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-d226e9c5ef08bf09a181a1ccbceca0b346cc905473ca6bf8e5569e03324ccc1f | What is the length of the shortest curve that divides the area of an equilateral triangle with side length $1$ unit into $2$ nonoverlapping sections of equal area? | [
"\\dfrac{\\sqrt{2}}{2}"
] | [
"What is the length of the shortest curve that divides the area of an equilateral triangle with side length $1$ unit into $2$ nonoverlapping sections of equal area?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
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"cot": 15
} | 15 | 0 | 0 |
omr-ws-88bad7db79183724ccb6b5f8e9b5124a760ea85746d9e8dc418a7661ca0b2681 | What is the length of the shortest ladder that can be placed on the ground, pass over a 2 m high fence, and rest against the wall of a building, given that the distance between the fence and the building is 3 m? | [
"\\(\\left(2^{\\frac{2}{3}} + 3^{\\frac{2}{3}}\\right)^{\\frac{3}{2}}\\)"
] | [
"What is the length of the shortest ladder that can be placed on the ground, pass over a 2 m high fence, and rest against the wall of a building, given that the distance between the fence and the building is 3 m?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
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"cot": 9,
"genselect": 6,
"tir": 10
} | 25 | 0 | 0 |
omr-ws-b4b3f2614be53ac911c879f5ea38532497b5bb381d9e8f033746f1f5da779d17 | What is the length of the shortest ladder that will reach from the ground over an 8 ft tall fence to the wall of a building, given that the fence is 4 ft away from the building? | [
"4\\left(1 + \\sqrt[3]{4}\\right)^{3/2}"
] | [
"What is the length of the shortest ladder that will reach from the ground over an 8 ft tall fence to the wall of a building, given that the fence is 4 ft away from the building?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 3
} | 18 | 0 | 0 |
omr-ws-b6eb35980fd87eda3d8d8eab966cbf0cf48e9c8061ab089d36ca71e620493b01 | What is the length of the shortest path from vertex A to vertex B along the surface of a regular icosahedron with edge length 1, where A and B are opposite vertices? | [
"\\( \\frac{3\\sqrt{3}}{2} \\)"
] | [
"What is the length of the shortest path from vertex A to vertex B along the surface of a regular icosahedron with edge length 1, where A and B are opposite vertices?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a86811c7ba2fb06360684b4824b94dc998bddbae983591f9d1fe31ab2c7c5391 | What is the length of the shortest path that begins at the point $(-1,1)$, touches the $x$-axis, and then ends at a point on the parabola $(x-y)^2=2(x+y-4)$? | [
"\\(3\\sqrt{2}\\)"
] | [
"What is the length of the shortest path that begins at the point $(-1,1)$, touches the $x$-axis, and then ends at a point on the parabola $(x-y)^2=2(x+y-4)$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 17,
"tir": 24
} | 41 | 0 | 0 |
omr-ws-ae34de17410316614c4a1539353e7d4df9cc460a19f455b36b18f6aaf94028eb | What is the length of the shortest path that begins at the point $(-3, 7)$, touches the $x$-axis, and then ends at a point on the circle $(x - 5)^2 + (y - 8)^2 = 25$? | [
"12"
] | [
"What is the length of the shortest path that begins at the point $(-3, 7)$, touches the $x$-axis, and then ends at a point on the circle $(x - 5)^2 + (y - 8)^2 = 25$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-2e18930032f409f9d5af356e046f05deb0f7ff06b0093422f05fc098e9ee3868 | What is the length of the shortest path that begins at the point $(-3,7)$, touches the x-axis, and then ends at a point on the circle $(x-5)^2+(y-8)^2=25$? | [
"12"
] | [
"What is the length of the shortest path that begins at the point $(-3,7)$, touches the x-axis, and then ends at a point on the circle $(x-5)^2+(y-8)^2=25$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-bcced900a50095b8938ce30f28b02ae525e7ea5cdb97afccd5ecae77da0b522b | What is the length of the shortest road system that connects all four houses located at the corners of a square with side lengths of one mile? | [
"\\(1 + \\sqrt{3}\\)"
] | [
"What is the length of the shortest road system that connects all four houses located at the corners of a square with side lengths of one mile?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-f9b53d8f14b9bea52520595b2b57e7e94aff06628ce86e4f43186a6583eca8bb | What is the length of the shortest string of digits that contains every number from 1 to 999 as a substring? | [
"1000"
] | [
"What is the length of the shortest string of digits that contains every number from 1 to 999 as a substring?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5ab7eae851c91f8b14029984f183d38c8cd119e2fc0758024a9a126ef9247bbb | What is the length of the shortest string that contains all possible strings of size 9 with characters from the set {A, B, C, D, E, F, G, H, I} (repetitions are allowed)? | [
"387420497"
] | [
"What is the length of the shortest string that contains all possible strings of size 9 with characters from the set {A, B, C, D, E, F, G, H, I} (repetitions are allowed)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-78f1700fb2df5a7312d159c2ac46fec9adde90edf43811c63a62dbd91ca022ab | What is the length of the shortest thrilling sequence whose first term is $1$ and whose last term is $100$, where a thrilling sequence is defined as a finite sequence of integers such that the sum of any two consecutive terms is a power of $3$? | [
