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-79eb88b01b37e3149b6aed5659bff7971a0aad5fa57a2df937a49bf665008ea1 | Three rays \( h_1, h_2, h_3 \) emanate from a point \( O \) and are not all in the same plane. If for any three points \( A_1, A_2, A_3 \) on \( h_1, h_2, h_3 \) respectively, distinct from \( O \), the triangle \( A_1A_2A_3 \) is acute-angled, calculate the value of \( \cos^2(\theta_{12}) + \cos^2(\theta_{23}) + \cos^... | [
"\\dfrac{1}{3}"
] | [
"Three rays \\( h_1, h_2, h_3 \\) emanate from a point \\( O \\) and are not all in the same plane. If for any three points \\( A_1, A_2, A_3 \\) on \\( h_1, h_2, h_3 \\) respectively, distinct from \\( O \\), the triangle \\( A_1A_2A_3 \\) is acute-angled, calculate the value of \\( \\cos^2(\\theta_{12}) + \\cos^2... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-18b105890e62cb41f999e28854a13382020db4eedcb5486bf8f381777f5ab0b1 | Three rays emanate from a common point, dividing the plane into three angles, each less than $180^\circ$. A circle of radius $1$ is centered on each ray. Let $L_1$, $L_2$, and $L_3$ be the lengths of the segments of the projections of these circles onto the rays. Calculate the maximum possible value of $L_1 + L_2 + L_3... | [
"6"
] | [
"Three rays emanate from a common point, dividing the plane into three angles, each less than $180^\\circ$. A circle of radius $1$ is centered on each ray. Let $L_1$, $L_2$, and $L_3$ be the lengths of the segments of the projections of these circles onto the rays. Calculate the maximum possible value of $L_1 + L_2... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 2
} | 3 | 0 | 0 |
omr-ws-0b0c1869286f9aa095c80c767130067c81955d7479f1e462b736672b982ddb34 | Three real numbers $a$, $b$, $c$ are roots to the cubic $x^3 + ax^2 + bx + c$. Given that $b \neq -1$ and $c \neq 0$, compute $c(a+1)$. | [
"1"
] | [
"Three real numbers $a$, $b$, $c$ are roots to the cubic $x^3 + ax^2 + bx + c$. Given that $b \\neq -1$ and $c \\neq 0$, compute $c(a+1)$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4194366e6c722e51a9ed8bdb057f2d38a4003f1d86c63d012578d02fdf21d7af | Three real numbers $a$, $b$, and $c$ are given such that the fractional part of the product of every two of them is $\frac{1}{2}$. If $a = \sqrt{2} + \frac{1}{2}$, find the value of $b + c$. | [
"-\\sqrt{2}"
] | [
"Three real numbers $a$, $b$, and $c$ are given such that the fractional part of the product of every two of them is $\\frac{1}{2}$. If $a = \\sqrt{2} + \\frac{1}{2}$, find the value of $b + c$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-11e9cf62a85c9c6070f2d2718f58451fef7047c9e68e671cfa6a3523fc3cef58 | Three real numbers $a, b,$ and $c$ are generated uniformly and independently in the interval $[0, x]$, where $x > 0$. What is the probability that the circle with center $(a, b)$ and radius $c$ intersects the circle with center $(0, 0)$ and radius $c$? | [
"\\(\\frac{\\pi}{8}\\)"
] | [
"Three real numbers $a, b,$ and $c$ are generated uniformly and independently in the interval $[0, x]$, where $x > 0$. What is the probability that the circle with center $(a, b)$ and radius $c$ intersects the circle with center $(0, 0)$ and radius $c$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6887aa7033186863e888099eb616ee572a9c326f55e696339b14c3833149a2fc | Three real numbers $a,b,$ and $c$ satisfy the equations $a+b+c=2$, $ab+ac+bc=-7$ and $abc=-14$. What is the largest of the three numbers? Express your answer in simplest radical form. | [
"\\sqrt{7}"
] | [
"Three real numbers $a,b,$ and $c$ satisfy the equations $a+b+c=2$, $ab+ac+bc=-7$ and $abc=-14$. What is the largest of the three numbers? Express your answer in simplest radical form."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-12998ac374fed5dd7b2dd22704e0e113aaa165e1afde6c7b39d66380c996252b | Three real numbers $x, y, z$ are randomly chosen in the range $[0, 200]$. Compute the probability that $x^2 + y^2 < 10000$, $y^2 + z^2 < 10000$, and $x^2 + z^2 < 10000$. Express your answer to the nearest tenth of a percent. | [
"13.1 %"
] | [
"Three real numbers $x, y, z$ are randomly chosen in the range $[0, 200]$. Compute the probability that $x^2 + y^2 < 10000$, $y^2 + z^2 < 10000$, and $x^2 + z^2 < 10000$. Express your answer to the nearest tenth of a percent."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bcc6c129935aa4e25b4eef7abf74389131ad6fea45288c27dec3082a75fc297e | Three real numbers \( x, y, z \) are chosen randomly and independently between \( 0 \) and \( 1 \), inclusive. What is the probability that the inequality \[ 1 - z \leq 2 \cdot \max\left(|x - \tfrac{1}{2}|, |y - \tfrac{1}{2}|\right) \] is true? | [
"\\(\\frac{2}{3}\\)"
] | [
"Three real numbers \\( x, y, z \\) are chosen randomly and independently between \\( 0 \\) and \\( 1 \\), inclusive. What is the probability that the inequality\n\\[ 1 - z \\leq 2 \\cdot \\max\\left(|x - \\tfrac{1}{2}|, |y - \\tfrac{1}{2}|\\right) \\]\nis true?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-87a1a9f5381581978770e9dbb221a048090ccd2386eb0ce8c67c195da837047c | Three real numbers \(a, b, c\) satisfy the equations \(a + b + c = 3\), \(a^2 + b^2 + c^2 = 9\), and \(a^3 + b^3 + c^3 = 24\). Find \(a^4 + b^4 + c^4\). | [
"75"
] | [
"Three real numbers \\(a, b, c\\) satisfy the equations \\(a + b + c = 3\\), \\(a^2 + b^2 + c^2 = 9\\), and \\(a^3 + b^3 + c^3 = 24\\). Find \\(a^4 + b^4 + c^4\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e6cbf78ceed1f5392f6afb4102ba009bcef58d292be4709d3339397462255b15 | Three real numbers \(a\), \(b\), and \(c\) are chosen at random between 0 and 1. Calculate the value of \(P\), where \(P\) is the probability that \(a\), \(b\), and \(c\) can form the side lengths of a triangle. Express \(P\) as a common fraction. | [
"\\dfrac{1}{2}"
] | [
"Three real numbers \\(a\\), \\(b\\), and \\(c\\) are chosen at random between 0 and 1. Calculate the value of \\(P\\), where \\(P\\) is the probability that \\(a\\), \\(b\\), and \\(c\\) can form the side lengths of a triangle. Express \\(P\\) as a common fraction."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fb7ff8d2732b8e1f1cfc805aefe9fed4e6cca26a697fafdb2ccc01e8b7b612e0 | Three real numbers \(x\), \(y\), and \(z\) satisfy the following conditions: \[ \begin{cases} xyz > 0 \\ 1995x^3 = 1996y^3 = 1997z^3 \\ \sqrt[3]{1995x^2 + 1996y^2 + 1997z^2} = \sqrt[3]{1995} + \sqrt[3]{1996} + \sqrt[3]{1997} \end{cases} \] Find the value of \(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\). | [
"1"
] | [
"Three real numbers \\(x\\), \\(y\\), and \\(z\\) satisfy the following conditions:\n\\[\n\\begin{cases}\nxyz > 0 \\\\\n1995x^3 = 1996y^3 = 1997z^3 \\\\\n\\sqrt[3]{1995x^2 + 1996y^2 + 1997z^2} = \\sqrt[3]{1995} + \\sqrt[3]{1996} + \\sqrt[3]{1997}\n\\end{cases}\n\\]\nFind the value of \\(\\frac{1}{x} + \\frac{1}{y} ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-d5a34d1c6c5cf3af9f90b172041f9636e715e89247c9cf5ad030f5676bb90439 | Three real numbers are chosen between 0 and 10. What is the probability that their sum is less than 20? | [
"\\dfrac{5}{6}"
] | [
"Three real numbers are chosen between 0 and 10. What is the probability that their sum is less than 20?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-38e019685ee845c06c0be7223ff366c76041b16481a38067b6dbbb593a6bd15e | Three real numbers are randomly chosen between 0 and 10. What is the probability that their sum is less than 10? | [
"\\(\\frac{1}{6}\\)"
] | [
"Three real numbers are randomly chosen between 0 and 10. What is the probability that their sum is less than 10?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-33de830ba8765d4aab569e4d87293488ca3cba3977c388da0de3a64d3163da76 | Three real numbers are selected at random between $0$ and $1$. Given that at least one of the three numbers is less than $\frac{1}{2}$, find the probability that the sum of the three numbers is less than $1$. | [
"\\dfrac{4}{21}"
] | [
"Three real numbers are selected at random between $0$ and $1$. Given that at least one of the three numbers is less than $\\frac{1}{2}$, find the probability that the sum of the three numbers is less than $1$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fc9adb3235e7380e1476e24b3cb816b273c0436387ba19c86bfd9e44566cc072 | Three real numbers satisfy the following conditions: 1. The square of their sum equals the sum of their squares. 2. The product of the first two numbers is equal to the square of the third number. Find these numbers. | [
"\\( (0, 0, 0), (0, k, 0), (l, 0, 0) \\) for any real numbers \\( k \\) and \\( l \\)."
