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-803e51ec1b536d64742c80da7ab624aa9cfe67c8b4dbd7bfbbfbfbb077a39f37 | What is the absolute maximum/minimum of the function $20p - 2p^2$? | [
"50"
] | [
"What is the absolute maximum/minimum of the function $20p - 2p^2$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-faf2833a8d4a494a9162f51141b994dacd8536ccc09783a9cbeffad2ea8f5928 | What is the absolute value of the difference between the squares of 101 and 99? | [
"400"
] | [
"What is the absolute value of the difference between the squares of 101 and 99?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f0638eda36b2cdcabcd4562271ff9e281aa79ea87bc867fac3597b0e811dfd1d | What is the absolute value of the sum of the $2022$nd powers of the roots of the polynomial $x^{2022} + 3x^2 - 2x + 1$? | [
"2022"
] | [
"What is the absolute value of the sum of the $2022$nd powers of the roots of the polynomial $x^{2022} + 3x^2 - 2x + 1$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5eb8b7b3b09b0c9914c517ffc1e31d16d995a89fed0cb435fec08e680cac307e | What is the actual rate of divergence for the series \( \sum \frac{1}{n} \)? | [
"\\( \\ln(n) + \\gamma \\)"
] | [
"What is the actual rate of divergence for the series \\( \\sum \\frac{1}{n} \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 5,
"tir": 18
} | 30 | 0 | 0 |
omr-ws-8c38b0c59d90b3ade9bf639256f547022369511166c859527d5dd0c8821b0a9e | What is the acute angle between the main diagonals of an icosahedron? | [
"$\\theta \\doteq 63.435^\\circ$",
"\\arctan(2)"
] | [
"What is the acute angle between the main diagonals of an icosahedron?"
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 21,
"genselect": 4,
"tir": 25
} | 50 | 0 | 0 |
omr-ws-7fbb5923e759743dda1f56248373ce9816451f8e4cd29542cd7fa1a758d3360c | What is the acute angle of a rhombus if its side is the geometric mean of its diagonals? | [
"30*",
"30^\\circ"
] | [
"What is the acute angle of a rhombus if its side is the geometric mean of its diagonals?"
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 27,
"genselect": 6,
"tir": 12
} | 45 | 0 | 0 |
omr-ws-f5ddd2a617c181790e879c42b09c50195adca3bee7e6a1f0c72054908aab3213 | What is the algebra generated by the nilpotent matrices in $M_n(\mathbb{C})$? | [
"\\( M_n(\\mathbb{C}) \\)"
] | [
"What is the algebra generated by the nilpotent matrices in $M_n(\\mathbb{C})$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 15
} | 30 | 0 | 0 |
omr-ws-f2b20951e7b44e2a76c9aff819b39e52c2989575c153d8e410834438de72d50e | What is the algebraic condition for the plane \(Ax + By + Cz + 1 = 0\) not to intersect the ellipsoid \(\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1\)? | [
"\\((aA)^2 + (bB)^2 + (cC)^2 < 1\\)"
] | [
"What is the algebraic condition for the plane \\(Ax + By + Cz + 1 = 0\\) not to intersect the ellipsoid \\(\\frac{x^2}{a^2} + \\frac{y^2}{b^2} + \\frac{z^2}{c^2} = 1\\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 16,
"genselect": 5,
"tir": 1
} | 22 | 0 | 0 |
omr-ws-65ee2d688a5d62b2ac7a52709c470d19457cf26317bc541540b8b2b6e2e86716 | What is the algebraic double dual of $\mathbb{R}^{(\infty)}$ (sequences with almost every term zero)? | [
"\\( \\mathbb{R}^{\\mathbb{R}} \\)"
] | [
"What is the algebraic double dual of $\\mathbb{R}^{(\\infty)}$ (sequences with almost every term zero)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 1
} | 2 | 0 | 0 |
omr-ws-e28083dc2bcbca5f83d7cf355802086bc74b959d766506e74b28fe94ccd6faf1 | What is the altitude to the hypotenuse of a right triangle with legs 3 and 4? | [
"4.1176"
] | [
"What is the altitude to the hypotenuse of a right triangle with legs 3 and 4?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-678b141234dc02744a97e35d8e081a38565bcbcae97bb8d6a8c06815bed2fb97 | What is the amalgamated product of the homomorphisms \( a: \mathbb{Z} \times \mathbb{Z} \to \mathbb{Z} \) and \( b: \mathbb{Z} \times \mathbb{Z} \to \mathbb{Z} \), where \( a(m,n) = n \) and \( b(m,n) = km + pn \)? | [
"\\mathbb{Z}/k\\mathbb{Z}"
] | [
"What is the amalgamated product of the homomorphisms \\( a: \\mathbb{Z} \\times \\mathbb{Z} \\to \\mathbb{Z} \\) and \\( b: \\mathbb{Z} \\times \\mathbb{Z} \\to \\mathbb{Z} \\), where \\( a(m,n) = n \\) and \\( b(m,n) = km + pn \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-2b52558421133b14ad5276c8d3096546d86b59a58916fac178f469f166077316 | What is the amplitude of the equation $y = 4 \sin{(3x)} - 6 \cos{\left(2x + \frac{\pi}{4}\right)}$? | [
"5\\sqrt{2}"
] | [
"What is the amplitude of the equation $y = 4 \\sin{(3x)} - 6 \\cos{\\left(2x + \\frac{\\pi}{4}\\right)}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 3
} | 14 | 0 | 0 |
