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-50e7931d7889716e0a660914d973ca4bdefa519833ac8978b29b9a7b788cbab4 | With the restriction $0 < t < \pi/2$, find the largest $t$ such that $a(t) = 2$. | [
"\\arctan(2)"
] | [
"With the restriction $0 < t < \\pi/2$, find the largest $t$ such that $a(t) = 2$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 4
} | 13 | 0 | 0 |
omr-ws-babf28d60476baf1b72340105fabaf90cf917b5dbab3b8498b383c96236a6da0 | With the restriction $0 < t < \pi/2$, find the smallest $t$ such that $a(t) = 2$. | [
"\\dfrac{\\pi}{4}"
] | [
"With the restriction $0 < t < \\pi/2$, find the smallest $t$ such that $a(t) = 2$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-c7c1a8a5ed10fd87a13a92ace1642caba06e9f4006a9745490d0157ef4d667a5 | With the same settings as IMO 2022 Problem 1, we randomly generated an initial configuration of $n=2022$. Assume $k=2022$, and it takes $x$ operations to first reach the final configuration (the leftmost $n$ coins are all of the same type). Estimate $x^2$. | [
"1022121"
] | [
"With the same settings as IMO 2022 Problem 1, we randomly generated an initial configuration of $n=2022$. Assume $k=2022$, and it takes $x$ operations to first reach the final configuration (the leftmost $n$ coins are all of the same type). Estimate $x^2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 3
} | 7 | 0 | 0 |
omr-ws-3b81e316a2e42486deb83140de5db6f633916b3560979fa1290ca4476f1d2035 | With the three digits of the integer 142, there are six two-digit positive integers that can be formed. When the sum of those two-digit integers is divided by the sum of the three digits of the original three-digit integer, the result is 22. When the same thing is done with the \( k \) digits of a \( k \)-digit integer... | [
"7",
"\\( k = 7 \\)"
] | [
"With the three digits of the integer 142, there are six two-digit positive integers that can be formed. When the sum of those two-digit integers is divided by the sum of the three digits of the original three-digit integer, the result is 22. When the same thing is done with the \\( k \\) digits of a \\( k \\)-digi... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 2
} | 34 | 0 | 0 |
omr-ws-72111d0d672c74e1dd66b384f40e6ac7240cbd3466c1f537306911cab3588ff5 | With the wind blowing from the northeast, a certain sailboat can travel north at 5 knots, south at 12 knots, east at 6 knots, and west at 11 knots. How fast can the boat travel south with the same strength wind out of the northwest? | [
"7"
] | [
"With the wind blowing from the northeast, a certain sailboat can travel north at 5 knots, south at 12 knots, east at 6 knots, and west at 11 knots. How fast can the boat travel south with the same strength wind out of the northwest?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-40d7314e652f41c7c5eb750ee2315324e707ce9f8b1eb4f2d9e78bc4417f156c | With three different digits, all greater than $0$, six different three-digit numbers are formed. If the sum of these six numbers is $4218$, and the sum of the three largest numbers minus the sum of the three smallest numbers equals $792$, find the three digits. | [
"\\(8, 7, 4\\)"
] | [
"With three different digits, all greater than $0$, six different three-digit numbers are formed. If the sum of these six numbers is $4218$, and the sum of the three largest numbers minus the sum of the three smallest numbers equals $792$, find the three digits."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 8,
"tir": 28
} | 36 | 0 | 0 |
omr-ws-62b8b08cf2b1af643a4ace0a8b8124f2b4220628b4c14828d515bd38f8d68fa6 | With three different digits, six different three-digit numbers are formed. If these six numbers are added, the result is 4218. The sum of the three largest numbers minus the sum of the smallest three equals 792. Find the three digits. | [
"9, 8, 2"
] | [
"With three different digits, six different three-digit numbers are formed. If these six numbers are added, the result is 4218. The sum of the three largest numbers minus the sum of the smallest three equals 792. Find the three digits."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0cc4f4a2c9942f019ba6bdee7818d3bb58cff01b3bccfa5f71ec4072a0fdb0e8 | With three different digits, six-digit numbers are written that are multiples of $3$. One digit is in the unit's place, another in the hundred's place, and the third in the remaining places. If we subtract two units from the hundred's digit and add these to the unit's digit, the number remains composed of the same digi... | [
"222312"
] | [
"With three different digits, six-digit numbers are written that are multiples of $3$. One digit is in the unit's place, another in the hundred's place, and the third in the remaining places. If we subtract two units from the hundred's digit and add these to the unit's digit, the number remains composed of the same... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-cc19064cef23239efa77ad75dea30796f7d3479b90ec825bd08a900624c5a407 | With two equal pyramids with a square base and all their edges equal, a regular octahedron is formed by gluing the two bases together. On each edge of the octahedron, two points are marked that divide them into three equal segments. The 24 marked points are the vertices of a new polyhedron, which cuts out six small equ... | [
"204"
] | [
"With two equal pyramids with a square base and all their edges equal, a regular octahedron is formed by gluing the two bases together. On each edge of the octahedron, two points are marked that divide them into three equal segments. The 24 marked points are the vertices of a new polyhedron, which cuts out six smal... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dc0da8a01c0e35230b91904550a9223410e47f26b31b375214e8ad84f2c7687c | Within a convex quadrilateral \(ABCD\), let \(X\) be a point such that its isogonal conjugate exists. If \(\angle AXB = 120^\circ\) and \(\angle CXD = \theta\), find the value of \(\theta\). | [
"60"
] | [
"Within a convex quadrilateral \\(ABCD\\), let \\(X\\) be a point such that its isogonal conjugate exists. If \\(\\angle AXB = 120^\\circ\\) and \\(\\angle CXD = \\theta\\), find the value of \\(\\theta\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a02b41f1f779738253db8bee9b5038cfdb4c032b7c35ae5f3b7165757f147863 | Within a given right angle, a given right-angled isosceles triangle \(ABC\) (\(\angle C = 90^\circ\)) moves in such a way that point \(A\) slides along one side of the angle, point \(B\) slides along the other. Find all possible positions of point \(C\). | [
"y = x"
] | [
"Within a given right angle, a given right-angled isosceles triangle \\(ABC\\) (\\(\\angle C = 90^\\circ\\)) moves in such a way that point \\(A\\) slides along one side of the angle, point \\(B\\) slides along the other. Find all possible positions of point \\(C\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-315e291d7d5f12808338cf18ebe77053dd093d635ac670f1a8ceb1104d51451f | Within a group of 2009 people, every two people have exactly one common friend. Find the least value of the difference between the person with the maximum number of friends and the person with the minimum number of friends. | [