] | [
"Three real numbers satisfy the following conditions:\n1. The square of their sum equals the sum of their squares.\n2. The product of the first two numbers is equal to the square of the third number.\n\nFind these numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 9
} | 10 | 0 | 0 |
omr-ws-dc0037906f6908cf10c63280effb4fecb9eb49f00801ff7ae0c4302bb08afb89 | Three real sequences \((a_n)\), \((b_n)\), and \((c_n)\) are constructed as follows: - \(a_0 = a\), \(b_0 = b\), \(c_0 = c\) where \(a\), \(b\), and \(c\) are given real numbers. - \(a_{k+1} = a_k + \frac{2}{b_k + c_k}\), \(b_{k+1} = b_k + \frac{2}{c_k + a_k}\), \(c_{k+1} = c_k + \frac{2}{a_k + b_k}\) for all \(k\). Ca... | [
"6\\sqrt{51}"
] | [
"Three real sequences \\((a_n)\\), \\((b_n)\\), and \\((c_n)\\) are constructed as follows:\n- \\(a_0 = a\\), \\(b_0 = b\\), \\(c_0 = c\\) where \\(a\\), \\(b\\), and \\(c\\) are given real numbers.\n- \\(a_{k+1} = a_k + \\frac{2}{b_k + c_k}\\), \\(b_{k+1} = b_k + \\frac{2}{c_k + a_k}\\), \\(c_{k+1} = c_k + \\frac{... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-a433eeee9b9eb60e86fa31aed62d461080b1173e610702b027be3b0e2e9197a4 | Three rebel camps are located at the vertices of a triangle with side lengths less than or equal to 20 miles, 30 miles, and 45 miles respectively. What is the maximal area that the triangle can cover? | [
"238.81"
] | [
"Three rebel camps are located at the vertices of a triangle with side lengths less than or equal to 20 miles, 30 miles, and 45 miles respectively. What is the maximal area that the triangle can cover?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f7c5197ffd63be7813bb6142119133fc907243ae1e0a10fadfa82bde943c20cd | Three rectangles of dimension \(X \times 2\) and four rectangles of dimension \(Y \times 1\) form a rectangle of area \(3XY\), where \(X\) and \(Y\) are positive integers. What is the sum of all possible values of \(X\)? | [
"6"
] | [
"Three rectangles of dimension \\(X \\times 2\\) and four rectangles of dimension \\(Y \\times 1\\) form a rectangle of area \\(3XY\\), where \\(X\\) and \\(Y\\) are positive integers. What is the sum of all possible values of \\(X\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-d1546faffe555c758a6b58e7fb6c19ffd7d129b57b03b1794fb3c93b77d5a323 | Three rectangles, with dimensions \(3 \times 5\), \(4 \times 2\), and \(6 \times 4\), are each divided into unit squares which are alternately colored black and white like a checkerboard. Each rectangle is cut along one of its diagonals into two triangles. For each triangle, let \(m\) be the total black area and \(n\) ... | [
"\\dfrac{1}{2}"
] | [
"Three rectangles, with dimensions \\(3 \\times 5\\), \\(4 \\times 2\\), and \\(6 \\times 4\\), are each divided into unit squares which are alternately colored black and white like a checkerboard. Each rectangle is cut along one of its diagonals into two triangles. For each triangle, let \\(m\\) be the total black... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 4
} | 9 | 0 | 0 |
omr-ws-5171dd8993a418321ea300500415eab097ee6377a47be0bc9534afcdc02b8ed0 | Three recursive sequences \(a\), \(b\), and \(c\) are defined as follows: \(a_0 = 0\), \(b_0 = 4\), \(c_0 = 2\) and for all nonnegative integers \(n\), \(a_{n+1} = (n+4)a_n + b_n\), \(b_{n+1} = (n+3)b_n + 2c_n\), and \(c_{n+1} = (n+2)c_n\). Compute the greatest integer \(k\) such that \(43^k\) divides \(a_{2020}\). | [
"49"
] | [
"Three recursive sequences \\(a\\), \\(b\\), and \\(c\\) are defined as follows: \\(a_0 = 0\\), \\(b_0 = 4\\), \\(c_0 = 2\\) and for all nonnegative integers \\(n\\), \\(a_{n+1} = (n+4)a_n + b_n\\), \\(b_{n+1} = (n+3)b_n + 2c_n\\), and \\(c_{n+1} = (n+2)c_n\\). Compute the greatest integer \\(k\\) such that \\(43^k... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-445da5f4e70a2b36d32a32fc1d3d63346f13978748dd44b036d6fe521f03cf6e | Three recursive sequences, $a$, $b$, and $c$ are defined as follows: $a_0 = 0$, $b_0 = 4$, $c_0 = 2$ and, for all nonnegative integers $n$, $a_{n+1} = (n+4)a_n + b_n$, $b_{n+1} = (n+3)b_n + 2c_n$, and $c_{n+1} = (n+2)c_n$. Compute the greatest integer $k$ such that $43^k$ divides $a_{2020}$. | [
"49"
] | [
"Three recursive sequences, $a$, $b$, and $c$ are defined as follows: $a_0 = 0$, $b_0 = 4$, $c_0 = 2$ and, for all nonnegative integers $n$, $a_{n+1} = (n+4)a_n + b_n$, $b_{n+1} = (n+3)b_n + 2c_n$, and $c_{n+1} = (n+2)c_n$. Compute the greatest integer $k$ such that $43^k$ divides $a_{2020}$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-2817debe825082ac2a47ce846c3fb772d6627322d08848e22a606bf3d1ef922b | Three red balls, three yellow balls, three green balls, and three white balls are placed in a bag. Nine balls are drawn out without replacement. How many different color combinations are there? a) Directly, as stated. b) Indirectly, by thinking about the colors of the balls remaining in the bag. | [
"20"
] | [
"Three red balls, three yellow balls, three green balls, and three white balls are placed in a bag. Nine balls are drawn out without replacement. How many different color combinations are there?\na) Directly, as stated.\nb) Indirectly, by thinking about the colors of the balls remaining in the bag."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-49d08969aa84e43b4e7e82ad660be89583c68b2e001be549477252f18548fe47 | Three red beads, two white beads, and one blue bead are placed in a line in random order. What is the probability that no two neighboring beads are the same color? | [
"1/6",
"\\(\\frac{1}{6}\\)"
] | [
"Three red beads, two white beads, and one blue bead are placed in a line in random order. What is the probability that no two neighboring beads are the same color?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 23,
"genselect": 8,
"tir": 55
} | 86 | 0 | 0 |
omr-ws-624851f79089855cca2ea4131e3c0bf3362ad0c345d64281a4b446f64ae392d8 | Three red counters, four green counters, and five blue counters are placed in a row in random order. Find the probability that no two blue counters are adjacent. | [
"\\dfrac{7}{99}"
] | [
"Three red counters, four green counters, and five blue counters are placed in a row in random order. Find the probability that no two blue counters are adjacent."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3defde409fbb05d7d890633048b59346b6a04587bead0673da9cf9d1457493a7 | Three red dots and three blue dots are placed randomly in the plane, with no three dots collinear. Nine line segments are drawn, one from each red dot to each blue dot. What is the fewest possible number of intersections formed by these line segments, not counting intersections at the six dots? | [
"$0$\n\nIt is possible to have no intersections at all. If we draw six dots so that the conditions are fulfilled, the first dot can have 5 lines drawn from it with no intersections. For the remaining 4 segments, connect adjacent dots together so that no lines intersect.\n\nBasically, this would be equivalent to a h... | [
"Three red dots and three blue dots are placed randomly in the plane, with no three dots collinear. Nine line segments are drawn, one from each red dot to each blue dot. What is the fewest possible number of intersections formed by these line segments, not counting intersections at the six dots?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4f268b4a93347f62d5c15ad56201bb8655efee558e4b4392794308b2e71b9480 | Three relatively prime integers are the sides of a right triangle. If the smallest leg has length 28, find the sum of all possible values of the hypotenuse, or prove that the sum is undefined. | [
"250"
] | [
"Three relatively prime integers are the sides of a right triangle. If the smallest leg has length 28, find the sum of all possible values of the hypotenuse, or prove that the sum is undefined."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-5d4b2fe51936f52bbf00172f9ac05a7ad104a08e71c1ff4265ce6fca848d5c71 | Three relatively prime integers are the sides of a right triangle. If the smallest side of the triangle has length $28$, find the sum of all possible values of the hypotenuse. | [
"250"
] | [
"Three relatively prime integers are the sides of a right triangle. If the smallest side of the triangle has length $28$, find the sum of all possible values of the hypotenuse."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c9a1e3a233bc1090aeb22a680c1ca9db7c5d72219cdb9465c5f4e0c628c93280 | Three ribbons of the same width are laid to form equilateral triangles. The perimeter of the internal triangle is 16 cm, and the perimeter of each quadrangle formed by the crossing ribbons is 4 cm. Find the perimeter of the outer triangle. | [
"25"
] | [