"9"
] | [
"What is the length of the shortest thrilling sequence whose first term is $1$ and whose last term is $100$, where a thrilling sequence is defined as a finite sequence of integers such that the sum of any two consecutive terms is a power of $3$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a2b54f2138fd12ec1513f78e4def96441a010e65e05e4d8b61477902dbfb2888 | What is the length of the side of the smallest square that will enclose three non-overlapping discs, each with a radius of 1 unit? | [
"\\( 2 + \\frac{\\sqrt{6} + \\sqrt{2}}{2} \\)"
] | [
"What is the length of the side of the smallest square that will enclose three non-overlapping discs, each with a radius of 1 unit?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 2,
"tir": 1
} | 6 | 0 | 0 |
omr-ws-351e6721b8edc6a63444cae5c8e1e0afae55cb574a24a76aee9f8a1252107c82 | What is the length of the side of the smallest square that will enclose three non-overlapping disks, each with a radius of 1? | [
"\\(\\frac{4+\\sqrt{2}+\\sqrt{6}}{2}\\)"
] | [
"What is the length of the side of the smallest square that will enclose three non-overlapping disks, each with a radius of 1?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 6,
"tir": 1
} | 8 | 0 | 0 |
omr-ws-41cf75dfbacd4ff8545f38c7a41eb17cdc681e3a4b1a2e016e2b9053753115c5 | What is the length of the tangent segment from the origin to the circle that passes through the points (3,4), (6,8), and (5,13)? | [
"\\( 5\\sqrt{2} \\)"
] | [
"What is the length of the tangent segment from the origin to the circle that passes through the points (3,4), (6,8), and (5,13)?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-111ebb7453aee3e8f77f18c48faa548a3272568a93d1d7923bc601cab0a1a061 | What is the length of the transverse axis for the hyperbola given by the equation \( XY = 6 \)? | [
"\\( 4\\sqrt{3} \\)"
] | [
"What is the length of the transverse axis for the hyperbola given by the equation \\( XY = 6 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-47e388df2abf5e9209ce8e08b9a3365ae6da96bc3e4af18e705ef2c6e8074d4d | What is the likelihood of getting 3 or more heads in a row when a coin is tossed ten times? | [
"520/1024"
] | [
"What is the likelihood of getting 3 or more heads in a row when a coin is tossed ten times?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-e1fef4e77f970471778d0245963c2c95a2ba793ddde8331ed6e76adbb6291622 | What is the likelihood ratio test statistic for the exponential distribution given $H_0: \lambda = \lambda_0$ and $H_1: \lambda = \lambda_1$? | [
"\\( \\left(\\frac{\\lambda_0}{\\lambda_1}\\right) \\exp\\left(-(\\lambda_0 - \\lambda_1) \\sum_{k=1}^n x_k\\right) \\)"
] | [
"What is the likelihood ratio test statistic for the exponential distribution given $H_0: \\lambda = \\lambda_0$ and $H_1: \\lambda = \\lambda_1$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 3,
"tir": 14
} | 26 | 0 | 0 |
omr-ws-d83967b7773d748eb01989dd10cbb0962e530ae043e594a8e91a056341406d53 | What is the limit as $n \to \infty$ of the first coordinate of the centroid of the first "octant" of the unit ball in $n$ dimensions? | [
"0"
] | [
"What is the limit as $n \\to \\infty$ of the first coordinate of the centroid of the first \"octant\" of the unit ball in $n$ dimensions?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-b0f0ada118d1e67fa9e4e76b36a011786bf9a46abc571ce37af732e6de4e6ad6 | What is the limit as \( n \) approaches infinity of the integral from \( 0 \) to \( \frac{\pi}{3} \) of \( \frac{\sin^n(x)}{\sin^n(x) + \cos^n(x)} \)? | [
"\\dfrac{\\pi}{12}"
] | [
"What is the limit as \\( n \\) approaches infinity of the integral from \\( 0 \\) to \\( \\frac{\\pi}{3} \\) of \\( \\frac{\\sin^n(x)}{\\sin^n(x) + \\cos^n(x)} \\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-cb839e4938ce784e75b3ad5957998f0a4284e212de56f01dae35d9b394fe0361 | What is the limit as \( n \) approaches infinity of the sum from \( i=1 \) to \( n \) of \( \left(1 + \frac{2i}{n}\right)^2 \cdot \frac{2}{n} \)? | [
"\\(\\frac{26}{3}\\)"
] | [
"What is the limit as \\( n \\) approaches infinity of the sum from \\( i=1 \\) to \\( n \\) of \\( \\left(1 + \\frac{2i}{n}\\right)^2 \\cdot \\frac{2}{n} \\)?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-faf322f40cfe6daae893539debb11efa4fd03e123f5ea0fcca557a0436bad3ae | What is the limit of $\int(2x-1)$ as $x$ approaches $7/2$ from the right side? Note: The original post seems to be asking about the meaning of "int" in the context of the limit problem. However, "int" is typically not used in this context; it might be a typo or confusion with "integral" (denoted by $\int$). The problem... | [
"6"
] | [
"What is the limit of $\\int(2x-1)$ as $x$ approaches $7/2$ from the right side? \n\nNote: The original post seems to be asking about the meaning of \"int\" in the context of the limit problem. However, \"int\" is typically not used in this context; it might be a typo or confusion with \"integral\" (denoted by $\\i... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-ba6a6851cb9de61032a20ea727e437646cd65cf27cf5c706e60480cbc52134a3 | What is the limit of $\lim _{x\to \infty }\left(\frac{1^9+2^9+3^9+\cdots+n^9}{n^{10}}\right)$? | [
