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-96211acf2c2f65f07022d8ce260e31e87cbc583980b1e5baad480fc2b33fd2ae | What is the quickest way to decompose the fraction $$\frac{\left(k^{2}-\frac{1}{2}\right)}{\left(k^{2}+k+\frac{1}{2}\right)\left(k^{2}-k+\frac{1}{2}\right)}$$ into the form $$\frac{\left(k-\frac{1}{2}\right)}{k^{2}-k+\frac{1}{2}}-\frac{\left(k+\frac{1}{2}\right)}{\left(k^{2}+k+\frac{1}{2}\right)}?$$ | [
"\\frac{\\left(k-\\frac{1}{2}\\right)}{k^{2}-k+\\frac{1}{2}}-\\frac{\\left(k+\\frac{1}{2}\\right)}{k^{2}+k+\\frac{1}{2}}"
] | [
"What is the quickest way to decompose the fraction\n$$\\frac{\\left(k^{2}-\\frac{1}{2}\\right)}{\\left(k^{2}+k+\\frac{1}{2}\\right)\\left(k^{2}-k+\\frac{1}{2}\\right)}$$\ninto the form\n$$\\frac{\\left(k-\\frac{1}{2}\\right)}{k^{2}-k+\\frac{1}{2}}-\\frac{\\left(k+\\frac{1}{2}\\right)}{\\left(k^{2}+k+\\frac{1}{2}\\... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8b42eaa5af83bd03499b61868c397bb449e458754ae7246cfb5c0129f8b7348d | What is the quotient and remainder when \(3x^3 + 5x^2 - 4x + 1\) is divided by \(x - 2\)? | [
"Quotient \\(3x^2 + 11x + 18\\), Remainder \\(37\\)"
] | [
"What is the quotient and remainder when \\(3x^3 + 5x^2 - 4x + 1\\) is divided by \\(x - 2\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-0702b6da1f393a0dda678c61f555c7baf1d065b6a4d2b64458011b83799e8732 | What is the quotient when $1\natural 4\natural3$ is divided by $\natural1$, given that there is no remainder? | [
"303"
] | [
"What is the quotient when $1\\natural 4\\natural3$ is divided by $\\natural1$, given that there is no remainder?"
] | [
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-310805078da03bce593872b0022444443dce6c9ea05a310c8eda8f3e53c24b74 | What is the quotient when $8x^3+16x^2-7x+4$ is divided by $2x+5$? | [
"4x^2 -2x + \\frac{3}{2}"
] | [
"What is the quotient when $8x^3+16x^2-7x+4$ is divided by $2x+5$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3dfd5daa731b2370eb87395f82b6fba471c78db22accbc00f7031e47f8a8ace7 | What is the quotient when $f(x) = 2x^6 - 3x^5 - 6x^4 - 2x^2 + 3x + 6$ is divided by $x^2 - 1$? | [
"\\( 2x^4 - 3x^3 - 4x^2 - 3x - 6 \\)"
] | [
"What is the quotient when $f(x) = 2x^6 - 3x^5 - 6x^4 - 2x^2 + 3x + 6$ is divided by $x^2 - 1$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-455f406269c04a7ec3f26af89998ff7b089879a55ea373bbfe89df05de43e4fc | What is the r th term of the expansion of \((1-x)^{-4}\)? | [
"\\(\\binom{r+2}{3}x^{r-1}\\)"
] | [
"What is the r th term of the expansion of \\((1-x)^{-4}\\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 12
} | 26 | 0 | 0 |
omr-ws-c8c5653ba2884e0688db41e59ac9b53c1297ff53404a40963c1a2f6036193d19 | What is the radical axis of the circumcircle and the nine-point circle of triangle ABC? | [
"\\text{The orthic axis}"
] | [
"What is the radical axis of the circumcircle and the nine-point circle of triangle ABC?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-ca588307ca14e665a057881e7fc7b383d7d2155b787545121108c864d915d2cf | What is the radius of a circle inscribed in a rhombus with diagonals of length 10 and 24? | [
"\\( \\frac{60}{13} \\)",
"\\(\\frac{60}{13}\\)"
] | [
"What is the radius of a circle inscribed in a rhombus with diagonals of length 10 and 24?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 51
} | 82 | 0 | 0 |
omr-ws-52f9592b4cce014cd76a736503c871a96d9c6bfc79458182a72d6c57ec06eb8e | What is the radius of a cylinder with a volume of 12 cubic centimeters and the least possible surface area? Express your answer as a common fraction. | [
"\\( \\frac{\\sqrt[3]{6}}{\\sqrt[3]{\\pi}} \\)"
] | [
"What is the radius of a cylinder with a volume of 12 cubic centimeters and the least possible surface area? Express your answer as a common fraction."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-55dbcee5000778994ae141651c800618b6972f636659b754d01f4f5d41aa9b11 | What is the radius of convergence for the power series $\sum_{n=0}^{\infty} (3x-5)^n$? | [
"\\(\\frac{1}{3}\\)"
] | [
"What is the radius of convergence for the power series $\\sum_{n=0}^{\\infty} (3x-5)^n$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-de9503e2a1f20885f50b7570c9d65009f47aef108995110f39686c1da4642602 | What is the radius of convergence for the series $\sum_{n=1}^\infty 2^n(x+3)^{2n+1}$? | [
"\\( \\frac{1}{\\sqrt{2}} \\)"
] | [
"What is the radius of convergence for the series $\\sum_{n=1}^\\infty 2^n(x+3)^{2n+1}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-9c25c85296dd23c7fa7407a84020d96dedd827a6bacd619bc654d6ab66e02acf | What is the radius of convergence for the series $\sum_{n=1}^{\infty} \frac{(x-3)^n (n+1)}{2^n(2n+1)}$? | [
"2"
] | [
"What is the radius of convergence for the series $\\sum_{n=1}^{\\infty} \\frac{(x-3)^n (n+1)}{2^n(2n+1)}$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-2c41f3af65db3e975791223313e3710cd1524bd1de687b0d14e931cdb2d77400 | What is the radius of convergence of the power series $x + \sum_{n=2}^{\infty} \frac{(n!)^2}{2n!} x^n$? | [
"4"
] | [
"What is the radius of convergence of the power series $x + \\sum_{n=2}^{\\infty} \\frac{(n!)^2}{2n!} x^n$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-88e104acc5deaa8f87fca7a7fec1a1a91f021daede9911ab2460dc91a9df1433 | What is the radius of convergence of the series $1-\frac{x^2}{2}+\frac{x^4}{2^2}-\ldots$? | [