"Three ribbons of the same width are laid to form equilateral triangles. The perimeter of the internal triangle is 16 cm, and the perimeter of each quadrangle formed by the crossing ribbons is 4 cm. Find the perimeter of the outer triangle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 5
} | 10 | 0 | 0 |
omr-ws-b38f768748b5440301c6522c9d4c7644fdb7e261180345456d2f546db0d78ae4 | Three right circular cones each with base radius $15$ and height $20$ share the same vertex, and their bases are pairwise tangent. The least possible height of a fourth cone that contains the other three cones in its interior can be expressed in the form $\sqrt{m}+\sqrt{n}$, where $m$ and $n$ are positive integers. Wha... | [
"no_valid_answer_found"
] | [
"Three right circular cones each with base radius $15$ and height $20$ share the same vertex, and their bases are pairwise tangent. The least possible height of a fourth cone that contains the other three cones in its interior can be expressed in the form $\\sqrt{m}+\\sqrt{n}$, where $m$ and $n$ are positive intege... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a243620e0613282c6001a070aeff388ceb34912ad696c08d942002cc7802b132 | Three right-angled triangles have been placed in a halfplane determined by a line $\ell$, each with one leg lying on $\ell$. Assume that there is a line parallel to $\ell$ cutting the triangles into three segments of equal area. If each of the triangles is rotated so that its other leg lies on $\ell$, find the ratio of... | [
"\\dfrac{1}{3}"
] | [
"Three right-angled triangles have been placed in a halfplane determined by a line $\\ell$, each with one leg lying on $\\ell$. Assume that there is a line parallel to $\\ell$ cutting the triangles into three segments of equal area. If each of the triangles is rotated so that its other leg lies on $\\ell$, find the... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-dffa14737f1c616d9f4a10c0179d05323b9a7f08d0535304b65fee91297166eb | Three rods of lengths $1396$, $1439$, and $2018$ millimeters are hinged from one tip on the ground. What is the smallest value for the radius of the circle passing through the other three tips of the rods in millimeters? | [
"311"
] | [
"Three rods of lengths $1396$, $1439$, and $2018$ millimeters are hinged from one tip on the ground. What is the smallest value for the radius of the circle passing through the other three tips of the rods in millimeters?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 5
} | 15 | 0 | 0 |
omr-ws-d76cec4f2da5feacf4ad19f9fba17ef0b93e4e8ab08217597b7baf0cbfe504d9 | Three runners start running simultaneously from the same point on a $500$-meter circular track. They each run clockwise around the course maintaining constant speeds of $4.4$, $4.8$, and $5.0$ meters per second. The runners stop once they are all together again somewhere on the circular course. How many seconds do the ... | [
"2500"
] | [
"Three runners start running simultaneously from the same point on a $500$-meter circular track. They each run clockwise around the course maintaining constant speeds of $4.4$, $4.8$, and $5.0$ meters per second. The runners stop once they are all together again somewhere on the circular course. How many seconds do... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-2864fdb4494a4f77b637aa6bbb77d87e0317005f02aaeb946ba3805fcee63b3d | Three runners start running simultaneously from the same point on a 500-meter circular track. They each run clockwise around the course maintaining constant speeds of 4.4, 4.8, and 5.0 meters per second. The runners stop once they are all together again somewhere on the circular course. How many seconds do the runners ... | [
"2500"
] | [
"Three runners start running simultaneously from the same point on a 500-meter circular track. They each run clockwise around the course maintaining constant speeds of 4.4, 4.8, and 5.0 meters per second. The runners stop once they are all together again somewhere on the circular course. How many seconds do the run... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-83fdb981479de311065e44a609f51e2f61c4f683efd945a11b7e5b3b2b57d9ba | Three sailors were shipwrecked on an island and gathered bananas. Each night, one sailor divided the bananas into three equal shares, found one banana left over, fed it to a monkey, and hid his share. This happened three times. The next morning, they divided the remaining bananas and found one banana left over, which t... | [
"79"
] | [
"Three sailors were shipwrecked on an island and gathered bananas. Each night, one sailor divided the bananas into three equal shares, found one banana left over, fed it to a monkey, and hid his share. This happened three times. The next morning, they divided the remaining bananas and found one banana left over, wh... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-821826ac73e6e0cbba6d8f9bb1f6db9f2e180d99193642f5ff881ec8704a8d27 | Three schools \(A\), \(B\), and \(C\) each have five players denoted \(a_i\), \(b_i\), and \(c_i\) respectively. The tournament rules are: 1. Players from each school have matches in order with respect to indices, and defeated players are eliminated; the first match is between \(a_1\) and \(b_1\). 2. If \(y_j \in Y\) d... | [
"0"
] | [
"Three schools \\(A\\), \\(B\\), and \\(C\\) each have five players denoted \\(a_i\\), \\(b_i\\), and \\(c_i\\) respectively. The tournament rules are:\n1. Players from each school have matches in order with respect to indices, and defeated players are eliminated; the first match is between \\(a_1\\) and \\(b_1\\).... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-5f80313bfd73d7ae6d64008e5d19ce2a4b563435942985abbf16b6af2fdaec26 | Three schools \(A\), \(B\), and \(C\), each with five players denoted \(a_i\), \(b_i\), \(c_i\) respectively, take part in a chess tournament. The tournament is held following the rules: (i) Players from each school have matches in order with respect to indices, and defeated players are eliminated; the first match is b... | [
"0"
] | [
"Three schools \\(A\\), \\(B\\), and \\(C\\), each with five players denoted \\(a_i\\), \\(b_i\\), \\(c_i\\) respectively, take part in a chess tournament. The tournament is held following the rules:\n(i) Players from each school have matches in order with respect to indices, and defeated players are eliminated; th... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-73401ea44cf29262d4a96fb23b4551110e6cf6ce7656af07351e7335be23fa56 | Three schools compete in a chess tournament, each with five students. The first match is between $a_{1}$ from School $A$ and $b_{1}$ from School $B$. If a player loses, they are immediately eliminated. If a player $x_{i}$ from School $X$ beats a player $y_{j}$ from School $Y$, and there is a player remaining in School ... | [
"4"
] | [
"Three schools compete in a chess tournament, each with five students. The first match is between $a_{1}$ from School $A$ and $b_{1}$ from School $B$. If a player loses, they are immediately eliminated. If a player $x_{i}$ from School $X$ beats a player $y_{j}$ from School $Y$, and there is a player remaining in Sc... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-17859b6587fe76fba540c963e6c4e526411d18ec16e9028506265af58c945593 | Three schools each have 200 students. Every student has at least one friend in each school. There exists a set $E$ of 300 students (among the 600) such that for any school $S$ and any two students $x, y \in E$ but not in $S$, the number of friends in $S$ of $x$ and $y$ are different. Find the minimum number of ways to ... | [
"100"
] | [
"Three schools each have 200 students. Every student has at least one friend in each school. There exists a set $E$ of 300 students (among the 600) such that for any school $S$ and any two students $x, y \\in E$ but not in $S$, the number of friends in $S$ of $x$ and $y$ are different. Find the minimum number of wa... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-00647cb7d2e836509cd3719d3b1ba4294ac5123c846673cc1a28ed38d029f2a2 | Three schools each have \( n \) students. Each student knows a total of \( n + 1 \) students at the other two schools. Let \( S(n) \) be the minimum number of triplets (one student from each school) who know each other. Find \( S(10) \). | [
"75"
] | [
"Three schools each have \\( n \\) students. Each student knows a total of \\( n + 1 \\) students at the other two schools. Let \\( S(n) \\) be the minimum number of triplets (one student from each school) who know each other. Find \\( S(10) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-708d0efa4c4c715dadb60740c09a3a3326121b255dfce8a6ad2ef94c2b162212 | Three secants $A-B-C$, $A-N-F$, and $A-D-E$ are drawn from an external point $A$ to a circle, with points $B, C, D, E, N, F$ lying on the circle. Given that $\angle CAF = \angle FAE = 60^\circ$, calculate the value of $AB + AC + AD + AE - AN - AF$. | [
