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-42a202ce1a8aebffbce9f2b2a00f493d328da738cba98987d66df8cdadec5dce | Three points $A, B, M$ are taken on a plane. A point $C$ taken on the line segment $AB$ is such that $| AC | : | CD | = k$. Express the vector $\overrightarrow{MC}$ in terms of $\overrightarrow{MA}$ and $\overrightarrow{MB}$. | [
"$\\overrightarrow{MC}=\\frac{1}{k+1}\\overrightarrow{MA} +\\frac{k}{k+1}\\overrightarrow{MB}$"
] | [
"Three points $A, B, M$ are taken on a plane. A point $C$ taken on the line segment $AB$ is such that $| AC | : | CD | = k$. Express the vector $\\overrightarrow{MC}$ in terms of $\\overrightarrow{MA}$ and $\\overrightarrow{MB}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-72c0e9d2528ee186687553d863dea59834ba98d4141b53a66c1fb4d20a27e556 | Three points $X$, $Y$, and $Z$ are randomly placed on the sides of a square such that $X$ and $Y$ are always on the same side of the square. The probability that $\triangle{XYZ}$ contains the center of the square can be written as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. What is $a+b$? (... | [
"5"
] | [
"Three points $X$, $Y$, and $Z$ are randomly placed on the sides of a square such that $X$ and $Y$ are always on the same side of the square. The probability that $\\triangle{XYZ}$ contains the center of the square can be written as $\\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. What is $... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 7
} | 9 | 0 | 0 |
omr-ws-df2fcbf73cfbebff889e5b5990ca3244d2adca5341997d59e2174f33f0ec06d8 | Three points A, B, C are on a circle with radius 1. Find the maximum and minimum values of $ \overrightarrow{AB} \cdot \overrightarrow{AC} $. | [
"Maximum = 4, Minimum = -\\(\\frac{1}{2}\\)"
] | [
"Three points A, B, C are on a circle with radius 1. Find the maximum and minimum values of $ \\overrightarrow{AB} \\cdot \\overrightarrow{AC} $."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 20
} | 20 | 0 | 0 |
omr-ws-4e3fff8c5914d8be06ab1cb34a39c20a6e477cf77093fa701a292bb0c7bac0d0 | Three points A, B, C are selected randomly on the circumference of a circle. Find the probability that A, B, C lie on a semicircle. | [
"\\(\\frac{3}{4}\\)"
] | [
"Three points A, B, C are selected randomly on the circumference of a circle. Find the probability that A, B, C lie on a semicircle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-a1b63dc1dcaea71723bac40d27cd14513a917d5576059588134e16cb9712a94b | Three points A, B, and C are selected at random from the circumference of a circle. Find the probability that the points lie on a semicircle. | [
"\\(\\frac{3}{4}\\)"
] | [
"Three points A, B, and C are selected at random from the circumference of a circle. Find the probability that the points lie on a semicircle."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 21
} | 36 | 0 | 0 |
omr-ws-a9ae4535bf2877557a847ce0aa395291232954043ed09bb6e51488fd50dd1b6c | Three points A, B, and C are selected at random on the circumference of a circle. Find the probability that the points lie on a semicircle. | [
"\\(\\frac{3}{4}\\)"
] | [
"Three points A, B, and C are selected at random on the circumference of a circle. Find the probability that the points lie on a semicircle."
