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-d581be763ca604fa9932ae8a29b4d32feb3d3b45756aabe79e952cd2cae2c379 | What is the maximum possible difference between two consecutive even heavenly numbers less than 10000? An even number is called heavenly if all its digits are even. | [
"1112"
] | [
"What is the maximum possible difference between two consecutive even heavenly numbers less than 10000? An even number is called heavenly if all its digits are even."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-a6c28630540331976ab6c10891f10cfb701cad19bac4e90192bc246df1c86c04 | What is the maximum possible integral value of a root for the polynomial equation \( x^4 - 2x^2 + ax + b = 0 \) given that it has four distinct real roots? | [
"1"
] | [
"What is the maximum possible integral value of a root for the polynomial equation \\( x^4 - 2x^2 + ax + b = 0 \\) given that it has four distinct real roots?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-3acdd7eb231454a36e3f5c396481e7a44666ac904e812baaf2ff0288d52f0d52 | What is the maximum possible length of a sequence of numbers such that the sum of any 19 consecutive numbers is negative, whereas the sum of any 22 consecutive numbers is positive? | [
"40"
] | [
"What is the maximum possible length of a sequence of numbers such that the sum of any 19 consecutive numbers is negative, whereas the sum of any 22 consecutive numbers is positive?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 1
} | 10 | 0 | 0 |
omr-ws-2a7d418e7379891fce0f16327797d2a9041ea404fc59176ee7291f630e34c133 | What is the maximum possible number of circles that can be drawn on a plane such that each pair of circles intersects at exactly one point, and no point is common to three or more circles? | [
"3"
] | [
"What is the maximum possible number of circles that can be drawn on a plane such that each pair of circles intersects at exactly one point, and no point is common to three or more circles?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6b084156a273e08f76e8c5df0ff12050c77396af8d343f842c26a0ca25b515de | What is the maximum possible number of edges in a graph with $n$ vertices that contains no cycle of even length? | [
"\\(\\lfloor 3(n-1)/2 \\rfloor\\)"
] | [
"What is the maximum possible number of edges in a graph with $n$ vertices that contains no cycle of even length?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 45,
"genselect": 6,
"tir": 11
} | 62 | 0 | 0 |
omr-ws-b08d47ed2b9909d59eb058c4fc1019478fe7ce0926b594b709be2d4724e52f4e | What is the maximum possible number of edges in a graph with $n$ vertices which contains exactly one triangle? | [
"\\left\\lfloor \\frac{(n - 3)^2}{4} \\right\\rfloor + n"
] | [
"What is the maximum possible number of edges in a graph with $n$ vertices which contains exactly one triangle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 6
} | 16 | 0 | 0 |
omr-ws-adcd9a7ef186c5edccbb0254f1dceea5efaf66cf2a136ad06a017b0d770a6b09 | What is the maximum possible number of elements in a subset of distinct integers from 1 to 3000, such that no integer in the subset is double another? | [
"2000"
] | [
"What is the maximum possible number of elements in a subset of distinct integers from 1 to 3000, such that no integer in the subset is double another?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-5b5124069e0068da12fbc6cda9ffaa4b4c8ad33b7287a25b41501f90da849a9e | What is the maximum possible number of internal right angles in a polygon having 2025 sides? (Note: A 270 degree internal angle does not count as an internal right angle.) | [
"1351"
] | [
"What is the maximum possible number of internal right angles in a polygon having 2025 sides? (Note: A 270 degree internal angle does not count as an internal right angle.)"
] | [
"aops_c6_high_school_olympiads"
] | [
"genselect"
] | {
"genselect": 8
} | 8 | 0 | 0 |
omr-ws-a43fdbb068e27529ff6f8e2555ee45cb161455eb3cd7ded6606d6f2a0f893fa6 | What is the maximum possible number of intersection points between two circles and two lines? | [
"11"
] | [
"What is the maximum possible number of intersection points between two circles and two lines?"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 34
} | 34 | 0 | 0 |
omr-ws-797a9554bc5e4b64c55e207e400fc2301919c7fc5127f798f7e225788b870ab6 | What is the maximum possible number of pieces into which a sphere can be divided by 10 planes? | [
"176"
] | [
"What is the maximum possible number of pieces into which a sphere can be divided by 10 planes?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 4
} | 27 | 0 | 0 |
omr-ws-d67c0dc6c36461fac1e880f19837d685c34774430eb2dc3b3fbd957b0985facf | What is the maximum possible number of points of intersection between an edge of a tetrahedron and a face of a cube, given that none of the edges of either solid is coplanar with a face of the other? | [
"24"
] | [
"What is the maximum possible number of points of intersection between an edge of a tetrahedron and a face of a cube, given that none of the edges of either solid is coplanar with a face of the other?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 1
} | 2 | 0 | 0 |
omr-ws-b03438622372a45649ab610f9dd45a4c83562da959e76d63a95162db99ad5f6f | What is the maximum possible number of regions, including the exterior region, that a sheet of paper can be divided into by the sides of three triangles? | [
"20"
] | [