"no_valid_answer_found"
] | [
"Three secants $A-B-C$, $A-N-F$, and $A-D-E$ are drawn from an external point $A$ to a circle, with points $B, C, D, E, N, F$ lying on the circle. Given that $\\angle CAF = \\angle FAE = 60^\\circ$, calculate the value of $AB + AC + AD + AE - AN - AF$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2b7d141a794e59dffe0a4afd52323805951dfc65f7d11ecbb69054c3313aefef | Three sectors are chosen at random from circle $C$, having angles $\frac{\pi}{10}, \frac{2 \pi}{10}, \frac{3 \pi}{10}$ respectively. What is the probability that these sectors have no points (other than the centre) in common? | [
"\\dfrac{49}{60}"
] | [
"Three sectors are chosen at random from circle $C$, having angles $\\frac{\\pi}{10}, \\frac{2 \\pi}{10}, \\frac{3 \\pi}{10}$ respectively. What is the probability that these sectors have no points (other than the centre) in common?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 8
} | 12 | 0 | 0 |
omr-ws-abb6f7399abe71d8161c55449bc8a39b42b9aa63fc6b1f2a89d061ed0c78e74a | Three seismograph stations are arranged in a collinear formation and detect an earthquake. The distances from the epicenter to the stations are 207 miles, 253 miles, and 299 miles, respectively. How many miles apart are the two closest stations? | [
"46"
] | [
"Three seismograph stations are arranged in a collinear formation and detect an earthquake. The distances from the epicenter to the stations are 207 miles, 253 miles, and 299 miles, respectively. How many miles apart are the two closest stations?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5cc895211df179028ee6acc77adc21ae66fb29c85c1d9d58d4894566e1cfa14e | Three semicircles of radius 1 are constructed on the diameter AB of a semicircle of radius 2. The centers of the small semicircles divide AB into four line segments of equal length. What is the area of the shaded region that lies within the large semicircle but outside the smaller semicircles? | [
"\\(\\frac{7\\pi}{6} - \\frac{\\sqrt{3}}{2}\\)"
] | [
"Three semicircles of radius 1 are constructed on the diameter AB of a semicircle of radius 2. The centers of the small semicircles divide AB into four line segments of equal length. What is the area of the shaded region that lies within the large semicircle but outside the smaller semicircles?"
] | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 7
} | 7 | 0 | 0 |
omr-ws-bf7c6228e7ace9614778f462eb6b28f4f31ee0d274e8badd4b9703835387f19b | Three semicircles of radius 1 are constructed on the diameter \( AB \) of a semicircle of radius 2. The centers of the small semicircles divide \( \overline{AB} \) into four line segments of equal length. What is the area of the shaded region that lies within the large semicircle but outside the smaller semicircles? | [
"\\( \\frac{7\\pi}{6} - \\frac{\\sqrt{3}}{2} \\)"
] | [
"Three semicircles of radius 1 are constructed on the diameter \\( AB \\) of a semicircle of radius 2. The centers of the small semicircles divide \\( \\overline{AB} \\) into four line segments of equal length. What is the area of the shaded region that lies within the large semicircle but outside the smaller semic... | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 7
} | 7 | 0 | 0 |
omr-ws-8cde3975d1f1a5a7b8c424c7d737e8492c36cf0b9836ed5a2533fe2b69a842cd | Three sequences ${a_n},{b_n},{c_n}$ satisfy the following conditions: - $a_1=2,\,b_1=4,\,c_1=5$ - $\forall n,\; a_{n+1}=b_n+\frac{1}{c_n}, \, b_{n+1}=c_n+\frac{1}{a_n}, \, c_{n+1}=a_n+\frac{1}{b_n}$ Find the smallest positive integer $n$ such that $max(a_n,b_n,c_n) > 10$. | [
"19"
] | [
"Three sequences ${a_n},{b_n},{c_n}$ satisfy the following conditions:\n- $a_1=2,\\,b_1=4,\\,c_1=5$\n- $\\forall n,\\; a_{n+1}=b_n+\\frac{1}{c_n}, \\, b_{n+1}=c_n+\\frac{1}{a_n}, \\, c_{n+1}=a_n+\\frac{1}{b_n}$\n\nFind the smallest positive integer $n$ such that $max(a_n,b_n,c_n) > 10$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8b32c301cdeeeb2a86df8a5f6257c07f6330c62297d680ca7f39ddaef6753c79 | Three sets of numbers \( A, B, \) and \( C \) are defined as powers of \( \frac{1}{2}, \frac{1}{3}, \) and \( \frac{1}{4} \) starting with the zeroeth power. Define two elements of two sets \( A_i \) and \( B_n \) to be corresponding if \( i = n \). Now, if we take each element of each set and multiply it by elements f... | [
"\\(\\frac{68}{23}\\)"
] | [
"Three sets of numbers \\( A, B, \\) and \\( C \\) are defined as powers of \\( \\frac{1}{2}, \\frac{1}{3}, \\) and \\( \\frac{1}{4} \\) starting with the zeroeth power. Define two elements of two sets \\( A_i \\) and \\( B_n \\) to be corresponding if \\( i = n \\). Now, if we take each element of each set and mul... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-2e4fb1e16665c0867571f8d563be7520d942e4c03fd45bb4d88b39e17a6c823c | Three sets of residues modulo \( n = 4k \) are given, with each set consisting of exactly \( n/2 = 2k \) elements. It is known that any shift of any of these three sets intersects any other set of the given trio by exactly \( k \) elements. Find the sum of all possible values of \( k \) for \( n \leq 100 \). | [
"30"
] | [
"Three sets of residues modulo \\( n = 4k \\) are given, with each set consisting of exactly \\( n/2 = 2k \\) elements. It is known that any shift of any of these three sets intersects any other set of the given trio by exactly \\( k \\) elements. Find the sum of all possible values of \\( k \\) for \\( n \\leq 100... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 7
} | 11 | 0 | 0 |
omr-ws-602782bfeb775d8df149edefb63d4b912b08618383ccb24d7d56872ee918bdbf | Three shepherds cyclically alternate shearing the same herd of sheep, following these rules: (i) Every day a sheep can be shorn on one side only; (ii) Every day at least one sheep must be shorn; (iii) No two days the same group of sheep can be shorn. Can any shepherd shear the sheep in such a way to ensure they avoid b... | [
"\\text{No}"
] | [
"Three shepherds cyclically alternate shearing the same herd of sheep, following these rules:\n(i) Every day a sheep can be shorn on one side only;\n(ii) Every day at least one sheep must be shorn;\n(iii) No two days the same group of sheep can be shorn.\nCan any shepherd shear the sheep in such a way to ensure the... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-88d7a971140e8d7651f797594e6587b9afa7fe682230d801dc043bca1cfd37fd | Three shooters have probabilities of hitting a target of \( \frac{1}{7} \), \( \frac{1}{6} \), and \( \frac{2}{13} \) respectively. What is the probability that exactly one of them will hit the target if they fire simultaneously? | [
"\\( \\frac{181}{546} \\)"
] | [
"Three shooters have probabilities of hitting a target of \\( \\frac{1}{7} \\), \\( \\frac{1}{6} \\), and \\( \\frac{2}{13} \\) respectively. What is the probability that exactly one of them will hit the target if they fire simultaneously?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-3fcd0365e8631c44d3d61fb04b9c61312e353e30508ae6e79a2a3195eabc1fe1 | Three siblings, Anne, Branwell, and Charlotte, each made a list of books they had read from a book club's catalogue. Each list had the same number of books. Anne's and Branwell's lists had 17 books in common, Anne's and Charlotte's lists had 8 books in common, Branwell's and Charlotte's lists had 9 books in common, and... | [
"55"
] | [
"Three siblings, Anne, Branwell, and Charlotte, each made a list of books they had read from a book club's catalogue. Each list had the same number of books. Anne's and Branwell's lists had 17 books in common, Anne's and Charlotte's lists had 8 books in common, Branwell's and Charlotte's lists had 9 books in common... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-341025e9cde58654f83ae2045faa281991d26e0359414d76698f125a2c7c0a5e | Three sides of a non-convex quadrilateral have the same length $1$. Specify the range of values for the length of the fourth side. | [
"\\(0 < \\text{length of the fourth side} < 3\\)"
] | [
"Three sides of a non-convex quadrilateral have the same length $1$. Specify the range of values for the length of the fourth side."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 5,
"tir": 3
} | 8 | 0 | 0 |
omr-ws-dfd79937c534796e7352249bc503829392c3f1e30da41fc00115da789e985fdb | Three sides of a quadrilateral have lengths 1, 2, and 5. The fourth side is an integer \( x \). What is the sum of all possible values of \( x \)? | [
"25"
] | [
