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-9efb3e510e7c3b099d64bad1777cf62e326d7ae9ddfadee58cd907328fc848ce | Three circles with the same radius pass through a common point. Let S be the set of points which are interior to at least two of the circles. How should the three circles be placed so that the area of S is minimized? | [
"\\text{The centers of the three circles form an equilateral triangle}"
] | [
"Three circles with the same radius pass through a common point. Let S be the set of points which are interior to at least two of the circles. How should the three circles be placed so that the area of S is minimized?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d9528997afaa93aa96ad735a0f487536d87aeed22f5d8c43f278db34475f5578 | Three circles, A, B, and C, are externally tangent to each other and internally tangent to a larger circle, D. Circle D has a radius of 2, and circle A has a radius of 1. Circle A intersects the center of circle D. If circles B and C are congruent, what is the radius of circle B? | [
"\\(\\frac{8}{9}\\)"
] | [
"Three circles, A, B, and C, are externally tangent to each other and internally tangent to a larger circle, D. Circle D has a radius of 2, and circle A has a radius of 1. Circle A intersects the center of circle D. If circles B and C are congruent, what is the radius of circle B?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-d3d7c3df50f304bbf96799705b308e65b3eebc0d1f9fc32c83613c78765b4bd7 | Three circles, \(C_1\), \(C_2\), and \(C_3\), are drawn in the plane such that each pair is externally tangent. Circle \(D\) is drawn externally tangent to all three, and circle \(E\) is internally tangent to all three. If \(D\) and \(E\) have the same center, find the ratio of the radius of circle \(D\) to the radius ... | [
"7 - 4\\sqrt{3}"
] | [
"Three circles, \\(C_1\\), \\(C_2\\), and \\(C_3\\), are drawn in the plane such that each pair is externally tangent. Circle \\(D\\) is drawn externally tangent to all three, and circle \\(E\\) is internally tangent to all three. If \\(D\\) and \\(E\\) have the same center, find the ratio of the radius of circle \... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 5
} | 11 | 0 | 0 |
omr-ws-9faf434a5173b5975e2a2c5027225b3e02c10ff5e1d0ac51fe2b2e9a8bffc753 | Three circles, each of diameter 1, are drawn such that each is tangential to the others. A square enclosing these three circles is drawn so that two adjacent sides of the square are tangents to one of the circles, and the square is as small as possible. The side length of this square is given by $a+\frac{\sqrt{b}+\sqrt... | [
"145"
] | [
"Three circles, each of diameter 1, are drawn such that each is tangential to the others. A square enclosing these three circles is drawn so that two adjacent sides of the square are tangents to one of the circles, and the square is as small as possible. The side length of this square is given by $a+\\frac{\\sqrt{b... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6ad473ce8ba0dc0fbdbf4e209791bb37e1e8ce26a98e0770531f6b9269589d14 | Three circles, each of radius $3$, are drawn with centers at $(14, 92)$, $(17, 76)$, and $(19, 84)$. A line passing through $(17,76)$ is such that the total area of the parts of the three circles to one side of the line is equal to the total area of the parts of the three circles to the other side of it. What is the ab... | [
"24"
] | [
"Three circles, each of radius $3$, are drawn with centers at $(14, 92)$, $(17, 76)$, and $(19, 84)$. A line passing through $(17,76)$ is such that the total area of the parts of the three circles to one side of the line is equal to the total area of the parts of the three circles to the other side of it. What is t... | [
"MATH_training_set",
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 31
} | 31 | 0 | 0 |
omr-ws-12e891f97bcc452818bda6c0eef022de37d9b20afcf13e5ecf919a44c5b51fd7 | Three circles, each of radius $3$, are drawn with centers at $(14, 92)$, $(17, 76)$, and $(19, 84)$. A line passing through $(17,76)$ is such that the total area of the parts of the three circles to one side of the line is equal to the total area of the parts of the three circles to the other side of it. What is the ab... | [
"024"
] | [
"Three circles, each of radius $3$, are drawn with centers at $(14, 92)$, $(17, 76)$, and $(19, 84)$. A line passing through $(17,76)$ is such that the total area of the parts of the three circles to one side of the line is equal to the total area of the parts of the three circles to the other side of it. What ... | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 5
} | 20 | 0 | 0 |
omr-ws-c548038b30f02de4bdc04fe233709fd2e71afd3301de5b3953feafb31ce92ced | Three circles, each of radius 3, are drawn with centers at $(14,92)$, $(17,76)$, and $(19,84)$. A line passing through $(17,76)$ divides the total area of the three circles into two equal parts. What is the absolute value of the slope of this line? | [
"24"
] | [
"Three circles, each of radius 3, are drawn with centers at $(14,92)$, $(17,76)$, and $(19,84)$. A line passing through $(17,76)$ divides the total area of the three circles into two equal parts. What is the absolute value of the slope of this line?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-ce71b0c5fdb7d749c0c4f49a6cc05f44c2f7b4917db308dbea621c269516b375 | Three circles, each of radius \(r\), are placed on a plane so that the center of each circle lies on a point of intersection of the other two circles. The region \(R\) consists of all points inside or on at least one of these three circles. Find the area of \(R\). | [
"\\(r^2\\left(\\sqrt{3} + \\frac{3}{2} \\pi\\right)\\)"
] | [
"Three circles, each of radius \\(r\\), are placed on a plane so that the center of each circle lies on a point of intersection of the other two circles. The region \\(R\\) consists of all points inside or on at least one of these three circles. Find the area of \\(R\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-58a5419f1968770f14134baf1534f683cad284d34aeff07211750493b8565f55 | Three circles, each with a radius of 7 cm, are tangent to each other. Consider the triangle formed by joining their centers. Calculate the area inside this triangle but outside the three circles, rounded to the nearest hundredths, using $\pi=3.14$. | [
"7.94"
] | [
"Three circles, each with a radius of 7 cm, are tangent to each other. Consider the triangle formed by joining their centers. Calculate the area inside this triangle but outside the three circles, rounded to the nearest hundredths, using $\\pi=3.14$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 26
} | 41 | 0 | 0 |
omr-ws-f800cdd4b9e7baa55cef8c3054258d48a27f1f5232ba086ea23157b60fde76dd | Three circles, each with radius 1, are such that the center of each circle lies on the circumference of the other two circles. What is the area of the intersection of these three circles? | [
"\\dfrac{\\pi - \\sqrt{3}}{2}"
] | [
"Three circles, each with radius 1, are such that the center of each circle lies on the circumference of the other two circles. What is the area of the intersection of these three circles?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-121f13a59f982c0af7e0dd10e27025216b9220097c8c34a77bc8e04d5d3aaf79 | Three circles, each with radius 1, are tangent to each other. Let the centers of the circles be \( A \), \( B \), and \( C \), and let the point of tangency between the circles be \( D \). Find the area of triangle \( ABC \). | [
"\\(\\frac{\\sqrt{3}}{4}\\)"
] | [
"Three circles, each with radius 1, are tangent to each other. Let the centers of the circles be \\( A \\), \\( B \\), and \\( C \\), and let the point of tangency between the circles be \\( D \\). Find the area of triangle \\( ABC \\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6bfbe7a126f8972dc739e679014019831df7ed32f70f220ab230ae62348a992b | Three circles, each with radius 6, intersect so that each circle passes through the centers of the other two circles. Find the area of the region that is the intersection of the interiors of the three circles. | [