"2006"
] | [
"Within a group of 2009 people, every two people have exactly one common friend. Find the least value of the difference between the person with the maximum number of friends and the person with the minimum number of friends."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 1
} | 33 | 0 | 0 |
omr-ws-01cb445d70df6ebcc528f4021bdc6955dda93e5fa843b5490fcb09684221e37c | Within a group of 2022 students, each student is assigned a number from 1 to 2022. Let $c_i$ be the number of friends of the $i$-th student. Find the largest possible value of the sum \[c_1 + c_2 + \dots + c_{2022}.\] | [
"202200"
] | [
"Within a group of 2022 students, each student is assigned a number from 1 to 2022. Let $c_i$ be the number of friends of the $i$-th student. Find the largest possible value of the sum \\[c_1 + c_2 + \\dots + c_{2022}.\\]"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-007b4a17576460146ba8b452f50a8a350dce8fea25a44bf9a9e6bb3fd4e05fc0 | Within a group of 2022 students, some pairs of them are friends, and friendship is always mutual. We know that every non-empty subset $S$ of the students satisfies the following property: there exists a student in $S$ which is friends with at most 100 members of $S$. Find the smallest positive integer $N$ such that it ... | [
"10505"
] | [
"Within a group of 2022 students, some pairs of them are friends, and friendship is always mutual. We know that every non-empty subset $S$ of the students satisfies the following property: there exists a student in $S$ which is friends with at most 100 members of $S$. Find the smallest positive integer $N$ such tha... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4a42ecafd92d8ebf2d273f3946e3ea063fb42ae6253e376a14a8224e34148e41 | Within a table of $m$ rows and $n$ columns, a box is marked at the intersection of the $p$th row and the $q$th column. How many of the rectangles formed by the boxes of the table contain the marked box? | [
"\\( p \\cdot (m - p + 1) \\cdot q \\cdot (n - q + 1) \\)"
] | [
"Within a table of $m$ rows and $n$ columns, a box is marked at the intersection of the $p$th row and the $q$th column. How many of the rectangles formed by the boxes of the table contain the marked box?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-6c42c9e0430bd75c8f45e38fdfcc827ebb860cba897c1306104da76d99b0b72e | Within a unit circle \( \Gamma \), \( N \geq 3 \) segments are marked such that the sum of all these \( N \) lengths is equal to \( 2 \sqrt{N} \). Let \( r \) be the radius of the largest concentric circle to \( \Gamma \) that intersects at least two of these segments. Find the maximum possible value of \( r^2 \). | [
""
] | [
"Within a unit circle \\( \\Gamma \\), \\( N \\geq 3 \\) segments are marked such that the sum of all these \\( N \\) lengths is equal to \\( 2 \\sqrt{N} \\). Let \\( r \\) be the radius of the largest concentric circle to \\( \\Gamma \\) that intersects at least two of these segments. Find the maximum possible val... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 1 |
omr-ws-ec077710bfb34c228d80770d1db685aabbb88a99cc55ebb932e61390235e2366 | Within an acute triangle \(ABC\), a point \(S\) is given such that \(\angle SAB = \angle SBC = \angle SCA\). The lines \(AS\), \(BS\), and \(CS\) intersect the circumcircles of triangles \(SBC\), \(SCA\), and \(SAB\) at points \(A_1\), \(B_1\), and \(C_1\) respectively. Calculate the minimum possible value of \(\frac{P... | [
"2"
] | [
"Within an acute triangle \\(ABC\\), a point \\(S\\) is given such that \\(\\angle SAB = \\angle SBC = \\angle SCA\\). The lines \\(AS\\), \\(BS\\), and \\(CS\\) intersect the circumcircles of triangles \\(SBC\\), \\(SCA\\), and \\(SAB\\) at points \\(A_1\\), \\(B_1\\), and \\(C_1\\) respectively. Calculate the min... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 2
} | 6 | 0 | 0 |
omr-ws-6cc41833b8aa8315e66097f9120c4edfb79ad9ba1cd45c3e934771bc4c4aac5b | Within an arithmetic progression of length 2005, find the number of arithmetic subprogressions of length 501 that do not contain the 1000-th term of the progression. | [
"1849"
] | [
"Within an arithmetic progression of length 2005, find the number of arithmetic subprogressions of length 501 that do not contain the 1000-th term of the progression."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-31252baa4ad90d649d5dd322e1722d7ccd17f5abc7804cd88bd8e76e0fe821fb | Within an equilateral triangle of side length 15, there are 111 points. Determine the minimum number of points that can always be covered by a circle of diameter $\sqrt{3}$, where part of the circle may lie outside the triangle. | [
"3"
] | [
"Within an equilateral triangle of side length 15, there are 111 points. Determine the minimum number of points that can always be covered by a circle of diameter $\\sqrt{3}$, where part of the circle may lie outside the triangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-25b0def01f63ef8a808740215871f7f5cea9ba3e936cdceee5522f1620569f32 | Within what limits can the largest angle of a triangle change if the vertices are at the touchpoints of the circle inscribed in the triangle with the sides? | [
"$[60^\\circ ; 90^\\circ )$"
] | [
"Within what limits can the largest angle of a triangle change if the vertices are at the touchpoints of the circle inscribed in the triangle with the sides?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 8,
"tir": 15
} | 34 | 0 | 0 |
omr-ws-1d78a5971b65fd95403baf88df47492848e7cbae5c9c1449fd8e4211353674ab | Without calculating the exact value, determine how many times each digit appears in the number \( x \) where \( x = 3^{2457} + 1 \). | [
"\\{132, 109, 123, 117, 111, 108, 113, 111, 126, 123\\}"
] | [
"Without calculating the exact value, determine how many times each digit appears in the number \\( x \\) where \\( x = 3^{2457} + 1 \\)."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 4
} | 4 | 0 | 0 |
omr-ws-cc75a7d2e83239374cc110f424975ac958793e433d5ff375932b7f18b5592d41 | Without further calculation, sketch the set of all points satisfying $\cos y = \frac{1}{2} \sin x$ for $- \pi \leq x \leq \pi$ and $- \pi \leq y \leq \pi$. | [
"\\text{The solution is the set of curves described by } y = \\pm \\arccos\\left(\\frac{1}{2} \\sin x\\right) \\text{ and } y = \\pm \\left(\\pi - \\arccos\\left(\\frac{1}{2} |\\sin x|\\right)\\right) \\text{ within the given domain.}"
] | [
"Without further calculation, sketch the set of all points satisfying $\\cos y = \\frac{1}{2} \\sin x$ for $- \\pi \\leq x \\leq \\pi$ and $- \\pi \\leq y \\leq \\pi$."