"Three sides of a quadrilateral have lengths 1, 2, and 5. The fourth side is an integer \\( x \\). What is the sum of all possible values of \\( x \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-58d51f56909258f2688fa1f80b0f3f449bd9352557d3296573b04b9b969c0bc7 | Three sides of a quadrilateral have lengths 1, 2, and 5. The fourth side is an integer, \( x \). What is the sum of all the possible values of \( x \)? | [
"25"
] | [
"Three sides of a quadrilateral have lengths 1, 2, and 5. The fourth side is an integer, \\( x \\). What is the sum of all the possible values of \\( x \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0671e8f46ec6dc90dec2ac1d5e2ba68b0562ef5ef22bbf3e2f2033adcb169c6f | Three sides of a quadrilateral have the same length, and they include angles measuring 60 and 70 degrees. What is the measure of the largest angle in the quadrilateral? | [
"145 degrees"
] | [
"Three sides of a quadrilateral have the same length, and they include angles measuring 60 and 70 degrees. What is the measure of the largest angle in the quadrilateral?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 8,
"tir": 9
} | 23 | 0 | 0 |
omr-ws-68d9999fba6faad06c584f4f40cf78c0f2ba664562916f6ae0ed4bbc2288c20f | Three sides of a regular hexagon are chosen at random, and their midpoints are connected. Find the probability that the resulting triangle is a right triangle. | [
"\\(\\frac{3}{5}\\)"
] | [
"Three sides of a regular hexagon are chosen at random, and their midpoints are connected. Find the probability that the resulting triangle is a right triangle."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 29
} | 42 | 0 | 0 |
omr-ws-05b3cfb7b94f8689aba27f22a09dbdd1e57e6fd6fc4c5761949a229ea77598c9 | Three sides of a tetrahedron are right-angled triangles with the right angle at their common vertex. The areas of these sides are $A$, $B$, and $C$. Find the total surface area of the tetrahedron. | [
"\\( A + B + C + \\sqrt{A^2 + B^2 + C^2} \\)"
] | [
"Three sides of a tetrahedron are right-angled triangles with the right angle at their common vertex. The areas of these sides are $A$, $B$, and $C$. Find the total surface area of the tetrahedron."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 18,
"tir": 16
} | 34 | 0 | 0 |
omr-ws-2fcddc66f7fad38e7f75e95b223154f0a9a6d537ee6d513913ef2a9e2d4c8638 | Three sides of a triangle touch the parabola \( y^2 = 4ax \) and two of its vertices lie on the parabola \( y^2 = 4b(x+c) \). Find the locus of the third vertex. | [
"y^2 = \\dfrac{4(2b - a)^2}{a} \\left( x + \\dfrac{4bc}{a} \\right)"
] | [
"Three sides of a triangle touch the parabola \\( y^2 = 4ax \\) and two of its vertices lie on the parabola \\( y^2 = 4b(x+c) \\). Find the locus of the third vertex."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 6
} | 17 | 0 | 0 |
omr-ws-1e1eafb4be8755c8bf1ba616412138dfcfd158badd322897274f34620d8c7d5f | Three sides of quadrilateral ABCD have length 5. The diagonals of ABCD intersect at P. If AB, BC, CD, DA, AP, BP, CP, and DP are of integer length, compute the area of ABCD. | [
"24"
] | [
"Three sides of quadrilateral ABCD have length 5. The diagonals of ABCD intersect at P. If AB, BC, CD, DA, AP, BP, CP, and DP are of integer length, compute the area of ABCD."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 5
} | 13 | 0 | 0 |
omr-ws-0915ceb3c6bcd5e368c180ece60ec949fef9bb8736cd9ab7206f423110b58918 | Three silvered mirrors each pass $1/4$ of the light striking them head-on and reflect $3/4$ of it. They are set parallel to each other, and a beam of light is directed head-on at the trio. What fraction of the original light eventually gets through all three mirrors? | [
"\\dfrac{4}{49}"
] | [
"Three silvered mirrors each pass $1/4$ of the light striking them head-on and reflect $3/4$ of it. They are set parallel to each other, and a beam of light is directed head-on at the trio. What fraction of the original light eventually gets through all three mirrors?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 1
} | 13 | 0 | 0 |
omr-ws-d5dbc39cac4de42d73f53d408123030f578cc75bafac035399f4a0093ef132cc | Three similar isosceles triangles share a common vertex opposite their bases, forming a convex hexagon with their other vertices. Let the midpoints of the opposite sides of this hexagon be connected to form three lines. Calculate the ratio of the distance from the common vertex of the triangles to the point of concurre... | [
"no_valid_answer_found"
] | [
"Three similar isosceles triangles share a common vertex opposite their bases, forming a convex hexagon with their other vertices. Let the midpoints of the opposite sides of this hexagon be connected to form three lines. Calculate the ratio of the distance from the common vertex of the triangles to the point of con... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-23f6b281aeecfeffeeb20a1582b6faf4e2ae48ffc49351b5b5a811036c9c8e82 | Three skaters whose rates, taken in some order, form a geometric progression, start out at the same time on a skating circuit. After a time, the second one overtakes the first, covering 400 meters more. The third skater covers the same distance that the first did when the latter was overtaken by the second during a tim... | [
"0.6 km/min"
] | [
"Three skaters whose rates, taken in some order, form a geometric progression, start out at the same time on a skating circuit. After a time, the second one overtakes the first, covering 400 meters more. The third skater covers the same distance that the first did when the latter was overtaken by the second during ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dd0d3ecb7fd062612d61f58ae5b36594dbe8d4e470f13a2085c8c8816882cf38 | Three slices of bread are to be toasted under a grill. The grill can hold two slices at once but only one side is toasted at a time. It takes 30 seconds to toast one side of a piece of bread, 5 seconds to put a piece in or take a piece out, and 3 seconds to turn a piece over. What is the shortest time in which the thre... | [
"136 seconds"
] | [
"Three slices of bread are to be toasted under a grill. The grill can hold two slices at once but only one side is toasted at a time. It takes 30 seconds to toast one side of a piece of bread, 5 seconds to put a piece in or take a piece out, and 3 seconds to turn a piece over. What is the shortest time in which the... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 4,
"tir": 1
} | 15 | 0 | 0 |
omr-ws-77ef2ae9a9d23a16d3b958c58549f628c59627cc3760ccf20682f24fbd8db1c9 | Three small congruent circles are mutually tangent, and a large circle is tangent to all three small circles. Point \( P \) is on the large circle, and \( PA \), \( PB \), and \( PC \) are tangents to the three small circles. If the radius of each small circle is \( r \) and the radius of the large circle is \( R \), f... | [
"2"
] | [
"Three small congruent circles are mutually tangent, and a large circle is tangent to all three small circles. Point \\( P \\) is on the large circle, and \\( PA \\), \\( PB \\), and \\( PC \\) are tangents to the three small circles. If the radius of each small circle is \\( r \\) and the radius of the large circl... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 2
} | 6 | 0 | 0 |
omr-ws-faf48307bb0b09dc753d1f300e0058db6679d4608f8a7f002e26127555b5801a | Three smash brothers, Vismay, Andrew, and Konwoo, are located at vertices $V$, $A$, and $K$ of a tetrahedron $KAVS$. Each second, each brother must move to a different vertex of the tetrahedron. If three smash brothers meet at the same vertex or in the middle of a path, they SMASH. Find the expected number of seconds b... | [
"\\dfrac{243}{14}"
] | [
"Three smash brothers, Vismay, Andrew, and Konwoo, are located at vertices $V$, $A$, and $K$ of a tetrahedron $KAVS$. Each second, each brother must move to a different vertex of the tetrahedron. If three smash brothers meet at the same vertex or in the middle of a path, they SMASH. Find the expected number of seco... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 4
} | 6 | 0 | 0 |
omr-ws-5daba1b696240fdb37e9aae43eff4fee5aee704a2cd87a55c03f0a5ae2c96f01 | Three snails stand 60 cm apart from each other and crawl towards the one clockwise to it at a speed of 5 cm/min. How long do the snails take to reach each other? | [
"8 minutes"
] | [
"Three snails stand 60 cm apart from each other and crawl towards the one clockwise to it at a speed of 5 cm/min. How long do the snails take to reach each other?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 42,
"genselect": 7,
"tir": 6
} | 55 | 0 | 0 |
omr-ws-33dacddf3410125d7ec9139e72c036237512f5d4a91bb0aa34142bb121522731 | Three snails stand 60 cm apart from each other and crawl towards the one clockwise to it at a speed of 5 cm/min. How many revolutions around the center do they make before they meet? | [
"Infinite"
] | [