"\\( 18(\\pi - \\sqrt{3}) \\)"
] | [
"Three circles, each with radius 6, intersect so that each circle passes through the centers of the other two circles. Find the area of the region that is the intersection of the interiors of the three circles."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 28
} | 42 | 0 | 0 |
omr-ws-a697462e928ccbebc160070ea680949fbbdd79f1c68e88220fe584c8d796bcc1 | Three circular arcs $\gamma_1, \gamma_2,$ and $\gamma_3$ connect points $A$ and $C$ in the same half-plane defined by line $AC$, with $\gamma_2$ between $\gamma_1$ and $\gamma_3$. Point $B$ lies on segment $AC$. Let $h_1, h_2,$ and $h_3$ be three rays starting at $B$ in the same half-plane, with $h_2$ between $h_1$ and... | [
"l_3"
] | [
"Three circular arcs $\\gamma_1, \\gamma_2,$ and $\\gamma_3$ connect points $A$ and $C$ in the same half-plane defined by line $AC$, with $\\gamma_2$ between $\\gamma_1$ and $\\gamma_3$. Point $B$ lies on segment $AC$. Let $h_1, h_2,$ and $h_3$ be three rays starting at $B$ in the same half-plane, with $h_2$ betwee... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 4
} | 8 | 0 | 0 |
omr-ws-0ad4b2fa14553eac4f4593763b9cd543d3577f772cd96ee6124b66f02891457b | Three circular arcs $\gamma_1, \gamma_2,$ and $\gamma_3$ connect the points $A$ and $C$ in the same half-plane defined by line $AC$ such that arc $\gamma_2$ lies between arcs $\gamma_1$ and $\gamma_3$. Point $B$ lies on the segment $AC$. Let $h_1, h_2$, and $h_3$ be three rays starting at $B$, lying in the same half-pl... | [
"2"
] | [
"Three circular arcs $\\gamma_1, \\gamma_2,$ and $\\gamma_3$ connect the points $A$ and $C$ in the same half-plane defined by line $AC$ such that arc $\\gamma_2$ lies between arcs $\\gamma_1$ and $\\gamma_3$. Point $B$ lies on the segment $AC$. Let $h_1, h_2$, and $h_3$ be three rays starting at $B$, lying in the s... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-86945474d772e289984c8d2551cc73bec237d735ad7d531fa151db57a2c5d6ce | Three circular arcs \( w_1, w_2, w_3 \) with common endpoints \( A \) and \( B \) are on the same side of the line \( AB \); \( w_2 \) lies between \( w_1 \) and \( w_3 \). Two rays emanating from \( B \) intersect these arcs at \( M_1, M_2, M_3 \) and \( K_1, K_2, K_3 \), respectively. If \( \frac{M_1M_2}{M_2M_3} = \f... | [
"\\dfrac{3}{2}"
] | [
"Three circular arcs \\( w_1, w_2, w_3 \\) with common endpoints \\( A \\) and \\( B \\) are on the same side of the line \\( AB \\); \\( w_2 \\) lies between \\( w_1 \\) and \\( w_3 \\). Two rays emanating from \\( B \\) intersect these arcs at \\( M_1, M_2, M_3 \\) and \\( K_1, K_2, K_3 \\), respectively. If \\( ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-37d1f53253cc0a164406a9f3030864f55fd9e0d6d9c3c8725cbba58b21785255 | Three clever monkeys divide a pile of bananas. The first monkey takes some bananas from the pile, keeps three-fourths of them, and divides the rest equally between the other two. The second monkey takes some bananas from the pile, keeps one-fourth of them, and divides the rest equally between the other two. The third m... | [
"408"
] | [
"Three clever monkeys divide a pile of bananas. The first monkey takes some bananas from the pile, keeps three-fourths of them, and divides the rest equally between the other two. The second monkey takes some bananas from the pile, keeps one-fourth of them, and divides the rest equally between the other two. The th... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-fec2a4487bdb613f172a1d100ffb1b8c4e54e978da7e1a34353addd0f9d95bbd | Three clever monkeys divide a pile of bananas. The first monkey takes some bananas from the pile, keeps three-fourths of them, and divides the rest equally between the other two. The second monkey takes some bananas from the pile, keeps one-fourth of them, and divides the rest equally between the other two. The third m... | [
"408"
] | [
"Three clever monkeys divide a pile of bananas. The first monkey takes some bananas from the pile, keeps three-fourths of them, and divides the rest equally between the other two. The second monkey takes some bananas from the pile, keeps one-fourth of them, and divides the rest equally between the other two. The th... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-37f719cc9e67e64be6fea1637c80b6966e3677af9c4906f3177cc7d634d9ac65 | Three clever monkeys divide a pile of bananas. The first monkey takes some bananas from the pile, keeps three-fourths of them, and divides the rest equally between the other two. The second monkey takes some bananas from the pile, keeps one-fourth of them, and divides the rest equally between the other two. The third m... | [
"408"
] | [
"Three clever monkeys divide a pile of bananas. The first monkey takes some bananas from the pile, keeps three-fourths of them, and divides the rest equally between the other two. The second monkey takes some bananas from the pile, keeps one-fourth of them, and divides the rest equally between the other two. The th... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5b17960f32b7d87e5679b3c0f7d7ba2bbec4ad6809b8cd24d4e63140e0115ee9 | Three clever monkeys divide a pile of bananas. The first monkey takes some bananas from the pile, keeps three-fourths of them, and divides the rest equally between the other two. The second monkey takes some bananas from the pile, keeps one-fourth of them, and divides the rest equally between the other two. The third m... | [
"408"
] | [
"Three clever monkeys divide a pile of bananas. The first monkey takes some bananas from the pile, keeps three-fourths of them, and divides the rest equally between the other two. The second monkey takes some bananas from the pile, keeps one-fourth of them, and divides the rest equally between the other two. The th... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-089dc9ae79dd1faae7f252c78f70bcb55ce6c01bcaaa44a3501e64c9efc158f8 | Three clever monkeys divide a pile of bananas. The first monkey takes some bananas from the pile, keeps three-fourths of them, and divides the rest equally between the other two. The second monkey takes some bananas from the pile, keeps one-fourth of them, and divides the rest equally between the other two. The third m... | [
"408"
] | [
"Three clever monkeys divide a pile of bananas. The first monkey takes some bananas from the pile, keeps three-fourths of them, and divides the rest equally between the other two. The second monkey takes some bananas from the pile, keeps one-fourth of them, and divides the rest equally between the other two. The th... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-691e9a8772937caf2c7cbf9a063a90a5a24f0f0fdf05816554d23ea53a21e49e | Three clocks are set to the correct time at 00:00 on January 1, 2013. The next day, the first clock shows the correct time, the second clock is six minutes late, and the third clock is ten minutes behind. If the clocks run continuously and at constant speeds, what is the first date when all three clocks will show the s... | [
"January 1, 2015"
] | [
"Three clocks are set to the correct time at 00:00 on January 1, 2013. The next day, the first clock shows the correct time, the second clock is six minutes late, and the third clock is ten minutes behind. If the clocks run continuously and at constant speeds, what is the first date when all three clocks will show ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-65293af07e27994b49f6c677f15c64a8497aeaa0c745c48609936d7fce0494da | Three cockroaches run along a circle in the same direction, starting simultaneously from a point $S$. Cockroach $A$ runs twice as slow as $B$, and three times as slow as $C$. Points $X$ and $Y$ on segment $SC$ are such that $SX = XY = YC$. The lines $AX$ and $BY$ meet at point $Z$. Find the locus of the centroids of tr... | [
"The origin."