] | [
"aops_c7_college_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 3
} | 4 | 0 | 0 |
omr-ws-e72a2394b3d9ab6285ce130be96c6f38b83c9ebdf43cf4b45498d8c9de7e738e | Without much calculation, determine the locus for \( z \) for each of the following: 1. \( \text{Re}\left(\frac{z-1-i}{z+1-i}\right)=0 \) 2. \( \text{Im}\left(\frac{z-1-i}{z+1-i}\right)=0 \) | [
"\\( |z - i| = 1 \\) and \\( \\Im\\{z\\} = 1 \\)"
] | [
"Without much calculation, determine the locus for \\( z \\) for each of the following:\n1. \\( \\text{Re}\\left(\\frac{z-1-i}{z+1-i}\\right)=0 \\)\n2. \\( \\text{Im}\\left(\\frac{z-1-i}{z+1-i}\\right)=0 \\)"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 15
} | 17 | 0 | 0 |
omr-ws-f58b6f95a0f449463bd2c5cbf14517eaf81385d493e82190ddd531e80054892c | Without overlapping, hexagonal tiles are placed inside an isosceles right triangle of area $1$ whose hypotenuse is horizontal. The tiles are similar to a given figure, but are not necessarily all the same size. The longest side of each tile is parallel to the hypotenuse of the triangle, and the horizontal side of lengt... | [
"2"
] | [
"Without overlapping, hexagonal tiles are placed inside an isosceles right triangle of area $1$ whose hypotenuse is horizontal. The tiles are similar to a given figure, but are not necessarily all the same size. The longest side of each tile is parallel to the hypotenuse of the triangle, and the horizontal side of ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 3
} | 15 | 0 | 0 |
omr-ws-444fd64486e88844962add5c64467de954291220544f4fe273d3aa4d232f86dd | Without overlapping, hexagonal tiles are placed inside an isosceles right triangle of area \( 1 \) whose hypotenuse is horizontal. The tiles are similar to the figure below, but are not necessarily all the same size. The longest side of each tile is parallel to the hypotenuse of the triangle, and the horizontal side of... | [
"2"
] | [
"Without overlapping, hexagonal tiles are placed inside an isosceles right triangle of area \\( 1 \\) whose hypotenuse is horizontal. The tiles are similar to the figure below, but are not necessarily all the same size. The longest side of each tile is parallel to the hypotenuse of the triangle, and the horizontal ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 2
} | 14 | 0 | 0 |
omr-ws-5974829dee733c6b4d43e327b572f420305b047cae98e64e7db5033e50264574 | Without solving the equation, find the sum of the squares of the roots of the equation \((x^2 + 2x)^2 - 5(x^2 + 2x) + 3 = 0\). | [
"10"
] | [
"Without solving the equation, find the sum of the squares of the roots of the equation \\((x^2 + 2x)^2 - 5(x^2 + 2x) + 3 = 0\\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-21f5b9f0da5238e157e495000de584a2f14896ab89d395d16c209dd068d9e121 | Without the aid of a calculator, compare the value of \( A = \sqrt{2008} + \sqrt{2009} + \sqrt{2010} \) and \( B = \sqrt{2005} + \sqrt{2007} + \sqrt{2015} \). | [
"\\( A \\ge B \\)"
] | [
"Without the aid of a calculator, compare the value of \\( A = \\sqrt{2008} + \\sqrt{2009} + \\sqrt{2010} \\) and \\( B = \\sqrt{2005} + \\sqrt{2007} + \\sqrt{2015} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 1
} | 6 | 0 | 0 |
omr-ws-c2e7439f9ab4aea2ce1dd6f71441bb4434d53a762ff4a81bf2977265e7132a17 | Without the use of a calculator, find the integer whose square root is nearest to \((\sqrt{2})^{(\sqrt{3})^{\sqrt{5}}}\). | [
"11"
] | [
"Without the use of a calculator, find the integer whose square root is nearest to \\((\\sqrt{2})^{(\\sqrt{3})^{\\sqrt{5}}}\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 19
} | 34 | 0 | 0 |
omr-ws-2727d5ef3d5694e7d8dc3327661aa89a36e989b536f4122412c70bc999f5d043 | Without using a calculator, compute $1003^2-997^2-1001^2+999^2$. | [
"8000"
] | [
"Without using a calculator, compute $1003^2-997^2-1001^2+999^2$."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ee77f5105e785369e72178492b785f24c226127b5472c5c2881c73c79f58ddcf | Without using a calculator, determine the value of $\sin\left(\frac{\pi}{14}\right) + 6\sin^2\left(\frac{\pi}{14}\right) - 8\sin^4\left(\frac{\pi}{14}\right)$. | [
"\\(\\frac{1}{2}\\)"
] | [
"Without using a calculator, determine the value of $\\sin\\left(\\frac{\\pi}{14}\\right) + 6\\sin^2\\left(\\frac{\\pi}{14}\\right) - 8\\sin^4\\left(\\frac{\\pi}{14}\\right)$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 25
} | 40 | 0 | 0 |
omr-ws-80df46237366bc8b4a3b2cd8021ad559f515d2bb09625cf126778bdb2cfd3880 | Without using a calculator, determine which is greater: $\sin 28^\circ$ or $\tan 21^\circ$. | [
"\\(\\sin 28^\\circ\\) is greater."
] | [
"Without using a calculator, determine which is greater: $\\sin 28^\\circ$ or $\\tan 21^\\circ$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-5d1aa8a74afbcc765c7181a7cd2dc5df48b30363e47df45d7136035d7faf5d0b | Without using a calculator, determine which number is greater: \( |\sqrt[3]{5} - \sqrt{3}| - \sqrt{3} \) or \( 0.01 \). | [
"\\( |\\sqrt[3]{5} - \\sqrt{3}| - \\sqrt{3} \\) is greater."
] | [
"Without using a calculator, determine which number is greater: \\( |\\sqrt[3]{5} - \\sqrt{3}| - \\sqrt{3} \\) or \\( 0.01 \\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d3146b286fb37d1de699007e1de9e94144c3b7501fa36dacb871122969ff152c | Without using a calculator, determine which number is greater: \(29^{200} \cdot 2^{151}\) or \(5^{279} \cdot 3^{300}\). | [
"\\(29^{200} \\cdot 2^{151}\\) is greater."
