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-5a5d79c0c62d6a7205090903005ad42942f9d6944b3a600cb834b71a3c110704 | Three different numbers are chosen such that when each of the numbers is added to the average of the remaining two, the numbers 65, 69, and 76 result. Find the average of the three original numbers. | [
"35"
] | [
"Three different numbers are chosen such that when each of the numbers is added to the average of the remaining two, the numbers 65, 69, and 76 result. Find the average of the three original numbers."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 18
} | 33 | 0 | 0 |
omr-ws-fa71be7155c4f1fe5343d7cd7a1f4c70fee5a60f260e45ad4f837286741e395b | Three different oblique ellipses $\mathcal{E}_1, \mathcal{E}_2, \mathcal{E}_3$ and three different straight ellipses $\mathcal{F}_1, \mathcal{F}_2, \mathcal{F}_3$ are such that for every permutation $(i,j,k)$ of the numbers $(1,2,3)$, the ellipses $\mathcal{E}_i, \mathcal{E}_j, \mathcal{F}_k$ have 4 common points. Let ... | [
"1"
] | [
"Three different oblique ellipses $\\mathcal{E}_1, \\mathcal{E}_2, \\mathcal{E}_3$ and three different straight ellipses $\\mathcal{F}_1, \\mathcal{F}_2, \\mathcal{F}_3$ are such that for every permutation $(i,j,k)$ of the numbers $(1,2,3)$, the ellipses $\\mathcal{E}_i, \\mathcal{E}_j, \\mathcal{F}_k$ have 4 commo... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-c29b804fc0514ae05e1c5ed8cb5de3cbf701d15585fdb4f2b3fa25dac0b5c997 | Three different palindromic numbers between 10 and 1000 are chosen such that their sum is also a palindrome. What is the maximum possible value of the product of these three numbers? - The numbers cannot start with 0. - A palindrome is a number that reads the same backward as forward. Examples: 11, 101, 292. | [
"538657232"
] | [
"Three different palindromic numbers between 10 and 1000 are chosen such that their sum is also a palindrome. What is the maximum possible value of the product of these three numbers?\n\n- The numbers cannot start with 0.\n- A palindrome is a number that reads the same backward as forward. Examples: 11, 101, 292."
... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1a9c0336e6ef317d31f801175ed19545862e22d45b85c4e9e319a46ec6ebe657 | Three different points \( A, B, \) and \( C \) lie on a circle with center \( M \) such that \( |AB| = |BC| \). Point \( D \) is inside the circle such that \( \triangle BCD \) is equilateral. Let \( F \) be the second intersection of \( AD \) with the circle. If the radius of the circle is 10, calculate the length of ... | [
"10"
] | [
"Three different points \\( A, B, \\) and \\( C \\) lie on a circle with center \\( M \\) such that \\( |AB| = |BC| \\). Point \\( D \\) is inside the circle such that \\( \\triangle BCD \\) is equilateral. Let \\( F \\) be the second intersection of \\( AD \\) with the circle. If the radius of the circle is 10, ca... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-3e8782d20310e758650dc512087b72addb254516918d6b1b45b3117c79459ea1 | Three different points are randomly selected from the vertices of a regular $2n$-gon. Let $p_n$ be the probability that the triangle formed by these points is acute-angled. Calculate $\lim_{n \to \infty} p_n$. | [
"\\(\\frac{1}{4}\\)"
] | [
"Three different points are randomly selected from the vertices of a regular $2n$-gon. Let $p_n$ be the probability that the triangle formed by these points is acute-angled. Calculate $\\lim_{n \\to \\infty} p_n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-bdd932e6d6fc5b5b24c2b2c27068bff4d2b3a92e71ca484d7dc029993bddbbff | Three different shades of blue and three different greens can be arranged in a 3x3 box such that no two offerings of the same color follow immediately straight or diagonally. How many ways can this be done? | [
"0"
] | [
"Three different shades of blue and three different greens can be arranged in a 3x3 box such that no two offerings of the same color follow immediately straight or diagonally. How many ways can this be done?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 2
} | 5 | 0 | 0 |
omr-ws-6cb1d38baeccf360025f1e8e5998f74b28f7da799be1a6e881e4f67b6e2c6ce5 | Three digits \( a, b, \) and \( c \) are selected from the set \( \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\} \), one at a time, with repetition allowed. What is the probability that \( a > b > c \)? | [
"\\(\\frac{2}{15}\\)"
] | [
"Three digits \\( a, b, \\) and \\( c \\) are selected from the set \\( \\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\\} \\), one at a time, with repetition allowed. What is the probability that \\( a > b > c \\)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4c17592079ae3d7383afdc7e230e979baf03ee173a421e9d01fb2bcb11c8cc77 | Three digits were written to the end of a positive integer \( A \). The resulting number was equal to the sum of numbers from 1 to \( A \). Find all possible values of \( A \). | [
"1999"
] | [
"Three digits were written to the end of a positive integer \\( A \\). The resulting number was equal to the sum of numbers from 1 to \\( A \\). Find all possible values of \\( A \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b9749e4e19db1dd21adba37f4cfa952b7214890fd0142fe1a38d0d5b5aa762e5 | Three disks of diameter \( d \) are touching a sphere at their centers. Each disk also touches the other two disks. Determine the radius \( R \) of the sphere such that the axis of the whole figure forms an angle of \( 60^\circ \) with the line connecting the center of the sphere to the point on the disks that is farth... | [
"\\( R = d \\left(\\sqrt{3} + \\frac{3}{2}\\right) \\)"
] | [
"Three disks of diameter \\( d \\) are touching a sphere at their centers. Each disk also touches the other two disks. Determine the radius \\( R \\) of the sphere such that the axis of the whole figure forms an angle of \\( 60^\\circ \\) with the line connecting the center of the sphere to the point on the disks t... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5e3d5e97b6e7b7c36c63847cee55e4c667ed67fedafdc4d4eff3aff36b9a65e0 | Three distinct circles $\omega_1$, $\omega_2$, and $\omega_3$ all pass through a common point $P$. There exist points $A$, $B$, and $C$ distinct from $P$ such that $A \in \omega_1 \cap \omega_2$, $B \in \omega_2 \cap \omega_3$, and $C \in \omega_3 \cap \omega_1$. Define $Q$ to be the reflection of $P$ across $BC$. It i... | [
"44"
] | [
"Three distinct circles $\\omega_1$, $\\omega_2$, and $\\omega_3$ all pass through a common point $P$. There exist points $A$, $B$, and $C$ distinct from $P$ such that $A \\in \\omega_1 \\cap \\omega_2$, $B \\in \\omega_2 \\cap \\omega_3$, and $C \\in \\omega_3 \\cap \\omega_1$. Define $Q$ to be the reflection of $... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-94548d126736bf4239199fa285424ac0f9395e051dd0e84c1fe4d54cd6aaa557 | Three distinct circles $\omega_1, \omega_2, \omega_3$ intersect to form three common chords that are concurrent at a point $X$. Consider two distinct circles $\alpha_1$ and $\alpha_2$ which are internally tangent to all three circles $\omega_1, \omega_2, \omega_3$. Let $O_1$ and $O_2$ be the centers of $\alpha_1$ and $... | [
"(5, 6)"
] | [
"Three distinct circles $\\omega_1, \\omega_2, \\omega_3$ intersect to form three common chords that are concurrent at a point $X$. Consider two distinct circles $\\alpha_1$ and $\\alpha_2$ which are internally tangent to all three circles $\\omega_1, \\omega_2, \\omega_3$. Let $O_1$ and $O_2$ be the centers of $\\... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-51c4d10c0bbbe13205268c5fcd1c0075409c3987e763b50c7a9ff195816fd292 | Three distinct circles ($\Gamma_1, \Gamma_2, \Gamma_3$) intersect at exactly two distinct points $A$ and $B$. A straight line passing through $A$ intersects $\Gamma_1, \Gamma_2, \Gamma_3$ at points $D, E, F$ respectively. Lines $g_1, g_2, g_3$ are drawn tangent to $\Gamma_1, \Gamma_2, \Gamma_3$ at points $D, E, F$ resp... | [
"no_valid_answer_found"
] | [
"Three distinct circles ($\\Gamma_1, \\Gamma_2, \\Gamma_3$) intersect at exactly two distinct points $A$ and $B$. A straight line passing through $A$ intersects $\\Gamma_1, \\Gamma_2, \\Gamma_3$ at points $D, E, F$ respectively. Lines $g_1, g_2, g_3$ are drawn tangent to $\\Gamma_1, \\Gamma_2, \\Gamma_3$ at points ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-693b490faa6ecb4645f7483a81a8b9a4db553a938d9b4f9e677a48e93f3e4da7 | Three distinct integers \(a\), \(b\), and \(c\) are chosen at random. Calculate the sum of all possible distinct prime factors of the greatest common divisor of the expressions \(a^3 b - b^3 a\) and \(b^3 c - c^3 b\) for any such choice of \(a\), \(b\), and \(c\). | [