"\\(\\frac{1}{10}\\)"
] | [
"What is the limit of $\\lim _{x\\to \\infty }\\left(\\frac{1^9+2^9+3^9+\\cdots+n^9}{n^{10}}\\right)$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-2dd9d1fa6b130afabc48988b061aecba67c3f8df6b2955466e3ffd96768e7b50 | What is the limit of $\sqrt{x-3}$ as $x$ approaches 3? | [
"0 (from the right side)"
] | [
"What is the limit of $\\sqrt{x-3}$ as $x$ approaches 3?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-a505e89121ccdbdd78b3b6df15f70f18f3e358cbb417e3cdeaf75a6285ee28a8 | What is the limit of \( \frac{A}{n} \) when \( n \) goes to infinity, where \( A = d(n!) \) is the number of digits of \( n! \)? | [
"\\infty"
] | [
"What is the limit of \\( \\frac{A}{n} \\) when \\( n \\) goes to infinity, where \\( A = d(n!) \\) is the number of digits of \\( n! \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e6b36954fa097ec29f117d1f1efa257eb2b69991a6d3ce656b6b3ff0d011508f | What is the limit of \( f(x) = 0^x \) as \( x \) approaches 0 (\( x \to 0 \))? | [
"undefined"
] | [
"What is the limit of \\( f(x) = 0^x \\) as \\( x \\) approaches 0 (\\( x \\to 0 \\))?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 3
} | 3 | 0 | 0 |
omr-ws-73413020f1a93e27a4f05022b48401deb8aded7e865d27b319a451b99106c7df | What is the limit of \( f(x) = 0^x \) as \( x \) approaches 0 from the left (\( x \to 0^- \))? | [
"undefined"
] | [
"What is the limit of \\( f(x) = 0^x \\) as \\( x \\) approaches 0 from the left (\\( x \\to 0^- \\))?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 8,
"tir": 30
} | 50 | 0 | 0 |
omr-ws-c9e509e42c755272107c4a051319fe2c577d3cc7c7408a041680854239d25c90 | What is the limit of \( f(x) = 2xe^{2x} \) as \( x \) approaches positive infinity? | [
"\\( +\\infty \\)"
] | [
"What is the limit of \\( f(x) = 2xe^{2x} \\) as \\( x \\) approaches positive infinity?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-e296fa68d01d26a5630a0f5d57b87d3d92144d8b1ea131adba3442c571a641b0 | What is the limit of \( f(x) \) as \( x \) approaches \( 0 \)? \[ f(x) = \frac{(2^x - 1)(7^x - 1)}{(3^x - 1)(5^x - 1)} \] | [
"\\(\\frac{\\ln 2 \\cdot \\ln 7}{\\ln 3 \\cdot \\ln 5}\\)"
] | [
"What is the limit of \\( f(x) \\) as \\( x \\) approaches \\( 0 \\)?\n\n\\[ f(x) = \\frac{(2^x - 1)(7^x - 1)}{(3^x - 1)(5^x - 1)} \\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-9180211633535715579e0c619a9864ba76da9e6cba4ff80a92c678a39857f8c0 | What is the limit of \( k = \frac{\text{number of trailing zeros of } n!}{n} \) as \( n \) approaches infinity? | [
"\\(\\frac{1}{4}\\)"
] | [
"What is the limit of \\( k = \\frac{\\text{number of trailing zeros of } n!}{n} \\) as \\( n \\) approaches infinity?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 32
} | 47 | 0 | 0 |
omr-ws-1547ca74e5bfb854557df0aabd4168b0423961bf52fff9d8509ace9961f843c3 | What is the limit of \( y \) as \( x \to 1 \) and \( n \to 0 \) for the function \( y = n \left( x + \frac{x^3}{3} + \frac{x^5}{5} + \cdots \right) \)? | [
"0"
] | [
"What is the limit of \\( y \\) as \\( x \\to 1 \\) and \\( n \\to 0 \\) for the function \\( y = n \\left( x + \\frac{x^3}{3} + \\frac{x^5}{5} + \\cdots \\right) \\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-7dac20316cd42d4caa8080a38ab7b9370dc392c062457d3bd40edbe62caea11a | What is the limit of \(\frac{1^n + 2^n + \cdots + n^n}{n^n}\) as \(n\) approaches infinity? | [
"\\( \\frac{1}{e-1} + 1 \\)"
] | [
"What is the limit of \\(\\frac{1^n + 2^n + \\cdots + n^n}{n^n}\\) as \\(n\\) approaches infinity?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-3a55d3f2c8a3bb821876725dd29d782454fb57a8eecde8a7d00e8ec09c1bee85 | What is the limit of \(\frac{e^{-1/(x^2+y^2)}}{x^4+y^4}\) as \((x, y)\) approaches \((0, 0)\)? | [
"0"
] | [
"What is the limit of \\(\\frac{e^{-1/(x^2+y^2)}}{x^4+y^4}\\) as \\((x, y)\\) approaches \\((0, 0)\\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-2b9504443f2c1208e72697f3caf25505339848efd07215a1e229893ba479a855 | What is the limit of \(\sqrt[n]{n!}\) as \(n\) tends to infinity? | [
"\\(\\infty\\)"
] | [
"What is the limit of \\(\\sqrt[n]{n!}\\) as \\(n\\) tends to infinity?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-cea2bf2208d5b3ce7b04e24951403e5ece994ccb0eb90d258cfa50331be43387 | What is the limit of the constant $c$ as $n$ approaches infinity, where the average number of steps the Euclidean Algorithm takes for a large $n$ and for each $m < n$ can be expressed in the form $c \cdot \ln n$? | [
"\\(\\frac{12}{\\pi^2} \\ln 2\\)"
] | [
"What is the limit of the constant $c$ as $n$ approaches infinity, where the average number of steps the Euclidean Algorithm takes for a large $n$ and for each $m < n$ can be expressed in the form $c \\cdot \\ln n$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-64886a34e71636ce8fcdb6227dee9ec6d52b68ba55c7b80bf567372705a25d9c | What is the limit of the density function \( \gamma_N(r) \) as \( N \to \infty \), denoted by \( \gamma_{\infty}(r) \)? | [
"\\( \\gamma_{\\infty}(r) = \\delta_1 \\) for \\( p = \\frac{1}{2} \\)."