] | [
"Without using a calculator, determine which number is greater: \\(29^{200} \\cdot 2^{151}\\) or \\(5^{279} \\cdot 3^{300}\\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b37db0809f27f8ee474ff3e7ae8048a525a5a3113a83fabe96a31e13f7be3f50 | Without using a calculator, find the largest prime factor of $15^4+2\times15^2+1-14^4$. | [
"211"
] | [
"Without using a calculator, find the largest prime factor of $15^4+2\\times15^2+1-14^4$."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-71473dbef0be2d201cb32d6168e141a39172887afba4b52586ec991153bb9f11 | Without using a calculator, which number is larger, \( S = 99!^{\frac{1}{99}} \) or \( T = 100!^{\frac{1}{100}} \)? | [
"\\( T > S \\)"
] | [
"Without using a calculator, which number is larger, \\( S = 99!^{\\frac{1}{99}} \\) or \\( T = 100!^{\\frac{1}{100}} \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-729d44ada11e478586336491caa5897b6ba56a1722895126b25766943879b3cf | Without using the result from part (a), find the general solution of \[ \sin x - \sin 3x + \sin 5x - \ldots + (-1)^{n+1} \sin (2n-1)x = 0. | [
"x = \\dfrac{k\\pi}{2n} \\text{ where } k \\in \\mathbb{Z} \\text{ and } k \\not\\equiv n \\pmod{2n}"
] | [
"Without using the result from part (a), find the general solution of\n\\[\n\\sin x - \\sin 3x + \\sin 5x - \\ldots + (-1)^{n+1} \\sin (2n-1)x = 0."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-e3ab8fee9f00ea0b47f128e31315201e3c02ba504b255bd918883f753a1c5795 | Wizard ? has a positive integer number of children, whose ages are positive integers. The product of their ages is Wizard ?'s age, and the sum of their ages is the number on the bus. Wizard ? knows the bus number and states that even with the age and number of children, the ages of the children cannot be uniquely deter... | [
"12"
] | [
"Wizard ? has a positive integer number of children, whose ages are positive integers. The product of their ages is Wizard ?'s age, and the sum of their ages is the number on the bus. Wizard ? knows the bus number and states that even with the age and number of children, the ages of the children cannot be uniquely ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 5
} | 6 | 0 | 0 |
omr-ws-d1fe3aa9c769e113d0c7f45ee3f79075aba3bff1491e53f71dfa7d6cb03c6242 | Woody Woodpecker pecked at a $17$m wooden pole until it cracked and the upper part fell, with the top hitting the ground $10$m from the foot of the pole. Since the upper part had not completely broken off, Woody pecked away where the pole had cracked. How far was Woody off the ground? | [
"\\( 5.5588 \\) meters"
] | [
"Woody Woodpecker pecked at a $17$m wooden pole until it cracked and the upper part fell, with the top hitting the ground $10$m from the foot of the pole. Since the upper part had not completely broken off, Woody pecked away where the pole had cracked. How far was Woody off the ground?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-29316e3b440cb13a9310074f66e648d4aa6012404c0a6306d120ae475d6bb890 | Work in \( \mathbb{R}^+ \). Suppose \( E \) and \( E^C \) are both unbounded subsets of \( \mathbb{R}^+ \). Let \( x, y \in E \) with \( 1 < x < y \). Define \( g(x, y) \) as the smallest value of \( y' \) such that there exists \( x' < y' \) in \( E^C \) with \( y < x' \) and \( x'y' \in E^C \). If no such \( x' \) an... | [
"0"
] | [
"Work in \\( \\mathbb{R}^+ \\). Suppose \\( E \\) and \\( E^C \\) are both unbounded subsets of \\( \\mathbb{R}^+ \\). Let \\( x, y \\in E \\) with \\( 1 < x < y \\). Define \\( g(x, y) \\) as the smallest value of \\( y' \\) such that there exists \\( x' < y' \\) in \\( E^C \\) with \\( y < x' \\) and \\( x'y' \\i... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 8
} | 13 | 0 | 0 |
omr-ws-28afdec7d491f4eba7c107b3577c8ae80b04b72d3c59aa1ced794a4df76c8dee | Worker $A$ takes $a$ more days than worker $B$ to complete a duty, and worker $B$ takes $b$ more days than worker $C$ to complete the same duty. When workers $A$ and $B$ work together, they complete the duty in the same time as worker $C$ working alone. Find the time it takes for each worker to complete the duty indivi... | [
"\\sqrt{b(a + b)} + a + b"
] | [
"Worker $A$ takes $a$ more days than worker $B$ to complete a duty, and worker $B$ takes $b$ more days than worker $C$ to complete the same duty. When workers $A$ and $B$ work together, they complete the duty in the same time as worker $C$ working alone. Find the time it takes for each worker to complete the duty i... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-809709c1329a0b47a442ee9ad7e554f3ad9e27cc37a0ea61b7e2c62182c39d6a | Workers $A$ and $B$ have been working the same number of days. If $A$ worked one day less and $B$ 7 days less then $A$ would earn 72 roubles and $B$ 64 roubles 80 kopecks. If, conversely, $A$ worked 7 days less and $B$ one day less $B$ would earn 32 roubles and 40 kopecks more than $A$. How much did in fact either work... | [
"75 roubles, 90 roubles"
] | [
"Workers $A$ and $B$ have been working the same number of days. If $A$ worked one day less and $B$ 7 days less then $A$ would earn 72 roubles and $B$ 64 roubles 80 kopecks. If, conversely, $A$ worked 7 days less and $B$ one day less $B$ would earn 32 roubles and 40 kopecks more than $A$. How much did in fact either... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-69c19c8aae9fc130e8c0009c9a219ac631e5e8a03dcf31eabfd5aa5358bae5b2 | Working 22 hours in the second week of June, Xenia was able to earn $\$$47.60 more than during the first week of June when she worked 15 hours. If her hourly wage was constant, how many dollars did she earn during the first two weeks of June? Express your answer to the nearest hundredth. | [
"\\$ 251.60"
] | [
"Working 22 hours in the second week of June, Xenia was able to earn $\\$$47.60 more than during the first week of June when she worked 15 hours. If her hourly wage was constant, how many dollars did she earn during the first two weeks of June? Express your answer to the nearest hundredth."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-48a7c37b95d01b079d9079e0ce88ef90438d8671f54ba0a899e3ab753eaa0c4a | Working 6 hours a day, how many nuts can 180 workers produce if they produce 52,500 bolts each day? | [