"\\(\\sqrt{2}\\)"
] | [
"What is the radius of convergence of the series $1-\\frac{x^2}{2}+\\frac{x^4}{2^2}-\\ldots$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-6a5321dced34950f7ba9edd0a47c2aa420a00d8d52915e45f930bad075e03877 | What is the radius of convergence of the series $\ln(1+x^{2}) = x^{2} - \frac{x^{4}}{2} + \frac{x^{6}}{3}- \frac{x^{8}}{4} + ...$? | [
"1"
] | [
"What is the radius of convergence of the series $\\ln(1+x^{2}) = x^{2} - \\frac{x^{4}}{2} + \\frac{x^{6}}{3}- \\frac{x^{8}}{4} + ...$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e50a01af1f4565d38830cb716b71202b161b92780dfce46484396bb091e178a9 | What is the radius of convergence of the series $\sum_{n=0}^{\infty} 16^n(z+i)^{4n}$? | [
"\\(\\frac{1}{2}\\)"
] | [
"What is the radius of convergence of the series $\\sum_{n=0}^{\\infty} 16^n(z+i)^{4n}$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-65cc6e8b07f40f90cea1772aab38bbcdbff508b6e7ecf2b1d0c34957127560d5 | What is the radius of convergence of the series $\sum_{n=1}^\infty 2^n(x+3)^{2n+1}$? | [
"\\( \\sqrt{2} \\)"
] | [
"What is the radius of convergence of the series $\\sum_{n=1}^\\infty 2^n(x+3)^{2n+1}$?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5090ace534e362e25f7da7f82d7aa7401f36ecea18f3599f8be2425affcf1aa2 | What is the radius of convergence of the series $\sum_{n=1}^{\infty} \frac{(xn)^n}{n!}$? | [
"\\(\\frac{1}{e}\\)"
] | [
"What is the radius of convergence of the series $\\sum_{n=1}^{\\infty} \\frac{(xn)^n}{n!}$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-beaf1b74a0aa5cc357e3a45ddb488a521d913a1b01806909e81866ab3e6e5894 | What is the radius of the ball that causes the greatest volume of water to spill out of a conical cup with height $h$ and semivertical angle $\theta$ when the ball is placed carefully inside the cup? | [
"\\dfrac{h \\sin \\theta}{1 + \\sin \\theta}"
] | [
"What is the radius of the ball that causes the greatest volume of water to spill out of a conical cup with height $h$ and semivertical angle $\\theta$ when the ball is placed carefully inside the cup?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-b046ab084651598251e3187e78864e90a98f351c5e8b2c3f072cd8873d0d4bdf | What is the radius of the circle formed by the intersection of the spheres given by the equations $\left(x+2\right)^{2}+\left(y+3\right)^{2}+\left(z+2\right)^{2}=25$ and $\left(x+2\right)^{2}+\left(y+3\right)^{2}+\left(z-4\right)^{2}=16$? | [
"\\(\\frac{\\sqrt{385}}{4}\\)"
] | [
"What is the radius of the circle formed by the intersection of the spheres given by the equations $\\left(x+2\\right)^{2}+\\left(y+3\\right)^{2}+\\left(z+2\\right)^{2}=25$ and $\\left(x+2\\right)^{2}+\\left(y+3\\right)^{2}+\\left(z-4\\right)^{2}=16$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-922c3e00ce52c74f54747a636b5ec89cb556ca266c600535a733b80b8c0c5b4c | What is the radius of the circle in which a regular decagon of side length 1 is inscribed? | [
"\\(\\frac{1 + \\sqrt{5}}{2}\\)"
] | [
"What is the radius of the circle in which a regular decagon of side length 1 is inscribed?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-75820ea9fb31a8b9108f244e73f99e37ea95be0ceae359e3f349990b7be59c93 | What is the radius of the circle inscribed in a triangle with sides of lengths 16, 17, and 17? | [
"\\(\\frac{24}{5}\\)"
] | [
"What is the radius of the circle inscribed in a triangle with sides of lengths 16, 17, and 17?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-bc841065aad2fcf86a69f2d1c270c1d0479896cc8ff6dbb63297d5cb0f6e1fff | What is the radius of the circle inscribed in the figure formed by a quarter circle of radius one and the two radii? | [
"\\(\\sqrt{2} - 1\\)"
] | [
"What is the radius of the circle inscribed in the figure formed by a quarter circle of radius one and the two radii?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-68bbf82f548a8bf72122fec27fd699b1892dfdee026d26952f15314cd1573623 | What is the radius of the circle inscribed in triangle $ABC$ if $AB = 22, AC=12,$ and $BC=14$? Express your answer in simplest radical form. | [
"\\sqrt{10}"
] | [
"What is the radius of the circle inscribed in triangle $ABC$ if $AB = 22, AC=12,$ and $BC=14$? Express your answer in simplest radical form."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-34303a27e74b35bfed21115d146aa2114da8d985a0d79d0b55600002ed104747 | What is the radius of the circle inscribed in triangle $ABC$ if $AB = 5, AC=6, BC=7$? Express your answer in simplest radical form. | [
"\\frac{2\\sqrt{6}}{3}"
] | [
"What is the radius of the circle inscribed in triangle $ABC$ if $AB = 5, AC=6, BC=7$? Express your answer in simplest radical form."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-019a74131894a2307244fde3ac136e21f1fce9bfbd3ca2d8ffd8315233c6cbc5 | What is the radius of the circle inscribed in triangle $ABC$ if $AB = AC=7$ and $BC=6$? Express your answer in simplest radical form. | [
"\\frac{3\\sqrt{10}}{5}"
] | [