] | [
"Three cockroaches run along a circle in the same direction, starting simultaneously from a point $S$. Cockroach $A$ runs twice as slow as $B$, and three times as slow as $C$. Points $X$ and $Y$ on segment $SC$ are such that $SX = XY = YC$. The lines $AX$ and $BY$ meet at point $Z$. Find the locus of the centroids ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 25,
"tir": 6
} | 31 | 0 | 0 |
omr-ws-b2cf9476c210165aba45982a3feb365852600d14eec46bb03c1e05b669bcbe81 | Three coins are placed at the origin of a Cartesian coordinate system. On one move, one removes a coin placed at some position $(x, y)$ and places three new coins at $(x+1, y)$, $(x, y+1)$, and $(x+1, y+1)$. After a sequence of moves, two coins are placed at the same point for the first time. Let $(a, b)$ be the coordi... | [
"1"
] | [
"Three coins are placed at the origin of a Cartesian coordinate system. On one move, one removes a coin placed at some position $(x, y)$ and places three new coins at $(x+1, y)$, $(x, y+1)$, and $(x+1, y+1)$. After a sequence of moves, two coins are placed at the same point for the first time. Let $(a, b)$ be the c... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-babd19059aab38fb7640a12c58060245d082269fa5814de7515d7736fd3ab70e | Three coins are tossed 72 times, and the number of heads obtained is recorded. At $\alpha = 0.05$, test the null hypothesis that the coins are fair and randomly tossed. - Use a goodness of fit test to determine if the coins are fair. - Use a run test to determine if the coins are randomly tossed. | [
"\\text{Fail to reject both null hypotheses}"
] | [
"Three coins are tossed 72 times, and the number of heads obtained is recorded. At $\\alpha = 0.05$, test the null hypothesis that the coins are fair and randomly tossed.\n- Use a goodness of fit test to determine if the coins are fair.\n- Use a run test to determine if the coins are randomly tossed."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-04f9a345b8d099138e8f6ed8cc4f055324563049d59a654d22113161013b28f7 | Three coins lie on integer points on the number line. A move consists of choosing and moving two coins, the first one 1 unit to the right and the second one 1 unit to the left. Under which initial conditions is it possible to move all coins to one single point? | [
"The sum of the initial positions of the three coins must be a multiple of three.",
"\\text{The sum of the positions is divisible by }3"
] | [
"Three coins lie on integer points on the number line. A move consists of choosing and moving two coins, the first one 1 unit to the right and the second one 1 unit to the left. Under which initial conditions is it possible to move all coins to one single point?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 22,
"genselect": 9,
"tir": 14
} | 45 | 0 | 0 |
omr-ws-306424f1603c4291a535a21e92925213ba7ebd1ca12479864bcfb7dc7fd56b4d | Three companies, A, B, and C, plan to build stations on a 1080 km loop railway. Company A will build three stations at 360 km intervals, Company B will build four stations at 270 km intervals, and Company C will build five stations at 216 km intervals. The stations need to be spaced so that the longest distance between... | [
"174KM"
] | [
"Three companies, A, B, and C, plan to build stations on a 1080 km loop railway. Company A will build three stations at 360 km intervals, Company B will build four stations at 270 km intervals, and Company C will build five stations at 216 km intervals. The stations need to be spaced so that the longest distance be... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 3
} | 3 | 0 | 0 |
omr-ws-560a10af89a3daaf84d6efea957b6c6db14f8ef737d2012f5c3bdd55e5bf8243 | Three complex numbers $a$, $b$, and $c$ not all equal satisfy: $$a^2=b+c$$ $$b^2=a+c$$ $$c^2=a+b$$ Determine the value of $|a|+|b|+|c|$. | [
"\\(2+\\sqrt{2}\\)"
] | [
"Three complex numbers $a$, $b$, and $c$ not all equal satisfy:\n$$a^2=b+c$$\n$$b^2=a+c$$\n$$c^2=a+b$$\nDetermine the value of $|a|+|b|+|c|$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-bc8c583ecfd29f05bac7bfed6089876be50e994ec618a0f4d1be8078cee48c6d | Three complex numbers $a, b, c$ satisfy $|a| = |b| = |c| = 1$ and $a^3 + b^3 + c^3 = 2abc$. If $a, b, c$ represent the vertices of a triangle in the complex plane, find the sum of all possible values of $|a - b|^2 + |b - c|^2 + |c - a|^2$. | [
"18"
] | [
"Three complex numbers $a, b, c$ satisfy $|a| = |b| = |c| = 1$ and $a^3 + b^3 + c^3 = 2abc$. If $a, b, c$ represent the vertices of a triangle in the complex plane, find the sum of all possible values of $|a - b|^2 + |b - c|^2 + |c - a|^2$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-5c72ca8e7c5663b7be6c18aac345861e8c7a494ce55500e16d8cebad67f0f2e9 | Three concentric circles are drawn with radii of 8 cm, 12 cm, and 20 cm. If a dart lands randomly on this target, what are the probabilities of it landing in each region? | [
"\\(\\frac{4}{25}, \\frac{1}{5}, \\frac{16}{25}\\)"
] | [
"Three concentric circles are drawn with radii of 8 cm, 12 cm, and 20 cm. If a dart lands randomly on this target, what are the probabilities of it landing in each region?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 30
} | 33 | 0 | 0 |
omr-ws-70b339bef7bf6bad49a0102ff5af1df4f2877bcad247ea8faab2fd621ce2b0a1 | Three concentric circles centered at \( O \) have radii of \( 1 \), \( 2 \), and \( 3 \). Points \( B \) and \( C \) lie on the largest circle. The region between the two smaller circles is shaded, as is the portion of the region between the two larger circles bounded by central angle \( \angle BOC \). Suppose the shad... | [
"108"
] | [
"Three concentric circles centered at \\( O \\) have radii of \\( 1 \\), \\( 2 \\), and \\( 3 \\). Points \\( B \\) and \\( C \\) lie on the largest circle. The region between the two smaller circles is shaded, as is the portion of the region between the two larger circles bounded by central angle \\( \\angle BOC \... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 17,
"tir": 3
} | 20 | 0 | 0 |
omr-ws-d1032328a9390e0c3a4d603587de280a7b060fa81cac488a4f22ae35e3685884 | Three concentric circles have radii $3$, $4$, and $5$. An equilateral triangle with one vertex on each circle has side length $s$. The largest possible area of the triangle can be written as $a+\frac{b}{c}\sqrt{d}$, where $a, b, c,$ and $d$ are positive integers, $b$ and $c$ are relatively prime, and $d$ is not divisib... | [
"41"
] | [
"Three concentric circles have radii $3$, $4$, and $5$. An equilateral triangle with one vertex on each circle has side length $s$. The largest possible area of the triangle can be written as $a+\\frac{b}{c}\\sqrt{d}$, where $a, b, c,$ and $d$ are positive integers, $b$ and $c$ are relatively prime, and $d$ is not ... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 4
} | 19 | 0 | 0 |
omr-ws-42a2e04d2cb9c22450224d87aeed82a1d69dc2a582125b979122c42aca1cd2fe | Three concentric circles have radii $3,$ $4,$ and $5.$ An equilateral triangle with one vertex on each circle has side length $s.$ The largest possible area of the triangle can be written as $a + \tfrac{b}{c} \sqrt{d},$ where $a,$ $b,$ $c,$ and $d$ are positive integers, $b$ and $c$ are relatively prime, and $d$ is not... | [
"041"
] | [
"Three concentric circles have radii $3,$ $4,$ and $5.$ An equilateral triangle with one vertex on each circle has side length $s.$ The largest possible area of the triangle can be written as $a + \\tfrac{b}{c} \\sqrt{d},$ where $a,$ $b,$ $c,$ and $d$ are positive integers, $b$ and $c$ are relatively prime, and ... | [
"MATH_training_set"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 16,
"genselect": 6,
"tir": 1
} | 23 | 0 | 0 |
omr-ws-9be7938e62d26412e28e13beca2e078a6b778deaa54336003ca17dfe68075bf8 | Three concentric circles with common center $O$ are cut by a common chord in successive points $A, B, C$. Tangents drawn to the circles at the points $A, B, C$ enclose a triangular region. If the distance from point $O$ to the common chord is equal to $p$, and $AB = 6$, $BC = 8$, $CA = 10$, calculate the area of the re... | [