] | [
"What is the limit of the density function \\( \\gamma_N(r) \\) as \\( N \\to \\infty \\), denoted by \\( \\gamma_{\\infty}(r) \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 1,
"tir": 1
} | 5 | 0 | 0 |
omr-ws-36923d90be45fc967259e272e1996890dad41fd2ebfe790075a0edf79e163c47 | What is the limit of the function $n\chi_{(0,\frac{1}{n})}(x)$ as $n \to \infty$? | [
"0"
] | [
"What is the limit of the function $n\\chi_{(0,\\frac{1}{n})}(x)$ as $n \\to \\infty$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-83e7b08f1463fbce20b4472c311993ab22b44b020ffcb411ddd2a92f3e3e283e | What is the limit of the sequence $\displaystyle\prod_{i = 1}^{n - 1} \left(\frac{k}{n^2} + 1\right)$ as $n$ approaches infinity? | [
"\\(\\sqrt{e}\\)"
] | [
"What is the limit of the sequence $\\displaystyle\\prod_{i = 1}^{n - 1} \\left(\\frac{k}{n^2} + 1\\right)$ as $n$ approaches infinity?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 5,
"tir": 9
} | 23 | 0 | 0 |
omr-ws-bc37b1860ae46bef1055d380ffe0bcc6a62f1020dd7e39a23f2881b4a62846bb | What is the limit of the sequence \( a(n) = \prod_{k=1}^{n} \left( \frac{2^k - 1}{2^k} \right) \)? | [
"\\prod_{k=1}^{\\infty} \\left(1 - \\frac{1}{2^k}\\right)"
] | [
"What is the limit of the sequence \\( a(n) = \\prod_{k=1}^{n} \\left( \\frac{2^k - 1}{2^k} \\right) \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-6ee7f618ada4116c705cd07f7f4904460b0cbb91002bd56de8afa3fa964177f2 | What is the limit of the sequence \( a_n = \left(1 - \frac{1}{2}\right)\left(1 - \frac{1}{2^2}\right)\left(1 - \frac{1}{2^3}\right) \cdots \left(1 - \frac{1}{2^n}\right) \)? Can we find a closed form for this limit? | [
"\\left( \\frac{1}{2}; \\frac{1}{2} \\right)_\\infty"
] | [
"What is the limit of the sequence \\( a_n = \\left(1 - \\frac{1}{2}\\right)\\left(1 - \\frac{1}{2^2}\\right)\\left(1 - \\frac{1}{2^3}\\right) \\cdots \\left(1 - \\frac{1}{2^n}\\right) \\)? Can we find a closed form for this limit?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-222a443b75b64d7d2872cceb19b4b80edabfa7e1c2a41f4f86d084a9d3436e76 | What is the limit of the sequence \( u_n = \ln(1 + \ln(2 + \ln(3 + \cdots + \ln(n-1 + \ln n) \cdots))) \)? | [
"0.82035986"
] | [
"What is the limit of the sequence \\( u_n = \\ln(1 + \\ln(2 + \\ln(3 + \\cdots + \\ln(n-1 + \\ln n) \\cdots))) \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 3,
"tir": 22
} | 27 | 0 | 0 |
omr-ws-21dd13095c0d062e267a6f6e8fca12eef3c6b48c14d226f5f1ec877bebc2462d | What is the limit of the sequence \( y_n = \int_{0}^{1} \int_{0}^{1} \cdots \int_{0}^{1} \frac{x_1^2 + \cdots + x_n^2}{x_1 + \cdots + x_n} \, dx_1 \cdots dx_n \) as \( n \) approaches infinity? | [
"\\(\\frac{2}{3}\\)"
] | [
"What is the limit of the sequence \\( y_n = \\int_{0}^{1} \\int_{0}^{1} \\cdots \\int_{0}^{1} \\frac{x_1^2 + \\cdots + x_n^2}{x_1 + \\cdots + x_n} \\, dx_1 \\cdots dx_n \\) as \\( n \\) approaches infinity?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-56f338d4cc25847e5a2e7f5b026c42f55c72d8ff6755b9056a74b67e741f2658 | What is the limit of the series $ \frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots+\frac{1}{p}+\cdots $? | [
"\\( +\\infty \\)"
] | [
"What is the limit of the series $ \\frac{1}{2}+\\frac{1}{3}+\\frac{1}{4}+\\cdots+\\frac{1}{p}+\\cdots $?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-ac20412abf6077dcbf0e70e2e17c8b2904479b760a38ca074d3fc0a93f99fe2c | What is the limit of the series \( u \) defined by \( u_n = 1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \dots + \frac{1}{\sqrt{n}} \) as \( n \) approaches infinity? | [
"\\( +\\infty \\)"
] | [
"What is the limit of the series \\( u \\) defined by \\( u_n = 1 + \\frac{1}{\\sqrt{2}} + \\frac{1}{\\sqrt{3}} + \\dots + \\frac{1}{\\sqrt{n}} \\) as \\( n \\) approaches infinity?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-f0fc44e9c50a823844ca5cf7d1623e6be7429470fd700c75490b6c2971026c6b | What is the limit, as \( x \) tends to \( 0^- \), of \( x^2 e^{1/\sin x} \)? Can this be accomplished by Taylor Series? The answer is \( \frac{1}{2} \). | [
"0"
] | [
"What is the limit, as \\( x \\) tends to \\( 0^- \\), of \\( x^2 e^{1/\\sin x} \\)? Can this be accomplished by Taylor Series? The answer is \\( \\frac{1}{2} \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-3ae0082972dc1d6f3cfceccbad4aaf4f024ea872cb621801e538f45c073c82ab | What is the limiting probability (for large \( n \)) that after a monkey randomly rearranges \( n \) distinct hats, there are two hats next to each other which were also next to each other in the original ordering? | [
"\\( 1 - \\frac{1}{e} \\)"
] | [
"What is the limiting probability (for large \\( n \\)) that after a monkey randomly rearranges \\( n \\) distinct hats, there are two hats next to each other which were also next to each other in the original ordering?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-b8f6fcd46901807b0bfaad5f39222ab7f8ae7cca33d676d24f4b2d9cbd9bd4a2 | What is the local linearization of \( e^{x^2} \) near \( x = 1 \)? | [
"e + 2e(x - 1)"
] | [
"What is the local linearization of \\( e^{x^2} \\) near \\( x = 1 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ea6a0c810ef3afb314f2a506721916949c914dc840a7f6c90934f58d7e175c11 | What is the locus of a point \( M \) outside a circle \( (O) \) such that the power of \( M \) with respect to \( (O) \) is constant? | [
"A circle centered at \\( O \\) with a constant radius \\( OM \\)."
