s$). Determine the coordina... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 7
} | 11 | 0 | 0 |
omr-ws-7b4e6114ec4119007c5ebc93faf43f593f88ceba4a4f60704c98763437b9d610 | There is an infinite region bounded by two parallel lines containing infinitely many non-overlapping (possibly tangent) circles of radius 1. Suppose that every line perpendicular to the boundary lines intersects or is tangent to at least two of the circles. Find the minimum possible distance between the boundary lines. | [
"\\(\\sqrt{2} + 3\\)"
] | [
"There is an infinite region bounded by two parallel lines containing infinitely many non-overlapping (possibly tangent) circles of radius 1. Suppose that every line perpendicular to the boundary lines intersects or is tangent to at least two of the circles. Find the minimum possible distance between the boundary l... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b9d263a65a3c57ea054fdb74b9898184c466d435aaf17b67a3cdf194de10ad5b | There is an infinite sequence of numbers where the average of the first \( k \) numbers is \( 2k + 7 \) for all positive integers \( k \). Find the 20th number in the sequence. | [
"85"
] | [
"There is an infinite sequence of numbers where the average of the first \\( k \\) numbers is \\( 2k + 7 \\) for all positive integers \\( k \\). Find the 20th number in the sequence."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-584de3f6bbbc54c5633dc7385ed231755e379c069b4ec1bd18bd780d704934ae | There is an integer $x$ such that $x^2 = k^3$, and $x^2 = k^5$, where $x>1$. Find $x$. | [
"1"
] | [
"There is an integer $x$ such that $x^2 = k^3$, and $x^2 = k^5$, where $x>1$. Find $x$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 1
} | 12 | 0 | 0 |
omr-ws-5820b40c6d386ebfec8b4e522e50bea3dd6061e77514d4ce39f686295a8608b3 | There is an integer in each cell of a $2m \times 2n$ table. We define the following operation: choose three cells forming an L-tromino (namely, a cell $C$ and two other cells sharing a side with $C$, one being horizontal and the other being vertical) and sum $1$ to each integer in the three chosen cells. Find a necessa... | [
"\\text{The total sum of all entries is divisible by } \\gcd(3, mn)"
] | [
"There is an integer in each cell of a $2m \\times 2n$ table. We define the following operation: choose three cells forming an L-tromino (namely, a cell $C$ and two other cells sharing a side with $C$, one being horizontal and the other being vertical) and sum $1$ to each integer in the three chosen cells. Find a n... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 3
} | 4 | 0 | 0 |
omr-ws-2225e3b68f28ed48fe8b44ef774e8d0ddf0f5f2801fd8e390b81bfd3df749e00 | There is an isosceles triangle \(ABC\) where \(AB = BC = 2\) and \(CA = 2\sqrt{3}\). If \(P\) is a point on the arc \(CA\) of the circumcircle of \(\triangle ABC\) such that \(PA = PC\), what is the length of segment \(BP\)? | [
"4"
] | [
"There is an isosceles triangle \\(ABC\\) where \\(AB = BC = 2\\) and \\(CA = 2\\sqrt{3}\\). If \\(P\\) is a point on the arc \\(CA\\) of the circumcircle of \\(\\triangle ABC\\) such that \\(PA = PC\\), what is the length of segment \\(BP\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a1ef84bc40998953ff3650b7e9ff472a2cb1f108bbc767d3780459cb64b1fa5e | There is an unlimited supply of congruent equilateral triangles made of colored paper. Each triangle is a solid color with the same color on both sides of the paper. A large equilateral triangle is constructed from four of these paper triangles as shown. Two large triangles are considered distinguishable if it is not p... | [
"336"
] | [
"There is an unlimited supply of congruent equilateral triangles made of colored paper. Each triangle is a solid color with the same color on both sides of the paper. A large equilateral triangle is constructed from four of these paper triangles as shown. Two large triangles are considered distinguishable if it i... | [
"MATH_training_set"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-d2c8488bc324ae03ed0647a10bedf18521c6878771aa0704d59970b644eda5f8 | There is one lock on the door and the correct key is among the six different ones you usually carry in your pocket. However, in haste, you’ve lost a key at random. What is the probability that you can still open the door? What is the probability that the first key you try will open the door? | [
"\\(\\frac{5}{6}\\) for the probability that you can still open the door and \\(\\frac{1}{6}\\) for the probability that the first key you try will open the door."