"\\dfrac{336}{p}"
] | [
"Three concentric circles with common center $O$ are cut by a common chord in successive points $A, B, C$. Tangents drawn to the circles at the points $A, B, C$ enclose a triangular region. If the distance from point $O$ to the common chord is equal to $p$, and $AB = 6$, $BC = 8$, $CA = 10$, calculate the area of t... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c31b22e568391f00ade0feb501b3f0b3807330f001fdc4f9484c5d5c1aba736f | Three concentric circles with radii 1, 2, and 3 are drawn. An equilateral triangle is placed such that each of its vertices lies on a different circle. What is the side length of the triangle? | [
"$\\sqrt{7}$"
] | [
"Three concentric circles with radii 1, 2, and 3 are drawn. An equilateral triangle is placed such that each of its vertices lies on a different circle. What is the side length of the triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-8306dd3d2530f83167b8544addcfc9e9228d146f6bcb60eb8c166117d6b99e68 | Three concentric hoops each have 20 beads, with 10 black and 10 white beads on each hoop. The beads are labeled from 1 to 20 counterclockwise. A match occurs at position \( i \) if all three beads at position \( i \) have the same color. Determine the maximum number of matches that can be achieved by sliding the beads ... | [
"20"
] | [
"Three concentric hoops each have 20 beads, with 10 black and 10 white beads on each hoop. The beads are labeled from 1 to 20 counterclockwise. A match occurs at position \\( i \\) if all three beads at position \\( i \\) have the same color. Determine the maximum number of matches that can be achieved by sliding t... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-fd3c0ff82f334240bff41c1cb09b06a896f514738887c51c574f34940d5f1d6e | Three concentric hoops each have 20 beads, with 10 black and 10 white beads on each hoop. The beads are labeled from 1 to 20. A match occurs at position \(i\) if all three beads at position \(i\) have the same color. Determine the maximum number of matches that can be achieved by sliding the beads around the hoops, and... | [
"10"
] | [
"Three concentric hoops each have 20 beads, with 10 black and 10 white beads on each hoop. The beads are labeled from 1 to 20. A match occurs at position \\(i\\) if all three beads at position \\(i\\) have the same color. Determine the maximum number of matches that can be achieved by sliding the beads around the h... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-d16af95185f3620501d506d21cee72dec65cc6f44a9155193292ed6dbc486f55 | Three concurrent lines $OX$, $OY$, and $OZ$ are cut by two parallel transversals at points $A$, $B$, $C$ and $P$, $Q$, $R$ respectively. If $AB = 6$, $BC = 9$, and $PQ = 4$, find the length of $QR$. | [
"6"
] | [
"Three concurrent lines $OX$, $OY$, and $OZ$ are cut by two parallel transversals at points $A$, $B$, $C$ and $P$, $Q$, $R$ respectively. If $AB = 6$, $BC = 9$, and $PQ = 4$, find the length of $QR$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fcb6bbd8cf9775bd555e3c6cf409ccab87ea1775fafe09c2b8db61112d88136d | Three cones have bases on the same plane, externally tangent to each other. The cones all face the same direction. Two of the cones have radii of $2$, and the other cone has a radius of $3$. The two cones with radii $2$ have height $4$, and the other cone has height $6$. Let $V$ be the volume of the tetrahedron with th... | [
"336"
] | [
"Three cones have bases on the same plane, externally tangent to each other. The cones all face the same direction. Two of the cones have radii of $2$, and the other cone has a radius of $3$. The two cones with radii $2$ have height $4$, and the other cone has height $6$. Let $V$ be the volume of the tetrahedron wi... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-9b0aeac0c0dec409ee4ff919ed8cd5b7a5283a35f33fe0d91883f8e775ae582b | Three cones have bases on the same plane, externally tangent to each other. The cones all face the same direction. Two of the cones have radii of 2, and the other cone has a radius of 3. The two cones with radii 2 have height 4, and the other cone has height 6. Let \( V \) be the volume of the tetrahedron with three of... | [
"336"
] | [
"Three cones have bases on the same plane, externally tangent to each other. The cones all face the same direction. Two of the cones have radii of 2, and the other cone has a radius of 3. The two cones with radii 2 have height 4, and the other cone has height 6. Let \\( V \\) be the volume of the tetrahedron with t... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3d668eb885ea7a6f58490d3ae3b856d6c69439b0852a981a41e9681a64125d59 | Three cones with a common apex and a height of $2$ and a base radius of $\sqrt{3}$ are in contact with each other externally. Two balls touch each other externally and all three cones. Find the ratio of the radii of the larger ball to the smaller ball. | [
"4:3"
] | [
"Three cones with a common apex and a height of $2$ and a base radius of $\\sqrt{3}$ are in contact with each other externally. Two balls touch each other externally and all three cones. Find the ratio of the radii of the larger ball to the smaller ball."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 4,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-1d2e1cdf4e05216dd19e3ea2e216ad75fbfd7828a3af98325bf338e6eb2837af | Three cones with apex $A$ touch each other externally. The first two cones are identical, and the third cone has an apex angle of $\frac{\pi}{4}$. All cones also touch a plane passing through point $A$ and lie on one side of it. Find the apex angle of the first two cones. (The apex angle of a cone is the angle between ... | [
"\\dfrac{7\\pi}{8}"
] | [
"Three cones with apex $A$ touch each other externally. The first two cones are identical, and the third cone has an apex angle of $\\frac{\\pi}{4}$. All cones also touch a plane passing through point $A$ and lie on one side of it. Find the apex angle of the first two cones. (The apex angle of a cone is the angle b... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-83b11c9e2210f86ac577f3fd018cd40469558f69d41060d67bf78f1fa8eda9c9 | Three cones with vertex \( A \) and generatrix \( \sqrt{8} \) touch each other externally. For two cones, the angle between the generatrix and the axis of symmetry is \( \frac{\pi}{6} \), and for the third it is \( \frac{\pi}{4} \). Find the volume of the pyramid \( O_1O_2O_3A \), where \( O_1, O_2, O_3 \) are the cent... | [
"$\\sqrt{\\sqrt{3}+1}$"
] | [
"Three cones with vertex \\( A \\) and generatrix \\( \\sqrt{8} \\) touch each other externally. For two cones, the angle between the generatrix and the axis of symmetry is \\( \\frac{\\pi}{6} \\), and for the third it is \\( \\frac{\\pi}{4} \\). Find the volume of the pyramid \\( O_1O_2O_3A \\), where \\( O_1, O_2... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 19,
"genselect": 6,
"tir": 5
} | 30 | 0 | 0 |
omr-ws-fa118f087d49784603c5c8eb8ff9bff9ae24c40b5412b780b9b2e176440e5bfb | Three congruent cakes are to be distributed among five people equally by cutting the cakes into pieces such that the size of the smallest piece is as large as possible. Find the maximal size for the smallest piece. | [
"\\(\\frac{1}{4}\\)"
] | [
"Three congruent cakes are to be distributed among five people equally by cutting the cakes into pieces such that the size of the smallest piece is as large as possible. Find the maximal size for the smallest piece."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-91f81e882344e8d153e255cc56c859986e6a90ce51cc3c9b5d9bf8b7da2a87d1 | Three congruent circles \( G_1 \), \( G_2 \), and \( G_3 \) have a common point \( P \). \( G_1 \) and \( G_2 \) intersect again at \( C \). \( G_2 \) and \( G_3 \) intersect again at \( A \). \( G_3 \) and \( G_1 \) intersect again at \( B \). If the radius of each circle is \( r \) and the distance from \( P \) to ea... | [
"\\dfrac{1}{2}"
] | [