] | [
"What is the locus of a point \\( M \\) outside a circle \\( (O) \\) such that the power of \\( M \\) with respect to \\( (O) \\) is constant?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-80a9137c56f1e6523ae3217f7b2c1611849407c03ce3178b137fa0b95e493842 | What is the locus of a point such that the sum of its distances from two given lines is constant? | [
"rhombus"
] | [
"What is the locus of a point such that the sum of its distances from two given lines is constant?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 28
} | 28 | 0 | 0 |
omr-ws-bfc1ef3e75c22b933dd2fffaeec83e7e7580d4ca3ed8a44e9ab1290e71278786 | What is the locus of all points $P$ such that if the vertices of the pedal triangle are labelled as $D, E, F$, then the lines $AD, BE, CF$ are concurrent? | [
"\\text{Darboux cubic}"
] | [
"What is the locus of all points $P$ such that if the vertices of the pedal triangle are labelled as $D, E, F$, then the lines $AD, BE, CF$ are concurrent?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2d1d554d37b8c00be6d442ee2c506b4e8a7b4b8c55021282e7d239b5cd069d19 | What is the locus of all points \( O \) such that the points of contact of the six tangents from \( O \) to three non-intersecting circles all lie on one circle? | [
"\\text{The radical center of the three circles}"
] | [
"What is the locus of all points \\( O \\) such that the points of contact of the six tangents from \\( O \\) to three non-intersecting circles all lie on one circle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-f2d7492aa4cd1fa52bb28d6015e045a23e1d6497e459404dace0f487a0c23184 | What is the locus of all points \( P \) inside a given triangle \( ABC \) such that there exists a triangle with side lengths \( PA \), \( PB \), and \( PC \)? | [
"\\text{The interior of triangle } ABC"
] | [
"What is the locus of all points \\( P \\) inside a given triangle \\( ABC \\) such that there exists a triangle with side lengths \\( PA \\), \\( PB \\), and \\( PC \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 2
} | 14 | 0 | 0 |
omr-ws-e70caacfb969077508ce5b626ba7cb6433e5f916cc053805855d92c729ddf263 | What is the locus of all points \( P \) with powers that are directly proportional with respect to two fixed circles \( \omega_1 \) and \( \omega_2 \)? | [
"\\text{A circle or a line}"
] | [
"What is the locus of all points \\( P \\) with powers that are directly proportional with respect to two fixed circles \\( \\omega_1 \\) and \\( \\omega_2 \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 6
} | 16 | 0 | 0 |
omr-ws-edcd1ae2fe0a3d8daf292244701aa1a7fdef8c7823e8f9b8221f346df97c42e9 | What is the locus of all points equidistant from two skew lines? | [
"hyperbolic paraboloid"
] | [
"What is the locus of all points equidistant from two skew lines?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-d5ab1b928683a601fe0f84f4fa1af59332c48c4848668fdcecf175608b59bfbe | What is the locus of point \( P \) such that its projections on the sides of quadrilateral \( ABCD \) are concyclic? | [
"\\text{a conic}"
] | [
"What is the locus of point \\( P \\) such that its projections on the sides of quadrilateral \\( ABCD \\) are concyclic?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 7
} | 18 | 0 | 0 |
omr-ws-fd4369f61e436b3702a072108b79ef5a7b6635dcd07b90918989fa2c181c6f8d | What is the locus of points A given a segment BC such that the directed angle formed by the internal angle bisector of $\angle BAC$ and the line BC is constant? In other words, find the locus of points A such that the internal angle bisectors of $\angle BAC$ are parallel to one another, where B and C are fixed. | [
"\\( x^{2} + \\left(t - \\frac{1}{t}\\right)xy - y^{2} - c^{2} = 0 \\) with \\( xy > 0 \\)"
] | [
"What is the locus of points A given a segment BC such that the directed angle formed by the internal angle bisector of $\\angle BAC$ and the line BC is constant? In other words, find the locus of points A such that the internal angle bisectors of $\\angle BAC$ are parallel to one another, where B and C are fixed."... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-556a06a795c5cad103982d0d23fe5aa339686437448adbe82cd8b07381d53811 | What is the locus of points equidistant from the sides of an angle? | [
"The angle bisector."
] | [
"What is the locus of points equidistant from the sides of an angle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-f73ab7b0be759c2c1e18062ae2000e7f30298c5ba392210392505c073a3c3123 | What is the locus of points equidistant from three non-collinear points? | [
"circumcenter of the triangle formed by the three points."
] | [
"What is the locus of points equidistant from three non-collinear points?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 1,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-c680b4d1eca079ced29706cfe652846d3407140eb20056b97d3e427847ed5b45 | What is the locus of points equidistant from two parallel lines? | [
"A third line, parallel to both and halfway between them."
] | [
"What is the locus of points equidistant from two parallel lines?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-c55114a634083026c24b38415669057a10f8e23e93f491b893ec6b3d0d3e9ef9 | What is the locus of points such that the sum of the distances from a fixed point and a fixed line is constant? | [
"Two parabolas given by the equations \\( q - (x - p)^2 - y^2 = \\pm x \\)",
"parabola"
] | [
"What is the locus of points such that the sum of the distances from a fixed point and a fixed line is constant?"