] | [
"There is one lock on the door and the correct key is among the six different ones you usually carry in your pocket. However, in haste, you’ve lost a key at random. What is the probability that you can still open the door? What is the probability that the first key you try will open the door?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 20
} | 20 | 0 | 0 |
omr-ws-2b45cfc06a61b99883cb9853a986b4b22f1b9421295fee0eb64b3bf1fcccedf4 | There is one value of $k$ for which the system of equations $r^2 + s^2 = t$ and $r + s + t = k$ will have exactly one real solution $(r, s, t)$. What is this value of $k$? | [
"-0.5"
] | [
"There is one value of $k$ for which the system of equations $r^2 + s^2 = t$ and $r + s + t = k$ will have exactly one real solution $(r, s, t)$. What is this value of $k$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-029952bd9ce7701220205f197d92d6355eee56b7bceb5c18057ca7caaab5efe8 | There is only one set of five prime numbers that form an arithmetic sequence with a common difference of 6. What is the sum of those five prime numbers? | [
"85"
] | [
"There is only one set of five prime numbers that form an arithmetic sequence with a common difference of 6. What is the sum of those five prime numbers?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c4e7358b56fe0c7c4410d861e46fc1e76e4796332858a62f19b33246f1d0a90c | There is only one value of $k$ for which the line $x=k$ intersects the graphs of $y=x^2+6x+5$ and $y=mx+b$ at two points which are exactly $5$ units apart. If the line $y=mx+b$ passes through the point $(1,6)$, and $b\neq 0$, find the equation of the line. Enter your answer in the form "$y = mx + b$". | [
"y=10x-4"
] | [
"There is only one value of $k$ for which the line $x=k$ intersects the graphs of $y=x^2+6x+5$ and $y=mx+b$ at two points which are exactly $5$ units apart. If the line $y=mx+b$ passes through the point $(1,6)$, and $b\\neq 0$, find the equation of the line. Enter your answer in the form \"$y = mx + b$\"."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5371250e225378c20b8e3b444da47bf9f155b4b08da2c2a22e77b9023bb321b8 | There is scrap of two grades of steel containing 5\% and 40\% of nickel. How much of each grade is required to obtain 140 tons of steel containing 30\% of nickel? | [
"40 tons of the first grade and 100 tons of the second grade"
] | [
"There is scrap of two grades of steel containing 5\\% and 40\\% of nickel. How much of each grade is required to obtain 140 tons of steel containing 30\\% of nickel?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7a027b1f332c463d3c086efdbd57f77e1d900265bb9f23b13067308962122014 | There is the number $1$ on the blackboard initially. The first step is to erase $1$ and write two nonnegative reals whose sum is $1$. Call the smaller number of the two $L_2$. For integer $k \ge 2$, the $k$-th step is to erase a number on the blackboard arbitrarily and write two nonnegative reals whose sum is the numbe... | [
"\\(\\frac{1405}{256}\\)"
] | [
"There is the number $1$ on the blackboard initially. The first step is to erase $1$ and write two nonnegative reals whose sum is $1$. Call the smaller number of the two $L_2$. For integer $k \\ge 2$, the $k$-th step is to erase a number on the blackboard arbitrarily and write two nonnegative reals whose sum is the... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 37,
"tir": 19
} | 56 | 0 | 0 |
omr-ws-e091931593a5c39701dd7d940ebde9d6b1644163d4e8c178ced1ec86184610df | There were $2n$, $n \ge 2$, teams in a tournament. Each team played against every other team once without draws. A team gets 0 points for a loss and gets as many points for a win as its current number of losses. For which $n$ can all the teams end up with the same non-zero number of points? | [
"\\(n = 2\\)"
] | [
"There were $2n$, $n \\ge 2$, teams in a tournament. Each team played against every other team once without draws. A team gets 0 points for a loss and gets as many points for a win as its current number of losses. For which $n$ can all the teams end up with the same non-zero number of points?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 4
} | 6 | 0 | 0 |
omr-ws-ffdda8c2477871e4d65a793172a6e5ea094ba4166ed7becf58af3c94f926e87e | There were $N$ people who took a survey about which entertainment from A, B, C they use, if any. Given the following information: - 50 people used B. - 13 people did not use C. - 74 people used at least two of A, B, C. - 61 people did not use A. What is the minimum number of $N$? | [
"86"
] | [
"There were $N$ people who took a survey about which entertainment from A, B, C they use, if any. Given the following information:\n- 50 people used B.\n- 13 people did not use C.\n- 74 people used at least two of A, B, C.\n- 61 people did not use A.\nWhat is the minimum number of $N$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 19,
"genselect": 6,
"tir": 2
} | 27 | 0 | 0 |
omr-ws-dd5c12236224fc832e08905f3dec8c05d807063919d9a04968fef480b7aa65ae | There were $n \ge 2$ teams in a tournament. Each team played against every other team once without draws. A team gets 0 points for a loss and gets as many points for a win as its current number of losses. For which $n$ can all the teams end up with the same number of points? | [
"\\( n \\) must be even."
] | [
"There were $n \\ge 2$ teams in a tournament. Each team played against every other team once without draws. A team gets 0 points for a loss and gets as many points for a win as its current number of losses. For which $n$ can all the teams end up with the same number of points?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8f64313400e3063f713b38270af6224dcddb5f2f02dc77de5de1713f012d598e | There were $n$ people standing in line to buy ice cream. Each person buys 1 stick of ice cream for $5$ dollars. The store only had $10$-dollar bills. There are $m$ people with $5$-dollar bills, and the rest have $10$-dollar bills. Determine the number of ways to queue so that the store does not have to stop to exchange... | [
"\\(\\binom{n}{m} - \\binom{n}{m+1}\\)"
] | [
"There were $n$ people standing in line to buy ice cream. Each person buys 1 stick of ice cream for $5$ dollars. The store only had $10$-dollar bills. There are $m$ people with $5$-dollar bills, and the rest have $10$-dollar bills. Determine the number of ways to queue so that the store does not have to stop to exc... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 21,
"genselect": 2,
"tir": 4
} | 27 | 0 | 0 |
omr-ws-3f616e8e7f29142ac864f28b1361d9cbf116183447bf92c7136d734c1778a63b | There were 10 boys and 10 girls at a party. Every boy likes a different positive number of girls, and every girl likes a different positive number of boys. Define the largest non-negative integer $n$ such that it is always possible to form at least $n$ disjoint pairs in which both members like each other. | [
"1"
] | [
"There were 10 boys and 10 girls at a party. Every boy likes a different positive number of girls, and every girl likes a different positive number of boys. Define the largest non-negative integer $n$ such that it is always possible to form at least $n$ disjoint pairs in which both members like each other."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 1
} | 3 | 0 | 0 |
omr-ws-99fb41fee6b221b64c373879e862bcbba3e1b4a2e3940b73bb4f3f8e499fd7a7 | There were 255 balls in a box, numbered from 1 to 255. N students each drew one ball. No student's number was exactly twice another student's number. What is the largest possible value of N? | [
"165"
] | [
"There were 255 balls in a box, numbered from 1 to 255. N students each drew one ball. No student's number was exactly twice another student's number. What is the largest possible value of N?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c9473295269c06b84a2cb517080cc82a8bd24027aa7251692068730f984a75cc | There were 36 participants at an event. Each participant shook hands with some others, but no two participants shook hands more than once. Each participant recorded the number of handshakes they made, and it was found that no two participants with the same number of handshakes had shaken hands with each other. Determin... | [
"546"
] | [
"There were 36 participants at an event. Each participant shook hands with some others, but no two participants shook hands more than once. Each participant recorded the number of handshakes they made, and it was found that no two participants with the same number of handshakes had shaken hands with each other. Det... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 77,
"genselect": 8,
"tir": 21
} | 106 | 0 | 0 |
omr-ws-89034cd20a12e0fdf8f6618d96808018cda85e1be2bbfd5f4a3afb3aab47d8ee | There were 45 hunters catching 2008 rabbits. Every fox caught 59 rabbits, every wolf 41 rabbits, and every bear 40 rabbits. How many foxes, wolves, and bears were there in the hunting company? | [
"10 foxes, 18 wolves, and 17 bears."