"Three congruent circles \\( G_1 \\), \\( G_2 \\), and \\( G_3 \\) have a common point \\( P \\). \\( G_1 \\) and \\( G_2 \\) intersect again at \\( C \\). \\( G_2 \\) and \\( G_3 \\) intersect again at \\( A \\). \\( G_3 \\) and \\( G_1 \\) intersect again at \\( B \\). If the radius of each circle is \\( r \\) an... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-90a753caf61c3c2413a516e02407fbd0096cee92a083c3e918f2a9d2aea6b84b | Three congruent circles \( G_1 \), \( G_2 \), and \( G_3 \) have a common point \( P \). \( G_1 \) and \( G_2 \) intersect again at \( C \). \( G_2 \) and \( G_3 \) intersect again at \( A \). \( G_3 \) and \( G_1 \) intersect again at \( B \). If the radius of each circle is \( r \), find the radius of the circumcircl... | [
"r"
] | [
"Three congruent circles \\( G_1 \\), \\( G_2 \\), and \\( G_3 \\) have a common point \\( P \\). \\( G_1 \\) and \\( G_2 \\) intersect again at \\( C \\). \\( G_2 \\) and \\( G_3 \\) intersect again at \\( A \\). \\( G_3 \\) and \\( G_1 \\) intersect again at \\( B \\). If the radius of each circle is \\( r \\), f... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-15f755e9beaf05dacdadba45699790d16cb917ead8761208ba959b70e466a97f | Three congruent circles are placed inside a unit square such that they do not overlap. What is the largest possible radius of one of these circles? | [
"\\(\\frac{\\sqrt{2}}{\\sqrt{3} + 1 + 2\\sqrt{2}}\\)"
] | [
"Three congruent circles are placed inside a unit square such that they do not overlap. What is the largest possible radius of one of these circles?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-525bb0f7a170311e73cfd41abe231ae5d3cca2a41a475736ce59ec5e5d2b767d | Three congruent circles have a common point $O$ and lie inside a given triangle $ABC$. Each circle touches a pair of sides of the triangle. If the coordinates of $A$, $B$, and $C$ are $(0,0)$, $(12,0)$, and $(3,9)$ respectively, and the radius of each circle is $r$, find the value of $r$ such that the distance between ... | [
"\\dfrac{36}{4 + 3\\sqrt{2} + \\sqrt{10}}"
] | [
"Three congruent circles have a common point $O$ and lie inside a given triangle $ABC$. Each circle touches a pair of sides of the triangle. If the coordinates of $A$, $B$, and $C$ are $(0,0)$, $(12,0)$, and $(3,9)$ respectively, and the radius of each circle is $r$, find the value of $r$ such that the distance bet... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f84cb3eb5072a1c65a09eca574a99399f8711a8cda17002e1416c819829f9bfc | Three congruent circles have a common point O and intersect pairwise at points A, B, and C. Let A'B'C' be the triangle containing the three circles such that each side of the triangle is tangent to exactly two of the circles. If the radius of each circle is 1, find the ratio of the area of triangle A'B'C' to the area o... | [
"9"
] | [
"Three congruent circles have a common point O and intersect pairwise at points A, B, and C. Let A'B'C' be the triangle containing the three circles such that each side of the triangle is tangent to exactly two of the circles. If the radius of each circle is 1, find the ratio of the area of triangle A'B'C' to the a... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-fb1e9b99abb8744744e4471f4fae569ea43e8c8647697900bdd024cd302847d9 | Three congruent circles have a common point O and lie inside a triangle such that each circle touches a pair of sides of the triangle. Let the incentre of the triangle be I and the circumcentre be C. If the distance from O to I is 12 units and the distance from I to C is 16 units, find the distance from O to C. | [
"20"
] | [
"Three congruent circles have a common point O and lie inside a triangle such that each circle touches a pair of sides of the triangle. Let the incentre of the triangle be I and the circumcentre be C. If the distance from O to I is 12 units and the distance from I to C is 16 units, find the distance from O to C."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-44bd99bfe9704378e3f3bb95143b365b782045c37584e1ce5370eb5564addfb9 | Three congruent circles have a common point \( O \) and lie inside a given triangle \( ABC \). Each circle touches a pair of sides of the triangle. If the incenter of the triangle is \( I \) and the circumcenter is \( O' \), calculate the ratio of the distance \( IO \) to the distance \( O'O \) when the triangle is equ... | [
"1"
] | [
"Three congruent circles have a common point \\( O \\) and lie inside a given triangle \\( ABC \\). Each circle touches a pair of sides of the triangle. If the incenter of the triangle is \\( I \\) and the circumcenter is \\( O' \\), calculate the ratio of the distance \\( IO \\) to the distance \\( O'O \\) when th... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 4
} | 11 | 0 | 0 |
omr-ws-62c6d0afa1538276e9d5c5aefe053e49332fdfb74d1773e174d1b5986a3d3cf9 | Three congruent circles intersect at a point H and again in pairs at points A, B, and C. If the radius of each circle is 5 units, find the area of triangle ABC. | [
"\\dfrac{75\\sqrt{3}}{4}"
] | [
"Three congruent circles intersect at a point H and again in pairs at points A, B, and C. If the radius of each circle is 5 units, find the area of triangle ABC."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a2a7f8cb6be5df7f4573c100f0ecbfbc82993d85960291209a8854e792e3cf0e | Three congruent circles intersect at a point \( H \) in pairs at \( A \), \( B \), and \( C \). If the radius of each circle is \( r \) and the distance from \( H \) to each side of \( \triangle ABC \) is \( d \), find the value of \( \frac{d}{r} \). | [
"\\dfrac{1}{2}"
] | [
"Three congruent circles intersect at a point \\( H \\) in pairs at \\( A \\), \\( B \\), and \\( C \\). If the radius of each circle is \\( r \\) and the distance from \\( H \\) to each side of \\( \\triangle ABC \\) is \\( d \\), find the value of \\( \\frac{d}{r} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-681631b9490ca96f5fae066d642ea8509dbe0e6bfaab6d99a5a5e786f12bca2b | Three congruent circles of radius $1$ intersect at a common point. A larger circle is circumscribed about them, tangent to each of them at one point. Consider the midpoint of one of the diameters of the smaller circles; call it $P$. What is the length of the arc along the bigger circle containing all the points that ar... | [
"\\( 4 \\arcsin{\\frac{1}{4}} \\)"
] | [
"Three congruent circles of radius $1$ intersect at a common point. A larger circle is circumscribed about them, tangent to each of them at one point. Consider the midpoint of one of the diameters of the smaller circles; call it $P$. What is the length of the arc along the bigger circle containing all the points th... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-374dfec1d9a6149d324ffd3cba64da217ae8adee488fb533ecf79342b2b7ba0b | Three congruent circles with centers \( M_1 \), \( M_2 \), and \( M_3 \) and radius \( r \) have a common point \( D \) on their circumference. These circles also intersect each other at three different points \( A \), \( B \), and \( C \). Given a circle with center \( S \) whose circumference passes through \( A \), ... | [
"1"
] | [
"Three congruent circles with centers \\( M_1 \\), \\( M_2 \\), and \\( M_3 \\) and radius \\( r \\) have a common point \\( D \\) on their circumference. These circles also intersect each other at three different points \\( A \\), \\( B \\), and \\( C \\). Given a circle with center \\( S \\) whose circumference p... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-12fd38e31510947d5b6ef9d45eb605211b9d52d4641c1d162905a875678539d3 | Three congruent circles with radii $6$ are drawn so that each passes through the centers of the other circles. Find the area of the region inside two of the circles but outside the last. | [
"\\(36\\sqrt{3} + 18\\pi\\)"
] | [
"Three congruent circles with radii $6$ are drawn so that each passes through the centers of the other circles. Find the area of the region inside two of the circles but outside the last."
] | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7e33f9ea02a6ff2ce5012b770f1c9335fab169a33683dd22f3380c1de12c7a7b | Three congruent circles, each with radius $2$ feet, are placed such that each circle intersects with the centers of the other two circles. The area of the shaded region, which is the area shared between all three of the circles, can be expressed as $m\pi + n\sqrt{q}$, where $m, n$, and $q$ are positive integers and $q$... | [
"7"
] | [
"Three congruent circles, each with radius $2$ feet, are placed such that each circle intersects with the centers of the other two circles. The area of the shaded region, which is the area shared between all three of the circles, can be expressed as $m\\pi + n\\sqrt{q}$, where $m, n$, and $q$ are positive integers ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-adaa1fa281d1006e110d0f086f7fd585803941acb97699ff7db4f447dff583ff | Three congruent cones share the same peak and are tangent to each other and to the same plane. What is the angle at the peak of these cones? | [
"\\(\\sin^{-1} \\left(\\dfrac{4\\sqrt{3}}{7}\\right)\\)"
] | [
"Three congruent cones share the same peak and are tangent to each other and to the same plane. What is the angle at the peak of these cones?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-44cf4d33f059e8266a8c89f1d032403237c04d4a4b84f5500755d678832d20af | Three congruent cones with height 115 and radius 10 are placed beside one another (stuck together by their bases) to hold a sphere in place. The top of the sphere does not exceed the tip of the cones. What is the radius of the sphere? | [
"\\dfrac{20\\sqrt{3}(\\sqrt{533} - 2)}{69}"
] | [
"Three congruent cones with height 115 and radius 10 are placed beside one another (stuck together by their bases) to hold a sphere in place. The top of the sphere does not exceed the tip of the cones. What is the radius of the sphere?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 7
} | 9 | 0 | 0 |
omr-ws-ff295fd8a7f2b33ba4d76371cb9784acae896bca985c548c43453a2bbffb8f48 | Three congruent externally tangent circles with radii $2$, and centers $A, B, C$ are drawn along with $\triangle ABC$ on the 2-dimensional plane. Another circle, $D$, is larger than and internally tangent to circles $A, B$, and $C$. Let $x$ be the total area of the region(s) inside circle $D$ formed by the overlap of a... | [
"$\\pi(\\frac{16}{\\sqrt{3}}-\\frac{8}{3})$"
] | [
"Three congruent externally tangent circles with radii $2$, and centers $A, B, C$ are drawn along with $\\triangle ABC$ on the 2-dimensional plane. Another circle, $D$, is larger than and internally tangent to circles $A, B$, and $C$. Let $x$ be the total area of the region(s) inside circle $D$ formed by the overla... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 5,
"tir": 17
} | 35 | 0 | 0 |
omr-ws-47d3cf5051e5877b31a9f13732f06cf97eb77369bc8f1241ad6241e207ad9e6b | Three congruent isosceles triangles $DAO$, $AOB$ and $OBC$ have $AD=AO=OB=BC=10$ and $AB=DO=OC=12$. These triangles are arranged to form trapezoid $ABCD$, as shown. Point $P$ is on side $AB$ so that $OP$ is perpendicular to $AB$. [asy] pair A, B, C, D, O, P; A= (6, 8); B=(18, 8); C=(24, 0); D=(0,0); O=(12,0); P=(12,8);... | [
"12"
] | [
"Three congruent isosceles triangles $DAO$, $AOB$ and $OBC$ have $AD=AO=OB=BC=10$ and $AB=DO=OC=12$. These triangles are arranged to form trapezoid $ABCD$, as shown. Point $P$ is on side $AB$ so that $OP$ is perpendicular to $AB$.\n\n[asy]\npair A, B, C, D, O, P;\nA= (6, 8);\nB=(18, 8);\nC=(24, 0);\nD=(0,0);\nO=(... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f8c6435209f9206fd19e779cbe6bce061c69749911790ae6c14d2103e842926b | Three congruent small circles with radius \( r \) are tangent to each other and to a larger circle with radius \( R \). Calculate the ratio between the gray area (the area of the larger circle not covered by the small circles) and the white area (the area covered by the small circles). | [
"\\(\\frac{4\\sqrt{3}-2}{9}\\)"
] | [
"Three congruent small circles with radius \\( r \\) are tangent to each other and to a larger circle with radius \\( R \\). Calculate the ratio between the gray area (the area of the larger circle not covered by the small circles) and the white area (the area covered by the small circles)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-80ca6b017f1b949abad022ca6fb69d05133c163cdc07d6f10e0d0456ef7c0a00 | Three conjurers perform a trick with a pack of cards numbered $1, 2, \ldots, 2n + 1$ where $n > 6$. A spectator takes one card and distributes the rest evenly between the first and second conjurers. Each conjurer selects an ordered pair of their cards and gives these pairs to the third conjurer. The third conjurer then... | [
"x \\equiv - (a + b + c + d) \\pmod{2n + 1}"
] | [
"Three conjurers perform a trick with a pack of cards numbered $1, 2, \\ldots, 2n + 1$ where $n > 6$. A spectator takes one card and distributes the rest evenly between the first and second conjurers. Each conjurer selects an ordered pair of their cards and gives these pairs to the third conjurer. The third conjure... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 4
} | 10 | 0 | 0 |
omr-ws-37a102283c946463bd663e95e75c24509292fe1f112de8ea8ea5ab2917976833 | Three consecutive chords of lengths 1, 2, and 3 are inscribed in a semicircle. What is the radius of the semicircle? | [
"1.1"
] | [
"Three consecutive chords of lengths 1, 2, and 3 are inscribed in a semicircle. What is the radius of the semicircle?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2c2cba6b1679c6a4067072db2db93b908439271832649b5fbb1bdb81314755ee | Three consecutive positive integers have the property that the squares of the smaller two integers differ by $10a + b$ while the squares of the smallest and largest integers differ by $10b + a$. If $a$ and $b$ are digits, what is the sum of the three integers? | [
"39"
] | [
"Three consecutive positive integers have the property that the squares of the smaller two integers differ by $10a + b$ while the squares of the smallest and largest integers differ by $10b + a$. If $a$ and $b$ are digits, what is the sum of the three integers?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 20
} | 35 | 0 | 0 |
omr-ws-9fb55bf052555d41afd9c384ac3f588cb16013292a61192550563507f928a4d9 | Three consecutive positive integers not exceeding $100$ are chosen at random. What is the probability that all three are composite? | [
"\\(\\frac{17}{49}\\)"
] | [
"Three consecutive positive integers not exceeding $100$ are chosen at random. What is the probability that all three are composite?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 25
} | 47 | 0 | 0 |
omr-ws-9dba705ef368b1f1e92d8c1c03bece6e4faa10666d7b98b5eb197d488ea93d32 | Three consecutive sides of an inscribed hexagon are each of length 3, and the remaining three sides are each of length 5. Find the radius of its circumscribed circle. | [
"\\(\\frac{7\\sqrt{3}}{3}\\)"
] | [
"Three consecutive sides of an inscribed hexagon are each of length 3, and the remaining three sides are each of length 5. Find the radius of its circumscribed circle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 27
} | 75 | 0 | 0 |
omr-ws-95faa071b510be8df2dfd2c40b4f389f6758ae4945d3a133468eab34032b4aff | Three consecutive two-digit numbers are written next to each other to form a six-digit number. Find all such numbers where the six-digit number is divisible by $17$. | [
"858687"
] | [
"Three consecutive two-digit numbers are written next to each other to form a six-digit number. Find all such numbers where the six-digit number is divisible by $17$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-62e975afa40e5b45c06676ae73f97f5b02929e4e764e36c4cc87e65cfd604e6e | Three contestants participate in a trivia game show with probabilities of answering a question correctly being 0.8, 0.9, and 0.75, respectively. If 10 questions are asked and every contestant answers every question, find the probability that one contestant answers exactly 7 questions correctly, while the other two give... | [
"0.365844295"
] | [
"Three contestants participate in a trivia game show with probabilities of answering a question correctly being 0.8, 0.9, and 0.75, respectively. If 10 questions are asked and every contestant answers every question, find the probability that one contestant answers exactly 7 questions correctly, while the other two... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 1,
"tir": 3
} | 7 | 0 | 0 |
omr-ws-b8867f99a2bf6ae6ea81ae9039ef0b96626cd7f92f4366f8eb8726faa99ae2a7 | Three coplanar circles with respective radii of 1, 2, and \( x \) are externally tangent to each other and internally tangent to a circle of radius 3. What is \( x \)? | [
"\\dfrac{6}{7}"
] | [
"Three coplanar circles with respective radii of 1, 2, and \\( x \\) are externally tangent to each other and internally tangent to a circle of radius 3. What is \\( x \\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0615e5a496dbdf0654b859bb8853f7dbe3441538d9107f341966bc855329dc10 | Three counters start at the origin of a coordinate plane. Every minute, each counter moves one unit in a direction parallel to an axis of the coordinate plane (all such moves are equally likely). After ten minutes, compute the expected value of the area of the triangle with vertices at the positions of the three counte... | [
"\\dfrac{5\\sqrt{3}}{\\sqrt{\\pi}}"
] | [
"Three counters start at the origin of a coordinate plane. Every minute, each counter moves one unit in a direction parallel to an axis of the coordinate plane (all such moves are equally likely). After ten minutes, compute the expected value of the area of the triangle with vertices at the positions of the three c... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-9ba27c64cb4156e5f77e077a49b31d8b2d98727efe5784ab3092d7c61073d1c6 | Three cows are tied to different vertices of a right-angled triangle garden with sides in geometric progression (GP). The cows can graze \( \frac{4}{5} \) of the garden with ropes of the same length. If a restricted garden is to be built in the form of the incircle of such a triangle, by how much must the length of the... | [
"\\( \\frac{a \\sqrt{\\frac{1 + \\sqrt{5}}{2}}}{1 + \\sqrt{\\frac{1 + \\sqrt{5}}{2}} + \\frac{1 + \\sqrt{5}}{2}} \\)"
] | [
"Three cows are tied to different vertices of a right-angled triangle garden with sides in geometric progression (GP). The cows can graze \\( \\frac{4}{5} \\) of the garden with ropes of the same length. If a restricted garden is to be built in the form of the incircle of such a triangle, by how much must the lengt... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-857846cb30df3c4161051726635b30f306b3519dc0dd0376d3bf8a3a94418cf0 | Three cubes of volume \( 1 \), \( 8 \), and \( 27 \) are glued together at their faces. What is the smallest possible surface area of the resulting configuration? | [
"78"
] | [
