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-a7ada9c5836229ff382e3d81f5af78d77f005ca0a3efd8386b585dd3dd6a10b0 | What is the rightmost digit (the units digit) of the product of all even numbers from 2 to 98 inclusive, excluding those ending in 0? | [
"6"
] | [
"What is the rightmost digit (the units digit) of the product of all even numbers from 2 to 98 inclusive, excluding those ending in 0?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-91b8da234f049d3b41c8ce77ee3a4ad535587ddb68701211bf9ac06c2809227b | What is the rightmost digit of \(1000000!\) (factorial of a million)? | [
"0"
] | [
"What is the rightmost digit of \\(1000000!\\) (factorial of a million)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7cc5ad4c647d187e9f9561c80dc9ad01e43cea55c459d37ac6a953dd80c85972 | What is the rightmost non-zero digit of \(1000000!\) (one million factorial)? | [
"6"
] | [
"What is the rightmost non-zero digit of \\(1000000!\\) (one million factorial)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-23e1cf5e464e8c628d2939b68042aa3278269738e1bd42d6beaf4f87f572758d | What is the rightmost non-zero digit of \(1000000!\)? | [
"4"
] | [
"What is the rightmost non-zero digit of \\(1000000!\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 6
} | 18 | 0 | 0 |
omr-ws-36696ae0d8c2d9308712f5b12606c649f7cecf87b81993ec29a6e1a4dec44a7a | What is the rightmost nonzero digit of $100!$? | [
"4"
] | [
"What is the rightmost nonzero digit of $100!$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 4,
"tir": 4
} | 18 | 0 | 0 |
omr-ws-060d89c54d0f67d42476760ee4b00bf7f292610140fc0210758525c752d66b31 | What is the rightmost nonzero digit of $1000000!$? | [
"2",
"4"
] | [
"What is the rightmost nonzero digit of $1000000!$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 6
} | 19 | 0 | 0 |
omr-ws-2d0335e3744b451ebf826c7c2ced0a34d90ef337b43e2a141cebcd1db18a7097 | What is the rightmost nonzero digit of $2006!$? | [
"6"
] | [
"What is the rightmost nonzero digit of $2006!$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 18,
"genselect": 8
} | 26 | 0 | 0 |
omr-ws-9d57e8253b65284ab424257ba0f72de4bf3becbc380da76ca9e9c4a59a5de669 | What is the rightmost nonzero digit of \(1\,000\,000!\)? | [
"4"
] | [
"What is the rightmost nonzero digit of \\(1\\,000\\,000!\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 3
} | 7 | 0 | 0 |
omr-ws-25fc11c98b342468a94ec0aa54818d88dbc86d9dd83d3814cc4bfff9b7514d10 | What is the rotation matrix for a rotation by π/6 clockwise around the y=x axis in 3D? | [
"\\(\\begin{pmatrix}\n\\dfrac{2+\\sqrt3}{4}&\\dfrac{2-\\sqrt3}{4}&-\\,\\dfrac{1}{2\\sqrt{2}}\\\\\n\\dfrac{2-\\sqrt3}{4}&\\dfrac{2+\\sqrt3}{4}&\\dfrac{1}{2\\sqrt{2}}\\\\\n\\dfrac{1}{2\\sqrt{2}}&-\\,\\dfrac{1}{2\\sqrt{2}}&\\dfrac{\\sqrt3}{\\,\\,2}\n\\end{pmatrix}\\)"
] | [
"What is the rotation matrix for a rotation by π/6 clockwise around the y=x axis in 3D?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 18
} | 33 | 0 | 0 |
omr-ws-a7d328ca424cdf2bb6e8ab1b423b9e0cdad21170e447331fd6f1a1a8b847a7eb | What is the rotation transformation of $\mathbb{R}^n$ by an angle $\alpha$, and what is its corresponding matrix representation? | [
"R_{i,j}(\\alpha) \\text{ with entries } [R_{i,j}(\\alpha)]_{k,l} = \\begin{cases} \\cos\\alpha & \\text{if } k=l=i \\text{ or } k=l=j, \\\\ -\\sin\\alpha & \\text{if } k=i \\text{ and } l=j, \\\\ \\sin\\alpha & \\text{if } k=j \\text{ and } l=i, \\\\ 1 & \\text{if } k=l \\neq i,j, \\\\ 0 & \\text{otherwise}. \\end... | [
"What is the rotation transformation of $\\mathbb{R}^n$ by an angle $\\alpha$, and what is its corresponding matrix representation?"
] | [
"aops_c7_college_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 5
} | 6 | 0 | 0 |
omr-ws-5bfef3a6694ea9b37c770066e33b56fc6b01982e227be416216371f24b4eeb1d | What is the row echelon form of the matrix with rows {(1,1,1,1), (-1,1,1,-1), (1,0,1,1), (1,1,0,1)}? | [
"\n\\begin{pmatrix}\n1 & 1 & 1 & 1 \\\\\n0 & 1 & 1 & 0 \\\\\n0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & 0\n\\end{pmatrix}\n"
] | [
"What is the row echelon form of the matrix with rows {(1,1,1,1), (-1,1,1,-1), (1,0,1,1), (1,1,0,1)}?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 7
} | 17 | 0 | 0 |
omr-ws-302bdc7ad64187a05e9111cd50be5179329d431fcb822aa8c30ed01dcace82ee | What is the rule for the total number of student phone calls needed to inform all students if each student calls \( n \) other students? | [
"S - 1"
] | [
"What is the rule for the total number of student phone calls needed to inform all students if each student calls \\( n \\) other students?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 5
} | 15 | 0 | 0 |
omr-ws-282a27367bfab94e8342b62f4f16e6bb704298c3e8e1cc5f73d251b901788093 | What is the rule for the total number of student phone calls needed to inform all students if each student calls three other students? | [
"N - 1"
] | [
"What is the rule for the total number of student phone calls needed to inform all students if each student calls three other students?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 1
} | 16 | 0 | 0 |
omr-ws-89d042acabdbd2b0f3940e8ab40b83ac74c37ccc0c546694903f2baaf373eeb2 | What is the second derivative of the function $f(x) = \sqrt{\tan{x}}$? | [
"\\(\\frac{(2 \\cos (2 x)-1) \\sec ^4(x)}{4 \\tan ^{\\frac{3}{2}}(x)}\\)"
] | [
"What is the second derivative of the function $f(x) = \\sqrt{\\tan{x}}$?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-a26fd724a42bd79eff7dc62df1e74f94e09c02ffa0d0b9cd0d4b1aeb2f4ea6ca | What is the second number in the row of Pascal's triangle that has 43 numbers? | [
"42"
] | [
"What is the second number in the row of Pascal's triangle that has 43 numbers?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-156803ff5f0c3849a269e4e222eb9fa9499db4d2809caa801515991f66e30667 | What is the second smallest four-digit number in Pascal's triangle? | [
"1001"
] | [
"What is the second smallest four-digit number in Pascal's triangle?"