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 20
} | 35 | 0 | 0 |
omr-ws-4b05b5b5d34cd8a7ec5f3a8d25cd0ce6c78bed8f8942a7dfd2dad6ca12c287ec | Three points A, B, and M lie on a circle. The distances from point M to the lines tangent to the circle at points A and B are equal to a and b, respectively. Find the distance from point M to the line AB. | [
"\\( \\sqrt{ab} \\)"
] | [
"Three points A, B, and M lie on a circle. The distances from point M to the lines tangent to the circle at points A and B are equal to a and b, respectively. Find the distance from point M to the line AB."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 3
} | 18 | 0 | 0 |
omr-ws-2891864bd4153b3e8d3c9c4ccded06b5604270fb891a8c9ca99a6dc2c432ea20 | Three points \( A \), \( B \), and \( C \) are located on a circle with radius \( r \). Given that \( AB - AC = r \) and \( AB \cdot AC = r \), find the measure of the angles \( \angle BAC \), \( \angle ABC \), and \( \angle ACB \). | [
"54^\\circ"
] | [
"Three points \\( A \\), \\( B \\), and \\( C \\) are located on a circle with radius \\( r \\). Given that \\( AB - AC = r \\) and \\( AB \\cdot AC = r \\), find the measure of the angles \\( \\angle BAC \\), \\( \\angle ABC \\), and \\( \\angle ACB \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-b033a204330d027a3d5617adbeb6f49de585512b7f00ecac7025f524e4e53c9f | Three points \( A \), \( B \), and \( C \) are selected randomly from \( \mathbb{R}^2 \), with each of the six coordinates independently and each following a standard normal distribution. Calculate the probability that point \( C \) lies inside the circle whose diameter is the segment \( AB \). Express your answer as a... | [
"\\dfrac{1}{4}"
] | [
"Three points \\( A \\), \\( B \\), and \\( C \\) are selected randomly from \\( \\mathbb{R}^2 \\), with each of the six coordinates independently and each following a standard normal distribution. Calculate the probability that point \\( C \\) lies inside the circle whose diameter is the segment \\( AB \\). Expres... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-dbde3d3863dd02c1b514ce300cafde89478c5a0154a92b6c477d283725c88dfe | Three points \( A \), \( B \), and \( T \) are fixed such that \( T \) lies on segment \( AB \), closer to point \( A \). Let \( AT = m \) and \( BT = n \) where \( m \) and \( n \) are positive integers. Construct circle \( O \) with a variable radius that is tangent to \( AB \) at \( T \). Let \( P \) be the point su... | [
"172"
] | [
"Three points \\( A \\), \\( B \\), and \\( T \\) are fixed such that \\( T \\) lies on segment \\( AB \\), closer to point \\( A \\). Let \\( AT = m \\) and \\( BT = n \\) where \\( m \\) and \\( n \\) are positive integers. Construct circle \\( O \\) with a variable radius that is tangent to \\( AB \\) at \\( T \... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-1a7f9d5c28e7eb9cf5c1f9c78d17d821cc4baf24b5e43e010daedde959020128 | Three points \( A(0, 1) \), \( B(-1, 0) \), and \( C(1, 0) \) are on the \( x \)-\( y \) plane. Denote \( PD \), \( PE \), and \( PF \) as the perpendiculars drawn from a point \( P \) (not on the lines \( AB \), \( BC \), or \( CA \)) to \( AB \), \( BC \), and \( CA \) respectively. Find the locus of \( P \) such tha... | [
"An ellipse"
] | [
"Three points \\( A(0, 1) \\), \\( B(-1, 0) \\), and \\( C(1, 0) \\) are on the \\( x \\)-\\( y \\) plane. Denote \\( PD \\), \\( PE \\), and \\( PF \\) as the perpendiculars drawn from a point \\( P \\) (not on the lines \\( AB \\), \\( BC \\), or \\( CA \\)) to \\( AB \\), \\( BC \\), and \\( CA \\) respectively.... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 5,
"tir": 1
} | 6 | 0 | 0 |
omr-ws-8f3fd4fbe3847275b9a3c9ce1662fcc3da04f010cbb47a5a4133631e5fad74f6 | Three points \( A, B, C \) are on a circle \( O \). Given \( AB = 18 \), \( \angle ABC = 59^\circ \), and \( \angle CAB = 3^\circ \). Points \( D \) and \( E \) lie on the tangent to \( O \) at \( A \) such that \( \angle DAC < 90^\circ \), \( DA = 12 \), \( AE = 18 \), and \( DE = 30 \). Let \( BD \cap O = K \), \( CE... | [
"36"
] | [