"5"
] | [
"Three distinct integers \\(a\\), \\(b\\), and \\(c\\) are chosen at random. Calculate the sum of all possible distinct prime factors of the greatest common divisor of the expressions \\(a^3 b - b^3 a\\) and \\(b^3 c - c^3 b\\) for any such choice of \\(a\\), \\(b\\), and \\(c\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-cb556bc880757c0594642d5cb2a0b706265f835c42a0cbef7d44acafe0412fb4 | Three distinct integers \(a\), \(b\), and \(c\) are chosen at random. Let \(P\) be the product of all distinct prime factors of \(a^3b - ab^3\) and \(a^3c - ac^3\). Find the value of \(P\). | [
"6"
] | [
"Three distinct integers \\(a\\), \\(b\\), and \\(c\\) are chosen at random. Let \\(P\\) be the product of all distinct prime factors of \\(a^3b - ab^3\\) and \\(a^3c - ac^3\\). Find the value of \\(P\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-39d0e113c09ecfd71cebb48e9ffc30bf30d11d95bd82f09a97cded6af6796e54 | Three distinct integers \(a\), \(b\), and \(c\) are chosen from \(0\) to \(1023 = 2^{10} - 1\) inclusive. What is the probability that \(f(a) + f(b) + f(c) = f(a + b + c)\)? | [
"\\dfrac{1}{1024}"
] | [
"Three distinct integers \\(a\\), \\(b\\), and \\(c\\) are chosen from \\(0\\) to \\(1023 = 2^{10} - 1\\) inclusive. What is the probability that \\(f(a) + f(b) + f(c) = f(a + b + c)\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-674eee70a0a300f417045a0eceacfd34e67b3b23def157e99feec54e363d3ac5 | Three distinct integers \(a\), \(b\), and \(c\) satisfy the following conditions: - \(abc = 17955\), - \(a\), \(b\), and \(c\) form an arithmetic sequence in that order, - \((3a + b)\), \((3b + c)\), and \((3c + a)\) form a geometric sequence in that order. What is the value of \(|a + b + c|\)? | [
"63"
] | [
"Three distinct integers \\(a\\), \\(b\\), and \\(c\\) satisfy the following conditions:\n- \\(abc = 17955\\),\n- \\(a\\), \\(b\\), and \\(c\\) form an arithmetic sequence in that order,\n- \\((3a + b)\\), \\((3b + c)\\), and \\((3c + a)\\) form a geometric sequence in that order.\nWhat is the value of \\(|a + b + ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 19
} | 34 | 0 | 0 |
omr-ws-dc1b22b97e52623e146546c2da421ca0eff89de56c75abac614eef21f6165140 | Three distinct integers are selected at random between $1$ and $2016$, inclusive. What is the probability $p$ that the product of the three integers is odd? | [
"\\( p < \\frac{1}{8} \\)"
] | [
"Three distinct integers are selected at random between $1$ and $2016$, inclusive. What is the probability $p$ that the product of the three integers is odd?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 2
} | 10 | 0 | 0 |
omr-ws-b922490db9fbdbcfa2b7eb191370ddd4d2fb2524e2a322b3f715d305fe97ab3c | Three distinct integers form a geometric progression with a sum of 1387, and the common ratio is an integer. Find the sum of all possible distinct values for the largest term of the geometric progression. | [
"4142"
] | [
"Three distinct integers form a geometric progression with a sum of 1387, and the common ratio is an integer. Find the sum of all possible distinct values for the largest term of the geometric progression."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-df00b71ab5deabe8c42925fec9adf1ca467beead739f885e775569e9e6f07e06 | Three distinct natural numbers $p$, $q$, and $r$ are such that the average of any non-empty subset of $\{p, q, r\}$ is a prime number. Find the minimal possible value of $p + q + r$. | [
"51"
] | [
"Three distinct natural numbers $p$, $q$, and $r$ are such that the average of any non-empty subset of $\\{p, q, r\\}$ is a prime number. Find the minimal possible value of $p + q + r$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 24
} | 45 | 0 | 0 |
omr-ws-569bea2386e226796f718de58b589b188d5011fbb46f982f9091e5190976ceee | Three distinct points $A$, $B$, and $N$ are marked on the line $l$, with $B$ lying between $A$ and $N$. For an arbitrary angle $\alpha \in (0,\frac{\pi}{2})$, points $C$ and $D$ are marked in the plane on the same side of $l$ such that $N$, $C$, and $D$ are collinear; $\angle NAD = \angle NBC = \alpha$; and $A$, $B$, $... | [
"A semicircle with radius $\\frac{AN \\cdot BN}{AN + BN}$."
] | [
"Three distinct points $A$, $B$, and $N$ are marked on the line $l$, with $B$ lying between $A$ and $N$. For an arbitrary angle $\\alpha \\in (0,\\frac{\\pi}{2})$, points $C$ and $D$ are marked in the plane on the same side of $l$ such that $N$, $C$, and $D$ are collinear; $\\angle NAD = \\angle NBC = \\alpha$; and... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-876b8b0396e2f08d50b4192e48c85cf5617513dae74ee29238c14b1cca422cec | Three distinct points $A, B$, and $C$ lie on a circle with center $O$, such that $\angle AOB$ and $\angle BOC$ each measure $60$ degrees. Point $P$ lies on chord $AC$ and triangle $BPC$ is drawn. If $P$ is chosen so that it maximizes the length of the altitude from $C$ to $BP$, determine the ratio $AP : PC$. | [
"1:2"
] | [
"Three distinct points $A, B$, and $C$ lie on a circle with center $O$, such that $\\angle AOB$ and $\\angle BOC$ each measure $60$ degrees. Point $P$ lies on chord $AC$ and triangle $BPC$ is drawn. If $P$ is chosen so that it maximizes the length of the altitude from $C$ to $BP$, determine the ratio $AP : PC$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 9
} | 56 | 0 | 0 |
omr-ws-b2c6802e4c07b4fbe5acfa71ae3d455df65efb6d35fbd524add69872dbd24de0 | Three distinct points $A, B$, and $C$ lie on a circle with center $O$. The triangles $AOB$, $BOC$, and $COA$ have equal area. What are the possible measures of the angles of triangle $ABC$? | [
"\\(60^\\circ\\)"
] | [
"Three distinct points $A, B$, and $C$ lie on a circle with center $O$. The triangles $AOB$, $BOC$, and $COA$ have equal area. What are the possible measures of the angles of triangle $ABC$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 1
} | 12 | 0 | 0 |
omr-ws-96e4d1af6329f86635dbeaf15d9170147699ab3b4efe26b74c6553c75020a2b4 | Three distinct points \( A \), \( B \), and \( C \) are fixed on a line in this order. Let \( \Gamma \) be a circle passing through \( A \) and \( C \) whose center does not lie on the line \( AC \). Denote by \( P \) the intersection of the tangents to \( \Gamma \) at \( A \) and \( C \). Suppose \( \Gamma \) meets th... | [
"2"
] | [
"Three distinct points \\( A \\), \\( B \\), and \\( C \\) are fixed on a line in this order. Let \\( \\Gamma \\) be a circle passing through \\( A \\) and \\( C \\) whose center does not lie on the line \\( AC \\). Denote by \\( P \\) the intersection of the tangents to \\( \\Gamma \\) at \\( A \\) and \\( C \\). ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d54e8c2e57e9449025bf5899eba4982427247c3292d692de30b4bac5b376d1a9 | Three distinct points \( A, B, C \) are found, in this order, on a line. Let \( G \) be a circle passing through \( A \) and \( C \) with its center outside the line \( AC \). Let \( P \) be the intersection point of the tangents to \( G \) drawn at points \( A \) and \( C \). Let \( Q \) be the intersection between \(... | [
"no_valid_answer_found"
] | [
"Three distinct points \\( A, B, C \\) are found, in this order, on a line. Let \\( G \\) be a circle passing through \\( A \\) and \\( C \\) with its center outside the line \\( AC \\). Let \\( P \\) be the intersection point of the tangents to \\( G \\) drawn at points \\( A \\) and \\( C \\). Let \\( Q \\) be th... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e771afb2d3c8ce15d4107ad0b56f800358b87956fb47952a2a41ca706fc85fe0 | Three distinct points \( A, B, C \) lie on a circle centered at point \( O \) with radius 1. If \( r \) is the inradius of \( \triangle ABC \), find the maximum possible value of \( r \). | [
"\\dfrac{1}{2}"
] | [
"Three distinct points \\( A, B, C \\) lie on a circle centered at point \\( O \\) with radius 1. If \\( r \\) is the inradius of \\( \\triangle ABC \\), find the maximum possible value of \\( r \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f2752678a21e78c17163e077683393edfe66ded9680b84a10a71d8588a9195ff | Three distinct points \( A, B, \) and \( N \) are marked on the line \( l \), with \( B \) lying between \( A \) and \( N \). For an arbitrary angle \( x \in (0, 90^\circ) \), points \( C \) and \( D \) are marked in the plane on the same side of \( l \) such that \( N, C, \) and \( D \) are collinear; \( \angle NAD = ... | [
"Circle with \\( MN \\) as the diameter."