"What is the radius of the circle inscribed in triangle $ABC$ if $AB = AC=7$ and $BC=6$? Express your answer in simplest radical form."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1a7b2460d2fb2b71809a0bb38d208d76030f45ea65bfa06583ff98b1ae5e5bbd | What is the radius of the circle passing through the center of the square $ABCD$ with side length $1$, its corner $A$, and the midpoint of its side $[BC]$? | [
"\\(\\frac{\\sqrt{10}}{4}\\)"
] | [
"What is the radius of the circle passing through the center of the square $ABCD$ with side length $1$, its corner $A$, and the midpoint of its side $[BC]$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-0da963fe1d4c7d33997f55f847ae85b2bf3387a0e248c711754ccf13e0f5adb6 | What is the radius of the circle tangent to the $x$-axis, the line $y=\sqrt{3}x$, and the circle $(x-10\sqrt{3})^2+(y-10)^2=25$? | [
"5"
] | [
"What is the radius of the circle tangent to the $x$-axis, the line $y=\\sqrt{3}x$, and the circle $(x-10\\sqrt{3})^2+(y-10)^2=25$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 4
} | 12 | 0 | 0 |
omr-ws-567a03db2b6b7449f76e55ea0f486c1ed13098141eb09e44de170c89fc0db311 | What is the radius of the circle that can be inscribed in an isosceles trapezoid with an area of $39$ $cm^2$ and a perimeter of $26$ $cm$? | [
"3 cm"
] | [
"What is the radius of the circle that can be inscribed in an isosceles trapezoid with an area of $39$ $cm^2$ and a perimeter of $26$ $cm$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-15347a5899b4337d144a1a5720a1d40cd42df7c97cf8e4ed059d99084f40ae41 | What is the radius of the circle which circumscribes a triangle with sides of length 2, 3, and 4? | [
"\\(\\frac{8\\sqrt{15}}{15}\\)"
] | [
"What is the radius of the circle which circumscribes a triangle with sides of length 2, 3, and 4?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-578f202bba59d80676bded963be83559e75bdb147ca70d7ff0ec73d6e6663c65 | What is the radius of the circle which circumscribes a triangle with sides of lengths $2, 3,$ and $4$? | [
"\\(\\frac{8\\sqrt{15}}{15}\\)"
] | [
"What is the radius of the circle which circumscribes a triangle with sides of lengths $2, 3,$ and $4$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-153d3f3c175d6f31bfe10a78ca4f30473c4767e992977b8733b83af8e6f2ea7e | What is the radius of the circumcircle of a triangle whose sides have lengths 29, 21, and 20? | [
"\\(\\frac{29}{2}\\)"
] | [
"What is the radius of the circumcircle of a triangle whose sides have lengths 29, 21, and 20?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 19
} | 34 | 0 | 0 |
omr-ws-5d563a6ca2c701954ec4821f4128be0a35053e5ca4652547b6db95d5fbe07cba | What is the radius of the inscribed circle in a 45-45-90 right triangle with a hypotenuse of 8 meters? | [
"\\( 4\\sqrt{2} - 4 \\)"
] | [
"What is the radius of the inscribed circle in a 45-45-90 right triangle with a hypotenuse of 8 meters?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-f3d5f5a01fd7d551562d2281e86c392cf7e1762e2d9d6053eaeba68cb1f94482 | What is the radius of the inscribed circle in a rhombus with side length 1 and an angle of 60 degrees? | [
"\\(\\frac{\\sqrt{3}}{4}\\)"
] | [
"What is the radius of the inscribed circle in a rhombus with side length 1 and an angle of 60 degrees?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-743b4a59c7083bc5a71e5b4462e33fec94e435f0c57f469230710d9199d4e3f4 | What is the radius of the largest circle that can be contained in a cube of side length 1? | [
"\\( \\frac{\\sqrt{6}}{4} \\)"
] | [
"What is the radius of the largest circle that can be contained in a cube of side length 1?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-b2b6865ec43a9cd73f68144fd1b216b5af23656f090aff781b8effd233fd4d4e | What is the radius of the largest circle that can be drawn inside a quarter circle of radius 1 cm? | [
"\\(\\sqrt{2} - 1\\)"
] | [
"What is the radius of the largest circle that can be drawn inside a quarter circle of radius 1 cm?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-71af859ad2747b90d130f1f925c11e0f085b0d2a65565d00f188ad192f5025b4 | What is the radius of the largest circle that can be drawn inside a quarter circle of radius 1 unit? | [
"\\(\\sqrt{2} - 1\\)"
] | [
"What is the radius of the largest circle that can be drawn inside a quarter circle of radius 1 unit?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-1c926cd61f1a7be0cb904fef8147576957c572fb905f33b59760571ae8ec33e4 | What is the radius of the largest circle that can enclose exactly three lattice points? | [
"\\dfrac{\\sqrt{5}}{2}"
] | [
"What is the radius of the largest circle that can enclose exactly three lattice points?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-97bce65783e379a1200ad16216969a58427d7f331932a47e0d4916f65733ea22 | What is the radius of the largest circle that can roll along inside the arc of the parabola \( y = x^2 \) and reach the bottom of the parabola, tangent to the origin, without getting stuck before getting there? | [
"\\( \\frac{1}{2} \\)"
] | [
"What is the radius of the largest circle that can roll along inside the arc of the parabola \\( y = x^2 \\) and reach the bottom of the parabola, tangent to the origin, without getting stuck before getting there?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 11