"Three points \\( A, B, C \\) are on a circle \\( O \\). Given \\( AB = 18 \\), \\( \\angle ABC = 59^\\circ \\), and \\( \\angle CAB = 3^\\circ \\). Points \\( D \\) and \\( E \\) lie on the tangent to \\( O \\) at \\( A \\) such that \\( \\angle DAC < 90^\\circ \\), \\( DA = 12 \\), \\( AE = 18 \\), and \\( DE = 3... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-097656f2bae1426cbc69811a7b1c18c947eb1029ae1630672fff59b316a5697b | Three points \( A, B, C \) lie on a given circle with center \( O \). Let \( P \) be the midpoint of the arc \( AB \) lying on the opposite side of \( AB \) to \( C \). Let \( Q \) and \( R \) be the feet of the perpendiculars from \( P \) to \( AC \) and \( BC \) respectively. Calculate the value of \( \frac{OQ^2 + OR... | [
"2"
] | [
"Three points \\( A, B, C \\) lie on a given circle with center \\( O \\). Let \\( P \\) be the midpoint of the arc \\( AB \\) lying on the opposite side of \\( AB \\) to \\( C \\). Let \\( Q \\) and \\( R \\) be the feet of the perpendiculars from \\( P \\) to \\( AC \\) and \\( BC \\) respectively. Calculate the ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-7290b9834f6dd52017bbc509617bf9181f9c0a9542ff4edf5d9716a79d8e2e6c | Three points \( A, B, C \) lie on a given circle with center \( O \). Let \( P \) be the midpoint of the arc \( AB \) lying on the opposite side of \( AB \) to \( C \). Let \( Q \) and \( R \) be the feet of the perpendiculars from \( P \) to \( AC \) and \( BC \), respectively. Calculate the value of \( \frac{OQ^2 + O... | [
"2"
] | [
"Three points \\( A, B, C \\) lie on a given circle with center \\( O \\). Let \\( P \\) be the midpoint of the arc \\( AB \\) lying on the opposite side of \\( AB \\) to \\( C \\). Let \\( Q \\) and \\( R \\) be the feet of the perpendiculars from \\( P \\) to \\( AC \\) and \\( BC \\), respectively. Calculate the... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-0deb00c559b1becf7f9876a8a6f7992a3eea04c6d3f7b59a3f0536a00b76082b | Three points \( A, B, C \) lie on a line in that order. A circle with center not on \( AC \) passes through points \( A \) and \( C \). The tangents to the circle at points \( A \) and \( C \) meet at a point \( P \). The segment \( PB \) cuts the circle at a point \( Q \). Let \( D \) be the intersection of the bisect... | [
"5"
] | [
"Three points \\( A, B, C \\) lie on a line in that order. A circle with center not on \\( AC \\) passes through points \\( A \\) and \\( C \\). The tangents to the circle at points \\( A \\) and \\( C \\) meet at a point \\( P \\). The segment \\( PB \\) cuts the circle at a point \\( Q \\). Let \\( D \\) be the i... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-dc2d9313de0e4f576fe840eb66d98012c24b14c3281718f31693de23381ea38c | Three points \( A, B, \) and \( C \) are marked on the edge of a circle so that \( \angle BAC = 60^\circ \), \( \angle ABC = 80^\circ \), and \( \angle ACB = 40^\circ \). We randomly choose a point \( X \) inside the circle. The probability that, among the points \( A, B, \) and \( C \), the farthest from \( X \) is \(... | [
"3"
] | [
"Three points \\( A, B, \\) and \\( C \\) are marked on the edge of a circle so that \\( \\angle BAC = 60^\\circ \\), \\( \\angle ABC = 80^\\circ \\), and \\( \\angle ACB = 40^\\circ \\). We randomly choose a point \\( X \\) inside the circle. The probability that, among the points \\( A, B, \\) and \\( C \\), the ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-44a737e8152a8fa64b29e022c3ae9f4ad473a4d9ce8b3fa79e64425375032fae | Three points \( A, B, \) and \( C \) are selected randomly on the circumference of a circle. Find the probability that the points lie on the semicircle. | [
"\\(\\frac{3}{4}\\)"
] | [
"Three points \\( A, B, \\) and \\( C \\) are selected randomly on the circumference of a circle. Find the probability that the points lie on the semicircle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-fa02fe81442cdc179d60d4ede318086558660f119a047e002664fd6935c1a110 | Three points \( P, Q, R \) move on the perimeter of a regular octagon \( A_1A_2 \cdots A_8 \) of side-length 1. (1) Find the maximum value of the area of \( \triangle PQR \). (2) If \( Q \) coincides with the vertex \( A_1 \) of the regular octagon and \( \angle PQR = 90^\circ \), then find the maximum value of the are... | [
"\\dfrac{2 + \\sqrt{2}}{2}"
] | [
"Three points \\( P, Q, R \\) move on the perimeter of a regular octagon \\( A_1A_2 \\cdots A_8 \\) of side-length 1.\n(1) Find the maximum value of the area of \\( \\triangle PQR \\).\n(2) If \\( Q \\) coincides with the vertex \\( A_1 \\) of the regular octagon and \\( \\angle PQR = 90^\\circ \\), then find the m... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-d5f8ba10818000c770aa88338cbaa51e669832fc5357961b8c9886bcb718effb | Three points \( U, P, Q \) lie on the curve parameterized by \( x = \tan \theta \) and \( y = \frac{1}{\cos \theta} \) for \( -\frac{\pi}{2} < \theta < \frac{\pi}{2} \). Let \( u, u-\lambda, u+\lambda \) (with \( u > 0 \)) be the abcissae of the points \( U, P, Q \), respectively. A line through the point \( U \) and p... | [