omr-ws-2c2e25d134cdf37f7ba9e3f06bc6f903a4689e60b9f9d56bec92cd1c3707f47e | What is the amplitude of the function \( f(x) = \sin(2x) + \cos(3x) - \tan(2x) \)? | [
"\\text{Undefined}"
] | [
"What is the amplitude of the function \\( f(x) = \\sin(2x) + \\cos(3x) - \\tan(2x) \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 32
} | 32 | 0 | 0 |
omr-ws-3a4cf1d7ff6ff79e361e181e55ccb2e066096d7ef69c8db2d7cffacfd4ec30ed | What is the angle $B$ in triangle $ABC$ if the distance between the feet of the altitudes drawn from vertices $A$ and $C$ is equal to half the radius of the circumscribed circle about triangle $ABC$? | [
"75^\\circ"
] | [
"What is the angle $B$ in triangle $ABC$ if the distance between the feet of the altitudes drawn from vertices $A$ and $C$ is equal to half the radius of the circumscribed circle about triangle $ABC$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-ff247041dddba5540d2977a0d0a722d7f0d19e2c7357c4434138a42cc0ec0d70 | What is the angle between a plane intersecting two parallel faces of a cube along their diagonals and a plane intersecting two other parallel faces of the cube along their diagonals? | [
"60^\\circ"
] | [
"What is the angle between a plane intersecting two parallel faces of a cube along their diagonals and a plane intersecting two other parallel faces of the cube along their diagonals?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c078e56a7d4e0b8f799fb19125299c7f199d211abbe55ba3eeaaa2a86a2f349e | What is the angle between the vectors \( v + w \) and \( v - w \) for any vectors \( v \) and \( w \)? | [
"\\( \\pi/2 \\)"
] | [
"What is the angle between the vectors \\( v + w \\) and \\( v - w \\) for any vectors \\( v \\) and \\( w \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 8,
"tir": 1
} | 9 | 0 | 0 |
omr-ws-8da527c7b0f214a42a33a0d491a1d854b5015859ef21943b03fab224b04feedb | What is the angle between two intersecting planes $\Pi_{1}: x = x_{0,1} + L_{1}\lambda$ and $\Pi_{2}: x = x_{0,2} + L_{2}\eta$ in $\mathbb{R}^{4}$, where $L_{1}, L_{2} \in \mathbb{R}^{4 \times 2}$? | [
"\\arccos\\left( \\sigma_{\\max}(Q_1^\\top Q_2) \\right)"
] | [
"What is the angle between two intersecting planes $\\Pi_{1}: x = x_{0,1} + L_{1}\\lambda$ and $\\Pi_{2}: x = x_{0,2} + L_{2}\\eta$ in $\\mathbb{R}^{4}$, where $L_{1}, L_{2} \\in \\mathbb{R}^{4 \\times 2}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 5
} | 17 | 0 | 0 |
omr-ws-3662eaa6a6f0fe2d187a4e00d81077dd65b83dee3d0b610e3fa0c54afff72245 | What is the angle formed between the lines $Ax + By + C = 0$ and $Rx + Sy + T = 0$? | [
"\\(\\theta = \\tan^{-1} \\left( \\left| \\frac{RA - BS}{AS + BR} \\right| \\right)\\)"
] | [
"What is the angle formed between the lines $Ax + By + C = 0$ and $Rx + Sy + T = 0$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 6,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-0bb9b086db2073c5b01f30d81634f84d588d738fe7ada6499de983a2488b7a0f | What is the angle formed by the diagonals of a cube? | [
"arccos 1/3"
] | [
"What is the angle formed by the diagonals of a cube?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-17d2e8909fba029108753d0dd1112266b87c52206d54b574dd14c656db54bdca | What is the angle formed by the lines \( y = 4x + 3 \) and \( y = 3x - 2 \)? | [
"\\(\\arctan\\left(\\frac{1}{13}\\right)\\)"
] | [
"What is the angle formed by the lines \\( y = 4x + 3 \\) and \\( y = 3x - 2 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 6,
"tir": 12
} | 32 | 0 | 0 |
omr-ws-b3a5f7eed2c991f5d4fb78f1c82bda8a4a970e63bc19b65d640a37bf3f5a95be | What is the angle of elevation of the sun when a 35-ft mast casts a 20-ft shadow? | [
"\\(\\tan^{-1}\\left(\\frac{7}{4}\\right) \\approx 60.255^\\circ\\)"
] | [
"What is the angle of elevation of the sun when a 35-ft mast casts a 20-ft shadow?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-7f9e30b3930098db9ba523f5e81fb059c54a57084db404ac76d3ccc14f86dddc | What is the angle of rotation from the parent function \(5x^2 + 8xy + 3y^2 + 4 = 0\)? | [
"\\dfrac{1}{2} \\arctan(4)"
] | [
"What is the angle of rotation from the parent function \\(5x^2 + 8xy + 3y^2 + 4 = 0\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6367a2aa5f61f8b3aa3bbe63737e30267c833fb417abfe9e68d7495aabdcfdac | What is the angle of rotation produced by the transformation $w = \frac{1}{z}$ at the point $z = i$? | [
"0"
] | [
"What is the angle of rotation produced by the transformation $w = \\frac{1}{z}$ at the point $z = i$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-f795bb1542bdd3f3fb6b506e807a4ff404fd3199443750de032021262cf3c7eb | What is the appropriate substitution for the integral \(\int \frac{\sqrt{x^2 - x^4}}{x^5} \, dx\)? | [
"\\( x = \\frac{1}{t} \\)"
] | [