} | 26 | 0 | 0 |
omr-ws-d31c3d3499977e7126721ea6bd88c1341e0225c661699e8781b99d700b05187c | What is the radius of the largest disc that can be covered by a finite number of closed discs of radius $1$ in such a way that each disc intersects at most three others? | [
"1 + \\dfrac{2}{\\sqrt{3}}"
] | [
"What is the radius of the largest disc that can be covered by a finite number of closed discs of radius $1$ in such a way that each disc intersects at most three others?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 5
} | 12 | 0 | 0 |
omr-ws-18333f8fc1c50c3a38ff8e375fdbc2288c7e396af58f091cb518cb1984047acd | What is the radius of the largest orange that can sit on a flat table in the space between three spherical melons, each with a radius of 9 cm, where each melon touches the other two? | [
"3"
] | [
"What is the radius of the largest orange that can sit on a flat table in the space between three spherical melons, each with a radius of 9 cm, where each melon touches the other two?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-6f5c177a1518e6057f31ff6bf6edc71a101367774bf1d2ea132ac6a86e8e36bb | What is the radius of the largest sphere that can fit inside an octahedron with every edge of length \( s \)? | [
"\\(\\frac{s\\sqrt{6}}{3}\\)"
] | [
"What is the radius of the largest sphere that can fit inside an octahedron with every edge of length \\( s \\)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b5c37452d678cfba3ce5a54d86e9f286ec60b117eb539dfb1d084cf56da3651a | What is the radius of the largest sphere that fits inside a cone with a base in the $xy$ plane with radius $1$ and center $(0,0,0)$, and a vertex at $(3,4,8)$? | [
"\\( 3 - \\sqrt{5} \\)"
] | [
"What is the radius of the largest sphere that fits inside a cone with a base in the $xy$ plane with radius $1$ and center $(0,0,0)$, and a vertex at $(3,4,8)$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 3
} | 4 | 0 | 0 |
omr-ws-29de597f1df00d1c576666673a9daf8e2f9dc9b6703e1f5dd9f56795e549f8f5 | What is the radius of the largest sphere that fits inside the tetrahedron whose vertices are the points $(0, 0, 0)$, $(1, 0, 0)$, $(0, 1, 0)$, $(0, 0, 1)$? | [
"\\(\\frac{3 - \\sqrt{3}}{6}\\)"
] | [
"What is the radius of the largest sphere that fits inside the tetrahedron whose vertices are the points $(0, 0, 0)$, $(1, 0, 0)$, $(0, 1, 0)$, $(0, 0, 1)$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-52ab1dc340c7766af52b0a6e2f380de79e38c86c7fe9acc97a5edd856d5b5039 | What is the radius of the largest sphere that will fit entirely within a square-based pyramid whose base has a side length of 1 and whose height is 3? | [
"$\\frac{\\sqrt{37}-1}{12}$"
] | [
"What is the radius of the largest sphere that will fit entirely within a square-based pyramid whose base has a side length of 1 and whose height is 3?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 25
} | 46 | 0 | 0 |
omr-ws-bc1a83963a681f94da075f4056586bfefec10921965b472f64f398a55093e85c | What is the radius of the largest spherical ball that can be placed on top of the center of a torus (donut) with an inner radius of 2 and an outer radius of 4, such that the ball still touches the horizontal plane? The torus is formed by revolving a circle centered at $(3, 0, 1)$ with radius $1$ about the $z$-axis. The... | [
"2.25"
] | [
"What is the radius of the largest spherical ball that can be placed on top of the center of a torus (donut) with an inner radius of 2 and an outer radius of 4, such that the ball still touches the horizontal plane? The torus is formed by revolving a circle centered at $(3, 0, 1)$ with radius $1$ about the $z$-axis... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 11,
"tir": 10
} | 21 | 0 | 0 |
omr-ws-6c667e64c36b741e81ff2160585763e2be46771cf25e24e560952448e1a6cbda | What is the radius of the semi-circle if the total area of a window, which consists of a semi-circle and a square, is 1 square meter? | [
"\\(\\frac{\\sqrt{2}}{\\sqrt{\\pi + 8}}\\)",
"\\(\\sqrt{\\frac{2}{\\pi + 8}}\\)"
] | [
"What is the radius of the semi-circle if the total area of a window, which consists of a semi-circle and a square, is 1 square meter?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 40
} | 72 | 0 | 0 |
omr-ws-daf5950c0637a005ff819525aeb3d7a16958696fa15d727054f2669db5c3a792 | What is the radius of the semicircle inscribed in a right triangle with sides 5, 12, and 13? | [
"\\(\\frac{10}{3}\\)"
] | [
"What is the radius of the semicircle inscribed in a right triangle with sides 5, 12, and 13?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 18,
"genselect": 6,
"tir": 9
} | 33 | 0 | 0 |
omr-ws-a0178c482f3e95962360f419e82e7d869168928768b8b4f8cbb26951462219b6 | What is the radius of the smallest circle that can contain an equilateral triangle of side length 2 with two semicircles of diameter 1 attached to one of its sides? | [
"\\(\\frac{6\\sqrt{3}+3}{11}\\)"
] | [
"What is the radius of the smallest circle that can contain an equilateral triangle of side length 2 with two semicircles of diameter 1 attached to one of its sides?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 5
} | 7 | 0 | 0 |