] | [
"There were 45 hunters catching 2008 rabbits. Every fox caught 59 rabbits, every wolf 41 rabbits, and every bear 40 rabbits. How many foxes, wolves, and bears were there in the hunting company?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 5,
"tir": 31
} | 36 | 0 | 0 |
omr-ws-e8b4d89fe135bdc8214f4ea1893983827306e73e06ac9becc1c15117861c80e2 | There were five (not necessarily distinct) real numbers written on a blackboard. For every pair of these numbers, Wendy calculated their sum and wrote the ten results: \(1, 2, 3, 5, 5, 6, 7, 8, 9, 10\). Determine all possible values of the product of the erased numbers. | [
"-144"
] | [
"There were five (not necessarily distinct) real numbers written on a blackboard. For every pair of these numbers, Wendy calculated their sum and wrote the ten results: \\(1, 2, 3, 5, 5, 6, 7, 8, 9, 10\\). Determine all possible values of the product of the erased numbers."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-a08198e412dd735b3311ad885a0751a1fe1a23539e786c2269a1bdd775d9741f | There were five friends one of whom had a monkey. Once they bought a bag of nuts and decided to share the nuts among themselves the next morning. At night one of them woke up. He divided the nuts into five equal parts, found that one extra nut remained after the division, gave it to the monkey, ate his part of the nuts... | [
"15621"
] | [
"There were five friends one of whom had a monkey. Once they bought a bag of nuts and decided to share the nuts among themselves the next morning. At night one of them woke up. He divided the nuts into five equal parts, found that one extra nut remained after the division, gave it to the monkey, ate his part of the... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-75e31f748af3e585247a4ce57f82a79b9d2b60894b6863d6c5bfe22e1ce7219b | There were some number of workers, and each would get daily wages for their work. If there were 6 fewer workers, each would get 1₹ more, and if there were 8 fewer workers, each would get 1.5₹ more. Find the number of workers and the daily wages for each worker. | [
"\\( w = 24 \\), \\( p = 3 \\)"
] | [
"There were some number of workers, and each would get daily wages for their work. If there were 6 fewer workers, each would get 1₹ more, and if there were 8 fewer workers, each would get 1.5₹ more. Find the number of workers and the daily wages for each worker."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 29
} | 29 | 0 | 0 |
omr-ws-7317f5da328bc838474d510c89294abed3a38b91b04a646e47821baa08b4e6b0 | There were ten points \(X_1, \ldots, X_{10}\) on a line in this particular order. An isosceles triangle was constructed on each segment \(X_1X_2, X_2X_3, \ldots, X_9X_{10}\) with the angle \(\alpha\) at its apex. It was observed that all the apexes of these triangles lie on a common semicircle with diameter \(X_1X_{10}... | [
"\\(45^\\circ\\)"
] | [
"There were ten points \\(X_1, \\ldots, X_{10}\\) on a line in this particular order. An isosceles triangle was constructed on each segment \\(X_1X_2, X_2X_3, \\ldots, X_9X_{10}\\) with the angle \\(\\alpha\\) at its apex. It was observed that all the apexes of these triangles lie on a common semicircle with diamet... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-24229c54a979c36acf8995165b61b843520e06f28da8d1f04ae6e4d6544627ad | There were two different copper alloys, the first containing 40 per cent less copper than the second. When these were melted together, the resulting alloy contained 36 per cent of copper. Determine the percentage of copper in the first and second alloys, if it is known that there were 6 kg of copper in the first alloy ... | [
"20\\% and 60\\%"
] | [
"There were two different copper alloys, the first containing 40 per cent less copper than the second. When these were melted together, the resulting alloy contained 36 per cent of copper. Determine the percentage of copper in the first and second alloys, if it is known that there were 6 kg of copper in the first a... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-44fd5be80444613e45fe799053279400b9708692702842cd955b4c30b2d0d7c2 | Theseus starts at the origin and takes a walk on the lattice points in the plane, starting with a move south to $(0, -1)$, eventually coming back to the origin without visiting any lattice point (besides the origin) more than once, and only visiting the origin at the start and end. Let $X$ be the number of times that T... | [
"1"
] | [
"Theseus starts at the origin and takes a walk on the lattice points in the plane, starting with a move south to $(0, -1)$, eventually coming back to the origin without visiting any lattice point (besides the origin) more than once, and only visiting the origin at the start and end. Let $X$ be the number of times t... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-09084ca2a0ed3bb614a8986d1ab522e93fdb2064fdf90eb0ecbde849d5f00a48 | Theseus starts at the point $(0, 0)$ in the plane. He can move one unit in any of the four cardinal directions. After a sequence of more than two moves, starting with a step one unit to the south (to point $(0, -1)$), Theseus returns to the point $(0, 0)$. He never visits any point other than $(0, 0)$ more than once, a... | [
"1"
] | [