] | [
"aops_c6_high_school_olympiads",
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 11,
"tir": 3
} | 28 | 0 | 0 |
omr-ws-c865a4222efdf456f720dde55b6f3a5d934e42961946589d33e582f638a08c3b | What is the locus of the centroid of an equilateral triangle inscribed in an ellipse? | [
"(0, 0)"
] | [
"What is the locus of the centroid of an equilateral triangle inscribed in an ellipse?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-2692c13bc2383a298279e340ebf47db3007353319952ffb125f20e50fefd05a4 | What is the locus of the point $z$ such that $\frac{z-i}{z-1}$ is purely imaginary? | [
"\\left(x - \\frac{1}{2}\\right)^2 + \\left(y - \\frac{1}{2}\\right)^2 = \\frac{1}{2}"
] | [
"What is the locus of the point $z$ such that $\\frac{z-i}{z-1}$ is purely imaginary?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-00c701acc072c7ed956a8a5603cecdac272ebf378e3432d4b94c6d8ac3e129fa | What is the locus of the points inside a triangle $ABC$ such that $DB + DC = DA$? | [
"\\emptyset"
] | [
"What is the locus of the points inside a triangle $ABC$ such that $DB + DC = DA$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 7
} | 16 | 0 | 0 |
omr-ws-cb9de86ec794cd1350987689b39d4ad49c2a5a833274755fc6d6cac86c5a6863 | What is the longest arithmetic progression contained in the sequence $1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots$? | [
"4"
] | [
"What is the longest arithmetic progression contained in the sequence $1, \\frac{1}{2}, \\frac{1}{3}, \\frac{1}{4}, \\ldots$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-8ee90c020f4c4d89866b8be570cd03515b08cccb88e5e5b36ac71b369af8e496 | What is the longest arithmetic progression of positive integers (both starting term and common difference are positive) such that none of its terms include the digit $9$? | [
"45"
] | [
"What is the longest arithmetic progression of positive integers (both starting term and common difference are positive) such that none of its terms include the digit $9$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 1
} | 10 | 0 | 0 |
omr-ws-c584f0c35fec6889ea660020b8b8d530c3659f7075fea14fffb74825413a90ac | What is the longest boat of length $c$ and negligible width that can pass through a right-angle bend in a river, where the banks are parallel and $a$ units apart, and the bend forms a square with the river flowing in across one side and out across an adjacent side? | [
"\\( 2a\\sqrt{2} \\)"
] | [
"What is the longest boat of length $c$ and negligible width that can pass through a right-angle bend in a river, where the banks are parallel and $a$ units apart, and the bend forms a square with the river flowing in across one side and out across an adjacent side?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 38,
"genselect": 8,
"tir": 14
} | 60 | 0 | 0 |
omr-ws-5e1f9eba1e55a869d91de7ec5b1b7ba9d3650a12f4de5059117df036055a2992 | What is the longest diagonal of a regular nonagon with side length 1? Leave your answer in trigonometric form. | [
"\\(\\frac{\\sin 80^\\circ}{\\sin 20^\\circ}\\)"
] | [
"What is the longest diagonal of a regular nonagon with side length 1? Leave your answer in trigonometric form."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-a5de6ca5b066caf44339631e72d6dc18a0eb2b6132f814573e54f95f49420073 | What is the longest length of a closed self-non-intersecting polyline that can be drawn along the grid lines of an $8 \times 8$ square with the four corner cells removed? | [
"\\(59\\)"
] | [
"What is the longest length of a closed self-non-intersecting polyline that can be drawn along the grid lines of an $8 \\times 8$ square with the four corner cells removed?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0540ae3c4000be4c076e88ab6c48447d62c6315d9d94acad82d6589892a0d3cd | What is the longest line you can place in the region enclosed by the functions $f(x) = \sqrt{x^2+1}$ and $h(x) = x^2 \sqrt{x^2+1}$? | [
"1.88247641"
] | [
"What is the longest line you can place in the region enclosed by the functions $f(x) = \\sqrt{x^2+1}$ and $h(x) = x^2 \\sqrt{x^2+1}$?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-af632c32b06f22e83d6d61d13474f80c72068f6edb8188c2fab0a4e3be9db9af | What is the longest possible sequence of distinct positive integers with the first member $1$ and the last member $31^a5^b1990^c$ such that each member is divisible by its predecessor? | [
"\\( a + b + 3c \\)"
] | [
"What is the longest possible sequence of distinct positive integers with the first member $1$ and the last member $31^a5^b1990^c$ such that each member is divisible by its predecessor?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-379ef36acf9c13e3bee67a138b0d5bbb3d30b8cabf58069073037461469ba759 | What is the lower bound for \( \frac{\varphi(n)}{n} \) when \( n \geq 10 \)? | [
"0"
] | [
"What is the lower bound for \\( \\frac{\\varphi(n)}{n} \\) when \\( n \\geq 10 \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 47
} | 47 | 0 | 0 |
omr-ws-594ad764292df665edfee2fd5c3652a9d5b39266d3978da6fddccf960604da8e | What is the lowest natural number \( n \) such that \( n^k \) ends in 2012 digits all equal to 1? | [
"\\dfrac{10^{2012} - 1}{9}"
] | [
"What is the lowest natural number \\( n \\) such that \\( n^k \\) ends in 2012 digits all equal to 1?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 6
} | 16 | 0 | 0 |
omr-ws-fa3e9a73619d6631de3e326f23dc5b8791b8dfe968dfd5310725fcc3131574b0 | What is the lowest positive integer $ n $ such that the equation $ \left[\frac{10^n}{x}\right] = 1989 $ has at least one positive integer solution for $ x $? Here, $ [z] $ denotes the integer part of $ z $. | [
"7"
] | [
"What is the lowest positive integer $ n $ such that the equation $ \\left[\\frac{10^n}{x}\\right] = 1989 $ has at least one positive integer solution for $ x $? Here, $ [z] $ denotes the integer part of $ z $."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 13
} | 36 | 0 | 0 |
omr-ws-25edb26be465a635e34f6b965ec2bacd576cfbcaacf48512a2d2f97f16ed91b9 | What is the lowest possible perimeter of a quadrilateral with an area of 2019 square units? | [
"\\(\\sqrt{2019} \\sqrt{5}\\)"
] | [