"What is the appropriate substitution for the integral \\(\\int \\frac{\\sqrt{x^2 - x^4}}{x^5} \\, dx\\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 1
} | 22 | 0 | 0 |
omr-ws-162ef01644928f085be7966cd4f999a0562e8cc5621ddd6e7a70630413edda24 | What is the approximate multiple value of the sides of a regular pentagon inscribed in a circle with radius $r=1$? | [
"\\( \\sqrt{\\frac{5-\\sqrt{5}}{2}} \\)"
] | [
"What is the approximate multiple value of the sides of a regular pentagon inscribed in a circle with radius $r=1$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 3,
"tir": 4
} | 17 | 0 | 0 |
omr-ws-387d58b15ee1014e1cf7542226139f6d5fa83d6f6086ddd7dcefaa294d0c5a9e | What is the approximate radius of the circle which is the intersection of the plane \(x + y + z = 1\) and the sphere \(x^2 + y^2 + z^2 = 1\)? | [
"0.8164"
] | [
"What is the approximate radius of the circle which is the intersection of the plane \\(x + y + z = 1\\) and the sphere \\(x^2 + y^2 + z^2 = 1\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-4d6622632b1402362360d6669e299a69d3a66077c5a3e3ac4654015195bd2146 | What is the approximate radius of the smallest circle that can be circumscribed about a 223-sided polygon with side lengths 1776, 1777, ..., 1998? | [
"66973"
] | [
"What is the approximate radius of the smallest circle that can be circumscribed about a 223-sided polygon with side lengths 1776, 1777, ..., 1998?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 6
} | 20 | 0 | 0 |
omr-ws-ffdc632c6b6c1edb478775d879a9a63724c3e5bb77879c4dd6aa280d0ba829f8 | What is the approximate value of the following integral? $$\frac{1}{2 \pi} \int_{-\pi}^{\pi} \frac{1}{2021} \left(\frac{\sin\left(\frac{2021x}{2}\right)}{\sin\left(\frac{x}{2}\right)}\right)^2 \frac{\sqrt{|x|+64}}{1 + x^2} \, dx$$ | [
"8"
] | [
"What is the approximate value of the following integral?\n$$\\frac{1}{2 \\pi} \\int_{-\\pi}^{\\pi} \\frac{1}{2021} \\left(\\frac{\\sin\\left(\\frac{2021x}{2}\\right)}{\\sin\\left(\\frac{x}{2}\\right)}\\right)^2 \\frac{\\sqrt{|x|+64}}{1 + x^2} \\, dx$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-e1be230c3ae7eae3db708a594968cf3f080e3e9a1eddc748c3a74726799e782f | What is the approximate value of the product $1^1 \cdot 2^2 \cdot 3^3 \cdots n^n$? | [
"\\( P \\approx A n^{n^2/2 + n/2 + 1/12} e^{-n^2/4} \\)"
] | [
"What is the approximate value of the product $1^1 \\cdot 2^2 \\cdot 3^3 \\cdots n^n$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 6,
"tir": 8
} | 21 | 0 | 0 |
omr-ws-06d9d1f71cfe4b483163d2dc723bbf33cb35644b91c045d5d32c16025d21e3f2 | What is the arc length of \( g(y) = y^{0.5} \) for \( y \in [0, 5] \)? | [
"6.33 units"
] | [
"What is the arc length of \\( g(y) = y^{0.5} \\) for \\( y \\in [0, 5] \\)?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2215884e3aabc8a71ca04effa7b90fac5bb5ee5c2984322c9cf4af56842df70a | What is the area bounded by the curves \( \arg(z) = \frac{\pi}{3} \), \( \arg(z) = \frac{2\pi}{3} \), and \( \arg(z - 2 - 2\sqrt{3}i) = \pi \) on the complex plane? | [
"\\(4\\sqrt{3}\\)"
] | [
"What is the area bounded by the curves \\( \\arg(z) = \\frac{\\pi}{3} \\), \\( \\arg(z) = \\frac{2\\pi}{3} \\), and \\( \\arg(z - 2 - 2\\sqrt{3}i) = \\pi \\) on the complex plane?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 25
} | 73 | 0 | 0 |
omr-ws-6a7da21204204cc1fbc72f6e1e13ac90ae24333ca9ffaea70709956a55cedbf5 | What is the area bounded by the lines $x=1$, $x=2$, $y=0$, and $y=mx+b$ if $m+b=4$ and $2m+b=7$? | [
"5.5"
] | [
"What is the area bounded by the lines $x=1$, $x=2$, $y=0$, and $y=mx+b$ if $m+b=4$ and $2m+b=7$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-4174e5b10bd5ccba88c2c28081580fef01715e41f320ec939aaf58f23c6ef36d | What is the area bounded by the region $|x+y|+|x-y|=4$, where $x$ and $y$ are real numbers? | [
"16"
] | [
"What is the area bounded by the region $|x+y|+|x-y|=4$, where $x$ and $y$ are real numbers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4a6e2895ccdd3165c0287fd17cec0a6954bc45e6f6eff54d74a92b5572c19873 | What is the area covered by the graph of $|||||x|-4|-3|-2|-1|+|||||y|-4|-3|-2|-1|<1$? | [
"4"
] | [
"What is the area covered by the graph of $|||||x|-4|-3|-2|-1|+|||||y|-4|-3|-2|-1|<1$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-d61507537d8d09c3f373e234e7137a9e88c4acbccd1a218138b5b6a26786fe70 | What is the area enclosed by the curve $|x+y-1|+|2x+y-1|=1$? | [
"2"
] | [
"What is the area enclosed by the curve $|x+y-1|+|2x+y-1|=1$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-718674b098bf7dc5efdfd24f5145763a5fefc7dde3d06d54d5666be50279cfda | What is the area enclosed by the equation $5x^{2}+y^{2}+3xy-x+4y-3=0$? | [