"Theseus starts at the point $(0, 0)$ in the plane. He can move one unit in any of the four cardinal directions. After a sequence of more than two moves, starting with a step one unit to the south (to point $(0, -1)$), Theseus returns to the point $(0, 0)$. He never visits any point other than $(0, 0)$ more than on... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a619c8d94574ffa7da008fecd61c0172ca567a7d867a4962b125f0c4bb5405d7 | Thetis is currently at the point $(1, -2)$ and needs to take Achilles to the river Styx, represented by the line $y = x + 1$, and then to the point $(2, 1)$. Find the shortest distance that Thetis can travel. | [
"\\(\\sqrt{17}\\)"
] | [
"Thetis is currently at the point $(1, -2)$ and needs to take Achilles to the river Styx, represented by the line $y = x + 1$, and then to the point $(2, 1)$. Find the shortest distance that Thetis can travel."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-118d4ae33a9ff751efab59ea455e7ebd7072fc0dd9ce7c369f15f850f04cb000 | Thirteen boys and $d$ girls participate in a math contest. The total number of points they earn is $d^2 + 10d + 17$. All students get the same integer number of points. Find the largest possible number of participants in this contest. | [
"56"
] | [
"Thirteen boys and $d$ girls participate in a math contest. The total number of points they earn is $d^2 + 10d + 17$. All students get the same integer number of points. Find the largest possible number of participants in this contest."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-121c6a9f2fec3cdad992b0e76f00261d05036e01195c1e03fecdba698524d934 | Thirteen boys and girls wait to take their seats in a row of 13 seats in a movie theater. The first person can choose any of the 13 seats. Each subsequent person must sit next to someone already seated. How many different ways can this be accomplished? | [
"4096"
] | [
"Thirteen boys and girls wait to take their seats in a row of 13 seats in a movie theater. The first person can choose any of the 13 seats. Each subsequent person must sit next to someone already seated. How many different ways can this be accomplished?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 32,
"genselect": 2
} | 34 | 0 | 0 |
omr-ws-ec77a640b5ecf39f29ab3544d9fb3a206aa170a3b5116006ef2627dc7a468094 | Thirteen hungry zombies are sitting at a circular table at a restaurant. They have five identical plates of zombie food. Each plate is either in front of a zombie or between two zombies. If a plate is in front of a zombie, that zombie and both of its neighbors can reach the plate. If a plate is between two zombies, onl... | [
"26"
] | [
"Thirteen hungry zombies are sitting at a circular table at a restaurant. They have five identical plates of zombie food. Each plate is either in front of a zombie or between two zombies. If a plate is in front of a zombie, that zombie and both of its neighbors can reach the plate. If a plate is between two zombies... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 5
} | 7 | 0 | 0 |
omr-ws-a62365529f4701d24c9595706404615645a253e838b11d64746e2f48fddf6316 | Thirteen hungry zombies are sitting at a circular table. There are 5 identical plates of zombie food. Each plate is either in front of a zombie or between two zombies. If a plate is in front of a zombie, that zombie and both of its neighbors can reach the plate. If a plate is between two zombies, only those two zombies... | [
"26"
] | [
"Thirteen hungry zombies are sitting at a circular table. There are 5 identical plates of zombie food. Each plate is either in front of a zombie or between two zombies. If a plate is in front of a zombie, that zombie and both of its neighbors can reach the plate. If a plate is between two zombies, only those two zo... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 5,
"tir": 4
} | 12 | 0 | 0 |
omr-ws-11b4f6d79f92a033f3e52e995fcb546ec0f12ecf64d59c15c725ff98b01a00b2 | Thirteen students, four of whom are named Bob, are going out to a math conference. They have three distinct cars, two of which will hold $4$ students and the other will hold $5$. If the students randomly scramble into the cars, what is the probability that every car has at least one Bob in it? | [
"\\(\\frac{30}{61}\\)"
] | [
"Thirteen students, four of whom are named Bob, are going out to a math conference. They have three distinct cars, two of which will hold $4$ students and the other will hold $5$. If the students randomly scramble into the cars, what is the probability that every car has at least one Bob in it?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8e4deafaaac467691109b351eadf70d6c02376e5eb4853ba0b20bfa66917864a | Thirty children were polled about their likes or dislikes of "Friends" or "Will and Grace". The number of children who like "Friends" is twice the number who like "Will and Grace". The number who dislike "Friends" equals the number who like "Will and Grace". There are four children who like neither. Find the number of ... | [
"16"
] | [
"Thirty children were polled about their likes or dislikes of \"Friends\" or \"Will and Grace\". The number of children who like \"Friends\" is twice the number who like \"Will and Grace\". The number who dislike \"Friends\" equals the number who like \"Will and Grace\". There are four children who like neither. Fi... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-0ca54cc97d6f2dafdc58c6f2770d2b31270559bf4a3b4d1ec321fe87dce6e7f2 | Thirty numbers are arranged on a circle such that each number equals the absolute difference of its two neighbors. Given that the sum of the numbers is 2000, determine the numbers. | [
"\\(100, 0, 100\\) repeated 10 times."