"What is the lowest possible perimeter of a quadrilateral with an area of 2019 square units?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-843d6eba59b48c23b7a1b9daac85bc6d0d047bd82ec9b704553c7133df7276b4 | What is the lowest robotic number (a positive integer that starts and ends with a 1, and every number in between is either a 1 or a 0) whose prime factorization contains the digit 8? | [
"101011"
] | [
"What is the lowest robotic number (a positive integer that starts and ends with a 1, and every number in between is either a 1 or a 0) whose prime factorization contains the digit 8?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-5b1811bd04671cecaa6462355cacb57a86dafe440a8e9e6282b3c66910a427a9 | What is the lowest value that the sum of the digits of the number \(2771^n\) can take, where \(n\) is a natural non-zero number? | [
"17"
] | [
"What is the lowest value that the sum of the digits of the number \\(2771^n\\) can take, where \\(n\\) is a natural non-zero number?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 8
} | 55 | 0 | 0 |
omr-ws-508bd57736cad4b4d79d28fb35538c7702fe3dd12249cf5e48191990c2b59a7e | What is the map \( \tilde{H}_n(S^n) \longrightarrow H_n(S^n, S^n \setminus \{x_1, \ldots, x_m\}) \longleftarrow H_n(\sqcup_i U_i, \sqcup_i (U_i \setminus \{x_i\})) \)? | [
"\\mathbb{Z} \\xrightarrow{\\Delta} \\mathbb{Z}^m"
] | [
"What is the map \\( \\tilde{H}_n(S^n) \\longrightarrow H_n(S^n, S^n \\setminus \\{x_1, \\ldots, x_m\\}) \\longleftarrow H_n(\\sqcup_i U_i, \\sqcup_i (U_i \\setminus \\{x_i\\})) \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 1
} | 8 | 0 | 0 |
omr-ws-7428a8edc16514ffdb57448b72a097d2fb132e3a0c91545e78da0c0ff9dc9500 | What is the map \( \tilde{H}_n(S^n) \longrightarrow H_n(S^n, S^n \setminus \{x_1\}) \longleftarrow H_n(U_1, U_1 \setminus \{x_1\}) \)? | [
"\\mathbb{Z}"
] | [
"What is the map \\( \\tilde{H}_n(S^n) \\longrightarrow H_n(S^n, S^n \\setminus \\{x_1\\}) \\longleftarrow H_n(U_1, U_1 \\setminus \\{x_1\\}) \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 2
} | 16 | 0 | 0 |
omr-ws-641a735487da734c9d31f9a7ab5964b542aaca513df74318a98abd87d96876b2 | What is the matrix for the transformation that reflects points over the line \( x + y = 0 \)? | [
"\\(\\begin{bmatrix} 0 & -1 \\\\ -1 & 0 \\end{bmatrix}\\)"
] | [
"What is the matrix for the transformation that reflects points over the line \\( x + y = 0 \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-6d6752669e3dc153aed976112f5a9382df7b768c7f4f4c648dad7e010855d480 | What is the matrix representation for a reflection through the origin? | [
"\\( \\begin{pmatrix} \\cos(\\theta) & \\sin(\\theta) \\\\ \\sin(\\theta) & -\\cos(\\theta) \\end{pmatrix} \\)"
] | [
"What is the matrix representation for a reflection through the origin?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 4
} | 4 | 0 | 0 |
omr-ws-0284c37c09942a05fefcee02e4ea3f53ad8ee17ad249d8d08d8e116b6f290b0a | What is the maximal area bounded by a closed broken line constructed along the lines of a grid with a total length of $2014$ cell sides? | [
"253512"
] | [
"What is the maximal area bounded by a closed broken line constructed along the lines of a grid with a total length of $2014$ cell sides?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 25
} | 40 | 0 | 0 |
omr-ws-6eb98cb064429f6d008a3d0ca982a7bf2f4ff43d6d76f483577797f1448a856f | What is the maximal area of a triangle whose sides \(a\), \(b\), and \(c\) satisfy the inequality \(0 \leq a \leq 1 \leq b \leq 2 \leq c \leq 3\)? | [
"1"
] | [
"What is the maximal area of a triangle whose sides \\(a\\), \\(b\\), and \\(c\\) satisfy the inequality \\(0 \\leq a \\leq 1 \\leq b \\leq 2 \\leq c \\leq 3\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9dd393ab8d5381737fe01ddad89c17e204e7de20a44dc02513d95c589e388b52 | What is the maximal area that a convex polygon $P$ with exactly 2020 integral coordinates inside can attain? | [
"4042"
] | [
"What is the maximal area that a convex polygon $P$ with exactly 2020 integral coordinates inside can attain?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-152e2960ff69f5334106ebcec689b0abea0cd0030da5630997e5bc48a31cfb00 | What is the maximal cardinality of a set of disjoint subsets of \(\mathbb{R}^2\), each homeomorphic to the closure of the set \(\{(x, y) : 0 < x \leq 1, y = \sin(1/x)\}\)? | [
"\\aleph_0"
] | [
"What is the maximal cardinality of a set of disjoint subsets of \\(\\mathbb{R}^2\\), each homeomorphic to the closure of the set \\(\\{(x, y) : 0 < x \\leq 1, y = \\sin(1/x)\\}\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7df05b1e7f1c45020e7211e2f49547523591d6e86c15ef43c3771445b51e871e | What is the maximal dimension of a linear subspace \( V \) of the vector space of real \( n \times n \) matrices such that for all \( A, B \in V \), we have \( \text{trace}(AB) = 0 \)? | [
"\\(\\frac{n(n-1)}{2}\\)"
] | [
"What is the maximal dimension of a linear subspace \\( V \\) of the vector space of real \\( n \\times n \\) matrices such that for all \\( A, B \\in V \\), we have \\( \\text{trace}(AB) = 0 \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 4
} | 19 | 0 | 0 |
omr-ws-e0a20977caea48f2468ed129710a97e5efe1cefe755775049cacd8c98fb3c690 | What is the maximal dimension of a subalgebra of $M_n(K)$? | [
"n^2 - n + 1"
] | [
"What is the maximal dimension of a subalgebra of $M_n(K)$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 6
} | 17 | 0 | 0 |
omr-ws-2fd9703ee994f9166a6fefce88b65757fdfc69743533cceb2628b02c5200b664 | What is the maximal dimension of a subspace $F$ of $M_n(\mathbb{R})$ such that the only non-invertible matrix in $F$ is the null matrix $O$? | [
"n"
] | [
"What is the maximal dimension of a subspace $F$ of $M_n(\\mathbb{R})$ such that the only non-invertible matrix in $F$ is the null matrix $O$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 3
} | 5 | 0 | 0 |
omr-ws-486a26ece1f2bffcc25cc209995d1bca6e04d062fee944f827e9277d25efa65f | What is the maximal dimension of a subspace $F$ of $M_{2n+1}(\mathbb{R})$ in which the determinant of every matrix is nonnegative? | [
"2n(2n + 1)"
] | [