] | [
"Three distinct points \\( A, B, \\) and \\( N \\) are marked on the line \\( l \\), with \\( B \\) lying between \\( A \\) and \\( N \\). For an arbitrary angle \\( x \\in (0, 90^\\circ) \\), points \\( C \\) and \\( D \\) are marked in the plane on the same side of \\( l \\) such that \\( N, C, \\) and \\( D \\) ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-95c93e655f5511d623d65a60cccf96bd6980cc1d5224ec4b218e5318e9fb5de8 | Three distinct points \(A, B,\) and \(C\) are collinear in that order. Let \(\omega\) be a circle passing through \(A\) and \(C\) whose center does not lie on line \(\overline{AC}\). Denote by \(P\) the intersection of the tangents to \(\omega\) at \(A\) and \(C\). Suppose that \(\omega\) meets segment \(PB\) at \(Q\).... | [
"4\\sqrt{15} - 12"
] | [
"Three distinct points \\(A, B,\\) and \\(C\\) are collinear in that order. Let \\(\\omega\\) be a circle passing through \\(A\\) and \\(C\\) whose center does not lie on line \\(\\overline{AC}\\). Denote by \\(P\\) the intersection of the tangents to \\(\\omega\\) at \\(A\\) and \\(C\\). Suppose that \\(\\omega\\)... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 5
} | 9 | 0 | 0 |
omr-ws-1495bf3b105299a4ca3ee9e7ba693bab776f1be3dd40042eedd9202b4be1221a | Three distinct points are chosen on a regular nonagon. What is the probability that the triangle formed by these vertices is scalene? | [
"\\(\\frac{4}{7}\\)"
] | [
"Three distinct points are chosen on a regular nonagon. What is the probability that the triangle formed by these vertices is scalene?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e639bcde9562c9112d09b3b4467ee1f4afb7988419a665e46e1368a289a689ac | Three distinct points are chosen uniformly at random from the vertices of a regular $2018$-gon. What is the probability that the triangle formed by these points is a right triangle? | [
"\\(\\frac{3}{2017}\\)"
] | [
"Three distinct points are chosen uniformly at random from the vertices of a regular $2018$-gon. What is the probability that the triangle formed by these points is a right triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-2278ba6e5a2197451c9b22d74191ada65f1655cb4b222deccc2baacd857dcac5 | Three distinct points are independently randomly selected on a circle. Find the probability that the triangle with these three points as vertices is an obtuse triangle. | [
"\\( \\frac{3}{4} \\)"
] | [
"Three distinct points are independently randomly selected on a circle. Find the probability that the triangle with these three points as vertices is an obtuse triangle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-81c34bc0fd2c84c8bc967dbe302a956076f09b1cdde726adc04baf2e01f9ac51 | Three distinct points are randomly chosen from the vertices of a regular polygon with nine sides. What is the probability that these points are the vertices of a scalene triangle? | [
"\\dfrac{9}{14}"
] | [
"Three distinct points are randomly chosen from the vertices of a regular polygon with nine sides. What is the probability that these points are the vertices of a scalene triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9159488b0474526a4bbc599cc1ea732fd98720ad4737914d4663a416aac722d8 | Three distinct points are randomly selected on the perimeter of a regular pentagon. A point is then randomly chosen inside the pentagon. What is the probability that the point chosen inside the pentagon lies within the triangle formed by connecting the three selected points on the perimeter? | [
"\\dfrac{5 + \\sqrt{5}}{20}"
] | [
"Three distinct points are randomly selected on the perimeter of a regular pentagon. A point is then randomly chosen inside the pentagon. What is the probability that the point chosen inside the pentagon lies within the triangle formed by connecting the three selected points on the perimeter?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 6
} | 8 | 0 | 0 |
omr-ws-edf0219554b7c18d329949b25e2f90457378c1ffbc4a4b2f5f477f9da6fd5fda | Three distinct points are randomly selected on the perimeter of a regular pentagon. A point is then randomly chosen inside the pentagon. What is the probability that this point lies inside the triangle formed by connecting the three perimeter points? | [
"\\dfrac{1}{4}"
] | [
"Three distinct points are randomly selected on the perimeter of a regular pentagon. A point is then randomly chosen inside the pentagon. What is the probability that this point lies inside the triangle formed by connecting the three perimeter points?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 2
} | 7 | 0 | 0 |
omr-ws-c49d1ce9cbbf045241cc9e310666af5e4ea5f7f2b5e0936f69cf303f4c8c21bb | Three distinct positive integers are such that the sum of any two is exactly divisible by the third. The largest of these integers is 36. What are the two smaller integers? | [
"12 and 24"
] | [
"Three distinct positive integers are such that the sum of any two is exactly divisible by the third. The largest of these integers is 36. What are the two smaller integers?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 29
} | 30 | 0 | 0 |
omr-ws-9df0b7a394fa3996b21150249a69dc2ac0bb3caf2dc58d80a657c12475b73a45 | Three distinct positive integers are written on the blackboard. When three real numbers \(a\), \(b\), and \(c\) are written on the blackboard, consider the process of simultaneously changing the three numbers to \(\frac{b + c}{2}\), \(\frac{c + a}{2}\), and \(\frac{a + b}{2}\). When this process is repeated 2021 times,... | [
"\\(3 \\cdot 2^{2021} + 3\\)"
] | [
"Three distinct positive integers are written on the blackboard. When three real numbers \\(a\\), \\(b\\), and \\(c\\) are written on the blackboard, consider the process of simultaneously changing the three numbers to \\(\\frac{b + c}{2}\\), \\(\\frac{c + a}{2}\\), and \\(\\frac{a + b}{2}\\). When this process is ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 20,
"tir": 2
} | 22 | 0 | 0 |
omr-ws-307d933299f2c6eaffc62c69bdb4f497c0c3802b838d248e1a2d2bb8208261f1 | Three distinct positive numbers \( x, y, z \) are chosen so that the equality \( x^2yz + xy^2z^2 - 4xyz + x + yz = 0 \) holds. How many of these numbers are greater than 1? | [
"1"
] | [
"Three distinct positive numbers \\( x, y, z \\) are chosen so that the equality \\( x^2yz + xy^2z^2 - 4xyz + x + yz = 0 \\) holds. How many of these numbers are greater than 1?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-959e4858e242e389601b59e1071807e6b74998957f0af0a58924f2d6c39c5b6a | Three distinct positive real numbers \( A \), \( B \), and \( C \) are chosen at random with \( C > B > A \). What is the expected value of \( \frac{A+B}{C} \)? | [
"1"
] | [
"Three distinct positive real numbers \\( A \\), \\( B \\), and \\( C \\) are chosen at random with \\( C > B > A \\). What is the expected value of \\( \\frac{A+B}{C} \\)?"