"52500"
] | [
"Working 6 hours a day, how many nuts can 180 workers produce if they produce 52,500 bolts each day?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-97d4b4b1109ed4eeed70609ac57770b848773553aed1d5afefb5f7c49e88631d | Working with modulo 9, find the missing digit \( x \) in the equation \( 2784x = x \cdot 5569 \). | [
"5"
] | [
"Working with modulo 9, find the missing digit \\( x \\) in the equation \\( 2784x = x \\cdot 5569 \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bc4696731d3054a5d0a8e3897e93ffc46d01225eae4abcee320c7d4739dd924c | Woulard forms an 8-letter word by picking each letter from the set $\{w, o, u\}$ with equal probability. The score of a word is the nonnegative difference between the number of distinct occurrences of the three-letter word "uwu" and the number of distinct occurrences of the three-letter word "uwu". If the expected scor... | [
"1"
] | [
"Woulard forms an 8-letter word by picking each letter from the set $\\{w, o, u\\}$ with equal probability. The score of a word is the nonnegative difference between the number of distinct occurrences of the three-letter word \"uwu\" and the number of distinct occurrences of the three-letter word \"uwu\". If the ex... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d4fd55b8e76ebf9bcd184391e0d1a549d54efff33121c7bcbf18e6b98a4a4ed6 | Wowyok is in a house with 300 rooms, starting in room number 8. For any room number \( i \), he can advance to room \( i+1 \) or \( i-1 \) with a 1/2 chance each. If he enters room number 1, he must leave the house. If he enters room 300, he wins a cash prize. Find the probability of winning the cash prize. | [
"\\dfrac{7}{299}"
] | [
"Wowyok is in a house with 300 rooms, starting in room number 8. For any room number \\( i \\), he can advance to room \\( i+1 \\) or \\( i-1 \\) with a 1/2 chance each. If he enters room number 1, he must leave the house. If he enters room 300, he wins a cash prize. Find the probability of winning the cash prize."... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0af5df97d41e472e38d0990c54bef10901c4b4fdc9e19386e9a63db4306c15e4 | Write $ \omega_{\alpha}$ as a disjoint union of two sets, each of cardinality $ \aleph_{\alpha}$. | [
"\\omega_{\\alpha} = A \\cup B \\text{ where } A \\cap B = \\emptyset \\text{ and } |A| = |B| = \\aleph_{\\alpha}"
] | [
"Write $ \\omega_{\\alpha}$ as a disjoint union of two sets, each of cardinality $ \\aleph_{\\alpha}$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-107110dba51db6938ed1bc03e8ec86e05246f8c2026a2a0fdf5ee522ef4b4ee9 | Write $ \sqrt{25 - 12\sqrt{2}}$ in the form $ a + b\sqrt{c}$. Show your work. | [
"\\( 3 - 2\\sqrt{2} \\)"
] | [
"Write $ \\sqrt{25 - 12\\sqrt{2}}$ in the form $ a + b\\sqrt{c}$. Show your work."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 1,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-36c7b86ebf92a39c4f8736f58f2a9becef64fe64876fe17424cfc8a9c4c67261 | Write $$\sum_{k = 0}^{n} \frac{1}{k+1}\dbinom{n}{k}z^k$$ without the sigma notation. | [
"\\(\\frac{(1 + z)^{n+1} - 1}{(n+1)z}\\)"
] | [
"Write $$\\sum_{k = 0}^{n} \\frac{1}{k+1}\\dbinom{n}{k}z^k$$ without the sigma notation."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 17,
"tir": 10
} | 27 | 0 | 0 |
omr-ws-c37012421aab981a314e433012bbe2de2b71a8b18e6c696024e55aa5dedbad37 | Write $(-5)^5\div5^3+3^{4}-6^{1}$ as an integer. | [
"50"
] | [
"Write $(-5)^5\\div5^3+3^{4}-6^{1}$ as an integer."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1f1f87b1e633663da15fb7bb5eaed717187dcb73945eead70eeb013f062c77ac | Write $(5m)^4 + 2016(mn)^2 + (12n)^4$ as the sum of two perfect squares. | [
"\\((24m^2)^2 + (7m^2 + 144n^2)^2\\)"
] | [
"Write $(5m)^4 + 2016(mn)^2 + (12n)^4$ as the sum of two perfect squares."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 4,
"tir": 21
} | 38 | 0 | 0 |
omr-ws-c49339446f693837ab1b3c8a9581baaf7a7c638c8a91e5b4f02cb8fa40b43314 | Write $(x^2-yz)^3+(y^2-zx)^3+(z^2-xy)^3-3(x^2-yz)(y^2-zx)(z^2-xy)$ as the square of a polynomial. | [
"\\( (x^3 + y^3 + z^3 - 3xyz)^2 \\)"
] | [
"Write $(x^2-yz)^3+(y^2-zx)^3+(z^2-xy)^3-3(x^2-yz)(y^2-zx)(z^2-xy)$ as the square of a polynomial."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 19
} | 40 | 0 | 0 |
omr-ws-6364b7205dacc1d24e3f6bda54ae6801d1cc1549ce9172a053b8153b2ed3c9e3 | Write $0.\overline{43}$ as a simplified fraction. | [
"\\frac{43}{99}"
] | [
"Write $0.\\overline{43}$ as a simplified fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-977901c8f8c1201386796e27f49262b7d37f0029879d174d7858339602604eeb | Write $1 + \sin{\theta}$ as a product. | [
"\\( 2 \\cos^2\\left(\\frac{\\pi}{4} - \\frac{\\theta}{2}\\right) \\)"
] | [
"Write $1 + \\sin{\\theta}$ as a product."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-230b096cc88c4188c9e366e08a61d46175ba9ec88a60c5073bb1226ed1d8687a | Write $1.5\overline{2189}$ as a fraction. Note: only 2189 is repeating. | [
"\\(\\frac{6917}{4545}\\)"
] | [
"Write $1.5\\overline{2189}$ as a fraction. Note: only 2189 is repeating."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-0ffa335a795a88847b147d69fabe27a16ad67355778a3fedd8d1d6ac6a010234 | Write $102$ as the sum of the largest number of distinct primes. | [
"9"
] | [
"Write $102$ as the sum of the largest number of distinct primes."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 3
} | 4 | 0 | 0 |
omr-ws-8c474415ff559c421a3748d494a6eb841d6fe336bcc51dec34d6f870b2a9df98 | Write $11+7i$ as a product of irreducible elements in the ring $\mathbb{Z}[i]$. | [
"\\( (1+i)(1+2i)(1-4i) \\)"
] | [
"Write $11+7i$ as a product of irreducible elements in the ring $\\mathbb{Z}[i]$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 1,
"tir": 5
} | 8 | 0 | 0 |
omr-ws-576db671475a28c2528d85393610b3f3495fa743760161ec6379aad5346d68bc | Write $2013$ as a sum of $m$ prime numbers. What is the smallest value of $m$? | [
"2"
] | [