] | [
"Thirty numbers are arranged on a circle such that each number equals the absolute difference of its two neighbors. Given that the sum of the numbers is 2000, determine the numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 5,
"tir": 30
} | 43 | 0 | 0 |
omr-ws-a4992673d1c29be432c3fbbc49e414b3bb38370273bab248bdf18f52d7d45584 | Thirty people sit at a round table. Each of them is either smart or dumb. Each of them is asked: "Is your neighbor to the right smart or dumb?" A smart person always answers correctly, while a dumb person can answer both correctly and incorrectly. It is known that the number of dumb people does not exceed $F$. What is ... | [
"14"
] | [
"Thirty people sit at a round table. Each of them is either smart or dumb. Each of them is asked: \"Is your neighbor to the right smart or dumb?\" A smart person always answers correctly, while a dumb person can answer both correctly and incorrectly. It is known that the number of dumb people does not exceed $F$. W... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-431391e18d20dbcff247cd21745bddaca3655e2ba1425fa72a978eb9b3e496d5 | Thirty people sit at a round table. Each person is either smart or dumb. A smart person always answers correctly when asked if their neighbor to the right is smart or dumb, while a dumb person can answer either correctly or incorrectly. Given that the number of dumb people does not exceed $F$, what is the largest possi... | [
"8"
] | [
"Thirty people sit at a round table. Each person is either smart or dumb. A smart person always answers correctly when asked if their neighbor to the right is smart or dumb, while a dumb person can answer either correctly or incorrectly. Given that the number of dumb people does not exceed $F$, what is the largest ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d936cd075b35793c299f0844cb2884396434e240582d1c50bf445f6f12ce0bbf | Thirty people stand on a circle with a radius of 1 unit. The distance between any two people must be more than 1 unit to avoid a fine. The fine for each person is equal to the number of people within 1 unit of them. Calculate the minimum total fine (in dollars) that has to be paid by all 30 people. | [
"60"
] | [
"Thirty people stand on a circle with a radius of 1 unit. The distance between any two people must be more than 1 unit to avoid a fine. The fine for each person is equal to the number of people within 1 unit of them. Calculate the minimum total fine (in dollars) that has to be paid by all 30 people."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-e0b74a7bdb4525930a986692bb89d1d88728348040196d006ff2ce188356ec13 | Thirty people stand on a circle with a radius of 1 unit. The distance between any two people must be more than 1 unit to avoid a fine. The fine for each person is equal to the number of people within 1 unit of them. Calculate the minimum total fine (in dollars) that has to be paid by all 30 people. (Assume the people a... | [
"30"
] | [
"Thirty people stand on a circle with a radius of 1 unit. The distance between any two people must be more than 1 unit to avoid a fine. The fine for each person is equal to the number of people within 1 unit of them. Calculate the minimum total fine (in dollars) that has to be paid by all 30 people. (Assume the peo... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-17e29d0d37c3cc5adbb96472b426dfdb936aa987f51fba1e1faa3e761781126c | Thirty rays with a common origin are constructed on a plane. Consider all angles between any two of these rays. Let $N$ be the number of acute angles among these angles. Find the smallest possible value of $N$. | [
"135"
] | [
"Thirty rays with a common origin are constructed on a plane. Consider all angles between any two of these rays. Let $N$ be the number of acute angles among these angles. Find the smallest possible value of $N$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 8
} | 12 | 0 | 0 |
omr-ws-6fb1a9c5953e6693e287b522f7ba04f82cc0e7842a0397ecbec39eb11651aedd | Thirty rays with a common origin are constructed on a plane. Consider all angles between any two of these rays. Let $N$ be the number of acute angles among these angles. Find the smallest value of $N$. | [
"210"
] | [
"Thirty rays with a common origin are constructed on a plane. Consider all angles between any two of these rays. Let $N$ be the number of acute angles among these angles. Find the smallest value of $N$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-cd00ad4c170f0a2758732f092a98279c4a1aec8cf4f8cc7f941d239bd392cd79 | Thirty students are seated in a classroom. Each student randomly chooses a different seat twice. Find the expected value of the number of pairs of students who traded seats, meaning each student in the pair returned to their original seat after the two seat changes. | [
"\\(\\frac{435}{841}\\)"
] | [
"Thirty students are seated in a classroom. Each student randomly chooses a different seat twice. Find the expected value of the number of pairs of students who traded seats, meaning each student in the pair returned to their original seat after the two seat changes."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-da58e05d7485978369f83d15cd1630c4cbc83ac65f036d35dd2ceb46839100bb | Thirty students with distinct heights stand in a line. The expected number of ordered triples of three students $(A, B, C)$ such that $A$ is directly in front of $B$, $B$ is directly in front of $C$, and the three students' heights are strictly increasing from front to back can be written as $\frac{a}{b}$ where $a$ and... | [
"017"
] | [
"Thirty students with distinct heights stand in a line. The expected number of ordered triples of three students $(A, B, C)$ such that $A$ is directly in front of $B$, $B$ is directly in front of $C$, and the three students' heights are strictly increasing from front to back can be written as $\\frac{a}{b}$ where $... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d4320e9f189e614c224273e573670bcc0470393cfea9deb6d4b48510df9f081c | Thirty-four countries participated in a jury session of the IMO, each represented by a leader and a deputy leader. Before the meeting, some participants exchanged handshakes, but no team leader shook hands with their deputy. After the meeting, the leader of the Illyrian team asked every other participant the number of ... | [