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 31
} | 31 | 0 | 0 |
omr-ws-1106a861c1a299b63e57f94b7dae09b3d0dfa10153a138a40fe792d5ab47d594 | Three distinct real numbers $\alpha, \beta, \gamma$ satisfy the following equations: \[ \begin{cases} \alpha + \beta + \gamma = 12 \\ \alpha^2 + \beta^2 + \gamma^2 = 12(10 - k) \\ \alpha^3 + \beta^3 + \gamma^3 = -6(k^2 + 40k - 212) \end{cases} \] Find the range of $k$. | [
"\\( k \\in (-18-12\\sqrt{3}, -18+12\\sqrt{3}) \\setminus \\{-2\\} \\)"
] | [
"Three distinct real numbers $\\alpha, \\beta, \\gamma$ satisfy the following equations:\n\\[\n\\begin{cases}\n\\alpha + \\beta + \\gamma = 12 \\\\\n\\alpha^2 + \\beta^2 + \\gamma^2 = 12(10 - k) \\\\\n\\alpha^3 + \\beta^3 + \\gamma^3 = -6(k^2 + 40k - 212)\n\\end{cases}\n\\]\nFind the range of $k$."
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 8,
"tir": 15
} | 30 | 0 | 0 |
omr-ws-aebc2ae1633eb79b93273d8d204cc4a929ea1818107e295e948cc3c1830a9c41 | Three distinct real numbers \(a, x, y\) satisfy the equation \(\sqrt{a(x-a)} + \sqrt{a(y-a)} = \sqrt{x-a} - \sqrt{a-y}\). Find the value of \(\frac{3x^2 + xy - y^2}{x^2 - xy + y^2}\). | [
"\\(\\frac{1}{3}\\)"
] | [
"Three distinct real numbers \\(a, x, y\\) satisfy the equation \\(\\sqrt{a(x-a)} + \\sqrt{a(y-a)} = \\sqrt{x-a} - \\sqrt{a-y}\\). Find the value of \\(\\frac{3x^2 + xy - y^2}{x^2 - xy + y^2}\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-58ee432d24aacc981f84de0d08254cfbfd78b03bfad496e88bfd49f1605f842a | Three distinct segments are chosen at random among the segments whose end-points are the vertices of a regular 12-gon. What is the probability that the lengths of these three segments are the three side lengths of a triangle with positive area? | [
"\\(\\frac{223}{286}\\)",
"\\dfrac{223}{286}"
] | [
"Three distinct segments are chosen at random among the segments whose end-points are the vertices of a regular 12-gon. What is the probability that the lengths of these three segments are the three side lengths of a triangle with positive area?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 28,
"genselect": 2,
"tir": 20
} | 50 | 0 | 0 |
omr-ws-ab10c6493f94344e913a599935d21fcfc913d85d5a175d91004de836be3d6508 | Three distinct segments are chosen at random among the segments whose endpoints are the vertices of a regular 12-gon. What is the probability that the lengths of these three segments are the three side lengths of a triangle with positive area? | [
"\\dfrac{223}{286}"
] | [
"Three distinct segments are chosen at random among the segments whose endpoints are the vertices of a regular 12-gon. What is the probability that the lengths of these three segments are the three side lengths of a triangle with positive area?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 1
} | 15 | 0 | 0 |
omr-ws-06f1e17b9cf44afb31164eebb168e76a348409ce62698330c0e31fa2051fc175 | Three distinct unit squares are selected uniformly and randomly from a $4 \times 4$ grid of unit squares. If the probability that each unit square shares a side with at least one of the two other unit squares is $\frac{m}{n}$ where $m,n$ are coprime integers, find the sum of the digits of $n$. | [
"5"
] | [
"Three distinct unit squares are selected uniformly and randomly from a $4 \\times 4$ grid of unit squares. If the probability that each unit square shares a side with at least one of the two other unit squares is $\\frac{m}{n}$ where $m,n$ are coprime integers, find the sum of the digits of $n$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 10,
"tir": 2
} | 12 | 0 | 0 |
omr-ws-38ccbcc8b081cb24794ebd7f40b67aceab212e9cc10c914175c8719d269ac78d | Three distinct vertices are chosen at random from the vertices of a given regular polygon of $(2n+1)$ sides. If all such choices are equally likely, what is the probability that the center of the given polygon lies in the interior of the triangle determined by the three chosen random points? | [
"\\frac{n+1}{4n-2}"
] | [
"Three distinct vertices are chosen at random from the vertices of a given regular polygon of $(2n+1)$ sides. If all such choices are equally likely, what is the probability that the center of the given polygon lies in the interior of the triangle determined by the three chosen random points?"
] | [
"MATH_training_set"
] | [
"genselect"
] | {
"genselect": 7
} | 7 | 0 | 0 |
omr-ws-15d330f66db6c40419b15fd092f592be0d6c863a0017c80cf92e0556b6f13421 | Three distinct vertices are chosen at random from the vertices of a given regular polygon with $(2n+1)$ sides. What is the probability that the center of the polygon lies in the interior of the triangle determined by the three chosen vertices? | [
"\\(\\frac{n+1}{2(2n-1)}\\)"
] | [
"Three distinct vertices are chosen at random from the vertices of a given regular polygon with $(2n+1)$ sides. What is the probability that the center of the polygon lies in the interior of the triangle determined by the three chosen vertices?"
] | [
"aops_c6_high_school_olympiads"
] | [
"genselect"
] | {
"genselect": 4
} | 4 | 0 | 0 |
omr-ws-9b9059bd44800c2b16f20fcadebd887ab269c4cc2da90b4aa3957233bdb957e5 | Three distinct vertices are chosen at random from the vertices of a regular polygon with \( n \) sides. What is the probability that the center of the polygon lies in the interior of the triangle determined by the three chosen vertices? | [
"\\(\\begin{cases} \\dfrac{n-4}{4(n-1)} & \\text{for } n \\text{ even}; \\\\ \\dfrac{n+1}{4(n-2)} & \\text{for } n \\text{ odd} \\end{cases}\\)"
] | [
"Three distinct vertices are chosen at random from the vertices of a regular polygon with \\( n \\) sides. What is the probability that the center of the polygon lies in the interior of the triangle determined by the three chosen vertices?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 1
} | 4 | 0 | 0 |
omr-ws-a55a4c0b7bbb152c5f5169520a848efacb5511e43acfa5801cda0997633c859a | Three distinct vertices are randomly selected from a regular $(2n+1)$-sided polygon. What is the probability that the triangle formed by these vertices contains the center of the polygon? | [
"$\\frac{n+1}{4n-2}$"
] | [
"Three distinct vertices are randomly selected from a regular $(2n+1)$-sided polygon. What is the probability that the triangle formed by these vertices contains the center of the polygon?"