] | [
"MATH_training_set"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 2
} | 3 | 0 | 0 |
omr-ws-188129067ffef8198fb5b8d08d8e8389ca886f82835ff7c3d3ec344cf5080174 | What is the second smallest positive integer that leaves a remainder of 2 when divided by 5 and leaves a remainder of 3 when divided by 7? | [
"52"
] | [
"What is the second smallest positive integer that leaves a remainder of 2 when divided by 5 and leaves a remainder of 3 when divided by 7?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-6318374d99654c2bb18fe8509ac1e22d4cc3cdad889920415681d235d663ff37 | What is the second year after 2007 where each digit is prime and the year itself is also prime? | [
"2357"
] | [
"What is the second year after 2007 where each digit is prime and the year itself is also prime?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-7bc591d2ad59c137ed886b51c6bf6e8a7d2a0a4627c87e2f96f28337112102b3 | What is the second-last digit of the product of the odd numbers less than 1000? | [
"7"
] | [
"What is the second-last digit of the product of the odd numbers less than 1000?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-60649cbaad76766f5ef2af2f136b2877b20bd53b8a1749b72541321b808fe4af | What is the series expansion of $ \sqrt{9 - x} $ at $ x = \infty $? | [
"\\( i \\cdot x^{1/2} - \\frac{9}{2} i \\cdot x^{-1/2} - \\frac{81}{8}i \\cdot x^{-3/2} + \\ldots \\)"
] | [
"What is the series expansion of $ \\sqrt{9 - x} $ at $ x = \\infty $?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-421f17a7af02475fa7a8ce57c7cca82ba0d38c46c2d967976cab5d460b65bdd6 | What is the series expansion of $\frac{\ln (1+x)}{1-x}$? | [
"\\sum_{n=1}^\\infty \\left( \\sum_{k=1}^n \\frac{(-1)^{k+1}}{k} \\right) x^n"
] | [
"What is the series expansion of $\\frac{\\ln (1+x)}{1-x}$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-97158d66a863797e7e269f52bf104523e53457cd55237ab59c68f704ea0a2fac | What is the set $(\mathbb{Z}/2\mathbb{Z})(\zeta _n)$ where $\zeta _n = e^{2i\pi /n}$ is a primitive root of unity? | [
"$(\\mathbb{Z}/2\\mathbb{Z})(\\zeta_n) = \\{a_0 + a_1\\zeta_n + \\cdots + a_{d-1}\\zeta_n^{d-1} : a_i \\in \\mathbb{Z}/2\\mathbb{Z}\\}$, where $d$ is the degree of the minimal polynomial of $\\zeta_n$ over $\\mathbb{Z}/2\\mathbb{Z}$."
] | [
"What is the set $(\\mathbb{Z}/2\\mathbb{Z})(\\zeta _n)$ where $\\zeta _n = e^{2i\\pi /n}$ is a primitive root of unity?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 3,
"tir": 3
} | 10 | 0 | 0 |
omr-ws-2fc1c3215cf5ac6cc6d751592f5664f85d222ed7194a6da05902f00d7fdefa38 | What is the set $(\mathbb{Z}/2\mathbb{Z})[\zeta _n]$ where $\zeta _n = e^{2i\pi /n}$ is a primitive root of unity? | [
"$(\\mathbb{Z}/2\\mathbb{Z})[\\zeta_n] = \\{a_0 + a_1\\zeta_n + a_2\\zeta_n^2 + \\cdots + a_{k-1}\\zeta_n^{k-1} \\mid a_i \\in \\mathbb{Z}/2\\mathbb{Z}\\}$ where $k$ is the degree of the minimal polynomial of $\\zeta_n$ over $\\mathbb{Z}/2\\mathbb{Z}$."