"What is the maximum possible number of regions, including the exterior region, that a sheet of paper can be divided into by the sides of three triangles?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-fb382ba8453992242f6db5e5b5e8f6493c46a2ccece44cfb4cc33685e09bcf33 | What is the maximum possible number of roots a funnynomial of the form \[ F(x) = a_{2023}x|x|^{2022} + a_{2022}x|x|^{2021} + \cdots + a_1x|x|^0 + a_0x|x|^{-1} \] can have, where \(a_0, a_1, \ldots, a_{2023}\) are real numbers, not all zero? A root of a funnynomial \(F\) is a real number \(x \neq 0\) such that \(F(x) = ... | [
"4046"
] | [
"What is the maximum possible number of roots a funnynomial of the form \n\\[ F(x) = a_{2023}x|x|^{2022} + a_{2022}x|x|^{2021} + \\cdots + a_1x|x|^0 + a_0x|x|^{-1} \\]\ncan have, where \\(a_0, a_1, \\ldots, a_{2023}\\) are real numbers, not all zero? A root of a funnynomial \\(F\\) is a real number \\(x \\neq 0\\) ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4a518b5b96c9191755ee10c5e0eef1b6b446ca702632a749b64f4b9047e711af | What is the maximum possible number of sides of a resulting cross-section when a cube is cut into two pieces with a single straight cut? | [
"6"
] | [
"What is the maximum possible number of sides of a resulting cross-section when a cube is cut into two pieces with a single straight cut?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-4fbf85f857eab58fe1d04498428c38318b26424ce138ed119c477adf22f2c8e0 | What is the maximum possible number of symmetry axes that a set of $n \ge 2$ distinct points can have in the plane? | [
"\\( n \\)"
] | [
"What is the maximum possible number of symmetry axes that a set of $n \\ge 2$ distinct points can have in the plane?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9e0daff19e0db2b37b7df49df586d25cf5dcec32e9ba98d2c1c49cc859629481 | What is the maximum possible number of terms when 2010 is written as a sum of consecutive positive or even integers? | [
"2010"
] | [
"What is the maximum possible number of terms when 2010 is written as a sum of consecutive positive or even integers?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 6
} | 10 | 0 | 0 |
omr-ws-09f454a567f5e44d52bdefacfc245a0902c1e4b5782724d5661b10d80a0d0690 | What is the maximum possible number of tri-state areas in a country with fifty states, where states must be contiguous and states touching only at corners do not count as bordering? | [
"96"
] | [
"What is the maximum possible number of tri-state areas in a country with fifty states, where states must be contiguous and states touching only at corners do not count as bordering?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 1
} | 13 | 0 | 0 |
omr-ws-7e085aebfd94575d76d56eb4eb09ef1e4d2e496632b5db3e0a2f07277e25bd80 | What is the maximum possible number of vertices of a regular polygon that can lie on the surface of a cube, given that the plane of the polygon does not contain any of the cube’s faces? | [
"6"
] | [
"What is the maximum possible number of vertices of a regular polygon that can lie on the surface of a cube, given that the plane of the polygon does not contain any of the cube’s faces?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-873834cd9c7929f717da69cba4db9b700a1ef56f97c4d4be185af37dac140ca8 | What is the maximum possible perimeter of an isosceles triangle if all three sides are integers less than 10? | [
"26"
] | [
"What is the maximum possible perimeter of an isosceles triangle if all three sides are integers less than 10?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-53f39ab22c44881cb7fcbd40ab67ff8afe01dd0c1448ffc209443baf221ebe7f | What is the maximum possible product of $n$ positive integers whose sum is 19? | [
"252"
] | [
"What is the maximum possible product of $n$ positive integers whose sum is 19?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-adfe9b470cd2636782e6c17b9f274b6ee6396bf69864e072b4c1a91952f2f57c | What is the maximum possible score that cannot be achieved in an infinitely long game of football, given that the possible scores are 3, 6, 7, and 8? | [
"5"
] | [
"What is the maximum possible score that cannot be achieved in an infinitely long game of football, given that the possible scores are 3, 6, 7, and 8?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2e6a34d8cdf30c2fc73197974ffab58a1422ccd95244f9767b5c7e99ce756a2a | What is the maximum possible sum of four different positive integers whose product is 360? | [
"66"
] | [
"What is the maximum possible sum of four different positive integers whose product is 360?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-9ff8c94dfd831078c611126507d4a3e4f2d161a90d354aece4991508c8263338 | What is the maximum possible sum of the numbers in an $m \times n$ minesweeper grid? | [
"\\(\\max_{(p,q)\\in \\{ (m,n),(n,m)\\} } \\{ \\Big( m(n-1)+n(m-1)+(m-1)(n-1)\\Big) -\\Big( (p-1)(q-1)+(p-1)\\Big) \\}\\)"
] | [
"What is the maximum possible sum of the numbers in an $m \\times n$ minesweeper grid?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 2
} | 3 | 0 | 0 |
omr-ws-fd1f02c215fc2353828efe61c91b9233b8f377a2b4031205ecc0633ddcee9dca | What is the maximum possible sum of three different positive integers whose product is 2010? | [
"1008"
] | [
"What is the maximum possible sum of three different positive integers whose product is 2010?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-a07025e748e7b48000e38659a0aa5470775b7e9deb17e2d419a2889472e77864 | What is the maximum possible value of $k$ for which $2013$ can be written as a sum of $k$ consecutive positive integers? | [
"61"
] | [
"What is the maximum possible value of $k$ for which $2013$ can be written as a sum of $k$ consecutive positive integers?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-adfdeca88b29fc6889066d102874de783cdab6bd69accecbbeaf8f76da283c51 | What is the maximum possible value of $k$ for which $2013$ can be written as the sum of $k$ consecutive positive integers? | [