"Three cubes of volume \\( 1 \\), \\( 8 \\), and \\( 27 \\) are glued together at their faces. What is the smallest possible surface area of the resulting configuration?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-28e8576f0c9aa0891654a3a3dae60c3ba2e966b0e6d73d0c519acc6d57953bb1 | Three cubes with volumes $1$, $8$, and $27$ are glued together at their faces. What is the smallest possible surface area of the resulting configuration? | [
"72"
] | [
"Three cubes with volumes $1$, $8$, and $27$ are glued together at their faces. What is the smallest possible surface area of the resulting configuration?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-454588d5e67309bab085f28c2b058ca13e1389df1dd5ad850bfe6687d03c0776 | Three customers call for ski lessons. What is the probability that all three are assigned to the German instructor, given that the instructors are assigned randomly? | [
"\\(\\frac{1}{27}\\)"
] | [
"Three customers call for ski lessons. What is the probability that all three are assigned to the German instructor, given that the instructors are assigned randomly?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-336830c10217b4d3cb2458d42d33fd8dff2ded604589e7591a89395ee1328ffa | Three cyclists ride on a circular track of length 300 meters in the same direction at different constant speeds. A photographer wants to take a photo when all three cyclists are within a segment of the track of length $d$ meters. Find the minimum value of $d$ such that this is possible. | [
"75"
] | [
"Three cyclists ride on a circular track of length 300 meters in the same direction at different constant speeds. A photographer wants to take a photo when all three cyclists are within a segment of the track of length $d$ meters. Find the minimum value of $d$ such that this is possible."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f4376fa8b3761483cec09dbd57d055600ba564de23fc6b701f72c1006185a91a | Three cyclists start out simultaneously from the same place in one direction around a circular track 1 km in length. The rates of the cyclists form, in a certain order, an arithmetic progression with common difference 5 km/h. After some time, the second one catches up with the first, having made one extra circuit; 4 mi... | [
"20 km/h, 25 km/h, 15 km/h"
] | [
"Three cyclists start out simultaneously from the same place in one direction around a circular track 1 km in length. The rates of the cyclists form, in a certain order, an arithmetic progression with common difference 5 km/h. After some time, the second one catches up with the first, having made one extra circuit;... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-178b557a50e5f99106255cc8aa09061c5258baa1a064c0afc98f5db08a39aa8d | Three cyclists started from point $A$. The first cyclist left an hour earlier than the other two, who started together. Some time later the third cyclist caught up with the first, while the second cyclist overtook the first two hours alter the third cyclist had done so. Determine the ratio between the speeds of the fir... | [
"1:2"
] | [
"Three cyclists started from point $A$. The first cyclist left an hour earlier than the other two, who started together. Some time later the third cyclist caught up with the first, while the second cyclist overtook the first two hours alter the third cyclist had done so. Determine the ratio between the speeds of th... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-14bb2d021bbd73661d7fe5f0a2ba27b17f5a1dcd51747b9b33c6b8a25d7294f1 | Three cyclists started simultaneously on three parallel straight paths. Cyclists travel at constant speeds. $1$ second after the start, the triangle formed by the cyclists had an area of $5$ m$^2$. What area will such a triangle have $10$ seconds after the start? | [
"50"
] | [
"Three cyclists started simultaneously on three parallel straight paths. Cyclists travel at constant speeds. $1$ second after the start, the triangle formed by the cyclists had an area of $5$ m$^2$. What area will such a triangle have $10$ seconds after the start?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b8d261b7c39e7b38767669e9247febe95bf49858c1b87ad09ec6b893ab3b0a9c | Three cylinders with radii 3 cm, 4 cm, and 5 cm are placed in contact with each other with their axes parallel. An elastic band is stretched around the three cylinders so that the plane of the elastic band is perpendicular to the axes of the cylinders. Calculate the length of the part of the band in contact with the la... | [
"\\(10\\pi-5\\arccos \\tfrac{1}{4} - 5\\arccos \\tfrac{1}{9} - 5 \\arccos \\tfrac{2}{3}\\)"
] | [
"Three cylinders with radii 3 cm, 4 cm, and 5 cm are placed in contact with each other with their axes parallel. An elastic band is stretched around the three cylinders so that the plane of the elastic band is perpendicular to the axes of the cylinders. Calculate the length of the part of the band in contact with t... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0828a850ad850d40c45d5787ed1401df5e615a2a4a6ecc9b6dac4224985795b1 | Three diagonals are drawn in a regular hexagon of side length $1$, splitting the hexagon into several regions. What is the least possible area of such a region? | [
"\\dfrac{\\sqrt{3}}{12}"
] | [
"Three diagonals are drawn in a regular hexagon of side length $1$, splitting the hexagon into several regions. What is the least possible area of such a region?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 4
} | 15 | 0 | 0 |
omr-ws-5b068820c738ebc1122714d5de99243e40cc489692c6752f868645cc02567eaa | Three dice \(A\), \(B\), and \(C\) are rolled once in this order. What is the probability that the maximum value of the faces is 4, given that the minimum value of the faces is 2? | [
"\\(\\frac{12}{61}\\)"
] | [
"Three dice \\(A\\), \\(B\\), and \\(C\\) are rolled once in this order. What is the probability that the maximum value of the faces is 4, given that the minimum value of the faces is 2?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 29
} | 77 | 0 | 0 |
omr-ws-b10377bd5efa99b629d8a8e4f2d4604648e2f9e667681933256add58db6dca1b | Three dice are rolled at once. The three numbers appearing on the dice are permuted to form all distinct 3-digit numbers, and the sum of these 3-digit numbers is noted as the 'score'. What is the average score after an infinite number of throws? | [
"1791.4"
] | [
"Three dice are rolled at once. The three numbers appearing on the dice are permuted to form all distinct 3-digit numbers, and the sum of these 3-digit numbers is noted as the 'score'. What is the average score after an infinite number of throws?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 25
} | 46 | 0 | 0 |
omr-ws-ef31517b50d77f4fd41a0b62d291433a41984f16b8eea7994af9c6472f2c7a36 | Three dice are rolled. If the probability that the numbers on the dice respectively form an isosceles triangle but not an equilateral triangle is given by $\frac{a}{b}$ where $\operatorname{gcd}(a, b)=1$, then find $a+b$. | [
"17"
] | [
"Three dice are rolled. If the probability that the numbers on the dice respectively form an isosceles triangle but not an equilateral triangle is given by $\\frac{a}{b}$ where $\\operatorname{gcd}(a, b)=1$, then find $a+b$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7885f8d1996a72977ed2672fa733ec05a57f3d2125ff3f2186f71a01b82a495b | Three dice are thrown. What is the probability that one die shows an even number while the other two show prime numbers? | [
"\\(\\frac{3}{8}\\)"
] | [
"Three dice are thrown. What is the probability that one die shows an even number while the other two show prime numbers?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 1
} | 5 | 0 | 0 |
omr-ws-75b5c05b15d2185aad4c161cf30daa84a2bbefd9f21fb603d4aab1c6cc77495d | Three dice with faces numbered 1 through 6 are stacked as shown. Seven of the eighteen faces are visible, leaving eleven faces hidden (back, bottom, between). What is the total number of dots NOT visible in this view? | [
"41"
] | [
"Three dice with faces numbered 1 through 6 are stacked as shown. Seven of the eighteen faces are visible, leaving eleven faces hidden (back, bottom, between). What is the total number of dots NOT visible in this view?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fdfc1008a37910e3020b70c2fd2c8ab57227a55f31a82199fb7dea96957d6c60 | Three dice with faces numbered 1 through 6 are stacked as shown. Seven of the eighteen faces are visible, leaving eleven faces hidden(back, bottom, between). What is the total number of dots NOT visible in this view? [asy] /* AMC8 2000 #8 Problem */ draw((0,0)--(1,0)--(1.5,0.66)--(1.5,3.66)--(.5,3.66)--(0,3)--cycle); d... | [
"41"
] | [
"Three dice with faces numbered 1 through 6 are stacked as shown. Seven of the eighteen faces are visible, leaving eleven faces hidden(back, bottom, between). What is the total number of dots NOT visible in this view? [asy]\n/* AMC8 2000 #8 Problem */\ndraw((0,0)--(1,0)--(1.5,0.66)--(1.5,3.66)--(.5,3.66)--(0,3)--cy... | [
"MATH_training_set"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-359cda1c57a66da39ccce8746e8d7b9a8603ddb4d37d3493e4838611c970cbfe | Three different bowls are created to be used as hourglasses. The functions describing the bowls are (i) \(y = x^2\), (ii) \(y = \alpha x\), and (iii) a hemispherical bowl of radius \(a\) centered at \((0, a)\). In all three cases, the bowls are filled with liquid to an initial height \(h_0\) (where \(0 < h_0 \leq a\)),... | [
"a = \\dfrac{5}{7}"
] | [
"Three different bowls are created to be used as hourglasses. The functions describing the bowls are (i) \\(y = x^2\\), (ii) \\(y = \\alpha x\\), and (iii) a hemispherical bowl of radius \\(a\\) centered at \\((0, a)\\). In all three cases, the bowls are filled with liquid to an initial height \\(h_0\\) (where \\(0... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-f5ce5f59e1a4781a032f77ec5424c7a8617df0320f536e7a45eabb6b35bde511 | Three different circles are drawn on the plane without common points outside each other. External tangents are drawn for each pair. Let the intersection points of these external tangents be \(A\), \(B\), and \(C\). Calculate the value of \(\frac{AB \cdot BC}{AC}\) if the radii of the circles are 3, 4, and 5 units, resp... | [
"\\dfrac{100}{3}"
] | [
"Three different circles are drawn on the plane without common points outside each other. External tangents are drawn for each pair. Let the intersection points of these external tangents be \\(A\\), \\(B\\), and \\(C\\). Calculate the value of \\(\\frac{AB \\cdot BC}{AC}\\) if the radii of the circles are 3, 4, an... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 5
} | 11 | 0 | 0 |
omr-ws-1a9147e73097f5b882ec1ac844a7bbd2a14d5f786743ff765bcefd36e40ee55a | Three different dice are rolled three times. What is the probability that they show different numbers exactly two times? | [
"$\\frac{100}{243}$",
"\\(\\frac{100}{243}\\)"
] | [
"Three different dice are rolled three times. What is the probability that they show different numbers exactly two times?"