] | [
"What is the set $(\\mathbb{Z}/2\\mathbb{Z})[\\zeta _n]$ where $\\zeta _n = e^{2i\\pi /n}$ is a primitive root of unity?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 1,
"tir": 5
} | 11 | 0 | 0 |
omr-ws-131804b37556817c374acdc6171b9a69748f84de9df1c5fb199da75981a2ae69 | What is the set \( D = \{ |\sigma(x+1) - \sigma(x)| \mid x \in \mathbb{Z} \} \)? | [
"\\mathbb{N}_0"
] | [
"What is the set \\( D = \\{ |\\sigma(x+1) - \\sigma(x)| \\mid x \\in \\mathbb{Z} \\} \\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-d647256716f9bcaf3bf996de30ad779d876fa4ec1d8b9a6496ec2a23efda2d36 | What is the set of all $n$'s yielding at least one square number of the type $s_n$? (A number is of type $s_n$ if it consists of $n$ identical blocks written in consecutive order.) | [
"\\{1\\}"
] | [
"What is the set of all $n$'s yielding at least one square number of the type $s_n$? (A number is of type $s_n$ if it consists of $n$ identical blocks written in consecutive order.)"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a5c8de83816ed286f7fe6d703ead79ff3a26c62bd2bd1d9a8384096c64193094 | What is the set of all natural numbers $m$ for which there exists an integer $x$ such that $m \mid x^2 + 1$? | [
"\\( m = 2^a \\cdot p_1^{e_1} \\cdot p_2^{e_2} \\cdots p_k^{e_k} \\) where \\( a \\leq 1 \\) and each \\( p_i \\equiv 1 \\pmod{4} \\)"
] | [
"What is the set of all natural numbers $m$ for which there exists an integer $x$ such that $m \\mid x^2 + 1$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-dc58031ed8be89bd539997314be96fca04875954ce535d2566cc651ba03161f9 | What is the set of all points on a plane such that the sum of the reciprocals of the distances from two fixed points is constant? | [
"\\text{A quartic curve}"
] | [
"What is the set of all points on a plane such that the sum of the reciprocals of the distances from two fixed points is constant?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 8
} | 19 | 0 | 0 |
omr-ws-ea650d31dfe56cfb248d2f08fc5212f5092c58f6fef8643f393d72b7e24e823c | What is the set of all possible values of $ab + cd$ given four positive real numbers $a, b, c, d$ such that $ac + bd = 1$? | [
"\\( (0, \\infty) \\)"
] | [
"What is the set of all possible values of $ab + cd$ given four positive real numbers $a, b, c, d$ such that $ac + bd = 1$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 40,
"genselect": 8,
"tir": 16
} | 64 | 0 | 0 |
omr-ws-be1045f370cde701d401bf0b8b891c075de4f6e1d80de00d8b3feaed24183239 | What is the shape of convex quadrilateral \(ABCD\) given that there exist points \(M \in AB\), \(N \in AC\), \(P \in CD\), and \(Q \in DA\) such that \(AM = BN = CP = DQ\) and \(\vec{AM} + \vec{BN} + \vec{CP} + \vec{DQ} = \vec{0}\)? | [
"parallelogram"
] | [
"What is the shape of convex quadrilateral \\(ABCD\\) given that there exist points \\(M \\in AB\\), \\(N \\in AC\\), \\(P \\in CD\\), and \\(Q \\in DA\\) such that \\(AM = BN = CP = DQ\\) and \\(\\vec{AM} + \\vec{BN} + \\vec{CP} + \\vec{DQ} = \\vec{0}\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 5
} | 13 | 0 | 0 |
omr-ws-7d94984e767ba65083a8acf62c9083eb3cde996b9416e3d496a5c174c7f485c9 | What is the shape of the figure described by the equation \((x-y)^2 + (y-z)^2 + (z-x)^2 = 3\)? | [
"\\text{circular cylinder}"
] | [
"What is the shape of the figure described by the equation \\((x-y)^2 + (y-z)^2 + (z-x)^2 = 3\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-997d15a2de9054652c628b178933cd75377665ed4c22c021493562a28e43c436 | What is the shape of the graph of the equation $|x| + |y| = 4$? | [
"\\(\\text{square}\\)"
] | [
"What is the shape of the graph of the equation $|x| + |y| = 4$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-b07fa168f4d81c4238d7c6c9b4728984b12c25d68bda332f397804feae35f673 | What is the shape of the graph of the equation \(x^2 + y^2 = z^2\) in a 3-dimensional plane? | [
"Two cones joined at the origin and pointing away from each other."
] | [
"What is the shape of the graph of the equation \\(x^2 + y^2 = z^2\\) in a 3-dimensional plane?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 8,
"tir": 10
} | 22 | 0 | 0 |
omr-ws-e5ac91559ac5c5f9f287f4fa3a236d088e3dd75cd223a9822bf9de58e2c959b5 | What is the shape of the region whose boundary is given by the equation $(2x^2 + xy − y^2 + 6y − 8)(x − y) = 0$? | [
"Triangle"
] | [
"What is the shape of the region whose boundary is given by the equation $(2x^2 + xy − y^2 + 6y − 8)(x − y) = 0$?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 6
} | 28 | 0 | 0 |
omr-ws-a2254773c05a62c3b1a16c3cf17a3f31ded2d247de93d7798ea3027df13171a7 | What is the shape of those points in a plane whose distances to three non-collinear points add up to a constant value? | [
"3-ellipse"
] | [
"What is the shape of those points in a plane whose distances to three non-collinear points add up to a constant value?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7d7defedf8b1313733ed4df520e6d096e81158f8c94501b01ab50fc549c2f3e2 | What is the shortest distance Annie the ant can walk from vertex A to vertex G on a rectangular prism ABCDEFGH, where AB=4, BC=6, and AE=8? | [
"2\\sqrt{41}"
] | [
"What is the shortest distance Annie the ant can walk from vertex A to vertex G on a rectangular prism ABCDEFGH, where AB=4, BC=6, and AE=8?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-00405457d66f58076e5888fc57b3cd3395c3c15d8681440089a065701d247144 | What is the shortest distance Boris can travel to ensure that all 2016 flags placed at random points in a circle of length 2016 are collected, no matter how Petryk places them? | [
"2016"
] | [
"What is the shortest distance Boris can travel to ensure that all 2016 flags placed at random points in a circle of length 2016 are collected, no matter how Petryk places them?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-5531ec222a6b28d40682bc303f81e1d01419e3006147fce44aa96263a13a7894 | What is the shortest distance between line A, which passes through the points (18, -2, -9) and (-6, -10, 7), and line B, which passes through the points (-31, -15, 12) and (-55, -18, 24)? | [