"61"
] | [
"What is the maximum possible value of $k$ for which $2013$ can be written as the sum of $k$ consecutive positive integers?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 4
} | 35 | 0 | 0 |
omr-ws-8d28b36832287e38761ee13a560cd7a0bcc70685a5764aeb01febd35f420dfdb | What is the maximum possible value of $n \in \mathbb{Z}^{+}$ such that for any choice of seven distinct elements from $\{1, 2, \cdots, n\}$, there will exist two numbers $x$ and $y$ satisfying $1 < \frac{x}{y} \le 2$? | [
"126"
] | [
"What is the maximum possible value of $n \\in \\mathbb{Z}^{+}$ such that for any choice of seven distinct elements from $\\{1, 2, \\cdots, n\\}$, there will exist two numbers $x$ and $y$ satisfying $1 < \\frac{x}{y} \\le 2$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 8
} | 18 | 0 | 0 |
omr-ws-5cf61c49704c9708006b00c6266f41ed1ca9a7ce5f879b32ce47334efac441b8 | What is the maximum possible value of $n$ such that the first $n$ members of an increasing, infinite arithmetic sequence of positive integers do not contain the digit $9$ when written in base $10$? | [
"45"
] | [
"What is the maximum possible value of $n$ such that the first $n$ members of an increasing, infinite arithmetic sequence of positive integers do not contain the digit $9$ when written in base $10$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-73951d731cdf1aa959a236fa2df76acc6215421518b6707d40b30d36577d30f1 | What is the maximum possible value of $x + y$ if $(x, y)$ are integers that satisfy $x^{4042} + 2y^2 = 4y + 7$? | [
"4"
] | [
"What is the maximum possible value of $x + y$ if $(x, y)$ are integers that satisfy $x^{4042} + 2y^2 = 4y + 7$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a0f7fe8d02eecfc89c54e89532273f951471877e3b7c7b69674e9133b574af6e | What is the maximum possible value of \( k \) for which 2013 can be written as a sum of \( k \) consecutive positive integers? | [
"61"
] | [
"What is the maximum possible value of \\( k \\) for which 2013 can be written as a sum of \\( k \\) consecutive positive integers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-6340ff26a8111b474434ee57814da0ea61756908b2ebc751404744c79a6e2968 | What is the maximum possible value of \( k \) for which 2021 can be written as the sum of \( k \) consecutive positive integers? | [
"43"
] | [
"What is the maximum possible value of \\( k \\) for which 2021 can be written as the sum of \\( k \\) consecutive positive integers?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-54857551085c5267d0d62ecadee2d41a9336a8fdfd265ba81c2e5bf7393807c7 | What is the maximum possible value of \( m \) such that there exist \( m \) integers \( a_1, a_2, \ldots, a_m \) where all the decimal representations of \( a_1!, a_2!, \ldots, a_m! \) end with the same amount of zeros? | [
"5"
] | [
"What is the maximum possible value of \\( m \\) such that there exist \\( m \\) integers \\( a_1, a_2, \\ldots, a_m \\) where all the decimal representations of \\( a_1!, a_2!, \\ldots, a_m! \\) end with the same amount of zeros?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-822a8146350b6ce7158ebfcbd4b90e7e35e65883c58c39ce4ce0f6942496f8ca | What is the maximum possible value of \( y \) when \( y^2 + xy + x^2 = 1 \)? Provide both a non-calculus and a calculus-based method. | [
"\\(\\frac{2}{\\sqrt{3}}\\)"
] | [
"What is the maximum possible value of \\( y \\) when \\( y^2 + xy + x^2 = 1 \\)? Provide both a non-calculus and a calculus-based method."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-c928fd6a7f7fa41032d57a803e9c76948e646d0bea6a5fa67e59248262af7de7 | What is the maximum possible value of \(\min\{P(X>Y), P(Y>Z), P(Z>X)\}\), where \(X\), \(Y\), and \(Z\) are independent random variables? | [
"\\(\\frac{\\sqrt{5}-1}{2}\\)"
] | [
"What is the maximum possible value of \\(\\min\\{P(X>Y), P(Y>Z), P(Z>X)\\}\\), where \\(X\\), \\(Y\\), and \\(Z\\) are independent random variables?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 6
} | 15 | 0 | 0 |
omr-ws-be86e18998ea6637c5de6d261cce30eba8339b15742a3bb88855c684ae7c8c59 | What is the maximum possible value of a positive integer $n$, such that for any choice of seven distinct elements from $\{1,2,\ldots,n\}$, there exist two numbers $x$ and $y$ satisfying $1 < \frac{x}{y} \le 2$? | [
"126"
] | [
"What is the maximum possible value of a positive integer $n$, such that for any choice of seven distinct elements from $\\{1,2,\\ldots,n\\}$, there exist two numbers $x$ and $y$ satisfying $1 < \\frac{x}{y} \\le 2$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 45,
"genselect": 8
} | 53 | 0 | 0 |
omr-ws-390267b6c033b99ca83e724fbd80c4d8b3d9c6d1838423c6b81646b07b343a86 | What is the maximum possible value of a positive integer \( n \) such that for any choice of 7 distinct elements from \(\{1, 2, \ldots, n\}\), there will exist two numbers \( x \) and \( y \) satisfying \( 1 < \frac{x}{y} \leq 2 \)? | [
"\\( 2^7 - 2 \\)"
] | [
"What is the maximum possible value of a positive integer \\( n \\) such that for any choice of 7 distinct elements from \\(\\{1, 2, \\ldots, n\\}\\), there will exist two numbers \\( x \\) and \\( y \\) satisfying \\( 1 < \\frac{x}{y} \\leq 2 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 1
} | 22 | 0 | 0 |
omr-ws-df697a6486edf9e1bb9c946fddba4f0ad5d5e29574f8e4a01043554f56c93e52 | What is the maximum possible value of a positive integer \( n \), such that for any choice of seven distinct elements from \(\{1, 2, 3, \ldots, n\}\), there will exist two numbers \( x \) and \( y \) satisfying \( 1 < \frac{x}{y} \leq 2 \)? | [