"\\(\\frac{252 \\pi \\sqrt{11}}{121}\\)"
] | [
"What is the area enclosed by the equation $5x^{2}+y^{2}+3xy-x+4y-3=0$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 17
} | 38 | 0 | 0 |
omr-ws-1d50c4c9b18dad6a2b8d1aaa8bc451274a7407256e6c926e3cec7e7130d721d0 | What is the area enclosed by the graph $r=\frac{6}{3+\cos{\theta}}$? | [
"\\dfrac{27\\sqrt{2}}{8} \\pi"
] | [
"What is the area enclosed by the graph $r=\\frac{6}{3+\\cos{\\theta}}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-34bbb1cce5ce1522af2282e0934d278da5f0fcba150917463b9ef3f45fb76f59 | What is the area enclosed by the graph of $|3x|+|4y|=12$? | [
"24"
] | [
"What is the area enclosed by the graph of $|3x|+|4y|=12$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cb2488787de06bc46022a7a3f55aabccd18434ba4d25f1844521804e471353cf | What is the area enclosed by the graph of \(x^2 + y^2 = 4|3x + y| + 4|x + 3y|\)? | [
"\\(128\\pi-96\\text{arctan}\\left(\\frac43\\right)\\)"
] | [
"What is the area enclosed by the graph of \\(x^2 + y^2 = 4|3x + y| + 4|x + 3y|\\)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-275dee5a23ab906862a982e7f50aff65fc5c24e13fe329fc8033ceac76c44d5c | What is the area enclosed by the region defined by the equation $x^2+y^2+10x+24y=0$? | [
"169\\pi"
] | [
"What is the area enclosed by the region defined by the equation $x^2+y^2+10x+24y=0$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4f776f8ad02ee7c6f40a4bcd7d19aca744c9ef04954e509d853f505b015d0eb7 | What is the area enclosed by the region defined by the equation $x^2+y^2+12x+16y=0$? | [
"100\\pi"
] | [
"What is the area enclosed by the region defined by the equation $x^2+y^2+12x+16y=0$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b73307ac018792e77002a790ebf2b82ba3788cf6b007901663b195fb59ee7cf3 | What is the area enclosed by the region defined by the equation $x^2+y^2+6x+8y=0$? | [
"25\\pi"
] | [
"What is the area enclosed by the region defined by the equation $x^2+y^2+6x+8y=0$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e4201ff9abe6948d593977ccef4397df5ea1ca9c8c37c68a8e7779fad3a1c93e | What is the area enclosed by the region in the first quadrant bounded by the x-axis, the line \( y = 2x \), and the circles \( x^2 + y^2 = 20 \) and \( x^2 + y^2 = 30 \)? | [
"\\( 5 \\arctan(2) \\)"
] | [
"What is the area enclosed by the region in the first quadrant bounded by the x-axis, the line \\( y = 2x \\), and the circles \\( x^2 + y^2 = 20 \\) and \\( x^2 + y^2 = 30 \\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-f4e6eeee42b4e51286407d398da2bcd9d7bfd0ebaabbdd542993296d29ba2fcc | What is the area formed by the line \(2x - 7y = 7\sqrt{-\ln(e^{-4})}\) and the coordinate axes? | [
"\\(\\frac{1}{7}\\)"
] | [
"What is the area formed by the line \\(2x - 7y = 7\\sqrt{-\\ln(e^{-4})}\\) and the coordinate axes?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6119a176c593fc5a6e89fa28fdc6283c9d3874a6fd1f47c0fd9cb2d99ddf857f | What is the area in square feet and acres of a quadrilateral with side lengths 68', 50', 64', and 30'? | [
"0.06"
] | [
"What is the area in square feet and acres of a quadrilateral with side lengths 68', 50', 64', and 30'?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-baa6b6c76c3ebfcc7535a8adaab3cc2d428394d538ad948572e06e2623e9c503 | What is the area in square meters of the smallest circular rug that can contain both a 2x3 m rectangular rug and a 1x4 m rectangular rug without them overlapping? | [
"\\( \\frac{3145}{576} \\pi \\)"
] | [
"What is the area in square meters of the smallest circular rug that can contain both a 2x3 m rectangular rug and a 1x4 m rectangular rug without them overlapping?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b5098c9361817b79919d3013f7bc9462389c0f2115c3ff090f83c9e8f29e9077 | What is the area in the xy-plane bounded by $x^2 + \frac{y^2}{3} \le 1$ and $\frac{x^2}{3} + y^2 \le 1$? | [
"\\dfrac{2\\sqrt{3}}{3} \\pi"
] | [
"What is the area in the xy-plane bounded by $x^2 + \\frac{y^2}{3} \\le 1$ and $\\frac{x^2}{3} + y^2 \\le 1$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4ad3e8226851322e788cc3a5f24537611398288b33de258f344023627972c5c0 | What is the area of $\vartriangle ABI_a$ given the formula: \[ \vartriangle ABC = \vartriangle ABI_a + \vartriangle ACI_a - \vartriangle BCI_a = \frac{1}{2}b \cdot r_a + \frac{1}{2}c \cdot r_a - \frac{1}{2}a \cdot r_a? \] | [
"\\(\\frac{1}{2} c r_a\\)"
] | [
"What is the area of $\\vartriangle ABI_a$ given the formula:\n\\[\n\\vartriangle ABC = \\vartriangle ABI_a + \\vartriangle ACI_a - \\vartriangle BCI_a = \\frac{1}{2}b \\cdot r_a + \\frac{1}{2}c \\cdot r_a - \\frac{1}{2}a \\cdot r_a?\n\\]"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 5,