"\\(\\frac{59\\sqrt{5}}{15}\\)"
] | [
"What is the shortest distance between line A, which passes through the points (18, -2, -9) and (-6, -10, 7), and line B, which passes through the points (-31, -15, 12) and (-55, -18, 24)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bc8952fe605e4b256fac2a1acbce542f8dcd2d1c67cc9dd33d28e5fc5ee4253e | What is the shortest distance between the circle \(x^2 + (y + 6)^2 = 1\) and the parabola \(y^2 = 8xy\)? | [
"\\(2\\sqrt{2} - 1\\)"
] | [
"What is the shortest distance between the circle \\(x^2 + (y + 6)^2 = 1\\) and the parabola \\(y^2 = 8xy\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 2,
"tir": 17
} | 25 | 0 | 0 |
omr-ws-a5a86b2647832477f6ee0ca85f35bfc97f0744fa4af0803a1464dc48bf1d2307 | What is the shortest distance between the circles defined by $x^2-10x +y^2-4y-7=0$ and $x^2+14x +y^2+6y+49=0$? | [
"4"
] | [
"What is the shortest distance between the circles defined by $x^2-10x +y^2-4y-7=0$ and $x^2+14x +y^2+6y+49=0$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c2875f19ae2d7fd80e5e10ca157f13cfba21291e3a1ed58b95170ed0d3caa62b | What is the shortest distance between the curves $y=(x+1)^2$ and $x=y^2$? | [
"0.427592353666"
] | [
"What is the shortest distance between the curves $y=(x+1)^2$ and $x=y^2$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 6,
"tir": 18
} | 29 | 0 | 0 |
omr-ws-bf1d7de44a9e072c4156a5cafb4be27a48998c399415744a1d1ded7f93366a7c | What is the shortest distance between the curves $y=x^2$ and $y=\ln(x)$? | [
"0.534"
] | [
"What is the shortest distance between the curves $y=x^2$ and $y=\\ln(x)$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 6
} | 16 | 0 | 0 |
omr-ws-a14ebd903ff484bee28edbc30d5c1c31340375d7573d57688f5876f5b4ccf12f | What is the shortest distance between the line \(3x + 4y + 5 = 0\) and the curve \(\frac{27}{28}x^2 + \frac{9}{7}y^2 = 1\)? | [
"\\(\\frac{1}{15}\\)"
] | [
"What is the shortest distance between the line \\(3x + 4y + 5 = 0\\) and the curve \\(\\frac{27}{28}x^2 + \\frac{9}{7}y^2 = 1\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-efe066cec3a08189b131dcb0c04797f75feabda8ae69a9871ff2898f55413e4d | What is the shortest distance between the plane \(Ax + By + Cz + 1 = 0\) and the ellipsoid \(\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1\)? | [
"\\dfrac{\\max\\left(0, 1 - \\sqrt{A^2 a^2 + B^2 b^2 + C^2 c^2}\\right)}{\\sqrt{A^2 + B^2 + C^2}}"
] | [
"What is the shortest distance between the plane \\(Ax + By + Cz + 1 = 0\\) and the ellipsoid \\(\\frac{x^2}{a^2} + \\frac{y^2}{b^2} + \\frac{z^2}{c^2} = 1\\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 7
} | 19 | 0 | 0 |
omr-ws-641821004957f8f5dec6f379c25abb9c91324e970c630b9eb10a2fb403a421ff | What is the shortest distance between the plane \(Ax + By + Cz + 1 = 0\) and the ellipsoid \(\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1\)? You may find it convenient to use the notation \(h = (A^2 + B^2 + C^2)^{-\frac{1}{2}}\) and \(m = (a^2A^2 + b^2B^2 + c^2C^2)^{\frac{1}{2}}\). | [
"\\(\\frac{1}{(a^2A^2 + b^2B^2 + c^2C^2)^{\\frac{1}{2}}}\\)"
] | [
"What is the shortest distance between the plane \\(Ax + By + Cz + 1 = 0\\) and the ellipsoid \\(\\frac{x^2}{a^2} + \\frac{y^2}{b^2} + \\frac{z^2}{c^2} = 1\\)? You may find it convenient to use the notation \\(h = (A^2 + B^2 + C^2)^{-\\frac{1}{2}}\\) and \\(m = (a^2A^2 + b^2B^2 + c^2C^2)^{\\frac{1}{2}}\\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1cd1fba7def4d4e1d8d6c6e4fc1cc22f780c21ffcf79979b006c0b5667e537fb | What is the shortest distance between the point $(4,0)$ and the parabola $y^2=2x$? | [
"None of these."
] | [
"What is the shortest distance between the point $(4,0)$ and the parabola $y^2=2x$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4004930462f76cae42b451cc9fbe90975a644b2c696ad52392699e55bacfb6d4 | What is the shortest distance from the origin to the circle defined by $x^2-24x +y^2+10y +160=0$? | [
"10"
] | [
"What is the shortest distance from the origin to the circle defined by $x^2-24x +y^2+10y +160=0$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ae8b42066d3c542a8d492ffae6dc8f7c986cda7f3faf9e7bad1f5b4a849ef000 | What is the shortest distance from the point (5, 10) to the curve \( x^2 = 12y \)? | [
"5.13"
] | [
"What is the shortest distance from the point (5, 10) to the curve \\( x^2 = 12y \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-c34a8795014a46272a5b823f8dc0b568db3882e0b698a76ab28054a55069db8a | What is the shortest distance from town A to town Z in kilometers, given the one-way routes and distances between towns A, M, C, X, Y, and Z as shown in the figure below? [asy] /* AMC8 P14 2024, by NUMANA: BUI VAN HIEU */ import graph; unitsize(2cm); real r=0.25; // Define the nodes and their positions pair[] nodes = {... | [
"28"
] | [
"What is the shortest distance from town A to town Z in kilometers, given the one-way routes and distances between towns A, M, C, X, Y, and Z as shown in the figure below?\n\n[asy]\n/* AMC8 P14 2024, by NUMANA: BUI VAN HIEU */\nimport graph;\nunitsize(2cm);\nreal r=0.25;\n// Define the nodes and their positions\npa... | [
"c5_contests_amp_programs"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-f8726cca04be2ea0198619fd9eed84ae366a291d55ef926c99a1ce3b78946716 | What is the shortest distance, in units, between the point $(3, -4, 12)$ and the plane defined by the equation $10x - 6y - 15z = -49$? | [
"\\dfrac{77}{19}"
] | [
"What is the shortest distance, in units, between the point $(3, -4, 12)$ and the plane defined by the equation $10x - 6y - 15z = -49$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-87960c1c09c6593ee653313b3af9e197d0ba941e449f3e8c410d3751b0393a79 | What is the shortest interval of \( A = \log 2 \)? (shortest distance) | [
"0"
] | [
"What is the shortest interval of \\( A = \\log 2 \\)? (shortest distance)"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 2
} | 8 | 0 | 0 |
omr-ws-18473ff5b1f1e5f702c2bae9850e1275d441caddbceb03f8826c93d29112625e | What is the shortest length of a \( \theta \)-interval that can produce the graph of the equation \( r^2 = -\sin 2\theta \)? | [