"\\(\\frac{\\sqrt{3}}{4} - \\frac{1}{2\\sqrt{2}}\\)"
] | [
"Three points \\( U, P, Q \\) lie on the curve parameterized by \\( x = \\tan \\theta \\) and \\( y = \\frac{1}{\\cos \\theta} \\) for \\( -\\frac{\\pi}{2} < \\theta < \\frac{\\pi}{2} \\). Let \\( u, u-\\lambda, u+\\lambda \\) (with \\( u > 0 \\)) be the abcissae of the points \\( U, P, Q \\), respectively. A line ... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-d94130bc0d9c500230fab44a3993956e5655b9fc050847348ad00a8cdf6882e9 | Three points \(X\), \(Y\), and \(Z\) are taken on the sides \(BC\), \(CA\), and \(AB\) respectively of triangle \(ABC\) such that \(AX\), \(BY\), and \(CZ\) are concurrent. Let \(YZ\) meet \(BC\) at \(X'\), \(ZX\) meet \(AC\) at \(Y'\), and \(XY\) meet \(BA\) at \(Z'\). Calculate the product of the ratios \(\frac{BX'}{... | [
"1"
] | [
"Three points \\(X\\), \\(Y\\), and \\(Z\\) are taken on the sides \\(BC\\), \\(CA\\), and \\(AB\\) respectively of triangle \\(ABC\\) such that \\(AX\\), \\(BY\\), and \\(CZ\\) are concurrent. Let \\(YZ\\) meet \\(BC\\) at \\(X'\\), \\(ZX\\) meet \\(AC\\) at \\(Y'\\), and \\(XY\\) meet \\(BA\\) at \\(Z'\\). Calcul... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a54e5f3a55108a53dc1da06a99cb07c768f5864d00b423046ab127a3a7f0f8eb | Three points are chosen at random along the outline of a square. What is the probability that the triangle formed by these points will contain the center of the square? | [
"\\dfrac{1}{2}"
] | [
"Three points are chosen at random along the outline of a square. What is the probability that the triangle formed by these points will contain the center of the square?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-8f10406080b954813a4031223693f90de25cf1d996244c6f53dcd98b62cf2546 | Three points are chosen at random from a 3 x 4 grid of points. What is the probability that they are collinear? | [
"\\(\\frac{1}{11}\\)"
] | [
"Three points are chosen at random from a 3 x 4 grid of points. What is the probability that they are collinear?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 41,
"genselect": 7,
"tir": 30
} | 78 | 0 | 0 |
omr-ws-b954fc3c5e2948f243658f25282cf9ee417872d33d9e35329873b56fa69d1255 | Three points are chosen at random from the circumference of a circle to form a triangle. Three more points are chosen at random from the same circumference to form a second triangle. Find the probability that the area of the overlap of the two triangles is at least $\frac{1}{4}$ the area of the circle. | [
"\\dfrac{1}{4}"
] | [
"Three points are chosen at random from the circumference of a circle to form a triangle. Three more points are chosen at random from the same circumference to form a second triangle. Find the probability that the area of the overlap of the two triangles is at least $\\frac{1}{4}$ the area of the circle."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cb5273e56027840a9eb61555bf689c2daae40296a57b33eb0f368806563910ba | Three points are chosen at random in a plane and connected to form a triangle. What is the probability that the triangle contains at least one angle that measures between $80$ and $100$ degrees? | [
"\\dfrac{8}{27}"
] | [
"Three points are chosen at random in a plane and connected to form a triangle. What is the probability that the triangle contains at least one angle that measures between $80$ and $100$ degrees?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 7
} | 15 | 0 | 0 |
omr-ws-3d5d32e25f92ab2942f54311ed502dcdadbd3223c492c9c0bbffd60d9b1ffe4f | Three points are chosen at random on the circumference of the unit circle. What is the expected value of the area of the triangle formed when these points are connected? | [
"\\(\\frac{3}{2\\pi}\\)"
] | [
"Three points are chosen at random on the circumference of the unit circle. What is the expected value of the area of the triangle formed when these points are connected?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 18
} | 32 | 0 | 0 |