"tir": 3
} | 8 | 0 | 0 |
omr-ws-240364da035b575de16a97ec6db11bfbac6a6f8d104f16b597b45e6afb4f5f0a | What is the area of a convex octagon inscribed in a circle where four consecutive sides have length 3 and the other four sides have length 2? | [
"13 + 12\\sqrt{2}"
] | [
"What is the area of a convex octagon inscribed in a circle where four consecutive sides have length 3 and the other four sides have length 2?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3da4f99e0a9ba32da377089d8dfd15067227c793be9288fce9d475e8a27771f7 | What is the area of a convex pentagon if every diagonal cuts off a triangle of unit area? | [
"5"
] | [
"What is the area of a convex pentagon if every diagonal cuts off a triangle of unit area?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-18e13e2907d9a4d645e45a829d21dd919496f358199af19fb38588206ba9162f | What is the area of a convex pentagon inscribed in a circle with side lengths 16, 16, 16, 16, and 6? | [
"\\( 123 \\sqrt{7} \\)"
] | [
"What is the area of a convex pentagon inscribed in a circle with side lengths 16, 16, 16, 16, and 6?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 43,
"genselect": 7,
"tir": 13
} | 63 | 0 | 0 |
omr-ws-8eb95559b0cf00458ca4bc817048b28ed29609ab09f53b68bd6bd204aa323bcd | What is the area of a cyclic quadrilateral with side lengths 2, 5, 7, and 12? | [
"\\( 4\\sqrt{33} \\)"
] | [
"What is the area of a cyclic quadrilateral with side lengths 2, 5, 7, and 12?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-fb831b2e78ea592d399cb75e24288a87b6bb17da9a0fcd528fdeea75ec207bce | What is the area of a decagon that is perfectly inscribed in a circle with an area of $64\pi$? | [
"\\(5\\sqrt{5-\\sqrt{5}}\\)"
] | [
"What is the area of a decagon that is perfectly inscribed in a circle with an area of $64\\pi$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e8a1e6ac87707e75e9d59fbc874e13a2f7844182f39bc26a06dca4db87b8d06e | What is the area of a four-pointed star formed by joining the midpoints of each side of a square with side length 1 metre to the corners of the square? | [
"\\dfrac{1}{2}"
] | [
"What is the area of a four-pointed star formed by joining the midpoints of each side of a square with side length 1 metre to the corners of the square?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-2981ace42cf5c8ab464e330ad1028f401d4d3b6fc74654c810e1fe727d815a9b | What is the area of a hexagon inscribed in a circle with edge lengths proceeding clockwise as $1, 1, 1, 2, 2, 2$? | [
"\\(\\frac{13}{4}\\sqrt{3}\\)"
] | [
"What is the area of a hexagon inscribed in a circle with edge lengths proceeding clockwise as $1, 1, 1, 2, 2, 2$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 20
} | 35 | 0 | 0 |
omr-ws-6496577296406bdb4afd235504185d715d850d295d90aba4ff4592995bb5b23f | What is the area of a hexagon with all angles equal and sides of lengths 1, 2, 3, 4, 5, and 6 in that order? | [
"16\\sqrt{3}"
] | [
"What is the area of a hexagon with all angles equal and sides of lengths 1, 2, 3, 4, 5, and 6 in that order?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 4
} | 10 | 0 | 0 |
omr-ws-c325481038b4a3dfba442df65573a8c2d8fb86d5db6487bfa233f695805abef8 | What is the area of a hexagon with consecutive angle measures of 90°, 120°, 150°, 90°, 120°, and 150°, where all sides are 4 units in length? | [
"\\(24 + 8\\sqrt{3}\\)"
] | [
"What is the area of a hexagon with consecutive angle measures of 90°, 120°, 150°, 90°, 120°, and 150°, where all sides are 4 units in length?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 28
} | 51 | 0 | 0 |
omr-ws-79e78ab727cec06843292bdea4fb4a40e69c85496fb01c38e32284dd4be6cd26 | What is the area of a hexagon with sides of lengths 1, 2, 3, 4, 5, and 6 and equal angles? | [
"\\dfrac{65\\sqrt{3}}{4}"
] | [
"What is the area of a hexagon with sides of lengths 1, 2, 3, 4, 5, and 6 and equal angles?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 3
} | 14 | 0 | 0 |
omr-ws-fae896d82ece96e334b70666342c1e9ae517a7cc55cc919e1f7205bad23e6030 | What is the area of a pentagon with all angles equal to 108 degrees in terms of the side length \( x \)? | [
"\\( \\frac{x^2}{4} \\sqrt{25 + 10\\sqrt{5}} \\)"
] | [
"What is the area of a pentagon with all angles equal to 108 degrees in terms of the side length \\( x \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 19
} | 34 | 0 | 0 |
omr-ws-2a21c26f9a0d8abf813bae2c6827bcd15e259ef220ec51d43aeade46e573a113 | What is the area of a polygon with vertices at $(4,2)$, $(8,3)$, $(-1,-6)$, and $(5,-4)$? | [
"31.5"
] | [
"What is the area of a polygon with vertices at $(4,2)$, $(8,3)$, $(-1,-6)$, and $(5,-4)$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 8,