"What is the maximal dimension of a subspace $F$ of $M_{2n+1}(\\mathbb{R})$ in which the determinant of every matrix is nonnegative?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 6
} | 20 | 0 | 0 |
omr-ws-eb862716e1d6ec4b610426e2c27a734c3ff53e05d24ac19e872cf31832206205 | What is the maximal dimension of a subspace $F$ of $M_{2n}(\mathbb{R})$ in which the determinant of every matrix is nonnegative? | [
"\\( 2n^2 \\)"
] | [
"What is the maximal dimension of a subspace $F$ of $M_{2n}(\\mathbb{R})$ in which the determinant of every matrix is nonnegative?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 6,
"tir": 2
} | 19 | 0 | 0 |
omr-ws-2c2a5502e3f74826b068ec9cd9f0d578bd6c2d8e10f719b18db6bf716591b2f3 | What is the maximal dimension of a subspace of $M_n(K)$ such that any two matrices in that subspace commute? | [
"\\(\\left[ \\frac{n^2}{4} \\right] + 1\\)"
] | [
"What is the maximal dimension of a subspace of $M_n(K)$ such that any two matrices in that subspace commute?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f5373890ba00307ca5beeacaa5ca44aa0fbdf695ffc66e0754187d3729e47f10 | What is the maximal dimension of a subspace of diagonalizable matrices? | [
"\\(\\frac{n(n+1)}{2}\\)"
] | [
"What is the maximal dimension of a subspace of diagonalizable matrices?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 2,
"tir": 8
} | 21 | 0 | 0 |
omr-ws-218c42bdeb0499c0960b1655ebb70ae9ded6fccfc838b344e527b7c0afee6a39 | What is the maximal length of a word in an alphabet of $n$ letters, given that any two consecutive letters are different and the word cannot be reduced to a form $abab$ (with $a \neq b$) by removing letters? | [
"\\( 2n-1 \\)"
] | [
"What is the maximal length of a word in an alphabet of $n$ letters, given that any two consecutive letters are different and the word cannot be reduced to a form $abab$ (with $a \\neq b$) by removing letters?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-3a840bd713b7e6fb2a0566eaa70822cca285f6505bdf790b44876616218c47fe | What is the maximal length of a word in the set \( M \), where \( M \) is the set of words such that when two letters of the word are the same, all the letters between them are different? | [
"\\( 3n \\)"
] | [
"What is the maximal length of a word in the set \\( M \\), where \\( M \\) is the set of words such that when two letters of the word are the same, all the letters between them are different?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-79aa57fb930d063ef29d554444d29bdac181bc0501468c41f307644d3ec87a66 | What is the maximal number $n$ such that $n+1$ points in space in general position (i.e., no four points are coplanar) can always be separated by $n$ parallel equidistant planes? | [
"3"
] | [
"What is the maximal number $n$ such that $n+1$ points in space in general position (i.e., no four points are coplanar) can always be separated by $n$ parallel equidistant planes?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 6
} | 18 | 0 | 0 |
omr-ws-2136407dbcb8402b8b09ad67349187cce3c21132ab185728866fc6fad182e08b | What is the maximal number of $1 \times 1 \times 4$ bricks that can fit in a $6 \times 6 \times 6$ box without intersecting the bricks or leaving the boundaries of the box? | [
"54"
] | [
"What is the maximal number of $1 \\times 1 \\times 4$ bricks that can fit in a $6 \\times 6 \\times 6$ box without intersecting the bricks or leaving the boundaries of the box?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-e3606bfead520c2eff6a06ba8c159a98c1b8c51a250d821fcb1bfec3e6140063 | What is the maximal number of 1's in a nilpotent order $n$ matrix over the reals? | [
"\\( n^2 - n + 1 \\)"
] | [
"What is the maximal number of 1's in a nilpotent order $n$ matrix over the reals?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bd78eef9143d3721400b4db131b601153bf0024f8d353218a10082a22f87b48e | What is the maximal number of 1's that a binary matrix can have if its square is also a binary matrix? | [
"\\( \\frac{n^2}{4} + \\frac{n}{2} \\)"
] | [
"What is the maximal number of 1's that a binary matrix can have if its square is also a binary matrix?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-df27d214de3f9bf7b277e24491554b49cae5277987f0d0559936e8fbb7c34f19 | What is the maximal number of acute angles in a convex polygon? | [
"3"
] | [
"What is the maximal number of acute angles in a convex polygon?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d2dd2a7831bc320e43143f85c7966b53df51e7e835111715b691378d17bcac5b | What is the maximal number of castles that can be placed on a $7 \times 7$ board such that no two castles share a cell and all towers are on the diagonals of the board? A castle is defined as a $2 \times 2$ square with one of its cells occupied by a tower. | [
"6"
] | [
"What is the maximal number of castles that can be placed on a $7 \\times 7$ board such that no two castles share a cell and all towers are on the diagonals of the board? A castle is defined as a $2 \\times 2$ square with one of its cells occupied by a tower."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-aaa37caa52e41cd9608e1ab46d084f382eef9011e875364103d7d6d6197bc6c1 | What is the maximal number of circles of radius $1$ that can be fit in a square with area $n$? ($n \in \mathbb{R}^{+}_0$) | [
"\\left\\lfloor \\dfrac{\\sqrt{n}}{2} \\right\\rfloor^2"
] | [
"What is the maximal number of circles of radius $1$ that can be fit in a square with area $n$? ($n \\in \\mathbb{R}^{+}_0$)"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 5
} | 19 | 0 | 0 |
omr-ws-774e55a1095ffeedfdd63188cca2c859f61189432a49529c3c80c5fed640980d | What is the maximal number of colors that can be used to color the squares of a $2n \times 2n$ square board such that each row or column contains squares of at most $n$ colors? | [
"\\(n^2\\)"
] | [
"What is the maximal number of colors that can be used to color the squares of a $2n \\times 2n$ square board such that each row or column contains squares of at most $n$ colors?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 44,
"genselect": 6
} | 50 | 0 | 0 |
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