"126"
] | [
"What is the maximum possible value of a positive integer \\( n \\), such that for any choice of seven distinct elements from \\(\\{1, 2, 3, \\ldots, n\\}\\), there will exist two numbers \\( x \\) and \\( y \\) satisfying \\( 1 < \\frac{x}{y} \\leq 2 \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 45,
"genselect": 8
} | 53 | 0 | 0 |
omr-ws-310d88adb33d26fd88b985d00d232f51ab80f3e64d9ca31ca2edff5502c8a5ae | What is the maximum possible value of a positive integer \( n \), such that for any choice of seven distinct elements from \(\{1, 2, \ldots, n\}\), there exist two numbers \( x \) and \( y \) satisfying \( 1 < \frac{x}{y} \leq 2 \)? | [
"126"
] | [
"What is the maximum possible value of a positive integer \\( n \\), such that for any choice of seven distinct elements from \\(\\{1, 2, \\ldots, n\\}\\), there exist two numbers \\( x \\) and \\( y \\) satisfying \\( 1 < \\frac{x}{y} \\leq 2 \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 43,
"genselect": 6,
"tir": 1
} | 50 | 0 | 0 |
omr-ws-7fd164edb958c679548ed67ee718c8ce8941eddc72c15138bc1fe0c4c43ad1d4 | What is the maximum possible value of the area of a triangle with vertices \((a,0)\), \((b,0)\), and \((c,d)\), where \(1 + a - 2a^2 = 1 + b + 2b^2\) and \(1 + c - 2c^2 = d\) and \(d > 0\)? Express your answer as a common fraction. | [
"\\(\\frac{9}{16}\\)"
] | [
"What is the maximum possible value of the area of a triangle with vertices \\((a,0)\\), \\((b,0)\\), and \\((c,d)\\), where \\(1 + a - 2a^2 = 1 + b + 2b^2\\) and \\(1 + c - 2c^2 = d\\) and \\(d > 0\\)? Express your answer as a common fraction."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 22
} | 37 | 0 | 0 |
omr-ws-f4025e989d46f6571e8634a31ca851752de4c1ddc463a38d1cbb80ed1eefddc1 | What is the maximum possible value of the expression $\sqrt{a} + \sqrt{b} + \sqrt{c}$ for $a, b, c \in \mathbb{R}^+$ such that $a + 2b + 3c = 60$? | [
"$\\sqrt{110}$"
] | [
"What is the maximum possible value of the expression $\\sqrt{a} + \\sqrt{b} + \\sqrt{c}$ for $a, b, c \\in \\mathbb{R}^+$ such that $a + 2b + 3c = 60$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-eaa5b0f3be534f6025311be39696b8f24c5e040e61208e579969aff471f472ee | What is the maximum possible value of the greatest common divisor of two consecutive terms of the sequence $a_n = n! + n$, where $n \ge 0$? | [
"2"
] | [
"What is the maximum possible value of the greatest common divisor of two consecutive terms of the sequence $a_n = n! + n$, where $n \\ge 0$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8461a5d68c367a7c998de86385c4a96f760b76806a47c75354417381eaddc649 | What is the maximum possible value of the median number of cups of coffee bought per customer on a day when Sundollars Coffee Shop sells 477 cups of coffee to 190 customers, and every customer buys at least one cup of coffee? | [
"3.5"
] | [
"What is the maximum possible value of the median number of cups of coffee bought per customer on a day when Sundollars Coffee Shop sells 477 cups of coffee to 190 customers, and every customer buys at least one cup of coffee?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 13
} | 28 | 0 | 0 |
omr-ws-b9e3d89bb216f1257667ee50d3d0536899dfd5aa1439ff18cbf5876ae44c319b | What is the maximum possible value of the product of a list of positive integers, given that the sum of these numbers is 100? | [
"\\(4 \\cdot 3^{32}\\)"
] | [
"What is the maximum possible value of the product of a list of positive integers, given that the sum of these numbers is 100?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-0314023189da55dc4747cbd5422c45b5114fbabe4becc84a4c3b9f58a2570292 | What is the maximum possible volume of a ball inscribed in a cone, with the center of the ball on the axis of symmetry of the cone? | [
"\\dfrac{4}{3} \\pi \\left( \\dfrac{H R}{\\sqrt{H^2 + R^2} + R} \\right)^3"
] | [
"What is the maximum possible volume of a ball inscribed in a cone, with the center of the ball on the axis of symmetry of the cone?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 7
} | 17 | 0 | 0 |
omr-ws-36d7bd830f7833ee8b205fa3715518b630ff3fb13084cfd7a6b784439f749d0a | What is the maximum possible volume of a cylinder inscribed in a cone, with the cylinder having the same axis of symmetry as the cone? | [
"\\dfrac{4}{27} \\pi R^2 H"
] | [
"What is the maximum possible volume of a cylinder inscribed in a cone, with the cylinder having the same axis of symmetry as the cone?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-72fc17e41713128640c94b2d3cc2ea45ecff151d6a956182d7caa346b492e322 | What is the maximum possible volume of a set of points in space where the maximum distance between any two points is less than or equal to 1? | [
"\\(\\frac{\\pi}{6}\\)"
] | [
"What is the maximum possible volume of a set of points in space where the maximum distance between any two points is less than or equal to 1?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 18,
"tir": 29
} | 47 | 0 | 0 |
omr-ws-df3dac11608c2a128e5b425457e85f8437a20f4fbf8010e73f53ab5a620554c6 | What is the maximum power of 2 that divides $!2^5!$, where $!n! := 1!2! \cdots n!$? | [
"81"
] | [
"What is the maximum power of 2 that divides $!2^5!$, where $!n! := 1!2! \\cdots n!$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-401ea83c698cb4395bc087265d8aa736f82d58c981ea63950969f98bd66b03a3 | What is the maximum product of a collection of integers that sum to 2009? | [
"\\(3^{669} \\cdot 2\\)"
] | [