"\\(\\frac{\\pi}{2}\\)"
] | [
"What is the shortest length of a \\( \\theta \\)-interval that can produce the graph of the equation \\( r^2 = -\\sin 2\\theta \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-538879a407aca4cc7fd84ae707be046acf63c6e38eb47818245c5b1fee240130 | What is the shortest path a bug can take that starts from $(a, b)$ and wants to get to $(-c, -d)$ if it runs $n$ times faster in the second quadrant than any other, where $a$, $b$, $c$, and $d$ are positive? | [
"\\sqrt{\\left(a + \\frac{c}{n}\\right)^2 + (b + d)^2}"
] | [
"What is the shortest path a bug can take that starts from $(a, b)$ and wants to get to $(-c, -d)$ if it runs $n$ times faster in the second quadrant than any other, where $a$, $b$, $c$, and $d$ are positive?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 4
} | 16 | 0 | 0 |
omr-ws-800799f57ada218db6fad0de097273f814d59664a9ea42122486279aa32839f6 | What is the shortest path from point \( A = (3, 8) \) to point \( B = (6, 4) \) if it must touch a point on each of the lines \( y = \frac{1}{2}x \) and \( y = 3x \)? | [
"\\sqrt{197}"
] | [
"What is the shortest path from point \\( A = (3, 8) \\) to point \\( B = (6, 4) \\) if it must touch a point on each of the lines \\( y = \\frac{1}{2}x \\) and \\( y = 3x \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 3
} | 12 | 0 | 0 |
omr-ws-fe4990d23c5a3876d397761cc6314d08f4abc7cae26f0c81d4975f14a7bb628d | What is the shortest path from point \( A = (3, 8) \) to point \( B = (6, 4) \) if it must touch a point on the line \( y = \frac{1}{2}x \)? | [
"\\dfrac{\\sqrt{1145}}{5}"
] | [
"What is the shortest path from point \\( A = (3, 8) \\) to point \\( B = (6, 4) \\) if it must touch a point on the line \\( y = \\frac{1}{2}x \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e171f585bb5f701b4f3341e1297daae1ed30916b3d47085f87e4188ab77e474b | What is the shortest possible side length of a four-dimensional hypercube that contains a regular octahedron with side length 1? | [
"\\( \\sqrt{2} \\)"
] | [
"What is the shortest possible side length of a four-dimensional hypercube that contains a regular octahedron with side length 1?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 1,
"tir": 10
} | 15 | 0 | 0 |
omr-ws-df21a2ef8f5192a46cd09074608f30f02b0d4910b5d512260d03fcaea8d37483 | What is the shortest segment that bisects the area of a triangle? | [
"\\text{The median}"
] | [
"What is the shortest segment that bisects the area of a triangle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 2
} | 12 | 0 | 0 |
omr-ws-515178e20404d10105255bfb7e741acb1b09bd21c77c1a56a8fe32594dd810c2 | What is the shortest time necessary to get all the water from the highest water tank to the lowest water tank, given the following conditions: - The highest water tank is full with 1000 liters. - The highest water tank has 7 taps with a flow rate of 10 liters per hour each. - The second highest water tank has 3 taps wi... | [
"\\(\\frac{1840}{49}\\) hours"
] | [
"What is the shortest time necessary to get all the water from the highest water tank to the lowest water tank, given the following conditions:\n- The highest water tank is full with 1000 liters.\n- The highest water tank has 7 taps with a flow rate of 10 liters per hour each.\n- The second highest water tank has 3... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ab5f87fdd6307a88d671f1a8ebe6e5eeb8e2e3f5f4f1c74b16824247a95fb811 | What is the side length of a cube inscribed in a square pyramid with base side length 1? Note that a square pyramid is a pyramid with a square base and with all sides equal. | [
"\\sqrt{2} - 1"
] | [
"What is the side length of a cube inscribed in a square pyramid with base side length 1? Note that a square pyramid is a pyramid with a square base and with all sides equal."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-eab986811081e722bde2ddb64fb00406199521856ca0f341fc6a47c1a5b1a8a8 | What is the side length of an equilateral triangle $\text{ABC}$ for which there is a point $\text{P}$ inside it such that $\text{AP} = 6$, $\text{BP} = 8$, and $\text{CP} = 10$? | [
"\\( 2\\sqrt{25 + 12\\sqrt{3}} \\)"
] | [
"What is the side length of an equilateral triangle $\\text{ABC}$ for which there is a point $\\text{P}$ inside it such that $\\text{AP} = 6$, $\\text{BP} = 8$, and $\\text{CP} = 10$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-b3147926a4f37dc770fc3eaa579de35729056b28e2af2c437ea6aed309f01b9d | What is the side length of an equilateral triangle \(ABC\) for which there is a point \(P\) inside it such that \(AP = 6\), \(BP = 8\), and \(CP = 10\)? | [
"\\(2\\sqrt{25 + 16\\sqrt{3}}\\)"
] | [
"What is the side length of an equilateral triangle \\(ABC\\) for which there is a point \\(P\\) inside it such that \\(AP = 6\\), \\(BP = 8\\), and \\(CP = 10\\)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f0ff29db8882800c71a84e239d40210729818420868721b922bda6dd57f1796b | What is the side length of the largest equilateral triangle that can be inscribed in a regular pentagon with side length $1$? | [
"\\dfrac{1+\\sqrt{5}}{2}"
] | [
"What is the side length of the largest equilateral triangle that can be inscribed in a regular pentagon with side length $1$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-c6ffaf345a2639b8c9609f1a53d63c2165cbaa5846ffff5c8d4d67a53d0c749d | What is the side length of the largest equilateral triangle that can be inscribed within a square of side length 6 cm? Express your answer in simplest radical form. | [
"\\( 6\\sqrt{6} - 6\\sqrt{2} \\)"
] | [
"What is the side length of the largest equilateral triangle that can be inscribed within a square of side length 6 cm? Express your answer in simplest radical form."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-534de19d7faa2c4c73d84727f1c836edeede96eceaae9e3911eb4d70e4a771b5 | What is the side length of the largest square that can fit inside a cube of side length \( n \)? | [
"\\(\\frac{3\\sqrt{2}n}{4}\\)"
] | [
"What is the side length of the largest square that can fit inside a cube of side length \\( n \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 7