omr-ws-a9a9873aa08523a9b4bbcb9c11b779619c762b9e83df59e98a0bcd4a9b192d2f | What is the radius of the smallest circle that is tangent to the three lines $y = x$, $y = -x$, and $y = 2$? | [
"\\( 2\\sqrt{2} - 2 \\)"
] | [
"What is the radius of the smallest circle that is tangent to the three lines $y = x$, $y = -x$, and $y = 2$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-86b167be7f9e8f26652f56181a0bd8e3206cfca0833ed0c91e2e7dd11aa9a62c | What is the radius of the smallest sphere in which 4 spheres of radius 1 will fit? | [
"\\(1 + \\sqrt{1.5}\\)"
] | [
"What is the radius of the smallest sphere in which 4 spheres of radius 1 will fit?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-7fc3576db4f058f71e5ccf712256a7c7b8ae9030b641b5bedc5a1f97801556bc | What is the radius of the smallest sphere that can enclose four balls of radius 1, all of which are tangent to each other? | [
"\\(\\frac{4\\sqrt{6}+9}{9}\\)"
] | [
"What is the radius of the smallest sphere that can enclose four balls of radius 1, all of which are tangent to each other?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3b0e8dcb4f239225d994ec0caa2480f375fa16aa004e1984cb2f57bcbfa89d3c | What is the range (set of all possible real values) of the expression $\frac{5}{x^2 + |2x-4|}$, with $x \in \mathbb{R}$. | [
"\\left(0, \\dfrac{5}{3}\\right]"
] | [
"What is the range (set of all possible real values) of the expression $\\frac{5}{x^2 + |2x-4|}$, with $x \\in \\mathbb{R}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-f6d422f45150235927eea0840d3044eecc8ba6011fda89bac82f45d194f7bcf6 | What is the range of $arc \cot{x}$? | [
"\\(\\left(-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\right]\\)"
] | [
"What is the range of $arc \\cot{x}$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4b320606aa0187642069896a1498e5419c4621148080712f525a8f973980c532 | What is the range of $arc \csc{x}$? | [
"\\(\\left[-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\right] \\setminus \\{0\\}\\)"
] | [
"What is the range of $arc \\csc{x}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 31
} | 46 | 0 | 0 |
omr-ws-b44e8990ef5bb00cf598774eda6846c4d8c9e85a8ba50fb44f31337483a5b2bb | What is the range of \( f(x) = x \lfloor x \rfloor \)? | [
"\\(\\bigcup_{n \\in \\mathbb{Z}} [n^2, n(n+1))\\)"
] | [
"What is the range of \\( f(x) = x \\lfloor x \\rfloor \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 1
} | 2 | 0 | 0 |
omr-ws-4f0dc8f871bf3ff90eae1447d78d009f76bc19abb6bdcbc303db065779ba5ace | What is the range of \( z^{-1} \cos(z^{-1}) \) for \( 0 < |z| < \delta \), where \( \delta \) is any positive number? | [
"\\(\\mathbb{C}\\)"
] | [
"What is the range of \\( z^{-1} \\cos(z^{-1}) \\) for \\( 0 < |z| < \\delta \\), where \\( \\delta \\) is any positive number?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-5f91382a68cc9f6a9c8e69746b6909663fb467ad0f94df28bfdb7bacaf4909d9 | What is the range of possible lengths for the diagonals of a parallelogram with two 19-inch sides and two 23-inch sides? | [
"\\(4 \\text{ inches} \\leq \\text{diagonal} \\leq 42 \\text{ inches}\\)"
] | [
"What is the range of possible lengths for the diagonals of a parallelogram with two 19-inch sides and two 23-inch sides?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-a787902f08f917bf1747103cc62fe9f915ecbd628f124b4af3923f9d9834a341 | What is the range of the function $$G(x) = |x+1|-|x-1|~?$$Express your answer in interval notation. | [
"[-2,2]"
] | [
"What is the range of the function $$G(x) = |x+1|-|x-1|~?$$Express your answer in interval notation."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-299b15831af8582cb02d42c193a8526f83455ed65a66b7d1c48c3e716e256b38 | What is the range of the function $$r(x) = \frac{1}{(1-x)^2}~?$$ Express your answer in interval notation. | [
"(0,\\infty)"
] | [
"What is the range of the function $$r(x) = \\frac{1}{(1-x)^2}~?$$ Express your answer in interval notation."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bb0c1c112714d2a6a26534bc24a4c0bddb3e91e7ea01a9672fc400925913d19c | What is the range of the function $\frac{|x - 2|}{|x| - 2}$? | [
"\\((- \\infty, -1] \\cup [1, \\infty)\\)"
] | [
"What is the range of the function $\\frac{|x - 2|}{|x| - 2}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-28239e4f9e829b5fbe0998feb91948094f5a8b6c35f7dfaf7fbd3c89e7a472c3 | What is the range of the function $f(x) = \frac{1}{x^2}$? | [
"(0,\\infty)"
] | [
"What is the range of the function $f(x) = \\frac{1}{x^2}$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-162aee19fecbf9315bdfdc013fa7f04d89abbeff18fff86d223a03ce2d9cec01 | What is the range of the function $y = \frac{x^2 + 3x + 2}{x+1}$? (Express your answer using interval notation.) | [
"(-\\infty, 1)\\cup(1, \\infty)"
] | [
"What is the range of the function $y = \\frac{x^2 + 3x + 2}{x+1}$? (Express your answer using interval notation.)"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6051587c64265b0a2955152135048aa557bbf7b9c56f1abde164de1fd3da1a4e | What is the range of the function $y=\log_2 (\sqrt{\cos x})$ for $-90^\circ< x < 90^\circ$? | [
"(-\\infty,0]"
] | [