"Three snails stand 60 cm apart from each other and crawl towards the one clockwise to it at a speed of 5 cm/min. How many revolutions around the center do they make before they meet?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 8
} | 8 | 0 | 0 |
omr-ws-09b80167057252b3fd026395c88a384670640826f1cdc2c82d6631d02495345a | Three snails start simultaneously from the origin \( O \) and move along the abscissa axis \( Ox \) with constant speeds \( v_1 \), \( v_2 \), and \( v_3 \), respectively. Let \( A(t) \), \( B(t) \), and \( C(t) \) be points on the graph of the hyperbola \( y = \frac{1}{x} \) such that their abscissae are equal to the ... | [
" \\dfrac{ |(v_1 - v_2)(v_2 - v_3)(v_3 - v_1)| }{ 2 v_1 v_2 v_3 } "
] | [
"Three snails start simultaneously from the origin \\( O \\) and move along the abscissa axis \\( Ox \\) with constant speeds \\( v_1 \\), \\( v_2 \\), and \\( v_3 \\), respectively. Let \\( A(t) \\), \\( B(t) \\), and \\( C(t) \\) be points on the graph of the hyperbola \\( y = \\frac{1}{x} \\) such that their abs... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-65318e7c47589be0733833f86f8f7d84caf6fea17b5aad3e7756e81b58686e68 | Three solid spheres with radii \( 1 \), \( 2 \), and \( 3 \) are all tangent to a plane, and each sphere is also externally tangent to the other two spheres. The three points at which each sphere is tangent to the plane are the vertices of a triangle in that plane. Compute the area of the triangle. | [
"$\\sqrt{23}$"
] | [
"Three solid spheres with radii \\( 1 \\), \\( 2 \\), and \\( 3 \\) are all tangent to a plane, and each sphere is also externally tangent to the other two spheres. The three points at which each sphere is tangent to the plane are the vertices of a triangle in that plane. Compute the area of the triangle."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-0e46839c61ab00e429f8504ce49b8e9040f1e2c271eb999ea36b9b8f934e7f9a | Three spacecraft are landing on a spherical planet, with their positions chosen independently and uniformly over the entire surface. Two spacecraft can communicate with each other only if the angle between their positions relative to the center of the planet is at least $\frac{\pi}{2}$. What is the probability that the... | [
"\\(\\frac{2+\\pi}{4\\pi}\\)"
] | [
"Three spacecraft are landing on a spherical planet, with their positions chosen independently and uniformly over the entire surface. Two spacecraft can communicate with each other only if the angle between their positions relative to the center of the planet is at least $\\frac{\\pi}{2}$. What is the probability t... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c96b73c8667d314fbf8f55b3b56ab6064180e62fce1ae410ccd241f45124e175 | Three spaceships A, B, and C land at random on the surface of a spherical planet X, with their positions being independent and uniformly distributed. Spaceships A and B can communicate directly if the angle AOB is less than 90 degrees. Let \( P \) be the probability that A and B can keep in touch (either directly or vi... | [
"2\\pi - 1"
] | [
"Three spaceships A, B, and C land at random on the surface of a spherical planet X, with their positions being independent and uniformly distributed. Spaceships A and B can communicate directly if the angle AOB is less than 90 degrees. Let \\( P \\) be the probability that A and B can keep in touch (either directl... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-751590b55872c6274a4e79a6b1ec80d38fe76bea27dc40904058aa3283ffe7cf | Three spaceships, $A$, $B$, and $C$, are positioned randomly on the circumference of a circle. Each spaceship can communicate directly with another if the arc between them is less than or equal to $\frac{\pi}{2}$. Calculate the probability that there exists a communication path between any two spaceships, either direct... | [
"\\dfrac{3}{4}"
] | [
"Three spaceships, $A$, $B$, and $C$, are positioned randomly on the circumference of a circle. Each spaceship can communicate directly with another if the arc between them is less than or equal to $\\frac{\\pi}{2}$. Calculate the probability that there exists a communication path between any two spaceships, either... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d6a88655a2a0f2efad30740afba80476952ef83efddc3e13d3b28e79519b6574 | Three speed skaters have a friendly "race" on a skating oval. They all start from the same point and skate in the same direction, but with different speeds that they maintain throughout the race. The slowest skater does $1$ lap per minute, the fastest one does $3.14$ laps per minute, and the middle one does $L$ laps a ... | [
"16",
"59"
] | [
"Three speed skaters have a friendly \"race\" on a skating oval. They all start from the same point and skate in the same direction, but with different speeds that they maintain throughout the race. The slowest skater does $1$ lap per minute, the fastest one does $3.14$ laps per minute, and the middle one does $L$ ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-b774e57f021b02921e9da02f8369e9ba4f5b62926c4de9c7fca294c64e8f3a62 | Three spheres $S_1$, $S_2$, and $S_3$ are mutually externally tangent and have radii 2004, 3507, and 4676 respectively. Plane $P$ is tangent to $S_1$, $S_2$, and $S_3$ at $A$, $B$, and $C$ respectively. The area of triangle $ABC$ can be expressed as $m\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is not di... | [
"707"
] | [
"Three spheres $S_1$, $S_2$, and $S_3$ are mutually externally tangent and have radii 2004, 3507, and 4676 respectively. Plane $P$ is tangent to $S_1$, $S_2$, and $S_3$ at $A$, $B$, and $C$ respectively. The area of triangle $ABC$ can be expressed as $m\\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is ... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-bc144f4ba888db8f3dda3044cd71a3f593163c41c18e88ea3333d7768d2a8a75 | Three spheres $S_1$, $S_2$, and $S_3$ are mutually externally tangent with radii $2004$, $3507$, and $4676$ respectively. Plane $P$ is tangent to $S_1$, $S_2$, and $S_3$ at points $A$, $B$, and $C$ respectively. The area of triangle $ABC$ can be expressed as $m\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ ... | [
"456"
] | [
"Three spheres $S_1$, $S_2$, and $S_3$ are mutually externally tangent with radii $2004$, $3507$, and $4676$ respectively. Plane $P$ is tangent to $S_1$, $S_2$, and $S_3$ at points $A$, $B$, and $C$ respectively. The area of triangle $ABC$ can be expressed as $m\\sqrt{n}$, where $m$ and $n$ are positive integers an... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b3d38a4f07c727ec5b4ef83d21cedcdcf1d8652cb28119c9a494a56655e92278 | Three spheres \( S_1, S_2, S_3 \) are mutually externally tangent with radii \( 2004, 3507, 4676 \) respectively. Plane \( P \) is tangent to \( S_1, S_2, S_3 \) at points \( A, B, C \) respectively. The area of triangle \( ABC \) can be expressed as \( m \sqrt{n} \), where \( m \) and \( n \) are positive integers and... | [
"707"
] | [
"Three spheres \\( S_1, S_2, S_3 \\) are mutually externally tangent with radii \\( 2004, 3507, 4676 \\) respectively. Plane \\( P \\) is tangent to \\( S_1, S_2, S_3 \\) at points \\( A, B, C \\) respectively. The area of triangle \\( ABC \\) can be expressed as \\( m \\sqrt{n} \\), where \\( m \\) and \\( n \\) a... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 7,
"tir": 15
} | 35 | 0 | 0 |
omr-ws-29c2b1630b159b7ac34c880a6c7fe81c724b65b32f8052013bcc7911e0d0a48a | Three spheres \( s_1 \), \( s_2 \), and \( s_3 \) intersect along a common circle \( \omega \). Let \( A \) be an arbitrary point on \( \omega \). Ray \( AB \) intersects the spheres at points \( B_1 \), \( B_2 \), and \( B_3 \) respectively, and ray \( AC \) intersects the spheres at points \( C_1 \), \( C_2 \), and \... | [
"\\dfrac{1}{2}"
] | [
"Three spheres \\( s_1 \\), \\( s_2 \\), and \\( s_3 \\) intersect along a common circle \\( \\omega \\). Let \\( A \\) be an arbitrary point on \\( \\omega \\). Ray \\( AB \\) intersects the spheres at points \\( B_1 \\), \\( B_2 \\), and \\( B_3 \\) respectively, and ray \\( AC \\) intersects the spheres at point... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d5ece0b69c26fc1cc871329698f4fabe31f7b8740c5c3e31a2c71d6005f1dcc3 | Three spheres are tangent to a plane at the vertices of a right triangle with sides $6$, $8$, and $10$. The spheres are also tangent to each other. Find the radii of these spheres. | [
"\\( \\frac{15}{4}, \\frac{12}{5}, \\frac{20}{5} \\)"
] | [
"Three spheres are tangent to a plane at the vertices of a right triangle with sides $6$, $8$, and $10$. The spheres are also tangent to each other. Find the radii of these spheres."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ec90cd8d80b395fb9c59dd41160a36edca4e16428e9bd4514e05d61914b116c1 | Three spheres are tangent to a plane at the vertices of a triangle with sides $6$, $8$, and $10$, and are also tangent to each other. Find the radii of these spheres. | [