"33"
] | [
"Thirty-four countries participated in a jury session of the IMO, each represented by a leader and a deputy leader. Before the meeting, some participants exchanged handshakes, but no team leader shook hands with their deputy. After the meeting, the leader of the Illyrian team asked every other participant the numbe... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-dc870109aa0335444dc9b4e979bbdf38bdd37b2e1faca9e2707fa85ede1645bf | This sequence lists the perfect squares in increasing order: $$0, 1, 4, 9, 16, \ldots, a, 10^8, b, \ldots$$ Determine the value of $b - a$. | [
"5544"
] | [
"This sequence lists the perfect squares in increasing order: $$0, 1, 4, 9, 16, \\ldots, a, 10^8, b, \\ldots$$ Determine the value of $b - a$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-93cca4ebc1da570a8fb999584b076d05d2a235162788c5dccb3828515e332083 | This year, February $29$ fell on a Saturday. What is the next year in which February $29$ will be a Saturday? | [
"2048"
] | [
"This year, February $29$ fell on a Saturday. What is the next year in which February $29$ will be a Saturday?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 4,
"tir": 22
} | 30 | 0 | 0 |
omr-ws-f05c363a26157401cf94cb9b211fdb0e35624525fee136110015b4607def7105 | Thomas and Nils are playing a game with 2018 cards numbered from 1 to 2018. They take alternate turns, and on each turn, a player chooses a card and either keeps it or gives it to the other player. The game ends when all cards are gone, and the player with the sum of card numbers that results in a difference divisible ... | [
"1018081"
] | [
"Thomas and Nils are playing a game with 2018 cards numbered from 1 to 2018. They take alternate turns, and on each turn, a player chooses a card and either keeps it or gives it to the other player. The game ends when all cards are gone, and the player with the sum of card numbers that results in a difference divis... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-bc809055c4d600b15d1dd0e9716d2540d5525e1c4d1cfc8fefef4fdbd3ddabbe | Thomas and Nils are playing a game with 2020 cards numbered from 1 to 2020. They take alternate turns, and on each turn, a player chooses a card and either keeps it or gives it to the other player. The game ends when all cards are gone, and the player with the sum of card numbers that results in a difference divisible ... | [
"Nils"
] | [
"Thomas and Nils are playing a game with 2020 cards numbered from 1 to 2020. They take alternate turns, and on each turn, a player chooses a card and either keeps it or gives it to the other player. The game ends when all cards are gone, and the player with the sum of card numbers that results in a difference divis... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 4,
"tir": 6
} | 23 | 0 | 0 |
omr-ws-ec56b988ddbb57cd09077e2a37c5c34aa353fb7c4596f85f29476bc3cdcbb497 | Thomas is flipping coins in a game where he can win $1776n$ points if he flips $n$ consecutive heads before flipping a tail. The game ends when he flips a tail after flipping $n$ consecutive heads. If $e$ is the expected value of points that he can earn in this game, find $\frac{e}{2}$. | [
"888"
] | [
"Thomas is flipping coins in a game where he can win $1776n$ points if he flips $n$ consecutive heads before flipping a tail. The game ends when he flips a tail after flipping $n$ consecutive heads. If $e$ is the expected value of points that he can earn in this game, find $\\frac{e}{2}$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-a24c556fd33d05da0856e218c4a54e46823c9e50a552529e02cda7276fb2d768 | Thomas is inviting 7 guests to dine with him at a circular table. He hopes to sit next to Dolly but does not want Bill to sit next to Dolly. If everyone sits down at random, what is the probability that Thomas gets his wishes? | [
"\\(\\frac{4}{15}\\)"
] | [
"Thomas is inviting 7 guests to dine with him at a circular table. He hopes to sit next to Dolly but does not want Bill to sit next to Dolly. If everyone sits down at random, what is the probability that Thomas gets his wishes?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8be49eb78295b468111cf1bef9f8b82cda571143b21f111599933d1e3563fc28 | Thomas is receiving a scholarship amount that is an eight-digit number using the digits 0 through 7 without repetition and follows these criteria: - The number is divisible by 5. - The last four digits are prime numbers. - A three-digit number comprised of the last three digits in order is divisible by 11. - A four-dig... | [
"10463275"
] | [
"Thomas is receiving a scholarship amount that is an eight-digit number using the digits 0 through 7 without repetition and follows these criteria:\n- The number is divisible by 5.\n- The last four digits are prime numbers.\n- A three-digit number comprised of the last three digits in order is divisible by 11.\n- A... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-19baca9c4aada870d360e68d20ccad5a40cd1a391c6eafe21238f2e47ae02ae2 | Thomas is walking on a coordinate plane starting at the origin $(0,0)$. He walks to $(6,0)$, then turns left by $\alpha$ degrees and walks a fraction $x$ of the distance he just walked to get to $(9,4)$. He continues this process of turning left by $\alpha$ degrees and walking a fraction $x$ of the previous distance. D... | [
"\\(\\frac{96+144i}{25}\\)"
] | [
"Thomas is walking on a coordinate plane starting at the origin $(0,0)$. He walks to $(6,0)$, then turns left by $\\alpha$ degrees and walks a fraction $x$ of the distance he just walked to get to $(9,4)$. He continues this process of turning left by $\\alpha$ degrees and walking a fraction $x$ of the previous dist... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9f4c4482b81d03eb67636c49b35ef7b11c84aa4158cce883671383a8ccf5f960 | Thor and Loki play a game where Thor chooses an integer \( n_1 \ge 1 \), Loki chooses \( n_2 > n_1 \), Thor chooses \( n_3 > n_2 \), and so on. Let \( X \) be the union of the sets of integers in the intervals \([n_{2j-1}, n_{2j})\) for \( j \in \mathbb{N}^* \). Define \( s \) as the sum of \( \frac{1}{n^2} \) for all ... | [
"Loki has the winning strategy."