"Write $2013$ as a sum of $m$ prime numbers. What is the smallest value of $m$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8fcf6c423bc2925d51d9c086f09dcf4492b9af7d43ef67d1950b0e4f9ff98bc3 | Write $2014! = A \cdot 10^n$, with $A$ not divisible by $10$. Find the last digit of $A$. | [
"6"
] | [
"Write $2014! = A \\cdot 10^n$, with $A$ not divisible by $10$. Find the last digit of $A$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 6,
"tir": 4
} | 22 | 0 | 0 |
omr-ws-80c0996654d9f91240dd0e8c95b9a4ea99e99d546123493e14ed80361c27b199 | Write $2018^{2019} + 2018$ as a sum of two perfect squares. | [
"(13 \\cdot 2018^{1009} - 43)^2 + (43 \\cdot 2018^{1009} + 13)^2"
] | [
"Write $2018^{2019} + 2018$ as a sum of two perfect squares."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4f9e1711dedf4c7c8ae7b0cc6a6d5d0ae939452147023b637558bb331d7119f2 | Write $2\log_{2}40-\log_{4}125+\frac{1}{2}\log_{8}5$ in the form $p+q\log_{2}r$, where $p$ and $r$ are integers, and $q\in\mathbb{R}$. | [
"\\( 6 + \\frac{2}{3}\\log_{2}5 \\)"
] | [
"Write $2\\log_{2}40-\\log_{4}125+\\frac{1}{2}\\log_{8}5$ in the form $p+q\\log_{2}r$, where $p$ and $r$ are integers, and $q\\in\\mathbb{R}$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-269a55e8df339be2b10f0e92b7f9d050adfa6bc69cd656850cd02ab8d59a7683 | Write $2^{2010} - 1$ in base $2010$. | [
"1833"
] | [
"Write $2^{2010} - 1$ in base $2010$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 4
} | 8 | 0 | 0 |
omr-ws-edbb36dd9988f5c1ec300ed74348b83a05672bea817b4e5f1f00a711d88a3448 | Write $39$ as the sum of $6$ odd numbers. | [
"\\(3, 5, 7, 9, 15, 1\\)"
] | [
"Write $39$ as the sum of $6$ odd numbers."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f738033263d1e36f266754ed883d10da085610a3c25cb028b5ad2434bc7f7508 | Write $4+7\sqrt{2}$ as a product of irreducible elements in the ring $\mathbb{Z}[\sqrt{2}]$. | [
"\\((2+\\sqrt{2})(-3+5\\sqrt{2})\\)"
] | [
"Write $4+7\\sqrt{2}$ as a product of irreducible elements in the ring $\\mathbb{Z}[\\sqrt{2}]$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 1,
"tir": 2
} | 7 | 0 | 0 |
omr-ws-f3f702d795c4a7c5a650a2bd852f41962080252f768f2c2990f18bc35ee1aba6 | Write $4.3+3.88$ as a decimal. | [
"8.18"
] | [
"Write $4.3+3.88$ as a decimal."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c1a0abb3b92a600a3c10124351892ce4c2e0414df3508515f7b85fe3849463b7 | Write $8$ as an irrational number raised to an irrational number in at least three different ways. | [
"\\( 8 = \\left(2^{\\sqrt{3}}\\right)^{\\sqrt{3}}, 8 = \\left(2^{\\sqrt[3]{3}}\\right)^{\\sqrt[3]{9}}, 8 = \\left(2^{\\sqrt[4]{3}}\\right)^{\\sqrt[4]{27}} \\)"
] | [
"Write $8$ as an irrational number raised to an irrational number in at least three different ways."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 4
} | 4 | 0 | 0 |
omr-ws-4dbc6d6b2344787124f4fc63e191a81d2de9f81cb4185d04c72cd4eb1e62b811 | Write $Alt(T)$ and $Alt(S)$ in terms of elementary alternating tensors, where \[ T(x,y,z) = 2x_2 y_2 z_1 + x_1 y_5 z_4 \] \[ S(x,y) = x_1 y_3 + x_3 y_1 \] | [
"\\[ Alt(T)(x,y,z) = \\frac{1}{6} \\left( (2x_2 y_2 z_1 + x_1 y_5 z_4) - (2x_2 z_2 y_1 + x_1 z_5 y_4) - (2y_2 x_2 z_1 + y_1 x_5 z_4) + (2y_2 z_2 x_1 + y_1 z_5 x_4) + (2z_2 x_2 y_1 + z_1 x_5 y_4) - (2z_2 y_2 x_1 + z_1 y_5 x_4) \\right) \\]\n\\[ Alt(S)(x,y) = 0 \\]"
] | [
"Write $Alt(T)$ and $Alt(S)$ in terms of elementary alternating tensors, where\n\\[ T(x,y,z) = 2x_2 y_2 z_1 + x_1 y_5 z_4 \\]\n\\[ S(x,y) = x_1 y_3 + x_3 y_1 \\]"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 7
} | 7 | 0 | 0 |
omr-ws-c9f6d238da314acce44f3a56e8dd18ebe4fa8436c503339f961abb956e02464a | Write $\arcsin (\sin x)$ in the form $(-1)^m (x - n\pi),$ where $m$ and $n$ are integers. Determine the values of $m$ and $n$. | [
"(-1)^n (x - n\\pi)"
] | [
"Write $\\arcsin (\\sin x)$ in the form $(-1)^m (x - n\\pi),$ where $m$ and $n$ are integers. Determine the values of $m$ and $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 4
} | 8 | 0 | 0 |
omr-ws-581091307e65b2adc939dd26c7720bd0efa07fae953a240c01ec1264a4289ebf | Write $\cot^{-1} x$ in terms of $\arcsin$, $\arccos$, and $\arctan$. | [
"\\(\\cot^{-1} x = \\frac{\\pi}{2} - \\tan^{-1} x\\)"
] | [
"Write $\\cot^{-1} x$ in terms of $\\arcsin$, $\\arccos$, and $\\arctan$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 46,
"genselect": 8,
"tir": 19
} | 73 | 0 | 0 |
omr-ws-cc0d2903a4b53a51f9fd680de287282c8b2b5f364d3fe621ac0a953adcefcbf5 | Write $\csc^{-1} x$ in terms of $\arcsin$, $\arccos$, and $\arctan$. | [
"\\(\\sin^{-1} \\frac{1}{x}\\)"
] | [
"Write $\\csc^{-1} x$ in terms of $\\arcsin$, $\\arccos$, and $\\arctan$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-48735bd26d2134e9d7b84ea94c832eaa4477e9ad5fce3158d45cb84a1a84b270 | Write $\displaystyle{\frac{1}{\sqrt[5]{2} - 1} = a + b\sqrt[5]{2} + c\sqrt[5]{4} + d\sqrt[5]{8} + e\sqrt[5]{16}}$, with $a$, $b$, $c$, $d$, and $e$ integers. Find $a^2 + b^2 + c^2 + d^2 + e^2$. | [
"5"
] | [
"Write $\\displaystyle{\\frac{1}{\\sqrt[5]{2} - 1} = a + b\\sqrt[5]{2} + c\\sqrt[5]{4} + d\\sqrt[5]{8} + e\\sqrt[5]{16}}$, with $a$, $b$, $c$, $d$, and $e$ integers. Find $a^2 + b^2 + c^2 + d^2 + e^2$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-dea5c6149efc3ec5f2f847e9c3ed2efeb8421e71d4a13df6765347fb0660171e | Write $\frac{1}{10+7 \sqrt[3]{2}+4 \sqrt[3]{4}}$ in the form $a+b \sqrt[3]{2}+c \sqrt[3]{4}$, where $a, b, c \in \mathbb{Q}$. | [
"\\(\\frac{22}{131} - \\frac{19}{131}\\sqrt[3]{2} + \\frac{9}{262}\\sqrt[3]{4}\\)"
] | [
"Write $\\frac{1}{10+7 \\sqrt[3]{2}+4 \\sqrt[3]{4}}$ in the form $a+b \\sqrt[3]{2}+c \\sqrt[3]{4}$, where $a, b, c \\in \\mathbb{Q}$."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 30
} | 51 | 0 | 0 |
omr-ws-49cc0f07c579e82cdf235855405cd3078cadec44c7c2ca91c9fe2bf6003b5a62 | Write $\frac{2}{7}$ as a sum of distinct terms from $\left\{ \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots \right\}$. | [
"\\(\\frac{1}{4} + \\frac{1}{28}\\)"
] | [
"Write $\\frac{2}{7}$ as a sum of distinct terms from $\\left\\{ \\frac{1}{2}, \\frac{1}{3}, \\frac{1}{4}, \\ldots \\right\\}$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-7c34a2c9645010721ac1e168e183a86b719fe0e967fa76b7b3c32f2e089feb5b | Write $\frac{4019}{4074}$ as the sum of four different unit fractions. | [