"tir": 3
} | 22 | 0 | 0 |
omr-ws-65f473357e702cb2122edb8355865fd39f3c5f387714ddbb22907e5d056ee998 | What is the area of a polygon with vertices at: $(0,0)$, $(3,4)$, $(6,0)$, $(3,-4)$, and $(0,3)$? | [
"30"
] | [
"What is the area of a polygon with vertices at: $(0,0)$, $(3,4)$, $(6,0)$, $(3,-4)$, and $(0,3)$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1fb91c00308b9122d5935d81cb976db024440e1c66c3e857a39098b952195f6d | What is the area of a polygon with vertices at: (0,0), (3,4), (6,0), (3,-4), and (0,3)? | [
"28.5"
] | [
"What is the area of a polygon with vertices at: (0,0), (3,4), (6,0), (3,-4), and (0,3)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fd6573f53ff099109063df512dc15a8b9004096979bba1abd4b7d385ed381ab5 | What is the area of a regular n-gon inscribed inside a unit circle? | [
"\\(\\frac{n \\sin\\left(\\frac{2\\pi}{n}\\right)}{2}\\)"
] | [
"What is the area of a regular n-gon inscribed inside a unit circle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-6c91c034392aaf022873c51f38613d4dfe95c84b164b15658c62a68b2b3b002b | What is the area of a regular octagon with a side measuring 8 cm? | [
"\\( 128 (\\sqrt{2} + 1) \\)"
] | [
"What is the area of a regular octagon with a side measuring 8 cm?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-8034dc1fa0ba38b3c381f6094572b9497dcd311a4ccf61f3871f26d11e29a676 | What is the area of a regular pentagon circumscribed about a circle with radius 25? | [
"\\( 3125 \\cot 52 \\) or \\( 3125 (\\sin 54)(\\sin 36) \\)"
] | [
"What is the area of a regular pentagon circumscribed about a circle with radius 25?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2451743a8793df0d2e979af4132b2c5f5d66648c6254635425d0cc474aaa2433 | What is the area of a semicircle tangent to two adjacent sides of a square, where the endpoints of the diameter lie on the other two sides of the square? | [
"\\((3 - 2\\sqrt{2})\\pi\\)"
] | [
"What is the area of a semicircle tangent to two adjacent sides of a square, where the endpoints of the diameter lie on the other two sides of the square?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 11
} | 11 | 0 | 0 |
omr-ws-3285a204d58556e0675dab1a5c0cde2ed0376e59850b79c4da83dbce88f3cc5c | What is the area of a semicircular region tangent to two sides of a unit square, with endpoints of its diameter on the other two sides? | [
"\\((3-2\\sqrt{2})\\pi\\)"
] | [
"What is the area of a semicircular region tangent to two sides of a unit square, with endpoints of its diameter on the other two sides?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-29523bb35c5c17bf9b77e30b93ed762f3089b828c58f8411d21f0b4d993541d2 | What is the area of a semicircular region that is tangent to two sides of a unit square, with the endpoints of its diameter on the other two sides? | [
"\\(\\pi (3 - 2\\sqrt{2})\\)"
] | [
"What is the area of a semicircular region that is tangent to two sides of a unit square, with the endpoints of its diameter on the other two sides?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-6eb91f5c59ad3baf82b88ff957f45b44b6ddd827cd0f5f90c7512231cf1df48f | What is the area of a square inscribed in the ellipse given by the equation $16x^2 + 9y^2 = 144$? | [
"\\(\\frac{576}{25}\\)"
] | [
"What is the area of a square inscribed in the ellipse given by the equation $16x^2 + 9y^2 = 144$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 21
} | 34 | 0 | 0 |
omr-ws-f09fad06c9ed841233cbe403fcc36dd0fd1108d4faf428aa9223e7e9ff1a00e1 | What is the area of a square whose diagonal is one unit longer than the length of its side? | [
"\\(3 + 2\\sqrt{2}\\)"
] | [
"What is the area of a square whose diagonal is one unit longer than the length of its side?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-7b6338d971f0d2d74ed9feb4c48726e97f9d047c9bf54bc5c186cefd485c484e | What is the area of a trapezoid if the parallel bases are 10 and 24 and the sides are 13 and 15? | [
"204"
] | [
"What is the area of a trapezoid if the parallel bases are 10 and 24 and the sides are 13 and 15?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-35d39720eadbbd79c2a17bda5c3bf0ca2bc1d419fbc797486a3c105dc9cfa3bb | What is the area of a trapezoid with corner points (2,5), (5,3), (1,8), and (7,6)? | [
"\\dfrac{29}{2}"
] | [
"What is the area of a trapezoid with corner points (2,5), (5,3), (1,8), and (7,6)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2bbcf50ed8bbe69bd9f0ebd0f4bb4cb34e365502503a64e122bff2a21b83a8cf | What is the area of a trapezoid with sides of length 15, 17, 37, and 61, in clockwise order? | [
"468"
] | [