] | [
"Thor and Loki play a game where Thor chooses an integer \\( n_1 \\ge 1 \\), Loki chooses \\( n_2 > n_1 \\), Thor chooses \\( n_3 > n_2 \\), and so on. Let \\( X \\) be the union of the sets of integers in the intervals \\([n_{2j-1}, n_{2j})\\) for \\( j \\in \\mathbb{N}^* \\). Define \\( s \\) as the sum of \\( \\... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 6,
"tir": 13
} | 27 | 0 | 0 |
omr-ws-f01babb9f3305e108d5dcb575d6c2d282165b7b9142a75061b7c4ecffab54fcc | Three $A$s, three $B$s, and three $C$s are placed randomly in a $3 \times 3$ matrix. What is the probability that no row or column contains two identical letters? | [
"\\(\\frac{1}{140}\\)"
] | [
"Three $A$s, three $B$s, and three $C$s are placed randomly in a $3 \\times 3$ matrix. What is the probability that no row or column contains two identical letters?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-531c008831e19996e72a1b8bf298223d4fe23dc0c9b989a57340c05cf90d0675 | Three 'a's, three 'b's, and three 'c's are placed randomly in a $3 \times 3$ matrix. What is the probability that no row or column contains two of the same letter? Express this probability as $\frac{p}{q}$ where $p$ and $q$ are coprime, and find $p+q$. | [
"141"
] | [
"Three 'a's, three 'b's, and three 'c's are placed randomly in a $3 \\times 3$ matrix. What is the probability that no row or column contains two of the same letter? Express this probability as $\\frac{p}{q}$ where $p$ and $q$ are coprime, and find $p+q$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-116f954f23376debaf824cd0afae1d73f52028120b6ddec3feb2a8892e6e869d | Three (distinct) vertices of a regular 12-gon are chosen at random. What is the probability that no two of these vertices are adjacent to each other? Express your answer as a common fraction. | [
"\\dfrac{28}{55}"
] | [
"Three (distinct) vertices of a regular 12-gon are chosen at random. What is the probability that no two of these vertices are adjacent to each other? Express your answer as a common fraction."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b1fec9a86e369435e5ef5ed6107ccc429a88835c3034479b1837233b7aa7a38b | Three 12 cm $\times$ 12 cm squares are each cut into two pieces $A$ and $B$ by joining the midpoints of two adjacent sides. These six pieces are then attached to a regular hexagon to form a polyhedron. What is the volume (in $\text{cm}^3$) of this polyhedron? | [
"864"
] | [
"Three 12 cm $\\times$ 12 cm squares are each cut into two pieces $A$ and $B$ by joining the midpoints of two adjacent sides. These six pieces are then attached to a regular hexagon to form a polyhedron. What is the volume (in $\\text{cm}^3$) of this polyhedron?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 2,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-dc5c06cdf8526744b4084b50bcd4e7ec8fb09f0c29692d1196bce10c45bade69 | Three 12 cm $\times$ 12 cm squares are each cut into two pieces by joining the midpoints of two adjacent sides. These six pieces are then attached to a regular hexagon to form a polyhedron. What is the volume (in $\mathrm{cm}^3$) of this polyhedron? | [
"\\( 144\\sqrt{3} \\)"
] | [
"Three 12 cm $\\times$ 12 cm squares are each cut into two pieces by joining the midpoints of two adjacent sides. These six pieces are then attached to a regular hexagon to form a polyhedron. What is the volume (in $\\mathrm{cm}^3$) of this polyhedron?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b6c98ff2f410f01f0ec71f7e25df41f96f481f3aa386b4b8b3cd8db5991ad4cc | Three 3-digit numbers are made using the digits 1 to 9 exactly once. Suppose these 3 numbers are in the ratio 1:2:3. Find the minimum possible sum of these three numbers. | [
"192"
] | [
"Three 3-digit numbers are made using the digits 1 to 9 exactly once. Suppose these 3 numbers are in the ratio 1:2:3. Find the minimum possible sum of these three numbers."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8032370706100a2ff0e234da60d307b42e232df9fa866360e1a7030763c1b7ec | Three Jeeps, Jeep A, B, and C, are at base camp with full tanks of gas. They are to help Jeep A drive to an oasis deep into the desert with Jeeps B and C returning safely back to base camp and Jeep A remaining at the oasis. Jeep A and B travel together for a spell, then B refills A's gas tank. A then continues to the o... | [
"\\(\\frac{2600}{3}\\) miles"
] | [
"Three Jeeps, Jeep A, B, and C, are at base camp with full tanks of gas. They are to help Jeep A drive to an oasis deep into the desert with Jeeps B and C returning safely back to base camp and Jeep A remaining at the oasis. Jeep A and B travel together for a spell, then B refills A's gas tank. A then continues to ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 7,
"tir": 8
} | 23 | 0 | 0 |
omr-ws-028e7a052bb47ed3937e996ab1247c46fa1da45fb3bd37061dbcd42908962c75 | Three PEA administrators each have a 15-minute project to complete using a fileserver between 8 PM and 11 PM. Each administrator will call the fileserver only once and not after 10:45 PM. The fileserver can handle only one call at a time. The probability that all three administrators complete their tasks is a/b where a... | [
"2060"
] | [
"Three PEA administrators each have a 15-minute project to complete using a fileserver between 8 PM and 11 PM. Each administrator will call the fileserver only once and not after 10:45 PM. The fileserver can handle only one call at a time. The probability that all three administrators complete their tasks is a/b wh... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 5
} | 18 | 0 | 0 |
omr-ws-be05a2f760dd06f2f5361e8221e7978299e3590bd611f5e015a01e41d5141520 | Three Small Children would like to split up $9$ different flavored Sweet Candies evenly, so that each one of the Small Children gets $3$ Sweet Candies. However, three blind mice steal one of the Sweet Candies, so one of the Small Children can only get two pieces. How many fewer ways are there to split up the candies no... | [
"0"
] | [
"Three Small Children would like to split up $9$ different flavored Sweet Candies evenly, so that each one of the Small Children gets $3$ Sweet Candies. However, three blind mice steal one of the Sweet Candies, so one of the Small Children can only get two pieces. How many fewer ways are there to split up the candi... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-d0fdff9f7c29c9adff9f4e9fdf081a4f16509e5eea47be4346a94a49883bb93d | Three WOOTers have $a$, $b$, and $c$ coins, respectively. Every second, exactly one person gives one coin to exactly one other person. Find the expected value of the number of seconds before one of the WOOTers becomes bankrupt. | [