"\\(\\frac{1}{2} + \\frac{1}{3} + \\frac{1}{7} + \\frac{1}{97}\\)"
] | [
"Write $\\frac{4019}{4074}$ as the sum of four different unit fractions."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-2de1d73524bf724ef04b91d81fce6c68db980c5140197019a3ca1a8e71d35a30 | Write $\frac{5}{8}$ as a decimal. | [
"0.625"
] | [
"Write $\\frac{5}{8}$ as a decimal."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-be4e46bf1a0c2f512b17289a4f669730216bc7eeb8e6dd6c6b06f690cfd8dc3d | Write $\frac{\sin mx}{\sin x}$ in terms of $\cos x$. | [
"\\(U_{m-1}(\\cos x)\\) or \\((2\\cos x)^{m-1} - (m-2)(2\\cos x)^{m-3} + \\frac{(m-3)(m-4)}{2!} (2\\cos x)^{m-5} - \\cdots\\)"
] | [
"Write $\\frac{\\sin mx}{\\sin x}$ in terms of $\\cos x$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 2,
"tir": 10
} | 17 | 0 | 0 |
omr-ws-8fe2b783501594479d380e01bc043e349c692690a085c52cf9ce3af5fd69a09e | Write $\log(15)$ in terms of $x = \log(2)$, $y = \log(3)$, and $z = \log(7)$. | [
"\\( y + 1 - x \\)"
] | [
"Write $\\log(15)$ in terms of $x = \\log(2)$, $y = \\log(3)$, and $z = \\log(7)$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-06aec48ee922af1b977e5a1cbfa081114d36ea1875478733ef2f12163f6ae9cc | Write $\sec^{-1} x$ in terms of $\arcsin$, $\arccos$, and $\arctan$. | [
"\\(\\cos^{-1} \\frac{1}{x}\\)"
] | [
"Write $\\sec^{-1} x$ in terms of $\\arcsin$, $\\arccos$, and $\\arctan$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-680565e5477daf206b02bedef8848724b4de38a48f2752c34f7d208a37896f3e | Write $\sin \alpha + \sin \beta + \sin \gamma$ as a product, where $\alpha$, $\beta$, and $\gamma$ are angles in a triangle. | [
"\\(4 \\cos\\left(\\frac{\\alpha}{2}\\right) \\cos\\left(\\frac{\\beta}{2}\\right) \\cos\\left(\\frac{\\gamma}{2}\\right)\\)"
] | [
"Write $\\sin \\alpha + \\sin \\beta + \\sin \\gamma$ as a product, where $\\alpha$, $\\beta$, and $\\gamma$ are angles in a triangle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-ea9f6ef2574f8bd2e24a1333f67d16adb988115587817c6c1441b971635f35b6 | Write $\sqrt{2}$ in base 2. | [
"1.0110101000001001111001100110011111110..."
] | [
"Write $\\sqrt{2}$ in base 2."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-fc26744a416af7fca9c30543815b32b046aa12421f084598b17bbb5347559fed | Write $\sqrt{\frac{16}{25}+\frac{9}{4}}$ as a common fraction. | [
"\\frac{17}{10}"
] | [
"Write $\\sqrt{\\frac{16}{25}+\\frac{9}{4}}$ as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9413506c6d3c26cdfb77265150ce7ec714dec0a4ddd77031c2a14509ded3b812 | Write $f(x) = \sqrt{1 - \cos(x)}$. Maximise $f(a) + f(b) + f(c)$ given that $a + b + c = 2\pi$. | [
"$\\frac{3\\sqrt{6}}{2}$"
] | [
"Write $f(x) = \\sqrt{1 - \\cos(x)}$. Maximise $f(a) + f(b) + f(c)$ given that $a + b + c = 2\\pi$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-2d32acbceb6100ac561df4b5bd2531191be2cbcf1f0cff8ebd01fe3f6bfeff6f | Write $n$ natural numbers from $1$ to $n$ in ascending order from left to right. $A$ and $B$ play the following game: On each turn, a player deletes two adjacent numbers and replaces them with their sum or product. The game continues until only one number remains. If the final number is odd, $A$ wins; otherwise, $B$ wi... | [
"\\( n \\) is even."
] | [
"Write $n$ natural numbers from $1$ to $n$ in ascending order from left to right. $A$ and $B$ play the following game: On each turn, a player deletes two adjacent numbers and replaces them with their sum or product. The game continues until only one number remains. If the final number is odd, $A$ wins; otherwise, $... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 6,
"tir": 2
} | 8 | 0 | 0 |
omr-ws-395450e4bed528acb8c41702a3f7bbb6d778de3029e50e7696bfcfd74e8ecba9 | Write $x^2-3x+8$ as a quadratic expression of $x-1$. | [
"\\((x-1)^2 - (x-1) + 6\\)"
] | [
"Write $x^2-3x+8$ as a quadratic expression of $x-1$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-62f481149e101c2c95a5a3236a2c02040237b31312a09c68fe9897327b0af9c7 | Write $x^{10} + x^5 + 1$ as the product of two polynomials with integer coefficients. | [
"(x^2 + x + 1)(x^8 - x^7 + x^5 - x^4 + x^3 - x + 1)"
] | [
"Write $x^{10} + x^5 + 1$ as the product of two polynomials with integer coefficients."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-48ba71c66c9e4818b336773f97150d1ba939edae745ba8f15294390b994ed09c | Write (in ascending order) all positive integer numbers of three or four digits whose square ends with 4444. (Write 0 if you think there are no numbers with this feature) | [
"0"
] | [
"Write (in ascending order) all positive integer numbers of three or four digits whose square ends with 4444. (Write 0 if you think there are no numbers with this feature)"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5ebe9a79466e2ff06c5bed0863dfa1fbda15e6550b7386c98f6b7f558c3090ac | Write (in ascending order) all the positive integer numbers of three digits whose square ends with 444. (Write 0 if you think there are no numbers with this feature) | [
"462, 538, 962"
] | [
"Write (in ascending order) all the positive integer numbers of three digits whose square ends with 444. (Write 0 if you think there are no numbers with this feature)"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 20
} | 24 | 0 | 0 |
omr-ws-ea1da4395b57fc7c29398b76b65ff3cee100cb6b810a89af9bf61e8b375d6616 | Write -10 in base -3. | [
"1212_{-3}"
] | [
"Write -10 in base -3."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-30b70668ebea15e8eec2ffe8de3e3b3e27ea7ac43095901fa03d7b07d473d235 | Write 0.232 (with the 32 part repeating) as the quotient of two integers. | [
"\\(\\frac{23}{99}\\)"
] | [
"Write 0.232 (with the 32 part repeating) as the quotient of two integers."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-e0e94ed2433ceb852190dba22f5748dc3abaf2001b2b60851ac030b6a256f755 | Write 1,000,000 as a sum of a prime number and a perfect square. | [
"\\(1,000,000 = 1999 + 999^2\\)",
"\\(1999 + 999^2\\)"
] | [
"Write 1,000,000 as a sum of a prime number and a perfect square."