"What is the maximum product of a collection of integers that sum to 2009?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-c68dfdfe51e7c956c8d913e7ef7bed133636a986387dc3aab9a7b8a66fb3b4cd | What is the maximum product of a group of positive integers that add up to 40? | [
"\\(4 \\cdot 3^{12}\\)"
] | [
"What is the maximum product of a group of positive integers that add up to 40?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 23
} | 38 | 0 | 0 |
omr-ws-f6337642c0b55319efb1f21ddc180c277df56067b43bb7b1c1074d7bd45fafd6 | What is the maximum product of a set of positive integers that sum to 31? | [
"78732"
] | [
"What is the maximum product of a set of positive integers that sum to 31?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-d58d82ed5a86d3abe0692eb6407ba7591c099396570c090c73c171b9f40b156f | What is the maximum product of natural numbers whose sum is 22? | [
"2916"
] | [
"What is the maximum product of natural numbers whose sum is 22?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-b81957dab83a93ff4a749a066044abd174f3a67cd436842a46006804572e7e93 | What is the maximum product of some (not necessarily distinct) natural numbers whose sum is 100? | [
"7412080755407364"
] | [
"What is the maximum product of some (not necessarily distinct) natural numbers whose sum is 100?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 7,
"tir": 29
} | 36 | 0 | 0 |
omr-ws-b483f58016ef898603de92aa41d29e31e5fd0c6e78046f68c66ea19eeb1c0d18 | What is the maximum radius of 5 identical balls that can fit inside a cube with a side length of 50 cm? | [
"\\(\\frac{25\\sqrt{3}}{2}\\)"
] | [
"What is the maximum radius of 5 identical balls that can fit inside a cube with a side length of 50 cm?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4f4844ae58eaf4fbfe7eaa8c714786fe2b9214d3fc9e99c830d29617190de055 | What is the maximum radius of a circle that can be inscribed in the ellipse given by the equation $(x-1)^2 + 9y^2 = 1$? | [
"\\(\\frac{1}{3}\\)"
] | [
"What is the maximum radius of a circle that can be inscribed in the ellipse given by the equation $(x-1)^2 + 9y^2 = 1$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-3f1164442f774d3824315cdadb80e01f9e7096cffdd0d75f1e2f2a541a194d12 | What is the maximum radius of a circle that is tangent to the curves $y=e^{-x^2}$ and $y=-e^{-x^2}$ at two points each? | [
"\\( \\sqrt{\\frac{1-\\ln{\\frac{1}{2}}}{2}} \\)"
] | [
"What is the maximum radius of a circle that is tangent to the curves $y=e^{-x^2}$ and $y=-e^{-x^2}$ at two points each?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 5,
"tir": 25
} | 45 | 0 | 0 |
omr-ws-5630bb25beadf6e1628b06d10e728b686d0ba44c4cbd087212b6bd63246c981d | What is the maximum radius of a hemisphere inscribed in a cube with an edge length of 1? | [
"\\(3 - \\sqrt{6}\\)"
] | [
"What is the maximum radius of a hemisphere inscribed in a cube with an edge length of 1?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3f66eb8bd6fbf3026686f95b472dda8deec778a8b279d960a884173fc306e7aa | What is the maximum radius of four identical spheres that can be packed inside a regular tetrahedron of side length 1, given that the spheres may not be inside other spheres and need not be mutually tangent? | [
"\\(\\frac{-1 + 2\\sqrt{2}}{14}\\)"
] | [
"What is the maximum radius of four identical spheres that can be packed inside a regular tetrahedron of side length 1, given that the spheres may not be inside other spheres and need not be mutually tangent?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a115bd0f49d537630ba050f4788271cec579922da296a4cd4aa8f4fb4902f632 | What is the maximum radius of two equal solid spheres that can fit inside a regular tetrahedron with side length $l$, such that the spheres can switch their positions with rigid moves without overlapping? | [
"\\( \\frac{\\sqrt{6}}{18} \\cdot l \\)"
] | [
"What is the maximum radius of two equal solid spheres that can fit inside a regular tetrahedron with side length $l$, such that the spheres can switch their positions with rigid moves without overlapping?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-628cfdcead93dc893795ce794f788e0106ea2f24c3dd50219369ad79ecabaa8a | What is the maximum ratio of the area of the inner pentagon formed by the intersection of the diagonals of a convex pentagon ABCDE to the area of pentagon ABCDE? | [
"\\dfrac{7 - 3\\sqrt{5}}{2}"
] | [
"What is the maximum ratio of the area of the inner pentagon formed by the intersection of the diagonals of a convex pentagon ABCDE to the area of pentagon ABCDE?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 1
} | 10 | 0 | 0 |
omr-ws-70abd33b710038c109ea63fd9db8a432f5fb4e6760e16a3554c4112e505deba6 | What is the maximum size of a set such that the sum of any three elements of the set is a prime number? | [
"5"
] | [
"What is the maximum size of a set such that the sum of any three elements of the set is a prime number?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-61975c1d8ff5dba50d8391c4d86baff5e66dab456b2faadb20ef8987fbb5021a | What is the maximum size of a subset of vertices of a regular 60-gon such that no three vertices in the subset form an isosceles triangle? | [
"20"
] | [
"What is the maximum size of a subset of vertices of a regular 60-gon such that no three vertices in the subset form an isosceles triangle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 27
} | 27 | 0 | 0 |
omr-ws-335d12cac7fd3599275f6e71d9b6a976e85e1c2daa2a8f5bd3f2d010e082a64a | What is the maximum size of a sum free subset of the set $\{1, 2, \ldots, 2n-1\}$, where a set is called "sum free" if no two elements of the set add up to a third element of the set? | [