"\\dfrac{3abc}{a + b + c}"
] | [
"Three WOOTers have $a$, $b$, and $c$ coins, respectively. Every second, exactly one person gives one coin to exactly one other person. Find the expected value of the number of seconds before one of the WOOTers becomes bankrupt."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 7
} | 10 | 0 | 0 |
omr-ws-4fcc5850008ee4b1131b6013985900d8856b78db703c7d5a522aa85185b460ad | Three acute triangles are inscribed in the same circle with their vertices being nine distinct points. Let $A$, $B$, and $C$ be the chosen vertices from each of the three triangles, respectively, such that $\triangle ABC$ has each of its angles at most $90^\circ$. If the circumradius of the circle is $R$, find the maxi... | [
"9R^2"
] | [
"Three acute triangles are inscribed in the same circle with their vertices being nine distinct points. Let $A$, $B$, and $C$ be the chosen vertices from each of the three triangles, respectively, such that $\\triangle ABC$ has each of its angles at most $90^\\circ$. If the circumradius of the circle is $R$, find t... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4108ee2ddf3b3c9d2613840c287d319b378e6f310aa24d4a0b85e4fae96ce6ba | Three acute triangles are inscribed in the same circle with their vertices being nine distinct points. Let $A$, $B$, and $C$ be the chosen vertices from each triangle such that the triangle $\triangle ABC$ formed has each of its angles at most $90^\circ$. Calculate the maximum possible value of the sum of the angles of... | [
"180"
] | [
"Three acute triangles are inscribed in the same circle with their vertices being nine distinct points. Let $A$, $B$, and $C$ be the chosen vertices from each triangle such that the triangle $\\triangle ABC$ formed has each of its angles at most $90^\\circ$. Calculate the maximum possible value of the sum of the an... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-509caed7c7333d2332fe0e6945ece95e78b15af9c3afc6cc2f67f968a870aae3 | Three aliens launch a 3-day attack on 4 galaxies. Each day, each alien chooses a galaxy uniformly at random and destroys it. If the probability that every galaxy is destroyed by the end of the attack can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m,n$, what is $m+n$? | [
"1151"
] | [
"Three aliens launch a 3-day attack on 4 galaxies. Each day, each alien chooses a galaxy uniformly at random and destroys it. If the probability that every galaxy is destroyed by the end of the attack can be expressed as $\\frac{m}{n}$ for relatively prime positive integers $m,n$, what is $m+n$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e64652de2c15b423c2cd57d95cfd0237eed7490210522cc6eb97f8416c02d07b | Three alloys contain respectively 45\% tin and 55\% lead; 10\% bismuth, 40\% tin, and 50\% lead; and 30\% bismuth and 70\% lead. The alloys are to be combined to obtain a new alloy containing 15\% bismuth. What are the maximal and minimal percentages of lead that the new alloy might have? | [
"62.5\\% and 55\\%"
] | [
"Three alloys contain respectively 45\\% tin and 55\\% lead; 10\\% bismuth, 40\\% tin, and 50\\% lead; and 30\\% bismuth and 70\\% lead. The alloys are to be combined to obtain a new alloy containing 15\\% bismuth. What are the maximal and minimal percentages of lead that the new alloy might have? "
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-07255451b3ca8837a7768d108fe7688e582fbdf49228416bd790908fe686baee | Three aluminum cans can be recycled to make a new can. How many new cans can eventually be made from 243 aluminum cans? (Remember that the first new cans that are made can then be recycled into even newer cans!) Do not include the original 243 cans in your count. | [
"121"
] | [
"Three aluminum cans can be recycled to make a new can. How many new cans can eventually be made from 243 aluminum cans? (Remember that the first new cans that are made can then be recycled into even newer cans!) Do not include the original 243 cans in your count."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1a03b4e6b13764e87aa749c9dc5d445cf6477201757ecad73b9248ddb4115c07 | Three angle bisectors were drawn in a triangle, and the angles between them are $50^\circ$, $60^\circ$, and $70^\circ$. Find the angles of the original triangle. | [
"\\(40^\\circ, 60^\\circ, 80^\\circ\\)"
] | [
"Three angle bisectors were drawn in a triangle, and the angles between them are $50^\\circ$, $60^\\circ$, and $70^\\circ$. Find the angles of the original triangle."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 11
} | 11 | 0 | 0 |
omr-ws-b8f9fd38df7571a03f44dc5fc938f2f66dd5bceb36fa006cc7453b3cff2c8f10 | Three angles are in geometric progression with a common ratio of 2. They are all less than 90 degrees. The cosine of one angle is equal to the sum of the cosines of the other two angles. What is the product of the cosines of these three angles? | [
"\\(\\frac{1}{8}\\)"
] | [
"Three angles are in geometric progression with a common ratio of 2. They are all less than 90 degrees. The cosine of one angle is equal to the sum of the cosines of the other two angles. What is the product of the cosines of these three angles?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-6c6f610aa968e8f0b2c1ba990a1d30c50b0f1702aad43c2ce26d046f92acc8fc | Three ants are initially positioned at three vertices of a rectangle with side lengths $a$ and $b$. In each turn, one ant moves along a straight line parallel to the line determined by the other two stationary ants. Determine the coordinates of the final positions of the ants after a series of turns if they end up at t... | [
"\\left( 0, \\frac{b}{2} \\right)"
] | [
"Three ants are initially positioned at three vertices of a rectangle with side lengths $a$ and $b$. In each turn, one ant moves along a straight line parallel to the line determined by the other two stationary ants. Determine the coordinates of the final positions of the ants after a series of turns if they end up... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
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