"What is the range of the function $y=\\log_2 (\\sqrt{\\cos x})$ for $-90^\\circ< x < 90^\\circ$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a1ddf9c656a15be73b9942418d0ba0ed9a3d5569b5bf743275ac3d18111d0d9c | What is the range of the function \( f \) given the points \((-6, 3)\), \((-3, 0)\), \((0, 3)\), \((2, 4)\), \((6, 0)\), \((8, -2)\), \((10, 0)\), and \((11, 1)\)? | [
"\\([-2, 4]\\)"
] | [
"What is the range of the function \\( f \\) given the points \\((-6, 3)\\), \\((-3, 0)\\), \\((0, 3)\\), \\((2, 4)\\), \\((6, 0)\\), \\((8, -2)\\), \\((10, 0)\\), and \\((11, 1)\\)?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 4
} | 4 | 0 | 0 |
omr-ws-3bf86ae1f93989683730a10684bc3bc88153e2eee9f7faedb3bdc088e81771ca | What is the range of the function \( f(x) = 3x^2 + 5x - 7 \)? | [
"\\(\\left[-\\frac{202}{36}, \\infty\\right)\\)"
] | [
"What is the range of the function \\( f(x) = 3x^2 + 5x - 7 \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 4,
"tir": 28
} | 46 | 0 | 0 |
omr-ws-d6265e2c498c2565342709c24f57bd8c9b3b1b9000be972ad966234516620d04 | What is the range of the function \( f(x) = \sin^{-1}x + \cos^{-1}x + \tan^{-1}x \)? | [
"\\(\\left[\\frac{\\pi}{4}, \\frac{3\\pi}{4}\\right]\\)"
] | [
"What is the range of the function \\( f(x) = \\sin^{-1}x + \\cos^{-1}x + \\tan^{-1}x \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-1aa6fb79008b50aaed5bfaa27ca60411c26f8f56e48f4b0f55690121104ccf77 | What is the range of the function \( f(x) = \sqrt{x^2 + 2011} - x \)? | [
"\\( (0, \\infty) \\)"
] | [
"What is the range of the function \\( f(x) = \\sqrt{x^2 + 2011} - x \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-b100cb0d7c3da8584ad044262c5fd2b2b74c44fbab20afc1641cbf8527c52c0b | What is the range of the function \( f(x) = x^2 + \frac{1}{x^4} \)? | [
"\\([ \\sqrt[3]{2} + \\frac{1}{\\sqrt[3]{4}}, \\infty )\\)"
] | [
"What is the range of the function \\( f(x) = x^2 + \\frac{1}{x^4} \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 4,
"tir": 20
} | 37 | 0 | 0 |
omr-ws-8146eab9b9541f842ac2c010b3f30f807abebae95c845be593d132dba8144c4c | What is the range of the function \( y = \frac{x - 2}{x^2 + 2x - 8} \)? | [
"\\(\\mathbb{R} \\setminus \\{0\\}\\)"
] | [
"What is the range of the function \\( y = \\frac{x - 2}{x^2 + 2x - 8} \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 4,
"tir": 9
} | 20 | 0 | 0 |
omr-ws-0a6944354d518c1f4d0001bb2226820d993f2343a013491ef0bd297c57149063 | What is the range of the function \( y = \sin(\cos(\tan(\csc(\sec(\cot(x))))) \)? | [
"\\(\\left[\\sin(-1), \\sin(1)\\right]\\)"
] | [
"What is the range of the function \\( y = \\sin(\\cos(\\tan(\\csc(\\sec(\\cot(x))))) \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 19
} | 34 | 0 | 0 |
omr-ws-c91d53ee9c2b84501223d92808f4a51ff46cca2e327483353468dd1100da01b8 | What is the range of the mean and median of the set $\{48, 75, 32, 45, 62\}$? | [
"4.4"
] | [
"What is the range of the mean and median of the set $\\{48, 75, 32, 45, 62\\}$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 4
} | 17 | 0 | 0 |
omr-ws-809875153f1d0744e0de17f7182bdef0f3f9e53b968de802a2d8e2fe60bb84de | What is the range of the parametric equations given by \( x(t) = t^2 - 1 \), \( y(t) = t^2 + 1 \), and \( z(t) = 1 - t \)? | [
"\\left\\{ (x, y, z) \\mid y = x + 2 \\text{ and } x = (z - 1)^2 - 1 \\right\\}"
] | [
"What is the range of the parametric equations given by \\( x(t) = t^2 - 1 \\), \\( y(t) = t^2 + 1 \\), and \\( z(t) = 1 - t \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-7f62dfe9cfac51626b40165d2385f12d1561a88d1ba0a865717ccd3e52a1bd1a | What is the range of the parametric function defined by \( x(t) = t^2 - 1 \), \( y(t) = t^2 + 1 \), and \( z(t) = 1 - t \)? | [
"\\([-1, \\infty) \\times [1, \\infty) \\times \\mathbb{R}\\)"
] | [
"What is the range of the parametric function defined by \\( x(t) = t^2 - 1 \\), \\( y(t) = t^2 + 1 \\), and \\( z(t) = 1 - t \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 3,
"tir": 2
} | 12 | 0 | 0 |
omr-ws-9a33bf420350c8278e4720d92f0b27bf16b0c7f4d8274feac28678b596a47a89 | What is the range of the rational function \( f(x) = \frac{3x^2 + 2x - 1}{x^2 - 4} \)? | [
"\\( (-\\infty, 3) \\cup (3, \\infty) \\)"
] | [
"What is the range of the rational function \\( f(x) = \\frac{3x^2 + 2x - 1}{x^2 - 4} \\)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ebc5dbc81250507e9276f0e9dea0271e8db25345c09016eed4db41f2df87c2b4 | What is the range, in degrees, of $y = \arctan x$? | [
"\\([-90^\\circ, 90^\\circ]\\)"
] | [
"What is the range, in degrees, of $y = \\arctan x$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-66be5d5b4b9cc66d41c3ccce6aaccf439f7d6d795f670efb44edaa23024ceebd | What is the rate of change of \( f(x,y,z) \) in the direction normal to the cylinder \( x^2 + y^2 = 25 \) at the point \( (3, -4, 4) \)? | [
"10"
] | [
"What is the rate of change of \\( f(x,y,z) \\) in the direction normal to the cylinder \\( x^2 + y^2 = 25 \\) at the point \\( (3, -4, 4) \\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-82f40c842e1855369f9a49891ad22d4db893db1e552e46551b72677c81532380 | What is the rate of change of the function \( y = |x| \) at \( x = 0 \)? | [
"The rate of change of the function \\( y = |x| \\) at \\( x = 0 \\) is undefined."