} | 9 | 0 | 0 |
omr-ws-3dca79c1e87c5fddbeafed25f3b0f0064090d96de7e403903505bb4c24ff2c19 | What is the side length of the largest square that can fit inside a cube with side length \( n \)? | [
"\\( \\frac{3n\\sqrt{2}}{4} \\)"
] | [
"What is the side length of the largest square that can fit inside a cube with side length \\( n \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 7,
"tir": 1
} | 11 | 0 | 0 |
omr-ws-ebd46f1bdf8b554214b0cf17e5c6c6e3b2f538c515bcc1c9916e9fe34966cbe2 | What is the side length of the middle square in a unit square where the diagonal is covered by small golden squares with sides parallel to the unit square, and each neighbor's side length is either half or double of the previous square, with midpoints on the diagonal at distances $\dfrac{1}{2}, \dfrac{1}{4}, \dfrac{1}{... | [
"\\(\\frac{1}{3}\\)"
] | [
"What is the side length of the middle square in a unit square where the diagonal is covered by small golden squares with sides parallel to the unit square, and each neighbor's side length is either half or double of the previous square, with midpoints on the diagonal at distances $\\dfrac{1}{2}, \\dfrac{1}{4}, \\d... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 38,
"genselect": 7,
"tir": 7
} | 52 | 0 | 0 |
omr-ws-d523614807cb2b357e7d42f31992be565b246d9422215f1e328225128304f67e | What is the side length of the smallest square box that can fit 4 spheres each with radius $R=5$? | [
"20"
] | [
"What is the side length of the smallest square box that can fit 4 spheres each with radius $R=5$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-69f22f6071fd666df5677e896895438b7a6bf16d9cbac404b427d7b219c6dc5e | What is the sign of \( (a)^{b} \) where \( a \in \mathbb{Z^{-}} \) and \( b \) is irrational? For example, is \( (-3)^{\pi} \) positive or negative, or is it impossible to define? If the latter is true, why can \( 3^{\pi} \) be defined? | [
"undefined."
] | [
"What is the sign of \\( (a)^{b} \\) where \\( a \\in \\mathbb{Z^{-}} \\) and \\( b \\) is irrational? For example, is \\( (-3)^{\\pi} \\) positive or negative, or is it impossible to define? If the latter is true, why can \\( 3^{\\pi} \\) be defined?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-7c979eaa66b65081d58f4c4b6a5e75bb1104c346b61efe822343787c2d864803 | What is the signature of the symmetric bilinear form defined by the following matrix? \[ \left(\begin{matrix} 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -5\\ 0 & 0& 0 & -5 & 0\\ 0 & 0 & -5 & 0 & 0\\ 0 & -5 & 0 & 0 & 0\\ \end{matrix}\right) \] | [
"\\( n_+ = 3 \\), \\( n_- = 2 \\), \\( n_0 = 0 \\) or in terms of the signature, \\( \\text{signature} = n_+ - n_- = 3 - 2 = 1 \\)."
] | [
"What is the signature of the symmetric bilinear form defined by the following matrix?\n\n\\[\n\\left(\\begin{matrix} \n 1 & 0 & 0 & 0 & 0\\\\\n 0 & 0 & 0 & 0 & -5\\\\\n 0 & 0& 0 & -5 & 0\\\\\n 0 & 0 & -5 & 0 & 0\\\\\n 0 & -5 & 0 & 0 & 0\\\\\n\\end{matrix}\\right)\n\\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 18
} | 33 | 0 | 0 |
omr-ws-7dd37394d8c1e939c669e089a9b0b1277f39a76504918010fccbf8ae9480b24d | What is the significance of the quotients obtained by repeatedly dividing 58 by 2 and 3, and summing the truncated quotients? How do these sums relate to the number of trailing zeros in \(58!\) when expressed in base 12? | [
"27"
] | [
"What is the significance of the quotients obtained by repeatedly dividing 58 by 2 and 3, and summing the truncated quotients? How do these sums relate to the number of trailing zeros in \\(58!\\) when expressed in base 12?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-de4efd046a23629511e5d0a26a19ab640fa022b1617db0b51c5b17076b927a84 | What is the simplest example of a polynomial function \( f: \mathbb{R} \to \mathbb{R} \) whose Newton iteration function has a superattracting 2-cycle? | [
"x^4 - 2x^3 + x - 1"
] | [
"What is the simplest example of a polynomial function \\( f: \\mathbb{R} \\to \\mathbb{R} \\) whose Newton iteration function has a superattracting 2-cycle?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-caa245e87fcad87580ac4fb24cca0c1c832661c18312f79abe938bb4ae93d38a | What is the simplest example of a polynomial function \( f: \mathbb{R} \to \mathbb{R} \) whose Newton iteration function has an attracting 2-cycle? | [
"f(x) = x^3 - 2x + 2"
] | [
"What is the simplest example of a polynomial function \\( f: \\mathbb{R} \\to \\mathbb{R} \\) whose Newton iteration function has an attracting 2-cycle?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-10ae0117ee948e1e25647f400cecb751bc9b0ac36615b3b83c1fba35e83cb02a | What is the simplest way to find the derangement of 21 objects? Is there a better method for calculating derangements than using the formula \( n! \left[ 1 - \frac{1}{1!} + \frac{1}{2!} - \cdots + (-1)^n \frac{1}{n!} \right] \)? | [
"\\(\\left\\lfloor \\frac{21!}{e} + \\frac{1}{2} \\right\\rfloor\\)"
] | [
"What is the simplest way to find the derangement of 21 objects? Is there a better method for calculating derangements than using the formula \\( n! \\left[ 1 - \\frac{1}{1!} + \\frac{1}{2!} - \\cdots + (-1)^n \\frac{1}{n!} \\right] \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-c362b3d6fd9c62ca3253628519e4a5e116b878f3dc5161d3ed98e35a9fe55896 | What is the simplified value of the sum: $-1^{2004} + (-1)^{2005} + 1^{2006} -1^{2007}$? | [
"-2"
] | [
"What is the simplified value of the sum: $-1^{2004} + (-1)^{2005} + 1^{2006} -1^{2007}$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-eb8d13330394208eeae004bb06b472564343a04d55d5c8b8f17726c36ee4643e | What is the single digit obtained by repeatedly summing the digits of $5^{2009}$ until only one digit remains? | [
"2"
] | [
"What is the single digit obtained by repeatedly summing the digits of $5^{2009}$ until only one digit remains?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b268daf34cb6e6ec9580dd50a062b7fb8486e9a9545bca8ca41d33c576231275 | What is the sixth perfect number, and does it end in 28? | [
"8589869056, and it does not end in 28."