] | [
"aops_c5_contests_amp_programs"
] | [
"genselect"
] | {
"genselect": 7
} | 7 | 0 | 0 |
omr-ws-a491a15cbf4158ca153dfd1ebe2580be4edc89750d08744ede9fdc538a8c68c8 | Three distinct vertices of a cube are chosen at random. What is the probability that the plane determined by these three vertices contains points inside the cube? | [
"\\(\\frac{4}{7}\\)"
] | [
"Three distinct vertices of a cube are chosen at random. What is the probability that the plane determined by these three vertices contains points inside the cube?"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 30,
"tir": 58
} | 88 | 0 | 0 |
omr-ws-cf19852617c95ab69cdd98a958220256bdc4c2e4e7b35c4a9a7d1e3b24e1c1fa | Three doctors, four nurses, and three patients stand in a line in random order. The probability that there is at least one doctor and at least one nurse between each pair of patients is \(\frac{m}{n}\), where \(m\) and \(n\) are relatively prime positive integers. Find \(m + n\). | [
"153"
] | [
"Three doctors, four nurses, and three patients stand in a line in random order. The probability that there is at least one doctor and at least one nurse between each pair of patients is \\(\\frac{m}{n}\\), where \\(m\\) and \\(n\\) are relatively prime positive integers. Find \\(m + n\\)."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 5
} | 5 | 0 | 0 |
omr-ws-ff88eb1e91b1fe760d7333e2045851f56ae4523e0add7ca0bc0cf74c9ac9a445 | Three dogs stand at the vertices of an equilateral triangle with side length 1. Dog at vertex 1 runs towards dog at vertex 2, dog at vertex 2 runs towards dog at vertex 3, and dog at vertex 3 runs towards dog at vertex 1. They all run with the same speed. Find the distance each dog travels before they meet. | [
"\\(\\frac{2}{3}\\)"
] | [
"Three dogs stand at the vertices of an equilateral triangle with side length 1. Dog at vertex 1 runs towards dog at vertex 2, dog at vertex 2 runs towards dog at vertex 3, and dog at vertex 3 runs towards dog at vertex 1. They all run with the same speed. Find the distance each dog travels before they meet."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 20
} | 68 | 0 | 0 |
omr-ws-3b046a1bb243aa0e80bcf6c8a9067c3ddd32e817cd239797f97daed3339ddc86 | Three ducks, Huey, Dewey, and Louie, each waddle along different straight lines in the plane at their own constant speeds. Suppose at three distinct times, they are collinear. If the positions of Huey, Dewey, and Louie at these times are represented by the coordinates $(1, 2)$, $(3, 4)$, and $(5, 6)$ respectively, and ... | [
"27"
] | [
"Three ducks, Huey, Dewey, and Louie, each waddle along different straight lines in the plane at their own constant speeds. Suppose at three distinct times, they are collinear. If the positions of Huey, Dewey, and Louie at these times are represented by the coordinates $(1, 2)$, $(3, 4)$, and $(5, 6)$ respectively,... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 3
} | 7 | 0 | 0 |
omr-ws-8c56992a07110a377036589aa8c38b970cced4f6a17158146c33ec874ed1e982 | Three edges of a parallelepiped lie on three intersecting diagonals of the lateral faces of a triangular prism. Find the ratio of the volumes of the parallelepiped and the prism. | [
"\\dfrac{1}{3}"
] | [
"Three edges of a parallelepiped lie on three intersecting diagonals of the lateral faces of a triangular prism. Find the ratio of the volumes of the parallelepiped and the prism."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 1
} | 8 | 0 | 0 |
omr-ws-6fbc5888c790761ec93b87c442225a4fe3cd7a3c54ad8c17b0d9b2dbb0d7abf8 | Three ellipses are pairwise tangent with semimajor axis $4$ and semiminor axis $3$. If the area of the triangle formed by their centers can be written as $a\sqrt{b}$, where $a$ and $b$ are integers and $b$ is square-free, find $a + b$. | [
"15"
] | [
"Three ellipses are pairwise tangent with semimajor axis $4$ and semiminor axis $3$. If the area of the triangle formed by their centers can be written as $a\\sqrt{b}$, where $a$ and $b$ are integers and $b$ is square-free, find $a + b$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-0aaba6309e39bd764d6d4d8df3d630c92c8e6b94f37eedff0256879784eada76 | Three ellipses that are translated images of each other and mutually tangent are defined as shown, centered at points \(A, B, C\). If \(AB = 8\) and \(BC = 18\), find the area inside triangle \(ABC\) but outside any of the three ellipses. | [
"\\(\\frac{8\\sqrt{693} - 4\\sqrt{231}\\pi}{3}\\)"
] | [
"Three ellipses that are translated images of each other and mutually tangent are defined as shown, centered at points \\(A, B, C\\). If \\(AB = 8\\) and \\(BC = 18\\), find the area inside triangle \\(ABC\\) but outside any of the three ellipses."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 3,
"tir": 1
} | 6 | 0 | 0 |
omr-ws-b414afaaf2f8ce63da7e791c46ebd23f1db2b4a241e9fbe90f6b2786fddd07ef | Three empty jugs are on a table. Winnie the Pooh, Rabbit, and Piglet take turns placing walnuts in the jugs. Winnie the Pooh can place walnuts in the first or second jug, Rabbit in the second or third jug, and Piglet in the first or third jug. The player whose move results in exactly 1999 walnuts in total loses the gam... | [
"1999"
] | [
"Three empty jugs are on a table. Winnie the Pooh, Rabbit, and Piglet take turns placing walnuts in the jugs. Winnie the Pooh can place walnuts in the first or second jug, Rabbit in the second or third jug, and Piglet in the first or third jug. The player whose move results in exactly 1999 walnuts in total loses th... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-a2aea58b57a0576eaf845e222047e7599e69484eccd946f72cae19f4953a56dd | Three equal circles \( k_1, k_2, k_3 \) intersect non-tangentially at a point \( P \). Let \( A \) and \( B \) be the centers of circles \( k_1 \) and \( k_2 \), respectively. Let \( D \) and \( C \) be the intersections of \( k_3 \) with \( k_1 \) and \( k_2 \), respectively, which are different from \( P \). If the r... | [
"\\sqrt{3}"
] | [
"Three equal circles \\( k_1, k_2, k_3 \\) intersect non-tangentially at a point \\( P \\). Let \\( A \\) and \\( B \\) be the centers of circles \\( k_1 \\) and \\( k_2 \\), respectively. Let \\( D \\) and \\( C \\) be the intersections of \\( k_3 \\) with \\( k_1 \\) and \\( k_2 \\), respectively, which are diffe... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-232468ee82f5f054214e2da8f2e7ab0139bb6bf5f8f81768356f41c77912cfdf | Three equal circles are arranged so that each is tangential to the others. They are enclosed in a regular hexagon so that the hexagon is tangential to the circles. Find the radius \( r \) of the circles in terms of the side-length \( x \) of the hexagon. | [
"\\( \\frac{\\sqrt{3} \\, x}{4} \\)"
] | [
"Three equal circles are arranged so that each is tangential to the others. They are enclosed in a regular hexagon so that the hexagon is tangential to the circles. Find the radius \\( r \\) of the circles in terms of the side-length \\( x \\) of the hexagon."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 1,
"tir": 1
} | 8 | 0 | 0 |
omr-ws-1c32d4fce6c483e80920dd8313c49c51fdbfb07ffd0e6ea4064c3f35bbd6a27d | Three equal circles are externally tangent to a circle of radius $r$ and pairwise tangent to one another. Find the areas of the three curvilinear triangles formed by these circles. | [
"S=\\left[12+7\\sqrt{3}-\\left(\\frac{23}{6}+2\\sqrt{3}\\right)\\pi\\right]r^2"
] | [
"Three equal circles are externally tangent to a circle of radius $r$ and pairwise tangent to one another. Find the areas of the three curvilinear triangles formed by these circles."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d211953070e58a5be3d68c3315fcbc39b0b9603574ea87d0b60dcf9639ae98cc | Three equal circles are tangent to each other and a circle $\omega$ is tangent to all three of them. From a point $P$ on $\omega$, tangent segments $PA$, $PB$, and $PC$ are drawn to the other three circles. If the lengths of $PA$ and $PB$ are $7$ and $11$ respectively, find the length of $PC$. | [
"18"
] | [
"Three equal circles are tangent to each other and a circle $\\omega$ is tangent to all three of them. From a point $P$ on $\\omega$, tangent segments $PA$, $PB$, and $PC$ are drawn to the other three circles. If the lengths of $PA$ and $PB$ are $7$ and $11$ respectively, find the length of $PC$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-36bfd6651da65f9a672feb02e6d9d7d99b4d09b477bbee87b6531b3b140d33b3 | Three equal circles have a common point $M$ and intersect in pairs at points $A, B, C$. If the radius of each circle is $r$, find the area of triangle $ABC$ in terms of $r$. | [