"What is the area of a trapezoid with sides of length 15, 17, 37, and 61, in clockwise order?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-27c441fe80b60a576c1c7515b0cc215fcb8558cac0083e16bb6eb192fd15185d | What is the area of a triangle bounded by the lines \(X - 2Y = 12\), \(X + 3Y = 9\), and \(4X - Y = -16\) using Heron's Formula? | [
"\\dfrac{6877}{70}"
] | [
"What is the area of a triangle bounded by the lines \\(X - 2Y = 12\\), \\(X + 3Y = 9\\), and \\(4X - Y = -16\\) using Heron's Formula?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-410ab78c7e3f57d928c5f5de575e0fe2a703d5c8d147ad06b7773d477c8bf8c5 | What is the area of a triangle inscribed in a circle of radius \( R \), in terms of the angles \( A, B, \) and \( C \) of the triangle? | [
"\\( 2R^2 \\sin A \\sin B \\sin C \\)"
] | [
"What is the area of a triangle inscribed in a circle of radius \\( R \\), in terms of the angles \\( A, B, \\) and \\( C \\) of the triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 12
} | 27 | 0 | 0 |
omr-ws-ddfb0082c476c22b230b46ec305fad506046f0486afd7f15b33e9334595412f4 | What is the area of a triangle with altitudes of lengths $12$, $15$, and $20$? | [
"150"
] | [
"What is the area of a triangle with altitudes of lengths $12$, $15$, and $20$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 4
} | 19 | 0 | 0 |
omr-ws-c0a6d6ad2c5e41bdf4c5d9d2309200a90c0ba6b5ed3fdf9607b2ecbac11fa932 | What is the area of a triangle with median lengths 18, 21, and 24? | [
"243.728"
] | [
"What is the area of a triangle with median lengths 18, 21, and 24?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 5,
"tir": 7
} | 18 | 0 | 0 |
omr-ws-c7f1039af53e07c316fad40d40f2969dfa1c6d40b8e0d67c5416b5a596c047eb | What is the area of a triangle with side lengths $17$, $25$, and $26$? | [
"204"
] | [
"What is the area of a triangle with side lengths $17$, $25$, and $26$?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-83927cfe01a3b19cf3902cefdf8eb568a46be8f2b3f4a434632192bb7869e155 | What is the area of a triangle with side lengths 7, 17, and 19 using Heron's formula? | [
"59.22"
] | [
"What is the area of a triangle with side lengths 7, 17, and 19 using Heron's formula?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 17
} | 18 | 0 | 0 |
omr-ws-9a3ef43d1c203de086c2c2ac088ca6ef6f68710f55cc20c261a334a3ccd4f1f7 | What is the area of a triangle with vertices at the points (-8,5), (3,1), and (5,-4)? | [
"\\dfrac{47}{2}"
] | [
"What is the area of a triangle with vertices at the points (-8,5), (3,1), and (5,-4)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4cdac90f81c00efe20f057407c7827748b68d79cfbaa05677e1e49f4fb0218aa | What is the area of a triangle with vertices at the points \((1,3,4)\), \((-2,3,5)\), and \((1,-1,2)\)? | [
"7"
] | [
"What is the area of a triangle with vertices at the points \\((1,3,4)\\), \\((-2,3,5)\\), and \\((1,-1,2)\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3b8d1e3d38467015c9dc23227ffe1af42c323145a94f4113b7bee0c35f7f3f78 | What is the area of all 50 six-sided stars, each made by putting two equilateral triangles with a radius of one inch together, in square inches? | [
"50√3"
] | [
"What is the area of all 50 six-sided stars, each made by putting two equilateral triangles with a radius of one inch together, in square inches?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 8,
"tir": 19
} | 38 | 0 | 0 |
omr-ws-341a1245faeff994e087040d207e19b95f0ae1511cc11f21192dd613b70f75ff | What is the area of an acute triangle whose altitudes have lengths of $20$, $24$, and $30$? | [
"\\(\\frac{960\\sqrt{7}}{7}\\)"
] | [
"What is the area of an acute triangle whose altitudes have lengths of $20$, $24$, and $30$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-65311fcb8c0123d97dc0383d2e6261c68a409c2b129ae722f04611b26eef53ce | What is the area of an equiangular hexagon $ABCDEF$ with side lengths $AB = 1$, $BC = EF = 2$, and $CD = 3$? | [
"$\\frac{11\\sqrt{3}}{2}$"
] | [
"What is the area of an equiangular hexagon $ABCDEF$ with side lengths $AB = 1$, $BC = EF = 2$, and $CD = 3$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-07f41727a293109f2a5bbea69c73e9a6e60bff1b0651ca7a4f3ee1e242ca2008 | What is the area of an equiangular hexagon ABCDEF with side lengths AB = 1, BC = EF = 2, and CD = 3? | [
"\\(\\frac{33\\sqrt{3}}{4}\\)"
] | [
"What is the area of an equiangular hexagon ABCDEF with side lengths AB = 1, BC = EF = 2, and CD = 3?"
] | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-af11ad77a3392336335d12e2b6740c478e1ec3fd633b5118edb6cbf3c3e34220 | What is the area of an equiangular hexagon with side lengths of 6, 8, 12, 6, 8, and 12 in that order? | [
"\\( 108\\sqrt{3} \\)"
] | [
"What is the area of an equiangular hexagon with side lengths of 6, 8, 12, 6, 8, and 12 in that order?"
] | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 30
} | 78 | 0 | 0 |
omr-ws-8eba4cc130c1448000a720e1fcbebac84bb368af1de9b00aec9b547702d1eaac | What is the area of an equilateral hexagon inscribed in a circle with a radius of 20 cm? | [
"\\( 600 \\sqrt{3} \\)"
] | [
"What is the area of an equilateral hexagon inscribed in a circle with a radius of 20 cm?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-b5dfee76b459965ac6f1ab5adfde6a76afe739fe2e12ffd7c9350a4a469d170b | What is the area of an equilateral hexagon with side length 2 and interior angles that alternate between $90^\circ$ and $150^\circ$? | [
"\\(6 + 2\\sqrt{3}\\)"
] | [
"What is the area of an equilateral hexagon with side length 2 and interior angles that alternate between $90^\\circ$ and $150^\\circ$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 7,
"tir": 27
} | 49 | 0 | 0 |
omr-ws-228bfa8d070fa63d58f64392ca2e991faa1b53096cdb08c07dd465f4533cbf45 | What is the area of an equilateral triangle if the distances from one point inside the triangle to the three vertices are 7, 8, and 9? | [
"\\dfrac{\\sqrt{3}}{4} \\left(97 + 24\\sqrt{15}\\right)"
] | [
"What is the area of an equilateral triangle if the distances from one point inside the triangle to the three vertices are 7, 8, and 9?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-84775160b70238408e956cf1c3c97d7fb97a504f60b2eff82f25dcdef291caa1 | What is the area of an octagon with side length 10? | [
"\\( 200(1 + \\sqrt{2}) \\)"
] | [
"What is the area of an octagon with side length 10?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-e7db0e99a24063498710005a01c02be8b950454d33e68ff713a598902a89605d | What is the area of an octagon with side lengths $3, 3, 11, 11, 15, 15, 15,$ and $15$ cm that is inscribed in a circle? | [
"840"
] | [
"What is the area of an octagon with side lengths $3, 3, 11, 11, 15, 15, 15,$ and $15$ cm that is inscribed in a circle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-736f4132a68dd2f3df71046782039f95f35a2aba124290d6318855fbfdc47e09 | What is the area of intersection between the parabolas $y = -x^2 + 9$ and $y = x^2 + 4x + 1$? Note that the answer can be negative. | [
"\\( 24\\sqrt{5} / 3 \\) or approximately \\( 30 \\)"
] | [
"What is the area of intersection between the parabolas $y = -x^2 + 9$ and $y = x^2 + 4x + 1$? Note that the answer can be negative."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a8b59a4215673c7916f8604573e8fe4c4e3376286f9dbae224057aa64cf091de | What is the area of overlap between \( \Delta ACF \) and \( \Delta BCF \) in a regular hexagon \( ABCDEF \) with side length 2? | [
"\\dfrac{4\\sqrt{3}}{3}"
] | [
"What is the area of overlap between \\( \\Delta ACF \\) and \\( \\Delta BCF \\) in a regular hexagon \\( ABCDEF \\) with side length 2?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3015085c731abdb8f7dc2b317edcd4edd17e19bd983933ac0241484152d8458a | What is the area of quadrilateral $ABCD$ if $AB = 8$, $BC = 20$, $CD = 5\sqrt{2}$, $\angle ABC = 90^{\circ}$, and $\angle BCD = 45^{\circ}$? | [
"110"
] | [
"What is the area of quadrilateral $ABCD$ if $AB = 8$, $BC = 20$, $CD = 5\\sqrt{2}$, $\\angle ABC = 90^{\\circ}$, and $\\angle BCD = 45^{\\circ}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 7,
"tir": 6
} | 25 | 0 | 0 |
omr-ws-31e45dc2affacdfa03db0bff58e4e9522e6396b835d3d2f42e487e68f052869c | What is the area of the Koch snowflake, which is constructed by starting with an equilateral triangle of side length 1, repeatedly dividing each side into 3 equal parts, constructing an equilateral triangle extending outwards on the middle section, and deleting its base? | [
"\\(\\frac{2\\sqrt{3}}{5}\\)"
] | [
"What is the area of the Koch snowflake, which is constructed by starting with an equilateral triangle of side length 1, repeatedly dividing each side into 3 equal parts, constructing an equilateral triangle extending outwards on the middle section, and deleting its base?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-99603622bbeddae825601f110217b9ede7e62a0862e02249bb4bf30d674cf772 | What is the area of the bounded region formed by the graph of the equation \( y^2 + 2xy + 40|x| = 400 \)? | [
"800"
] | [
"What is the area of the bounded region formed by the graph of the equation \\( y^2 + 2xy + 40|x| = 400 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-0a42855bce608226cc09a4800f6d26e06ec35e5773a05e321dfb84596091ebd0 | What is the area of the circle circumscribed about a hexagon with sides 1, 1, 1, 1, 2, 2 (in that order)? | [
"\\(\\frac{7\\pi}{3}\\)"
] | [
"What is the area of the circle circumscribed about a hexagon with sides 1, 1, 1, 1, 2, 2 (in that order)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-04506e3844fc19f3ee4f171feaac7cae2d25ec376586fa27daddf7ad6acf87a2 | What is the area of the circle defined by $x^2-6x +y^2-14y +33=0$ that lies beneath the line $y=7$? | [
"\\frac{25\\pi}{2}"
] | [
"What is the area of the circle defined by $x^2-6x +y^2-14y +33=0$ that lies beneath the line $y=7$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7245cf577c7f292138cb5947e16a11886b845b2f2b0565295c7d3d7a408391d9 | What is the area of the circle inscribed in a triangle with a perimeter of 17 cm and a radius of 2 cm? | [
"\\( 4\\pi \\)"
] | [
"What is the area of the circle inscribed in a triangle with a perimeter of 17 cm and a radius of 2 cm?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-7decd9df24bba82895fb5e9e93be5036099e6e576c2208f860b7f26f7dc73122 | What is the area of the circle inscribed in an equilateral triangle with a circumcircle of radius 1? | [
"\\(\\frac{\\pi}{4}\\)"
] | [
"What is the area of the circle inscribed in an equilateral triangle with a circumcircle of radius 1?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
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