] | [
"What is the sixth perfect number, and does it end in 28?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 30
} | 34 | 0 | 0 |
omr-ws-c4bcbbb4133d9e3e392db9129c0ede7d16837a6c7588c3d9ad8b232179a12d91 | What is the size of the largest square that can fit inside a unit cube? | [
"\\(\\frac{3\\sqrt{2}}{4}\\)"
] | [
"What is the size of the largest square that can fit inside a unit cube?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 3,
"tir": 2
} | 8 | 0 | 0 |
omr-ws-c635d5af5dd68b9b9160083c6bd5c0b25c18e3b2b84a0423544e2e4c1edea2ed | What is the size of the largest subset \( S \) of \(\{1, 2, 3, \ldots, 50\}\) such that no pair of distinct elements of \( S \) has a sum divisible by 5? | [
"21"
] | [
"What is the size of the largest subset \\( S \\) of \\(\\{1, 2, 3, \\ldots, 50\\}\\) such that no pair of distinct elements of \\( S \\) has a sum divisible by 5?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-a1102182ea445210c1140375783e848646d7dc4b5212c31234a8e6e0ca72f898 | What is the size of the largest subset \( S \) of \(\{1, 2, 3, \ldots, 50\}\) such that no pair of distinct elements of \( S \) has a sum divisible by 7? | [
"23"
] | [
"What is the size of the largest subset \\( S \\) of \\(\\{1, 2, 3, \\ldots, 50\\}\\) such that no pair of distinct elements of \\( S \\) has a sum divisible by 7?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 33
} | 49 | 0 | 0 |
omr-ws-59a713b014d61082f2d06bb8c0caca05ea31c5331a9f134ebd5ad19563075dfd | What is the size of the largest subset \( S' \) of \( S = \{2^x 3^y 5^z : 0 \leq x, y, z \leq 4\} \) such that there are no distinct elements \( p, q \) in \( S' \) such that \( p \mid q \)? | [
"19"
] | [
"What is the size of the largest subset \\( S' \\) of \\( S = \\{2^x 3^y 5^z : 0 \\leq x, y, z \\leq 4\\} \\) such that there are no distinct elements \\( p, q \\) in \\( S' \\) such that \\( p \\mid q \\)?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 23
} | 46 | 0 | 0 |
omr-ws-2f058b6af2f16c4af484e2fca1c0f0a4e42ac6491741e5963a3ac4ab15450517 | What is the size of the largest subset \( S' \) of \( S = \{2^x3^y5^z : 0 \le x,y,z \le 4\} \) such that there are no distinct elements \( p, q \in S' \) with \( p \mid q \)? | [
"19"
] | [
"What is the size of the largest subset \\( S' \\) of \\( S = \\{2^x3^y5^z : 0 \\le x,y,z \\le 4\\} \\) such that there are no distinct elements \\( p, q \\in S' \\) with \\( p \\mid q \\)?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 46,
"genselect": 8,
"tir": 16
} | 70 | 0 | 0 |
omr-ws-f2efb50126091e1cd2acf7b25fb15f393ddf754be4ece267ab918900f5650216 | What is the size of the largest subset, $S$, of $\{1, 2, 3, \ldots, 50\}$ such that no pair of distinct elements of $S$ has a sum divisible by $7$? | [
"23"
] | [
"What is the size of the largest subset, $S$, of $\\{1, 2, 3, \\ldots, 50\\}$ such that no pair of distinct elements of $S$ has a sum divisible by $7$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 18,
"tir": 30
} | 48 | 0 | 0 |
omr-ws-327a102a06d0ebc82bcbb44ada2b75d8a732a63cae124924d7c6b8bb31658e91 | What is the size of the set $\{0^{42} \mod 101, 1^{42} \mod 101, 2^{42} \mod 101, \ldots, 100^{42} \mod 101\}$? | [
"51"
] | [
"What is the size of the set $\\{0^{42} \\mod 101, 1^{42} \\mod 101, 2^{42} \\mod 101, \\ldots, 100^{42} \\mod 101\\}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 23
} | 38 | 0 | 0 |
omr-ws-23544681579dda4c96f1144a6f60239b2ff089b7a44793f9996f31ccc1546173 | What is the slope of a line parallel to $2x+4y=-17$? Express your answer as a common fraction. | [
"-\\frac{1}{2}"
] | [
"What is the slope of a line parallel to $2x+4y=-17$? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3233acc62717c38ec628e713e38458f6f57c24952ec2004f99660119a4ca7343 | What is the slope of a line parallel to the line $2x - 4y = 9$? Express your answer as a common fraction. | [
"\\frac{1}{2}"
] | [
"What is the slope of a line parallel to the line $2x - 4y = 9$? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9e15aed1fd0e72e09cf22385ac76c58f2f561cc452a880f11aa41409d38399e4 | What is the slope of a line perpendicular to the line containing the points $(4,-7)$ and $(-5,-1)$? Express your answer as a common fraction. | [
"\\frac{3}{2}"
] | [
"What is the slope of a line perpendicular to the line containing the points $(4,-7)$ and $(-5,-1)$? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c958e3ff56908ef4c024eaa4ed425e33a66b7b0ee4895d2eb3edec4936df63a5 | What is the slope of the family of lines tangent to the curve \( y = x^2 \)? | [
"\\mathbb{R}"
] | [
"What is the slope of the family of lines tangent to the curve \\( y = x^2 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b078e22007895928f1b34d9f3d71e431d5f765eca369bfedfb38e7448205106e | What is the slope of the line $2y = -3x + 6$? | [
"-\\frac32"
] | [