"\\dfrac{3\\sqrt{3}}{4} r^2"
] | [
"Three equal circles have a common point $M$ and intersect in pairs at points $A, B, C$. If the radius of each circle is $r$, find the area of triangle $ABC$ in terms of $r$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d4f709d26d7b97742bb9d61b6eac6a5ba25587852be6efb618817f611f579702 | Three equal circles have a common point H and intersect each other pairwise at points A, B, and C. If the radius of each circle is 5 units, find the area of the circumcircle of triangle ABC. | [
"25\\pi"
] | [
"Three equal circles have a common point H and intersect each other pairwise at points A, B, and C. If the radius of each circle is 5 units, find the area of the circumcircle of triangle ABC."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3881964777e966d6de2a5b53c8e450af76ba73b6379df6108459a65afd4a5bb5 | Three equal circles of radius \( r \) are arranged such that each circle touches the other two. A smaller circle is inscribed in the gap formed between these three circles. Find the radius of the smaller circle. | [
"\\(\\frac{(2\\sqrt{3}-3)r}{3}\\)"
] | [
"Three equal circles of radius \\( r \\) are arranged such that each circle touches the other two. A smaller circle is inscribed in the gap formed between these three circles. Find the radius of the smaller circle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-9963005e83536db86e60df4a4cb8c1772fc51c5f461b9703c9ee51a56378b2b2 | Three equal circles of radius \( r \) each pass through the centers of the other two. What is the area of the region common to all three circles? | [
"\\( r^2 \\left( \\frac{\\pi - \\sqrt{3}}{2} \\right) \\)"
] | [
"Three equal circles of radius \\( r \\) each pass through the centers of the other two. What is the area of the region common to all three circles?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 15
} | 30 | 0 | 0 |
omr-ws-c2fc4ade58052a2ad96741a8185ed06fc7372032340c4b990da565105ba205f7 | Three equal circles pass through a given point \(H\) and meet one another two by two at points \(A, B, C\). If the radius of each circle is \(r\) and the distance from \(H\) to each of the points \(A, B,\) and \(C\) is \(d\), find the value of \(\frac{d}{r}\). | [
"1"
] | [
"Three equal circles pass through a given point \\(H\\) and meet one another two by two at points \\(A, B, C\\). If the radius of each circle is \\(r\\) and the distance from \\(H\\) to each of the points \\(A, B,\\) and \\(C\\) is \\(d\\), find the value of \\(\\frac{d}{r}\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ba742cb55b3c1fcf35c375155ca85e372f386229e598d517586141e85d227324 | Three equal circles tangent to one another are inscribed in a regular triangle, whose side is equal to $a$. Each of them is in contact with two sides of the given triangle. Determine the radii of the circles. | [
"$r=\\frac{a}{2\\left(\\sqrt 3+1\\right)}=\\frac{a\\left(\\sqrt 3-1\\right)}{4}$"
] | [
"Three equal circles tangent to one another are inscribed in a regular triangle, whose side is equal to $a$. Each of them is in contact with two sides of the given triangle. Determine the radii of the circles."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-299ded79e2c4515a052f02b296b9c1c4e591eaaa1ca25dbabde01c910226de1f | Three equal circles with a common point \( H \) intersect each other pairwise at points \( A \), \( B \), and \( C \). If the radius of each of the three given circles is \( r \), find the radius of the circumcircle of triangle \( ABC \) in terms of \( r \). | [
"r"
] | [
"Three equal circles with a common point \\( H \\) intersect each other pairwise at points \\( A \\), \\( B \\), and \\( C \\). If the radius of each of the three given circles is \\( r \\), find the radius of the circumcircle of triangle \\( ABC \\) in terms of \\( r \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-46f41df84fbd7d0afc56cd02c63e3c9cd6ced4a6427ad4d79fdef3f21c465060 | Three equal circles with radius \((2+\sqrt{2})3\sqrt{3}\) touch each other. Find the radius of another circle that touches all three of these circles. | [
"\\(6\\sqrt{3} + 3\\sqrt{6} + 6 + 3\\sqrt{2}\\)"
] | [
"Three equal circles with radius \\((2+\\sqrt{2})3\\sqrt{3}\\) touch each other. Find the radius of another circle that touches all three of these circles."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 2
} | 3 | 0 | 0 |
omr-ws-3e205e948ee47b7786f822e7c709c984d413eafed6ee5c969316a828e5364b2e | Three equal pairwise tangent spheres of radius $R$ lie on a plane. A fourth sphere is tangent to the plane and to each given sphere. Find the radius of the fourth sphere. | [
"\\(\\frac{R}{3}\\)"
] | [
"Three equal pairwise tangent spheres of radius $R$ lie on a plane. A fourth sphere is tangent to the plane and to each given sphere. Find the radius of the fourth sphere."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-be11f39bcce39bcece2c334ab070bf94c1bc454ef7790d086c719fc29652a2ee | Three equal right circular cones with angle $\alpha $ $(\alpha \le 2\pi /3)$ in the axial section have a common vertex and are externally tangent to one another along the generatrices $l_1$, $l_2$, $l_3$. Find the angle between $l_1$ and $l_2$. | [
"$2\\arcsin [1/2\\tan(\\alpha /2)]$"
] | [
"Three equal right circular cones with angle $\\alpha $ $(\\alpha \\le 2\\pi /3)$ in the axial section have a common vertex and are externally tangent to one another along the generatrices $l_1$, $l_2$, $l_3$. Find the angle between $l_1$ and $l_2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2f28206e7f384b4df82e9e514913847a95f5e1388cd6430b09337324a2df7b98 | Three equal spheres of radius $R$ tangent to one another lie on a plane $P$. A right circular cone with its base in $P$ is externally tangent to the spheres. Find the radius of the base of the cone if its altitude is equal to $qR$. | [
"r = \\frac{2\\sqrt{3}(q-1) - \\sqrt{9q^2 - 18q + 12}}{3(q-2)}R\n\nFor the specific case when \\( q = 2 \\):\nr = \\frac{\\sqrt{3}}{6}R\n\nSo, the final answer is:\n\\[\nr = \\begin{cases} \n\\frac{\\sqrt{3}}{6}R & \\text{if } q = 2, \\\\\n\\frac{2\\sqrt{3}(q-1) - \\sqrt{9q^2 - 18q + 12}}{3(q-2)}R & \\text{if } q \... | [
"Three equal spheres of radius $R$ tangent to one another lie on a plane $P$. A right circular cone with its base in $P$ is externally tangent to the spheres. Find the radius of the base of the cone if its altitude is equal to $qR$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-049205af0be71e643a93cc5c756245917da8681d4d7331d4c7112a911a142c44 | Three equally skilled players are participating in a 3-game tournament. If all 3 games have no winner, additional games are played until one player wins. The prize money is $20. Calculate the expected value of the prize money for one player. | [
"\\(\\frac{20}{3}\\)"
] | [
"Three equally skilled players are participating in a 3-game tournament. If all 3 games have no winner, additional games are played until one player wins. The prize money is $20. Calculate the expected value of the prize money for one player."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-f5b64009d8123a96e9876215509aaa0a215bf2e05d2a6e3f1d299cc425092f74 | Three equally spaced parallel lines intersect a circle, creating three chords of lengths $38, 38,$ and $34$. What is the distance between two adjacent parallel lines? | [
"6"
] | [
"Three equally spaced parallel lines intersect a circle, creating three chords of lengths $38, 38,$ and $34$. What is the distance between two adjacent parallel lines?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 74,
"genselect": 8,
"tir": 1
} | 83 | 0 | 0 |
omr-ws-e2865cf741aeda7bdd952686f5c5a6de6a008a318e071b51c2a0420b59fa9406 | Three equally spaced parallel lines intersect a circle, creating three chords of lengths $38, 38,$ and $34.$ What is the distance between two adjacent parallel lines? | [
"6"
] | [
"Three equally spaced parallel lines intersect a circle, creating three chords of lengths $38, 38,$ and $34.$ What is the distance between two adjacent parallel lines?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-6fdbb99b3fc7a4986b8258b714101ea2ee5cb5563b010b5fa003b7016dbc501f | Three equally spaced parallel lines intersect a circle, creating three chords of lengths $38,$ $38,$ and $34.$ What is the distance between two adjacent parallel lines? | [
"6"
] | [
"Three equally spaced parallel lines intersect a circle, creating three chords of lengths $38,$ $38,$ and $34.$ What is the distance between two adjacent parallel lines?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 6
} | 18 | 0 | 0 |