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 56
} | 88 | 0 | 0 |
omr-ws-ea41a15d9f5f75565bb3aeb265aa5f97567c34792cd4bfc2aa68bd59232242e2 | Write 12 natural numbers in a row such that the fourth number is 7, the twelfth number is 10, and the sum of any three consecutive numbers is 102. Identify the numbers in this sequence. | [
"\\( 7, 85, 10, 7, 85, 10, 7, 85, 10, 7, 85, 10 \\)"
] | [
"Write 12 natural numbers in a row such that the fourth number is 7, the twelfth number is 10, and the sum of any three consecutive numbers is 102. Identify the numbers in this sequence."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 16
} | 30 | 0 | 0 |
omr-ws-e7487289db13fe0150d4843c22f85e544c76f8b5b5169379e329283c31d7f225 | Write 16 different positive integers in a 4x4 grid such that the products of numbers in each row, each column, and the two diagonals are equal. Find the minimum possible value of the biggest number in the grid. | [
"28"
] | [
"Write 16 different positive integers in a 4x4 grid such that the products of numbers in each row, each column, and the two diagonals are equal. Find the minimum possible value of the biggest number in the grid."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 1
} | 2 | 0 | 0 |
omr-ws-f9efbf4ede3b77e3f29fa3389da759aaabcc4f0871c91ad2fe64b42d3ed065e3 | Write 1998 as a sum of distinct positive integers such that the decimal representations of any two of the summands contain the same digits. | [
"468, 846, 684"
] | [
"Write 1998 as a sum of distinct positive integers such that the decimal representations of any two of the summands contain the same digits."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-89f9468b20730af2e2ac1fe64d1c2a431599fec1fc538fbe13ef2b33d6c04000 | Write 2.24 as a simplified fraction. | [
"\\frac{56}{25}"
] | [
"Write 2.24 as a simplified fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9ed534a7b9f9aa36a01a442ac82429f7a7bebc9af8c9bbeb6fe9ee28a57eaf54 | Write 2023 distinct integers on the board. Perform the following operation consecutively: If two distinct numbers \(a\) and \(b\) exist such that \(a - b > 1\), then replace \(a\) with \(a - 1\) and \(b\) with \(b + 1\). If no such numbers exist, stop the operation. Let \(S\) be the sum of all the numbers on the board ... | [
"2047276"
] | [
"Write 2023 distinct integers on the board. Perform the following operation consecutively: If two distinct numbers \\(a\\) and \\(b\\) exist such that \\(a - b > 1\\), then replace \\(a\\) with \\(a - 1\\) and \\(b\\) with \\(b + 1\\). If no such numbers exist, stop the operation. Let \\(S\\) be the sum of all the ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3fb843fb9e18f3669cc9ca1b7d2f616f6b063d6424ab8a9a0c058961910b379b | Write 271 as the sum of positive integers to maximize their product. | [
"\\( 89 \\times 3 + 2 + 2 \\)"
] | [
"Write 271 as the sum of positive integers to maximize their product."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 2,
"tir": 16
} | 31 | 0 | 0 |
omr-ws-ecb0e3e982d35f61a6047381005c5f4288c9deeb40351359bb97b55a73dfecec | Write 580 as a sum of two squares. | [
"\\( 24^2 + 2^2 \\) or \\( 18^2 + 16^2 \\)"
] | [
"Write 580 as a sum of two squares."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 1,
"tir": 20
} | 33 | 0 | 0 |
omr-ws-894915a3d4cfe2481fb770f199e3f6bf9ed3f511f629ee03cbefbb475e57969e | Write \( (a-b)^2(b-c)^2(c-a)^2 \) as a function of \( \sigma_1 = a + b + c \), \( \sigma_2 = ab + bc + ca \), and \( \sigma_3 = abc \). | [
"\\( \\sigma_1^2\\sigma_2^2 + 18\\sigma_1\\sigma_2\\sigma_3 - 4\\sigma_1^3\\sigma_3 - 4\\sigma_2^3 - 27\\sigma_3^2 \\)"
] | [
"Write \\( (a-b)^2(b-c)^2(c-a)^2 \\) as a function of \\( \\sigma_1 = a + b + c \\), \\( \\sigma_2 = ab + bc + ca \\), and \\( \\sigma_3 = abc \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-532cbcc2bd95fba7da9eecb8eb04a6a8d6c0b754041c845b8c5e2f46ea0cdc87 | Write \( A = 16^{0.249999999\ldots} \) as a fraction in simplest form. | [
"2/1"
] | [
"Write \\( A = 16^{0.249999999\\ldots} \\) as a fraction in simplest form."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-f578e807b8faf412ef0a9baabcd11eb06e2bee7939fdf22be39d55054abdd112 | Write \( B_n(1) \) as \( I \times J(x) \times B_{n-2}(x,y) \), where \( I \) is a fixed interval for the variable \( x \), \( J \) is an interval for \( y \) dependent on \( x \), and \( B_{n-2}(x,y) \) is a ball in \( \mathbb{R}^{n-2} \) with a radius dependent on \( x \) and \( y \). | [
"[-1, 1] \\times \\left[-\\sqrt{1 - x^2}, \\sqrt{1 - x^2}\\right] \\times B_{n-2}\\left(\\sqrt{1 - x^2 - y^2}\\right)"
] | [
"Write \\( B_n(1) \\) as \\( I \\times J(x) \\times B_{n-2}(x,y) \\), where \\( I \\) is a fixed interval for the variable \\( x \\), \\( J \\) is an interval for \\( y \\) dependent on \\( x \\), and \\( B_{n-2}(x,y) \\) is a ball in \\( \\mathbb{R}^{n-2} \\) with a radius dependent on \\( x \\) and \\( y \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
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