"What is the slope of the line $2y = -3x + 6$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-35da4f2172d5e552dffaaa2de030d18de85411ce8c71e9ca34c802ae5634c2df | What is the slope of the line containing the midpoint of the segment with endpoints at (0, 0) and (2, 3) and the midpoint of the segment with endpoints at (5, 0) and (6, 3)? Express your answer in simplest form. | [
"0"
] | [
"What is the slope of the line containing the midpoint of the segment with endpoints at (0, 0) and (2, 3) and the midpoint of the segment with endpoints at (5, 0) and (6, 3)? Express your answer in simplest form."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ce0386b4fa10239b9e646e562de829d0cfdefe10f313a7d87eaae442fce97f9f | What is the slope of the line determined by any two solutions to the equation $\frac{2}{x}+\frac{3}{y} = 0$? Express your answer as a common fraction. | [
"- \\frac 32"
] | [
"What is the slope of the line determined by any two solutions to the equation $\\frac{2}{x}+\\frac{3}{y} = 0$? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-93c8fb854e04316d264be68715bc2fc304374bb594ecb28bea3ce1b0d908267e | What is the slope of the line passing through $(-3,5)$ and $(2,-5)$? | [
"-2"
] | [
"What is the slope of the line passing through $(-3,5)$ and $(2,-5)$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9957814861e2ee1e3bb8badf7b0429381b9c3c2ff809c67d07b8fbd720f2284a | What is the slope of the line perpendicular to $y=2x\sqrt{3}+8\sqrt{56}-3\sqrt{3}x$? | [
"\\dfrac{\\sqrt{3}}{3}"
] | [
"What is the slope of the line perpendicular to $y=2x\\sqrt{3}+8\\sqrt{56}-3\\sqrt{3}x$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-0ea167d0d618d0100d87444f87714b03af495bd6c61820305d6582c37289325c | What is the slope of the line tangent to the graph $x^2 + y^2 = 169$ at the point (-12, -5)? | [
"-12/5"
] | [
"What is the slope of the line tangent to the graph $x^2 + y^2 = 169$ at the point (-12, -5)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-bfa3ceb7d38ecde32d61acc3ecfd8c0bdaf29ba8c2570aebde2fd8b25461b316 | What is the slope of the line that bisects the angle formed by two lines with slopes 1 and 3? | [
"\\(\\frac{\\sqrt{5}+1}{2}\\)"
] | [
"What is the slope of the line that bisects the angle formed by two lines with slopes 1 and 3?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-2b4124f8684d4977cdd4495760d46cc6d1c6afc661a58b7159bc7016835395b7 | What is the slope of the line that bisects the angle formed by two lines with slopes 1 and 7? | [
"2"
] | [
"What is the slope of the line that bisects the angle formed by two lines with slopes 1 and 7?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b1c0ece03cb7b5d5dfbdbff91bd1745ff5b41bddc8b3a5e0bc7439ff97ac7ac2 | What is the slope of the line that is tangent to a circle at point (5,5) if the center of the circle is (3,2)? Express your answer as a common fraction. | [
"-\\frac{2}{3}"
] | [
"What is the slope of the line that is tangent to a circle at point (5,5) if the center of the circle is (3,2)? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c6c2d3d4f4e1488f458b6363a62924b2556be2fec647e494846e556f01465a32 | What is the slope of the spiral $r = \theta$ at $\theta = \pi/4$? | [
"\\(\\dfrac{4+\\pi}{4-\\pi}\\)"
] | [
"What is the slope of the spiral $r = \\theta$ at $\\theta = \\pi/4$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-b12aced76ee799e4d0966827964ba35d3ae2b010455bc59922de24ddc79cc737 | What is the slope of the tangent line to the curve $16x^2 - 9y^2 = 144$ at the point $P\left(3 \frac{3}{4}, 3\right)$? | [
"\\(\\frac{20}{9}\\)"
] | [
"What is the slope of the tangent line to the curve $16x^2 - 9y^2 = 144$ at the point $P\\left(3 \\frac{3}{4}, 3\\right)$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-c71eefd4a7494362d6c01e1f0fabdc764ad91bdfcd1f4a94117fe618ff0dd65f | What is the smallest $ n$ so that $ 71$ divides $ 7n + 1$? | [
"10"
] | [
"What is the smallest $ n$ so that $ 71$ divides $ 7n + 1$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-5ddff4857903c7c2c46bd29c1860389afc13776804d817de0dc3ffdc0059b6af | What is the smallest $\epsilon > 0$ for which, for all $N \in \mathbb{N}$, any rational number can be represented as $\frac{1}{a_1} + \dots + \frac{1}{a_n}$ with $a_i > (2-\epsilon)a_{i-1}$ and $a_i \in \mathbb{Z}$ for all $i$, and $a_1 > N$? | [
"1"
] | [
"What is the smallest $\\epsilon > 0$ for which, for all $N \\in \\mathbb{N}$, any rational number can be represented as $\\frac{1}{a_1} + \\dots + \\frac{1}{a_n}$ with $a_i > (2-\\epsilon)a_{i-1}$ and $a_i \\in \\mathbb{Z}$ for all $i$, and $a_1 > N$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-f0ebb38a494ae295d8382cde5291e9bc7e8bee94c324cc2840355be1ca9cbbef | What is the smallest $n > 1$ for which the average of the first $n$ (non-zero) squares is a square? | [
"337"
] | [
"What is the smallest $n > 1$ for which the average of the first $n$ (non-zero) squares is a square?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
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