] | [
"What is the rate of change of the function \\( y = |x| \\) at \\( x = 0 \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-b0eda06f2a075f29b5b5a3ac9134b220107dd2de42273bf52e3944a611167540 | What is the ratio $a:h$ so that the lateral area of a right pyramid with a square base of side $a$ and height $h$ is equal to the lateral area of a cube with side $h$? | [
"\\(\\sqrt{2\\sqrt{5}-2}\\)"
] | [
"What is the ratio $a:h$ so that the lateral area of a right pyramid with a square base of side $a$ and height $h$ is equal to the lateral area of a cube with side $h$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-bdb6cb7604d4efc1d5c6070cd3e9500a8c6940b62f8ba4796555647b4da1a24b | What is the ratio between the area of \( \bigtriangleup ABC \) and the area of \( \bigtriangleup XYZ \), where \( \bigtriangleup ABC \) is inscribed in a circle and the same circle is inscribed in \( \bigtriangleup XYZ \)? | [
"\\(\\frac{1}{4}\\)"
] | [
"What is the ratio between the area of \\( \\bigtriangleup ABC \\) and the area of \\( \\bigtriangleup XYZ \\), where \\( \\bigtriangleup ABC \\) is inscribed in a circle and the same circle is inscribed in \\( \\bigtriangleup XYZ \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-1389742057251a6577b0536f758aa667ea9f8b66cacff2c5ce1d865f2c28f34b | What is the ratio between the largest distance between two corners of a hypercube and the side length of the hypercube? | [
"2"
] | [
"What is the ratio between the largest distance between two corners of a hypercube and the side length of the hypercube?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-7b6637bf6485228a8d8f4d2c686d994e2889388909a224c7b6dbfa0ca5075c00 | What is the ratio of $x$ to $y$ if: $\frac{10x-3y}{13x-2y} = \frac{3}{5}$? Express your answer as a common fraction. | [
"\\frac{9}{11}"
] | [
"What is the ratio of $x$ to $y$ if: $\\frac{10x-3y}{13x-2y} = \\frac{3}{5}$? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c401bef6f3f23fcb20a706c8e7598aa40c01aacc70e67ccf8296067b928d3db7 | What is the ratio of the Manhattan length of the Euclidean circle to the Manhattan length of its diameter? | [
"4"
] | [
"What is the ratio of the Manhattan length of the Euclidean circle to the Manhattan length of its diameter?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 16,
"genselect": 2
} | 18 | 0 | 0 |
omr-ws-541903c4eef94028a0d504cfe591c4b4a2b9e97bbefff3d8f03dbbaef79e9ecd | What is the ratio of the area of a regular twelve-sided polygon to the area of a circle inscribed in it? | [
"\\(\\frac{12\\tan(15)}{\\pi}\\)"
] | [
"What is the ratio of the area of a regular twelve-sided polygon to the area of a circle inscribed in it?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-26573e1bfe6890b21ee905394779f2d954493a4921a8124f892ffe7c1d3f89ac | What is the ratio of the area of a square inscribed in a regular octagon to the area of the regular octagon? Express the final answer in simplified radicals. | [
"\\dfrac{\\sqrt{2}}{2}"
] | [
"What is the ratio of the area of a square inscribed in a regular octagon to the area of the regular octagon? Express the final answer in simplified radicals."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 4
} | 19 | 0 | 0 |
omr-ws-5cbb32a69981d4969a26d3a12395f8931b4aa4e683527596eb3b1db3d5226678 | What is the ratio of the area of a square inscribed in a semicircle to the area of a square inscribed in the entire circle? | [
"2:5",
"\\( \\frac{2}{5} \\)"
] | [
"What is the ratio of the area of a square inscribed in a semicircle to the area of a square inscribed in the entire circle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 60
} | 91 | 0 | 0 |
omr-ws-8792cfb52f73517ddf395baa74632b43aa71ef469a4701f615855afaa2a5d8a1 | What is the ratio of the area of a square inscribed in a semicircle with radius $r$ to the area of a square inscribed in a circle with radius $r$? Express your answer as a common fraction. | [
"\\dfrac{2}{5}"
] | [
"What is the ratio of the area of a square inscribed in a semicircle with radius $r$ to the area of a square inscribed in a circle with radius $r$? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8c5179dd7c9e3c1c85dd8317cc3d774d37d8e47d634a4994a4bb928b4e837602 | What is the ratio of the area of a square inscribed in a semicircle with radius \( r \) to the area of a square inscribed in a circle with radius \( r \)? | [
"\\(\\frac{2}{5}\\)"
] | [
"What is the ratio of the area of a square inscribed in a semicircle with radius \\( r \\) to the area of a square inscribed in a circle with radius \\( r \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-3c4055610bb0789663b4a599cfa81700ad42627a300b983a6ad8f6a60fa8bf58 | What is the ratio of the area of a square that circumscribes a circle to the area of a square that inscribes the same circle? | [
"2"
] | [