"\\(n\\)"
] | [
"What is the maximum size of a sum free subset of the set $\\{1, 2, \\ldots, 2n-1\\}$, where a set is called \"sum free\" if no two elements of the set add up to a third element of the set?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a789aff39702a53396dabb28a405a9ad24856433970aac4d16c807b14202ca46 | What is the maximum size of a sum-free subset of \( S = \{1, 2, \ldots, 2n+1\} \), where a sum-free set is defined as a set in which no two elements add up to another element in the set, and no element is double another element in the set? | [
"\\( n+1 \\)"
] | [
"What is the maximum size of a sum-free subset of \\( S = \\{1, 2, \\ldots, 2n+1\\} \\), where a sum-free set is defined as a set in which no two elements add up to another element in the set, and no element is double another element in the set?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5430289a9e88f9f94aa9e5b9f1e447aaf5ff81a921a5b322408956a23af64791 | What is the maximum sum of the dates of all Thursdays in November? The dates are summed without considering the month or year. For example, 11/6/1998 would just be 6. | [
"80"
] | [
"What is the maximum sum of the dates of all Thursdays in November? The dates are summed without considering the month or year. For example, 11/6/1998 would just be 6."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 5
} | 20 | 0 | 0 |
omr-ws-f4d52226f5366a0d2a822ca42f60a68fdf1f932b618ffd1dbf695ad8acedddea | What is the maximum total number of regions that three distinct lines can create within an equilateral heptagon by intersecting it, and possibly intersecting with each other? | [
"13"
] | [
"What is the maximum total number of regions that three distinct lines can create within an equilateral heptagon by intersecting it, and possibly intersecting with each other?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a4821dbc9a41e1811b1c0b560d2238fef94650dc17077e64f3dc45483be8c456 | What is the maximum transversable distance across an icosahedron from one vertex back to that vertex, given that each vertex is visited exactly once? | [
"12"
] | [
"What is the maximum transversable distance across an icosahedron from one vertex back to that vertex, given that each vertex is visited exactly once?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6d80b2d4886ba12e7be21d31f01cb56eb40d76fec993c29805a69e7ad26ade65 | What is the maximum two-digit prime number that is a divisor of $\frac{200!}{100! \cdot 100!}$? | [
"61"
] | [
"What is the maximum two-digit prime number that is a divisor of $\\frac{200!}{100! \\cdot 100!}$?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-01fdbb62f46eb9ae3157fe9a5f0e7160abcdc21065086e463d6caa9aa8c133cc | What is the maximum value among the sequence \(1, \sqrt{2}, \sqrt[3]{3}, \ldots\)? | [
"\\(3^{1/3}\\)"
] | [
"What is the maximum value among the sequence \\(1, \\sqrt{2}, \\sqrt[3]{3}, \\ldots\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-19bf0ce0c519bdd2c9e194aa7339fd0d5670ade38bb06eefd5c7f6b298d33d88 | What is the maximum value of $(11-x)\sqrt{22x-121}$ for $x \in \mathbb{R^+}$? | [
"\\(\\frac{121\\sqrt{3}}{9}\\)"
] | [
"What is the maximum value of $(11-x)\\sqrt{22x-121}$ for $x \\in \\mathbb{R^+}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-1503c4cc4045c0563dc11a315e9844b168b69c2bd2d48d9710b765fc6e1240f1 | What is the maximum value of $(a-c)^2+(b-d)^2$ where $a, b, c, d$ are real numbers which satisfy $a^2+b^2=c^2+d^2=25$? | [
"100"
] | [
"What is the maximum value of $(a-c)^2+(b-d)^2$ where $a, b, c, d$ are real numbers which satisfy $a^2+b^2=c^2+d^2=25$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fa079e37197d9c174e0c1980d14cc43aca9deb1226902f50accddac6e95d71b3 | What is the maximum value of $-4z^2+20z-6$? | [
"19"
] | [
"What is the maximum value of $-4z^2+20z-6$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-21a539869325cf71b4123a7fb694ede2549cdb5489820259664da6a321138947 | What is the maximum value of $N$ such that for any set of $m$ elements selected from the natural numbers $1$ to $N$, there are at least two subsets with equal sums? | [
"2^{m-1} - 1"
] | [
"What is the maximum value of $N$ such that for any set of $m$ elements selected from the natural numbers $1$ to $N$, there are at least two subsets with equal sums?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-837b3ba1f2b4098360c4e3f2b0278ec2d0f23763941d192ec60aa3a5fe2426e7 | What is the maximum value of $\cos \theta \sin \theta + \cos^2 \theta \tan \alpha$ in terms of $\alpha$? | [
"\\(\\frac{1 + \\sin \\alpha}{2 \\cos \\alpha}\\)"
] | [
"What is the maximum value of $\\cos \\theta \\sin \\theta + \\cos^2 \\theta \\tan \\alpha$ in terms of $\\alpha$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 6
} | 20 | 0 | 0 |
omr-ws-4558ce227da4d8f799b80d265e3d0d2d0837c8f054bb318546ec939338c73f24 | What is the maximum value of $\dfrac{x^2 + y^2 + z^2}{xy + yz + xz}$ for positive real numbers $x, y, z$? | [
"\\infty"
] | [
"What is the maximum value of $\\dfrac{x^2 + y^2 + z^2}{xy + yz + xz}$ for positive real numbers $x, y, z$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 7
} | 16 | 0 | 0 |
omr-ws-8989d5a0d2c3de8b90a19ea27971be7426cc1491f8187e1eef72c2795e9c2219 | What is the maximum value of $\frac{(2^t-3t)t}{4^t}$ for real values of $t?$ | [
"\\(\\frac{1}{12}\\)"
] | [
"What is the maximum value of $\\frac{(2^t-3t)t}{4^t}$ for real values of $t?$"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 26
} | 41 | 0 | 0 |