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 64,
"tir": 58
} | 122 | 0 | 0 |
omr-ws-e29b36f204e041ccca1a566e41b06ff9d7a5382a4222dd88a6f6beabd01b8079 | Three different digits were used to create three different three-digit numbers forming an arithmetic progression. (In each number, all the digits are different.) What is the largest possible difference in this progression? | [
"438"
] | [
"Three different digits were used to create three different three-digit numbers forming an arithmetic progression. (In each number, all the digits are different.) What is the largest possible difference in this progression?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-abd8436ddfd569ac5f061ab3dddced103a05a84af21aeeb7c9da90245511eb96 | Three different dredgers were at work, deepening the entrance channel to a port. The first dredger, working alone, would have taken 10 days longer to do the job; the second, working alone, would have required an extra 20 days, and the third dredger, working alone, would have required six times more time than needed for... | [
"The job can be done by the first dredger alone in 20 days, by the second, in 30 days, and by the third, in 60 days"
] | [
"Three different dredgers were at work, deepening the entrance channel to a port. The first dredger, working alone, would have taken 10 days longer to do the job; the second, working alone, would have required an extra 20 days, and the third dredger, working alone, would have required six times more time than neede... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4ae339caa67d9c736ab20d428adec6f08b1e038778ffac27ed5e6fdbb156b1d4 | Three different faces of a regular dodecahedron are selected at random and painted. What is the probability that there is at least one pair of painted faces that share an edge? | [
"\\(\\frac{10}{11}\\)"
] | [
"Three different faces of a regular dodecahedron are selected at random and painted. What is the probability that there is at least one pair of painted faces that share an edge?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 6,
"tir": 10
} | 25 | 0 | 0 |
omr-ws-bbe9949459b7ebe0c1035d8835a2b16a157d65da6a29750fc1037e1098175c16 | Three different integers from 1 to 11 inclusive are chosen at random. What is the probability that their sum is divisible by 3? | [
"\\(\\frac{19}{55}\\)"
] | [
"Three different integers from 1 to 11 inclusive are chosen at random. What is the probability that their sum is divisible by 3?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-f5121fe50ae2f458a817a1bb27e38ecaf66d520ec76e33f4d642be8bf9f73854 | Three different natural numbers $x$, $y$ and $z$ each have exactly three natural-number factors. How many factors does $x^{2}y^{3}z^{4}$ have? | [
"315"
] | [
"Three different natural numbers $x$, $y$ and $z$ each have exactly three natural-number factors. How many factors does $x^{2}y^{3}z^{4}$ have?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2a771c30282f8e301f2b16531e976382d7a687d8a39664a66deb04da44fe414c | Three different non-zero digits are used to form 6 different three-digit numbers. The sum of five of these numbers is 3231. What is the sixth number? | [
"765"
] | [
"Three different non-zero digits are used to form 6 different three-digit numbers. The sum of five of these numbers is 3231. What is the sixth number?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-911453d270abf262cc75428acf974dfbbed49fe7562f614a7f8a606463e3b34b | Three different non-zero natural numbers are written on a board. A step consists of choosing two numbers and replacing one of the chosen numbers with their arithmetic mean. After a sequence of steps, two of the numbers on the board become equal. Find the minimum number of steps required to achieve this if the initial n... | [
"1"
] | [
"Three different non-zero natural numbers are written on a board. A step consists of choosing two numbers and replacing one of the chosen numbers with their arithmetic mean. After a sequence of steps, two of the numbers on the board become equal. Find the minimum number of steps required to achieve this if the init... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-44fa6dc30a968adadb3b5fadf79c6638d7315422ff494d0ce2462e4e3f2b2d83 | Three different non-zero real numbers $a, b, c$ satisfy the equations $a + \frac{2}{b} = b + \frac{2}{c} = c + \frac{2}{a} = p$, where $p$ is a real number. Given that $a = 1$, find the value of $bc + 2p$. | [
"0"
] | [
"Three different non-zero real numbers $a, b, c$ satisfy the equations $a + \\frac{2}{b} = b + \\frac{2}{c} = c + \\frac{2}{a} = p$, where $p$ is a real number. Given that $a = 1$, find the value of $bc + 2p$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-973d0118023c4082273588f99866cc4daff7163d6cd233109085ae3381e552fb | Three different nonzero numbers $a$, $b$, and $c$ are given. Every quadratic trinomial whose coefficients are these three numbers in any order has an integral root. If $P(x)$ is one such trinomial with root $1$, find the sum of all possible values of $a+b+c$. | [
"0"
] | [
"Three different nonzero numbers $a$, $b$, and $c$ are given. Every quadratic trinomial whose coefficients are these three numbers in any order has an integral root. If $P(x)$ is one such trinomial with root $1$, find the sum of all possible values of $a+b+c$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5488904ae8dbc75e925020e42bea7309a679dd730898be9a32e4c26d3c229189 | Three different nonzero real numbers \(a, b, c\) satisfy the equations \(a + \frac{2}{b} = b + \frac{2}{c} = c + \frac{2}{a} = p\), where \(p\) is a real number. Find the value of \(abc + 2p\) given that \(a + b + c = 3p\). | [
"0"
] | [
"Three different nonzero real numbers \\(a, b, c\\) satisfy the equations \\(a + \\frac{2}{b} = b + \\frac{2}{c} = c + \\frac{2}{a} = p\\), where \\(p\\) is a real number. Find the value of \\(abc + 2p\\) given that \\(a + b + c = 3p\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-ec9846f6e475bfc2a8a640b4c8e90be06f83a6140822c06293003a6110228a55 | Three different numbers $a, b, c$ will be chosen one by one at random from $1, 2, 3, 4, \ldots, 10$ with respect to the order. The probability that $ab + c$ is even is ... | [
"\\dfrac{4}{9}"
] | [
"Three different numbers $a, b, c$ will be chosen one by one at random from $1, 2, 3, 4, \\ldots, 10$ with respect to the order. The probability that $ab + c$ is even is ..."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
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