"\\( \\frac{12}{5}, \\frac{15}{4}, \\frac{20}{3} \\)"
] | [
"Three spheres are tangent to a plane at the vertices of a triangle with sides $6$, $8$, and $10$, and are also tangent to each other. Find the radii of these spheres."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 14
} | 16 | 0 | 0 |
omr-ws-6689b5990c3dd05d64eaac209850e5f497af8ed0b0b4267e4b0a59d3f2852baa | Three spheres are tangent to a plane at the vertices of a triangle with sides $6, 8,$ and $10$, and are tangent to each other. Find the radii of these spheres. | [
"\\( a = \\frac{12}{5}, b = \\frac{15}{4}, c = \\frac{20}{3} \\)"
] | [
"Three spheres are tangent to a plane at the vertices of a triangle with sides $6, 8,$ and $10$, and are tangent to each other. Find the radii of these spheres."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 11
} | 15 | 0 | 0 |
omr-ws-17546496705ea51d1ad6fed3b687a289ee96839aed61f90cd2f04b4a0d1ca6bb | Three spheres are tangent to a plane at the vertices of a triangle with sides 6, 8, and 10, and are tangent to each other. Find the radii of these spheres. | [
"\\( \\frac{12}{5} \\), \\( \\frac{15}{4} \\), \\( \\frac{20}{3} \\)",
"\\( a = \\frac{20}{3}, b = \\frac{12}{5}, c = \\frac{15}{4} \\)",
"\\(\\left(\\frac{15}{4}, \\frac{12}{5}, \\frac{20}{3}\\right)\\)"
] | [
"Three spheres are tangent to a plane at the vertices of a triangle with sides 6, 8, and 10, and are tangent to each other. Find the radii of these spheres."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 6,
"tir": 59
} | 65 | 0 | 0 |
omr-ws-daaf6d7ebf6bb435479760c60448ac2d0c0d41d67a862a12cf9054a50847d175 | Three spheres are tangent to each other in pairs and tangent to the same plane. The points of contact with the plane form a triangle with sides of lengths 3, 6, and 4. What are the radii of the three spheres? | [
"\\( \\left( \\frac{9}{4}, 1, 4 \\right) \\)"
] | [
"Three spheres are tangent to each other in pairs and tangent to the same plane. The points of contact with the plane form a triangle with sides of lengths 3, 6, and 4. What are the radii of the three spheres?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 12,
"tir": 11
} | 23 | 0 | 0 |
omr-ws-ce5cea24bf24f6351f83e5b1561fa723afb6d4d4fb0a8cfe6eb4394bc84b7396 | Three spheres are tangent to one plane, to a straight line perpendicular to this plane, and in pairs to each other. The radius of the largest sphere is $1$. Within what limits can the radius of the smallest sphere vary? | [
"\\(0 < R_3 \\leq \\frac{1}{3}\\)"
] | [
"Three spheres are tangent to one plane, to a straight line perpendicular to this plane, and in pairs to each other. The radius of the largest sphere is $1$. Within what limits can the radius of the smallest sphere vary?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-12e0b9a15b5c2524170de7204686e4c2e3776e426e8a5d89a883e756f072b656 | Three spheres each of unit radius have centers P, Q, R such that the center of each sphere lies on the surface of the other two spheres. Let C denote the cylinder with cross-section PQR (the triangular lamina with vertices P, Q, R) and axis perpendicular to PQR. Let M denote the space which is common to the three spher... | [
"\\dfrac{13\\sqrt{3}}{144}"
] | [
"Three spheres each of unit radius have centers P, Q, R such that the center of each sphere lies on the surface of the other two spheres. Let C denote the cylinder with cross-section PQR (the triangular lamina with vertices P, Q, R) and axis perpendicular to PQR. Let M denote the space which is common to the three ... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-13c420588d59bce52f7d0a8ea7634531ad7f8d080b0ca7f0afa263bbd35a0f38 | Three spheres each of unit radius have centers \( P, Q, R \) such that the center of each sphere lies on the surface of the other two spheres. Let \( C \) denote the cylinder with cross-section \( PQR \) (the triangular lamina with vertices \( P, Q, R \)) and axis perpendicular to \( PQR \). Let \( M \) denote the spac... | [
"\\dfrac{\\pi}{12}"
] | [
"Three spheres each of unit radius have centers \\( P, Q, R \\) such that the center of each sphere lies on the surface of the other two spheres. Let \\( C \\) denote the cylinder with cross-section \\( PQR \\) (the triangular lamina with vertices \\( P, Q, R \\)) and axis perpendicular to \\( PQR \\). Let \\( M \\... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 1
} | 3 | 0 | 0 |
omr-ws-0a1e9ba3d9247d157277adc7da8d351453e4be40670c154d7dbdbb89d49ac4a9 | Three spheres have radii $144$, $225$, and $400$, are pairwise externally tangent to each other, and are all tangent to the same plane at points $A$, $B$, and $C$. Compute the area of triangle $ABC$. | [
"86400"
] | [
"Three spheres have radii $144$, $225$, and $400$, are pairwise externally tangent to each other, and are all tangent to the same plane at points $A$, $B$, and $C$. Compute the area of triangle $ABC$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-7fb5fba73661af4ca4409bd92519a5d8f6772d911055fdc8b40976fec7288cd7 | Three spheres of radius $10$ are placed on a table all touching each other. A fourth sphere of radius $10$ is placed so that it lies on top of the other three. The distance from the bottom of the fourth sphere to the table is $h$, and $h^2=\frac{a}{b}$, where $a$ and $b$ are positive, coprime integers. What is the valu... | [
"803"
] | [
"Three spheres of radius $10$ are placed on a table all touching each other. A fourth sphere of radius $10$ is placed so that it lies on top of the other three. The distance from the bottom of the fourth sphere to the table is $h$, and $h^2=\\frac{a}{b}$, where $a$ and $b$ are positive, coprime integers. What is th... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 48
} | 48 | 0 | 0 |
omr-ws-9b9eef2e67e9e9a061a49cd215a3f318794c2e31260569e7bd671965060127ea | Three spheres of radius $r$ lie on the base of a right circular cone. A fourth sphere of the same radius lies on the first three. Each of the four spheres touches the lateral surface of the cone and the three other spheres. Find the altitude of the cone. | [
"$\\frac{r\\left(3+2\\sqrt 2+\\sqrt 3\\right)}{\\sqrt 3}$"
] | [
"Three spheres of radius $r$ lie on the base of a right circular cone. A fourth sphere of the same radius lies on the first three. Each of the four spheres touches the lateral surface of the cone and the three other spheres. Find the altitude of the cone."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5e64384c497bf2365ca7aeb2ff3281805f2ebf1e60d5aabde09d764a54e8e56e | Three spheres of radius $r$ lie on the base of a right circular cone. On top of them lies a fourth sphere of radius $r$. Each of the four spheres is tangent to the lateral surface of the cone and to the three other spheres. Find the altitude of the cone. | [
"$r\\left(1+\\sqrt 3+2/3\\sqrt 6\\right)$"
] | [
"Three spheres of radius $r$ lie on the base of a right circular cone. On top of them lies a fourth sphere of radius $r$. Each of the four spheres is tangent to the lateral surface of the cone and to the three other spheres. Find the altitude of the cone."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1a36de2faeec21c721987ec74b02cc124da3bb5a96d745c49df553fadf9a7488 | Three spheres of radius 1 are placed on a horizontal table and inside a vertical hollow cylinder of height 2 units which is just large enough to surround them. What fraction of the internal volume of the cylinder is occupied by the spheres? | [
"$\\frac{42-24\\sqrt3}{1}$\n\nHowever, it's important to note that the expression should be simplified to a proper fraction. The expression given in the forum post seems to be the result of rationalizing the denominator but is not in its simplest form. The correct simplified fraction should be $\\frac{42-24\\sqrt3}... | [
"Three spheres of radius 1 are placed on a horizontal table and inside a vertical hollow cylinder of height 2 units which is just large enough to surround them. What fraction of the internal volume of the cylinder is occupied by the spheres?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 6,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-72e7492e1c780f31b1a264aef383f4675e43b82171200f6e7776b0138f99a85e | Three spheres of radius 10 are placed on a hemispherical dish. The tops of the spheres are exactly level with the top of the dish. What is the radius of the dish? | [
"\\(\\frac{10\\sqrt{21}}{3} + 10\\)"
] | [
"Three spheres of radius 10 are placed on a hemispherical dish. The tops of the spheres are exactly level with the top of the dish. What is the radius of the dish?"
] | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 2
} | 2 | 0 | 0 |
omr-ws-539e3dc389e519301441155825544d97d137f90a2ca53c2e1240a946eeffcd81 | Three spheres of radius 3 cm each are tangent to each other and to the base of a hemisphere. How long is the radius of the hemisphere? | [
"\\(2\\sqrt{3} + 6\\)"
] | [
"Three spheres of radius 3 cm each are tangent to each other and to the base of a hemisphere. How long is the radius of the hemisphere?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 6
} | 14 | 0 | 0 |
omr-ws-33e8d94475f3ca6d7c1e2d07c90e0bc9e0b059539f5a436211dbdf4013adc036 | Three spheres of radius \( R \) sit on a flat surface touching one another. A smaller sphere sits on the same surface, in the middle, and touches all three of the bigger spheres. What is the radius of the smaller sphere? | [
"\\dfrac{R}{3}"
] | [
"Three spheres of radius \\( R \\) sit on a flat surface touching one another. A smaller sphere sits on the same surface, in the middle, and touches all three of the bigger spheres. What is the radius of the smaller sphere?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2fbdc3bcd8b06d330ae1f54de74646ef4f5bab74f864627e45cf05c4e600c791 | Three spheres of radius one are placed in a triangle such that they are tangent to one another. A fourth sphere, also of radius one, is placed on top so that it is tangent to the other three. Find the radius of the sphere that passes through the centers of all four spheres. | [
"\\( \\frac{5\\sqrt{6}}{6} \\)"
] | [
"Three spheres of radius one are placed in a triangle such that they are tangent to one another. A fourth sphere, also of radius one, is placed on top so that it is tangent to the other three. Find the radius of the sphere that passes through the centers of all four spheres."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-074ac2ee2633e324dda52f59d9bc74ffb2e89300abc3d9f7741ca32e11c88c26 | Three spheres touch each other in pairs, each touching the plane $P$ and the line $L$ which is perpendicular to $P$. The radius of the larger sphere is $1$. Find the minimum radius of the smallest sphere. | [
"\\(\\frac{1}{3}\\)"
] | [
"Three spheres touch each other in pairs, each touching the plane $P$ and the line $L$ which is perpendicular to $P$. The radius of the larger sphere is $1$. Find the minimum radius of the smallest sphere."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ceee3b8c6a0473683e88d1dd9cd814cd21d0050951dbdd753e6a1e97d267c6f3 | Three spheres touch the plane of the triangle $ABC$ at its vertices and each sphere touches the other two. Find the radii of the spheres if the side $AB$ is $c$ long and the angles adjacent to it are $\alpha $ and $\beta $. | [
"$r_1=\\frac{c\\sin \\beta }{2\\sin \\alpha },\\ r_2=\\frac{c\\sin \\alpha }{2\\sin \\beta },\\ r_3=\\frac{c\\sin \\alpha \\sin \\beta }{2\\sin ^2 (\\alpha +\\beta )}$"
] | [
"Three spheres touch the plane of the triangle $ABC$ at its vertices and each sphere touches the other two. Find the radii of the spheres if the side $AB$ is $c$ long and the angles adjacent to it are $\\alpha $ and $\\beta $."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-59550a97135e7ba9959ebda0a1a61831d65da049df0ee9712280ca4946309c36 | Three spheres touch the plane of triangle $ABC$ at its vertices and touch each other in pairs. Find the radii of the spheres if we know the length $c$ of side $AB$ and the adjoining angles $A$ and $B$. | [
"$1/2c\\sin B\\, {\\rm cosec}\\, A$, $1/2c\\sin A\\, {\\rm cosec}\\, B$, $1/2c\\sin A\\sin B\\, {\\rm cosec}^2(A+B)$. Express the sides of the triangle in terms of the radii of the spheres"
] | [
"Three spheres touch the plane of triangle $ABC$ at its vertices and touch each other in pairs. Find the radii of the spheres if we know the length $c$ of side $AB$ and the adjoining angles $A$ and $B$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-edb246faf2f99434ae9df16de18723d293e4780736d13d0f69ded6cc88a8d914 | Three spheres with radii $11$, $13$, and $19$ are mutually externally tangent. A plane intersects the spheres in three congruent circles centered at $A$, $B$, and $C$, respectively, and the centers of the spheres all lie on the same side of this plane. Given that $AB^2 = 560$, find $AC^2$. | [
"756"
] | [
"Three spheres with radii $11$, $13$, and $19$ are mutually externally tangent. A plane intersects the spheres in three congruent circles centered at $A$, $B$, and $C$, respectively, and the centers of the spheres all lie on the same side of this plane. Given that $AB^2 = 560$, find $AC^2$."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 3
} | 51 | 0 | 0 |
omr-ws-5a51def1060dd0095f962a5f40ab88f0fec556e1df0ae56875da2bb712a2df56 | Three spheres with radii $11$, $13$, and $19$ are mutually externally tangent. A plane intersects the spheres in three congruent circles centered at $A$, $B$, and $C$, respectively, and the centers of the spheres all lie on the same side of this plane. Suppose that $AB^2 = 560$. Find $AC^2$. | [
"756"
] | [
"Three spheres with radii $11$, $13$, and $19$ are mutually externally tangent. A plane intersects the spheres in three congruent circles centered at $A$, $B$, and $C$, respectively, and the centers of the spheres all lie on the same side of this plane. Suppose that $AB^2 = 560$. Find $AC^2$."
] | [
"MATH_training_set",
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 3
} | 34 | 0 | 0 |
omr-ws-d16d6fbc4114b992c90c2b8480520ff7d166d3ef6e982fde555d1b9f9ca9549e | Three spheres with radii 1, 2, and 3 are placed on a table such that they are tangent to each other pairwise. A plate (a plane) is placed over the spheres such that it is tangent to each sphere. What is the angle between the plane of the plate and the plane of the table? | [
"\\arccos\\left(\\dfrac{5}{18}\\right)"
] | [
"Three spheres with radii 1, 2, and 3 are placed on a table such that they are tangent to each other pairwise. A plate (a plane) is placed over the spheres such that it is tangent to each sphere. What is the angle between the plane of the plate and the plane of the table?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 7
} | 20 | 0 | 0 |
omr-ws-063e9d126ac9146e5c2be636645239a58edef7dc4039422e6a10aa2af8d8a3b2 | Three spheres with radius 13 are placed on a table so that each sphere is externally tangent to the other two. Three more spheres, each externally tangent to the other two, are placed on top of the three on the table so that each sphere is tangent to four spheres. The smallest sphere that can surround the stack has rad... | [
"351"
] | [
"Three spheres with radius 13 are placed on a table so that each sphere is externally tangent to the other two. Three more spheres, each externally tangent to the other two, are placed on top of the three on the table so that each sphere is tangent to four spheres. The smallest sphere that can surround the stack ha... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.