omr-ws-faef0affce032bcd884797d4ff985f081b28acb29ad15ea48400ec7735ddbd49 | What is the maximum value of $\frac{2}{a^2+1}-\frac{2}{b^2+1}+\frac{3}{c^2+1}$ given that $abc + a + c = b$? | [
"\\dfrac{10}{3}"
] | [
"What is the maximum value of $\\frac{2}{a^2+1}-\\frac{2}{b^2+1}+\\frac{3}{c^2+1}$ given that $abc + a + c = b$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-6ba6544a4d5840000bcf2dd5b9361384d589272cac82eb877e219665938dc317 | What is the maximum value of $\left(\arcsin(x)\right)^3 + \left(\arccos(x)\right)^3$? | [
"\\(\\frac{7\\pi^3}{8}\\)"
] | [
"What is the maximum value of $\\left(\\arcsin(x)\\right)^3 + \\left(\\arccos(x)\\right)^3$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-787488be4f2fc8033d40c776e76b5438eb85a2e1449a4573fab7ddd19725d7b6 | What is the maximum value of $\sin^2(A) + \sin^2(B) + \sin^2(C)$ for angles $A$, $B$, and $C$ of a triangle? | [
"\\dfrac{9}{4}"
] | [
"What is the maximum value of $\\sin^2(A) + \\sin^2(B) + \\sin^2(C)$ for angles $A$, $B$, and $C$ of a triangle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0b39aad13c18d3772831b38771f5142e80916fed1de1a32a4fb1f2d6ada1dc99 | What is the maximum value of $\sin^4\theta + \cos^6\theta$? | [
"1"
] | [
"What is the maximum value of $\\sin^4\\theta + \\cos^6\\theta$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0213a48ee9c366c4b16f0104ef906c898e9c72ef030c76ef900e20a0965670f2 | What is the maximum value of $ab - \frac{b^2\pi}{4}$? Express the solution in terms of $a$ and $b$ where $b = ar$. | [
"\\(\\frac{a^2}{\\pi}\\)"
] | [
"What is the maximum value of $ab - \\frac{b^2\\pi}{4}$? Express the solution in terms of $a$ and $b$ where $b = ar$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 2,
"tir": 1
} | 13 | 0 | 0 |
omr-ws-95b99ce9ade1a0fa012826bdeeeaad346a8aa09dd2e47f95d3f62337316c1e3e | What is the maximum value of $c$ such that for every convex polyhedron $P$ with $n$ faces, there are at least $cn$ faces with the same number of edges? | [
"\\dfrac{1}{3}"
] | [
"What is the maximum value of $c$ such that for every convex polyhedron $P$ with $n$ faces, there are at least $cn$ faces with the same number of edges?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 2
} | 9 | 0 | 0 |
omr-ws-fc8f0c88ea71432239318867c68bd9a00a50667483c27840c883266018cbad65 | What is the maximum value of $f(x) = \ln(x^2-4x-3)$ over $x \in [-1,5]$? | [
"NOTA\n\nExplanation:\nThe function \\( f(x) = \\ln(x^2 - 4x - 3) \\) is not defined over the entire interval \\([-1, 5]\\) because the argument of the logarithm, \\( x^2 - 4x - 3 \\), must be positive. The quadratic \\( x^2 - 4x - 3 \\) is negative between its roots, which are \\( 2 - \\sqrt{7} \\) and \\( 2 + \\s... | [
"What is the maximum value of $f(x) = \\ln(x^2-4x-3)$ over $x \\in [-1,5]$?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fc58206d00e9fc6463bd5e489e96e7c010daf24c3e792b4cdaad60d82d36571a | What is the maximum value of $k$ such that if any 7 real numbers are selected from the set $[1, k]$, there always exists at least 3 numbers among the chosen 7 numbers that can form the sides of a triangle? | [
"12"
] | [
"What is the maximum value of $k$ such that if any 7 real numbers are selected from the set $[1, k]$, there always exists at least 3 numbers among the chosen 7 numbers that can form the sides of a triangle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 24,
"genselect": 8
} | 32 | 0 | 0 |
omr-ws-ae1c9eb0c6f9ee74f90504d850807802c8106cc6b43b28332b0a1bffc291e37b | What is the maximum value of $n$ for which there exists a set of distinct positive integers $k_1, k_2, \ldots, k_n$ such that $k_1^2 + k_2^2 + \cdots + k_n^2 = 2002$? | [
"17"
] | [
"What is the maximum value of $n$ for which there exists a set of distinct positive integers $k_1, k_2, \\ldots, k_n$ such that $k_1^2 + k_2^2 + \\cdots + k_n^2 = 2002$?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 45
} | 77 | 0 | 0 |
omr-ws-d270aa769bd0b56eb13cb5642b55d2210c2953183b09311ace2cdd131e4a56e5 | What is the maximum value of $n$ such that $n!$ can be written as the product of $(n-3)$ consecutive numbers? | [
"7"
] | [
"What is the maximum value of $n$ such that $n!$ can be written as the product of $(n-3)$ consecutive numbers?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-a30ed13b9d72c12657a9fbb0ce9ed7db0b99cab57a5fe5e4efaa841d173bdff4 | What is the maximum value of $n$ such that any subset of $(1, 2, 3, \ldots, 2015)$ with $(2015 - n)$ elements will contain at least one pair of co-prime numbers? | [
"1007"
] | [
"What is the maximum value of $n$ such that any subset of $(1, 2, 3, \\ldots, 2015)$ with $(2015 - n)$ elements will contain at least one pair of co-prime numbers?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e2b605f5a9d6538217bf3b61781092282a71a132a3f2a64cfe39b59683d793c1 | What is the maximum value of $n$ such that any subset of $\{1, 2, 3, \ldots, 2015\}$ with $(2015 - n)$ elements contains at least one pair of co-prime numbers? | [
"1007"
] | [
"What is the maximum value of $n$ such that any subset of $\\{1, 2, 3, \\ldots, 2015\\}$ with $(2015 - n)$ elements contains at least one pair of co-prime numbers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 17
} | 17 | 0 | 0 |
omr-ws-b823b67ba64cf14c5afcaedef9d381b24ca5659b1f791d3a4631bfee2f487e0c | What is the maximum value of $n$ such that any subset of $\{1, 2, \ldots, 2015\}$ with $(2015 - n)$ elements contains at least one pair of co-prime numbers? | [
"1007",
"1008"
] | [
"What is the maximum value of $n$ such that any subset of $\\{1, 2, \\ldots, 2015\\}$ with $(2015 - n)$ elements contains at least one pair of co-prime numbers?"