omr-ws-4c1381db3184ad64fb075c31805356b65b724d59dd49085b7175cf36acfdf0c5 | Three equally spaced parallel lines intersect a circle, creating three chords of lengths 38, 38, and 34. What is the distance between two adjacent parallel lines? | [
"6"
] | [
"Three equally spaced parallel lines intersect a circle, creating three chords of lengths 38, 38, and 34. What is the distance between two adjacent parallel lines?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 29,
"genselect": 8
} | 37 | 0 | 0 |
omr-ws-5055e15a9820767137eec0775cd64ff8bb1c8f6313913e462595d5b48270b313 | Three equilateral triangles $ABC_1, BCA_1,$ and $CAB_1$ are constructed externally on the sides of an arbitrary triangle $ABC$. The midpoints of the segments $A_1B_1, B_1C_1,$ and $C_1A_1$ are labeled $C_2, A_2,$ and $B_2$, respectively. Let $P$ be the point of intersection of the lines $AA_2$ and $BB_2$. Calculate the... | [
"2"
] | [
"Three equilateral triangles $ABC_1, BCA_1,$ and $CAB_1$ are constructed externally on the sides of an arbitrary triangle $ABC$. The midpoints of the segments $A_1B_1, B_1C_1,$ and $C_1A_1$ are labeled $C_2, A_2,$ and $B_2$, respectively. Let $P$ be the point of intersection of the lines $AA_2$ and $BB_2$. Calculat... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-15738ced0bec02a16ce7a51d287dfc172875e5bd76ef11d1f01dbe655f7db13c | Three equilateral triangles of equal size fit inside a unit square without overlapping. What is the greatest possible area of one of these triangles? | [
"2\\sqrt{3} - 3"
] | [
"Three equilateral triangles of equal size fit inside a unit square without overlapping. What is the greatest possible area of one of these triangles?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 2
} | 10 | 0 | 0 |
omr-ws-c2b8c09e77301ecdcb38b0b61bf985128392739b69abc5bb10d3f02bfd3cac18 | Three equilateral triangles of paper whose sides measure 13 cm were placed as shown in the figure in such a way that they form an equilateral triangle whose sides measure 24 cm. In the part a white equilateral triangle is formed in the center. What is the perimeter of that white triangle in cm? | [
"99"
] | [
"Three equilateral triangles of paper whose sides measure 13 cm were placed as shown in the figure in such a way that they form an equilateral triangle whose sides measure 24 cm. In the part a white equilateral triangle is formed in the center. What is the perimeter of that white triangle in cm?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d2a15a04d9a02089e8222a70d69c937a3679b7c856b38d23803005a2b3cb56ec | Three equilateral triangles with side length $c$ can completely cover an equilateral triangle with side length $1$. The triangles may overlap with each other. Find the minimum possible value of $c$. | [
"\\dfrac{2}{3}",
"\\dfrac{2}{\\sqrt{3}}"
] | [
"Three equilateral triangles with side length $c$ can completely cover an equilateral triangle with side length $1$. The triangles may overlap with each other. Find the minimum possible value of $c$."
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 23,
"genselect": 5
} | 28 | 0 | 0 |
omr-ws-838182df17772053736dab16162560982399680fe0128c99ce81145a8729e72b | Three events E1, E2, E3 must occur such that only one can happen. The odds are 7 to 4 against E1 and 5 to 3 against E2. Find the odds against E3. | [
"65 to 23"
] | [
"Three events E1, E2, E3 must occur such that only one can happen. The odds are 7 to 4 against E1 and 5 to 3 against E2. Find the odds against E3."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-e173fc32cc93b85484fe2f6908a0bed937e3937c4e867e01a413f14ba3680556 | Three excavators are busy digging a pit. The difference between the outputs of the first and third excavators is three times the difference between that of the third and second excavators. The first excavator does $\frac{4}{5}$ of the whole job within a period of time that would be needed for the second excavator alone... | [
"Three-fold"
] | [
"Three excavators are busy digging a pit. The difference between the outputs of the first and third excavators is three times the difference between that of the third and second excavators. The first excavator does $\\frac{4}{5}$ of the whole job within a period of time that would be needed for the second excavator... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e19568b057796730967c97da4618a70c406d686d5b9c2776a47ee18c7a7b479e | Three excavators had to dig a pit each: the first and the second excavator a pit of 800 m$^3$ and the third one, a pit of 400 m$^3$. The first and the second excavator, working together, removed three times as much earth per hour as the third one; the first and the third excavator began working simultaneously and the s... | [
"600 m$^3$"
] | [
"Three excavators had to dig a pit each: the first and the second excavator a pit of 800 m$^3$ and the third one, a pit of 400 m$^3$. The first and the second excavator, working together, removed three times as much earth per hour as the third one; the first and the third excavator began working simultaneously and ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6d467cc7bc574aa3536c45d063cf03406bc9cf97387189c82c1e02df538f0904 | Three excavators participated in digging a pit with the volume of 340 m$^3$. The first excavator removes 40 m$^3$ of earth per hour, the second excavator takes out $c$ m$^3$ less than the first, and the third one removes $2c$ m$^3$ more than the first. First the first and the second excavators worked simultaneously and... | [
"$c=9\\frac{11}{16}$"
] | [
"Three excavators participated in digging a pit with the volume of 340 m$^3$. The first excavator removes 40 m$^3$ of earth per hour, the second excavator takes out $c$ m$^3$ less than the first, and the third one removes $2c$ m$^3$ more than the first. First the first and the second excavators worked simultaneousl... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-256f314a195793b2a96c21e4b6ce1fd1376679d20818f37e705b9951052fd329 | Three experts each have a 60% accuracy rate in predicting the winner of a game. What is the probability that they are correct if all three experts concur in their prediction? | [
"\\(\\frac{27}{35}\\)"
] | [
"Three experts each have a 60% accuracy rate in predicting the winner of a game. What is the probability that they are correct if all three experts concur in their prediction?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-ed18f1a279ce4912227c1caeda9884b3b024a7c4f1565eef3b9dc560530a1fbd | Three faces of a rectangular prism meet at a vertex. The centers of these faces form a triangle with side lengths 4, 5, and 6. What is the square of the volume of the prism? | [
"48600"
] | [
"Three faces of a rectangular prism meet at a vertex. The centers of these faces form a triangle with side lengths 4, 5, and 6. What is the square of the volume of the prism?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-2488c453719a289376b0899207b7fd5d761763a3746a126828cff70506bcccb2 | Three faces of a right rectangular prism have areas of 48, 49 and 50 square units. What is the volume of the prism, in cubic units? Express your answer to the nearest whole number. | [
"343"
] | [
"Three faces of a right rectangular prism have areas of 48, 49 and 50 square units. What is the volume of the prism, in cubic units? Express your answer to the nearest whole number."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2e1a6f35fbca989118f46dc07c243495dbadf9acf7b68559db953ab09a686903 | Three fair 6-sided dice are tossed. What is the probability that the sum of the numbers on two of the dice equals the number on the third die? | [
"\\dfrac{5}{24}"
] | [
"Three fair 6-sided dice are tossed. What is the probability that the sum of the numbers on two of the dice equals the number on the third die?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0f344b5236b5a5e69896e9b9019a98d3a752df3f955a6f7ed8a6fb95cfa09189 | Three fair 6-sided dice are tossed. What is the probability that the sum of the numbers on two of the dice equals the third? | [
"\\(\\frac{5}{24}\\)"
] | [
"Three fair 6-sided dice are tossed. What is the probability that the sum of the numbers on two of the dice equals the third?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-37179c7b4c2d6db8895104c05b5668ef3a234af2e44614b336d0666144671239 | Three fair six-sided dice are rolled. The probability that there is exactly 1 one, at least 2 twos, or exactly 3 threes can be expressed as $\frac{m}{n}$. Find the value of $m+n$. | [
"77"
] | [