"What is the ratio of the area of a square that circumscribes a circle to the area of a square that inscribes the same circle?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-3af5174902d9174d05428d7f120f0461a2c18fa6143b8749d6964e99e3cd32cf | What is the ratio of the area of a triangle ABC to the area of the triangle whose sides are equal to the medians of triangle ABC? | [
"4:3"
] | [
"What is the ratio of the area of a triangle ABC to the area of the triangle whose sides are equal to the medians of triangle ABC?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-bd53ec048e7be8f241896cfd91632cbe73ab9e1e3e54b2ffbafc3d3d14abb42a | What is the ratio of the area of an equilateral triangle to the area of the largest rectangle that can be inscribed inside the triangle? | [
"2"
] | [
"What is the ratio of the area of an equilateral triangle to the area of the largest rectangle that can be inscribed inside the triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-f31da36efbea58d1c00532189de1421c99d8b3254d20e0ce6ae88884d75300f3 | What is the ratio of the area of the largest equilateral triangle that can be inscribed in a square to the area of the regular hexagon in which the square is inscribed? | [
"2 - \\sqrt{3}"
] | [
"What is the ratio of the area of the largest equilateral triangle that can be inscribed in a square to the area of the regular hexagon in which the square is inscribed?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-aee92ee72dc23e73e4f0d174305863c7e1a45d9f884404dc213a54cc114c8b4e | What is the ratio of the area of triangle \(ABC\) to the area of the entire regular decagon? Write the answer as a fraction of integers. | [
"\\(\\frac{1}{5}\\)"
] | [
"What is the ratio of the area of triangle \\(ABC\\) to the area of the entire regular decagon? Write the answer as a fraction of integers."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 4,
"tir": 8
} | 15 | 0 | 0 |
omr-ws-3ccf880a9f21bf5b69d00b4850e4385dc025110b4959f8234af05008cde330c6 | What is the ratio of the areas of the smaller pentagon formed by the diagonals to the area of the original regular pentagon? | [
"\\(\\frac{7-3\\sqrt{5}}{2}\\)"
] | [
"What is the ratio of the areas of the smaller pentagon formed by the diagonals to the area of the original regular pentagon?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 23,
"genselect": 5,
"tir": 19
} | 47 | 0 | 0 |
omr-ws-26038160b9aae8f52fb7a71bc569d0e6e74873a806e8221b1f79527f7c4cd166 | What is the ratio of the coefficients of the $x^2$ and $x^3$ terms in the expansion of $(2x-3y)^7$? | [
"\\(\\frac{-9}{10}\\)"
] | [
"What is the ratio of the coefficients of the $x^2$ and $x^3$ terms in the expansion of $(2x-3y)^7$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-0b021cf5750bac93dba96d158a46615950f81403f58e641dcfb398bcbee01a4e | What is the ratio of the numerical value of the area, in square units, of an equilateral triangle of side length 4 units to the numerical value of its perimeter, in units? Express your answer as a common fraction in simplest radical form. | [
"\\frac{\\sqrt{3}}{3}"
] | [
"What is the ratio of the numerical value of the area, in square units, of an equilateral triangle of side length 4 units to the numerical value of its perimeter, in units? Express your answer as a common fraction in simplest radical form."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-474e325d02880829a84ac2881786ed4c91760dac6b5293000d688ab2bd3cb3c8 | What is the ratio of the numerical value of the area, in square units, of an equilateral triangle of side length 8 units to the numerical value of its perimeter, in units? Express your answer as a common fraction in simplest radical form. | [
"\\frac{2\\sqrt{3}}{3}"
] | [
"What is the ratio of the numerical value of the area, in square units, of an equilateral triangle of side length 8 units to the numerical value of its perimeter, in units? Express your answer as a common fraction in simplest radical form."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c6565fedd1a298d9f423a06a678694fd00f85671bc9e59c573dc32d7bd290bcc | What is the ratio of the sides of a 15-75-90 triangle? | [
"\\(1:2+\\sqrt{3}:\\sqrt{6}+\\sqrt{2}\\)"
] | [
"What is the ratio of the sides of a 15-75-90 triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 2,
"tir": 4
} | 19 | 0 | 0 |
omr-ws-574b3c11729a3c0cb38dcb0649c490a582f3d86a690862399b1c535775337eab | What is the ratio of the sum of the squares of the factors of 17424 to the sum of the factors of 17424? | [
"8547"
] | [
"What is the ratio of the sum of the squares of the factors of 17424 to the sum of the factors of 17424?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-7046e7d1734e4544acd3c7508f0e2a0b879b806bfc23934fe9e863553ef6305f | What is the ratio of the volume of a cube with edge length six inches to the volume of a cube with edge length one foot? Express your answer as a common fraction. | [
"\\frac{1}{8}"
] | [
"What is the ratio of the volume of a cube with edge length six inches to the volume of a cube with edge length one foot? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
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