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 16,
"genselect": 8,
"tir": 2
} | 26 | 0 | 0 |
omr-ws-42834a2f5ff933cc9d7fb67dd52bc60804bf384373588be7299f21a17a50d4b0 | What is the maximum value of $n$ such that for some positive integer $a$ and for all $k=1,2,\ldots,n$, the numbers $a+2^k$ can be expressed as $xy(x+y)$ for some values $x,y \in \mathbb{N}$? | [
"3"
] | [
"What is the maximum value of $n$ such that for some positive integer $a$ and for all $k=1,2,\\ldots,n$, the numbers $a+2^k$ can be expressed as $xy(x+y)$ for some values $x,y \\in \\mathbb{N}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 8,
"tir": 7
} | 17 | 0 | 0 |
omr-ws-851bb00195c47e7831b133f64dda980f2a6c3e6aca3ec303eedf48a67cff97f0 | What is the maximum value of $s(m^4)$ such that $s(m) = 4$, where $m$ is a natural number and $s(x)$ denotes the sum of the digits of $x$? | [
"130"
] | [
"What is the maximum value of $s(m^4)$ such that $s(m) = 4$, where $m$ is a natural number and $s(x)$ denotes the sum of the digits of $x$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7034d6572f37e4bbe6a7eff425804083fe8a45c8982e0ddf953c7bee1e1120a3 | What is the maximum value of $x^2y^3$ if $x$ and $y$ are non-negative integers satisfying $x + y \le 9$? | [
"1029"
] | [
"What is the maximum value of $x^2y^3$ if $x$ and $y$ are non-negative integers satisfying $x + y \\le 9$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f59005b080776043e4884ada6345041a5a4296a55740a7e8e62d4ba3bdf8c8d2 | What is the maximum value of $|\sin z|$ in the closed unit disc? | [
"\\(\\sinh(1)\\)"
] | [
"What is the maximum value of $|\\sin z|$ in the closed unit disc?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-df3805e735c44843e6085887df91ccf6968bc900c71e4bee1c42dec14a470968 | What is the maximum value of $|\sqrt{x^2+4x+5}-\sqrt{x^2+2x+5}|^4$? | [
"4"
] | [
"What is the maximum value of $|\\sqrt{x^2+4x+5}-\\sqrt{x^2+2x+5}|^4$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-6f854d45ad5b95de77a9148170f8939fd4b2c89607e144ccbef3a1bc293c43c9 | What is the maximum value of $|z^3 - z + 2|$, where $z$ is a complex number with $|z| = 1$? | [
"\\(\\sqrt{13}\\)"
] | [
"What is the maximum value of $|z^3 - z + 2|$, where $z$ is a complex number with $|z| = 1$?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-83e1d335ab1bc38a07718582ad44c1062fd37d6c58eb57b9a15bd5631a29b22c | What is the maximum value of \( \cos^2 \left(\frac{x}{2}\right) \sin^2 x \)? | [
"\\(\\frac{16}{27}\\)"
] | [
"What is the maximum value of \\( \\cos^2 \\left(\\frac{x}{2}\\right) \\sin^2 x \\)?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-0e3db9991e68aac34cf76dea84d84a18b3c981d936d957e404be22a8b2a42492 | What is the maximum value of \( \prod_{1\leq i<j\leq n} (x_i - x_j) \) where \( x_1, \ldots, x_n \) are positive numbers that add up to 1? | [
"0"
] | [
"What is the maximum value of \\( \\prod_{1\\leq i<j\\leq n} (x_i - x_j) \\) where \\( x_1, \\ldots, x_n \\) are positive numbers that add up to 1?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-8d053668afeeeb6512c0ea6bf5d352b714257efec8a4620d066f9bbfae614821 | What is the maximum value of \( \sin^{\alpha}(\theta)\cos^{\beta}(\theta) \) and where does equality occur? (\( \alpha \neq \beta \)) | [
"\\(\\sqrt{\\frac{\\alpha^{\\alpha} \\beta^{\\beta}}{(\\alpha+\\beta)^{\\alpha+\\beta}}}\\) and \\(\\theta = \\arcsin{\\sqrt{\\frac{\\alpha}{\\alpha+\\beta}}}\\)"
] | [
"What is the maximum value of \\( \\sin^{\\alpha}(\\theta)\\cos^{\\beta}(\\theta) \\) and where does equality occur? (\\( \\alpha \\neq \\beta \\))"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 3,
"tir": 6
} | 13 | 0 | 0 |
omr-ws-5198bab72a46be1b9c8c3be190b66ae96d74fd678c5874fdeaf05b4effd6d182 | What is the maximum value of \( \sin{A} \cos{\frac{A}{3}} \cos{\frac{2A}{3}} \) for \( m(\angle BAC) \in (0, 135^\circ) \)? | [
"\\dfrac{3\\sqrt{3}}{8}"
] | [
"What is the maximum value of \\( \\sin{A} \\cos{\\frac{A}{3}} \\cos{\\frac{2A}{3}} \\) for \\( m(\\angle BAC) \\in (0, 135^\\circ) \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-b5fed326ea968b0b3f560cd853f9664d0750a198bc3403f0d8ec6e54dc5b0745 | What is the maximum value of \( a^2 + b^2 \), given that the equation \( x^4 + ax^3 + bx^2 + ax + 1 = 0 \) has real roots? | [
"52"
] | [
"What is the maximum value of \\( a^2 + b^2 \\), given that the equation \\( x^4 + ax^3 + bx^2 + ax + 1 = 0 \\) has real roots?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 8
} | 19 | 0 | 0 |
omr-ws-5052ae4dbe6151bea0b4716d8a4d9cd0821eb6f09700bd0888f480c18badaa2a | What is the maximum value of \( f(\ldots f(f(f(x_1 - x_2) - x_3) - x_4) \ldots - x_{2012}) \) where \( x_1, x_2, \ldots, x_{2012} \) are distinct integers in the set \(\{1, 2, 3, \ldots, 2012\}\) and \( f \) is the absolute value function? | [
"2012"
] | [
"What is the maximum value of \\( f(\\ldots f(f(f(x_1 - x_2) - x_3) - x_4) \\ldots - x_{2012}) \\) where \\( x_1, x_2, \\ldots, x_{2012} \\) are distinct integers in the set \\(\\{1, 2, 3, \\ldots, 2012\\}\\) and \\( f \\) is the absolute value function?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 4
} | 13 | 0 | 0 |
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