"Three fair six-sided dice are rolled. The probability that there is exactly 1 one, at least 2 twos, or exactly 3 threes can be expressed as $\\frac{m}{n}$. Find the value of $m+n$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-78b53c7636cc09fa52fb0b4739ca178e8a0eb2931f08f48852949195c88a5863 | Three fair six-sided dice are rolled. What is the probability that the values shown on two of the dice sum to the value shown on the remaining die? | [
"\\(\\frac{5}{24}\\)",
"\\dfrac{5}{24}"
] | [
"Three fair six-sided dice are rolled. What is the probability that the values shown on two of the dice sum to the value shown on the remaining die?"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 32
} | 64 | 0 | 0 |
omr-ws-0c499f6069b0b198cb4ae0ce3016f3f50b96d1f80f18dded4cd08e37a42eaf0a | Three fair six-sided dice, each numbered 1 through 6, are rolled. The probability that the three numbers that come up can be expressed as \( \frac{m}{n} \). Compute \( m+n \). | [
"109"
] | [
"Three fair six-sided dice, each numbered 1 through 6, are rolled. The probability that the three numbers that come up can be expressed as \\( \\frac{m}{n} \\). Compute \\( m+n \\)."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 7,
"tir": 25
} | 40 | 0 | 0 |
omr-ws-819b288888665f3b9bf07220a66934234bd5f06d861465d63c746b4c227c1798 | Three fair, standard six-sided dice are rolled. What is the probability that the sum of the numbers on the top faces is 18? Express your answer as a common fraction. | [
"\\frac{1}{216}"
] | [
"Three fair, standard six-sided dice are rolled. What is the probability that the sum of the numbers on the top faces is 18? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a3b6213e3a3351f4469032cd13efae8517e235a4ca1cecf66ac8b4935fbee614 | Three families sit down around a circular table, each person choosing their seat at random. One family has two members, while the other two families have three members. What is the probability that every person sits next to at least one person from a different family? | [
"\\dfrac{53}{70}"
] | [
"Three families sit down around a circular table, each person choosing their seat at random. One family has two members, while the other two families have three members. What is the probability that every person sits next to at least one person from a different family?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-469e3042a96999fb987a4ff887c77cb82f4865d80d949534aa2be3cf3daf7eb5 | Three farmers (Bob, Dick, and Tom) rented a pasture and a large horse barn for a fixed amount per month. They agreed to pay in proportion to the number of horses each held. For the first month, Bob had 18 horses while Dick and Tom had some. Dick paid $480 of rent that month. For the second month, each put in 2 more hor... | [
"1193.10"
] | [
"Three farmers (Bob, Dick, and Tom) rented a pasture and a large horse barn for a fixed amount per month. They agreed to pay in proportion to the number of horses each held. For the first month, Bob had 18 horses while Dick and Tom had some. Dick paid $480 of rent that month. For the second month, each put in 2 mor... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c199d62f568605252234db2367f17a02d1cb115ec436a5d9a936fdfbc1bc641e | Three farmers sell chickens at a market. One has 10 chickens, another has 16, and the third has 26. Each farmer sells at least one, but not all, of his chickens before noon, all farmers selling at the same price per chicken. Later in the day, each sells his remaining chickens at a reduced price. If each farmer received... | [
"\\( a = \\frac{15}{4}, b = \\frac{5}{4} \\)"
] | [
"Three farmers sell chickens at a market. One has 10 chickens, another has 16, and the third has 26. Each farmer sells at least one, but not all, of his chickens before noon, all farmers selling at the same price per chicken. Later in the day, each sells his remaining chickens at a reduced price. If each farmer rec... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 18
} | 19 | 0 | 0 |
omr-ws-4737871cf8b2e0bb044e2d7f89f5e99ba01f73061f3e1df0f5775676d26dd1a9 | Three farmers sell chickens at a market. One has 10 chickens, another has 16, and the third has 26. Each farmer sells at least one, but not all, of his chickens before noon, all farmers selling at the same price per chicken. Later in the day, each sells his remaining chickens, all again selling at the same reduced pric... | [
"1.25"
] | [
"Three farmers sell chickens at a market. One has 10 chickens, another has 16, and the third has 26. Each farmer sells at least one, but not all, of his chickens before noon, all farmers selling at the same price per chicken. Later in the day, each sells his remaining chickens, all again selling at the same reduced... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-62abe8a913a6cc737b4c46d59a455f49fe7c1608e012185208c4e3d44db62890 | Three farmers sold chickens at the local market. One farmer had 10 chickens, another had 16 chickens, and the last had 26 chickens. They agreed to sell all chickens at the same price initially and then lowered the price to the same lower price point by lunchtime. By the end of the day, each farmer collected $35 from th... | [
"\\( p = \\frac{15}{4}, q = \\frac{5}{4} \\)"
] | [
"Three farmers sold chickens at the local market. One farmer had 10 chickens, another had 16 chickens, and the last had 26 chickens. They agreed to sell all chickens at the same price initially and then lowered the price to the same lower price point by lunchtime. By the end of the day, each farmer collected $35 fr... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 14
} | 15 | 0 | 0 |
omr-ws-f142aa48f95e34fc2a2413a3721a24ec2f3d0b48b5336c0da1954c7d6fdb8056 | Three farmers sold chickens at the local market. The first farmer had 10 chickens, the second had 16 chickens, and the third had 26 chickens. They agreed to sell all their chickens at the same price initially and then lowered their prices to the same new price by lunchtime. By the end of the day, each farmer collected ... | [
"1.25"
] | [
"Three farmers sold chickens at the local market. The first farmer had 10 chickens, the second had 16 chickens, and the third had 26 chickens. They agreed to sell all their chickens at the same price initially and then lowered their prices to the same new price by lunchtime. By the end of the day, each farmer colle... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-6b4b59950da77157982585d62a27726d689bbf9898626d86bae4daceb5396edd | Three faucets fill a 100-gallon tub in 6 minutes. How long, in seconds, does it take six faucets to fill a 25-gallon tub? Assume that all faucets dispense water at the same rate. | [
"45"
] | [
"Three faucets fill a 100-gallon tub in 6 minutes. How long, in seconds, does it take six faucets to fill a 25-gallon tub? Assume that all faucets dispense water at the same rate."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f8d5f206cb24008e772c4c535686e1eb33ba7a2bcfa95bce082da273d08c89fb | Three fixed circles pass through points A and B. Let X be a variable point on the first circle different from A and B. The line AX intersects the other two circles at points Y and Z, with Y between X and Z. If the ratio XY/YZ is constant and equals k, find the value of k when the radii of the three circles are 5, 7, an... | [
"\\dfrac{8 + 10\\sqrt{2}}{17}"
] | [
"Three fixed circles pass through points A and B. Let X be a variable point on the first circle different from A and B. The line AX intersects the other two circles at points Y and Z, with Y between X and Z. If the ratio XY/YZ is constant and equals k, find the value of k when the radii of the three circles are 5, ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bd728f1b6e935f0a647eed74cb130b190fffa36f3897f008de58c3e63251d09c | Three fixed circles pass through the points $A$ and $B$. Let $X$ be a variable point on the first circle different from $A$ and $B$. The line $AX$ intersects the other two circles at $Y$ and $Z$ (with $Y$ between $X$ and $Z$). If the radii of the three circles are $r_1$, $r_2$, and $r_3$ respectively, and $AB = d$, fin... | [
"\\dfrac{\\sqrt{r_2^2 - \\dfrac{d^2}{4}} - \\sqrt{r_1^2 - \\dfrac{d^2}{4}}}{\\sqrt{r_3^2 - \\dfrac{d^2}{4}} - \\sqrt{r_2^2 - \\dfrac{d^2}{4}}}"
] | [
"Three fixed circles pass through the points $A$ and $B$. Let $X$ be a variable point on the first circle different from $A$ and $B$. The line $AX$ intersects the other two circles at $Y$ and $Z$ (with $Y$ between $X$ and $Z$). If the radii of the three circles are $r_1$, $r_2$, and $r_3$ respectively, and $AB = d$... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 8
} | 11 | 0 | 0 |
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