omr-ws-8c2322c270fd4c540c4d3d41ae2ddbb1428bb26352ede6c75473fd4144c492c6 | Three points are chosen at random, uniformly and independently, from within a unit square. Let \( A \) be the area of the triangle with vertices at those points. For \( \epsilon = 0.01 \), calculate the maximum possible value of \( \text{Prob}(A < \epsilon) \). Express your answer as a decimal rounded to four decimal p... | [
"0.0055"
] | [
"Three points are chosen at random, uniformly and independently, from within a unit square. Let \\( A \\) be the area of the triangle with vertices at those points. For \\( \\epsilon = 0.01 \\), calculate the maximum possible value of \\( \\text{Prob}(A < \\epsilon) \\). Express your answer as a decimal rounded to ... | [
"aops_c7_college_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 4
} | 5 | 0 | 0 |
omr-ws-07091e04e23777bf0ee393b6c283cb14903e3234c2f2d1b776ae4fb281f53338 | Three points are chosen independently and uniformly at random in the interval $[0,1]$. Determine the probability that the minimum distance between any two of these points is greater than a positive number $m$. | [
"$(1-2m)^3$ for $m \\leq \\frac{1}{2}$"
] | [
"Three points are chosen independently and uniformly at random in the interval $[0,1]$. Determine the probability that the minimum distance between any two of these points is greater than a positive number $m$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 20
} | 41 | 0 | 0 |
omr-ws-dd844d789bdacdb4d320a1e6313f3f86b24d2b9e30f9cc078edfaffce5b135fe | Three points are chosen independently and uniformly at random inside the unit circle. What is the expected area of the triangle formed by these three points? | [
"\\dfrac{35}{48\\pi}"
] | [
"Three points are chosen independently and uniformly at random inside the unit circle. What is the expected area of the triangle formed by these three points?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 8
} | 19 | 0 | 0 |
omr-ws-77cf2889fd671316cd3b126665c9ea59600a0a0183ccb9b92ea902fddafa9cd3 | Three points are chosen on the circumference of a circle. What is the probability that all three line segments connecting these points are less than the radius of the circle? | [
"\\(\\frac{1}{12}\\)"
] | [
"Three points are chosen on the circumference of a circle. What is the probability that all three line segments connecting these points are less than the radius of the circle?"
] | [
"aops_c5_contests_amp_programs"
] | [
"genselect"
] | {
"genselect": 5
} | 5 | 0 | 0 |
omr-ws-b97aaf3a5b55f70c8d072104c221a55bb8d9d6566b088c6d61dbaa5b5de9ae0c | Three points are chosen on the graph of the equation $x^3 + 39xy + y^3 - 2197 = 0$ such that the points form an equilateral triangle. The area of that equilateral triangle, in simplest radical form, can be expressed as $\frac{a\sqrt{b}}{c}$. Also, $a$ and $c$ are coprime. What is the value of $a + b + c$? | [
"512"
] | [
"Three points are chosen on the graph of the equation $x^3 + 39xy + y^3 - 2197 = 0$ such that the points form an equilateral triangle. The area of that equilateral triangle, in simplest radical form, can be expressed as $\\frac{a\\sqrt{b}}{c}$. Also, $a$ and $c$ are coprime. What is the value of $a + b + c$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 6
} | 27 | 0 | 0 |
omr-ws-2c02e3f81d55141c5a0336e6945942b87102f49a104e1cbc82748740c002519f | Three points are chosen randomly and independently on a circle. What is the probability that all three pairwise distances between the points are less than the radius of the circle? | [
"1/12",
"1/12\"\n\nThe answer to the problem we are looking at is explicitly stated in the first post.",
"\\(\\frac{1}{12}\\)",
"\\dfrac{1}{12}"
] | [
"Three points are chosen randomly and independently on a circle. What is the probability that all three pairwise distances between the points are less than the radius of the circle?"
] | [
"aops_c5_contests_amp_programs"
] | [
"genselect"
] | {
"genselect": 21
} | 21 | 0 | 0 |
omr-ws-5b8a675fd58ed718e20cc4db46ed93f1f28186c1bdda5724f7df27ae00459cb9 | Three points are chosen randomly from the interval [0,1]. Find the probability that the longest distance between any two of these points is less than or equal to 1/2. | [
"\\dfrac{1}{2}"
] | [
"Three points are chosen randomly from the interval [0,1]. Find the probability that the longest distance between any two of these points is less than or equal to 1/2."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8a9f166143f76f1492493a29ae8e7a4a77fc082a81e723b222dc3c66978f6bff | Three points are chosen randomly in a circle. What is the probability that the circumcircle of the triangle formed by these points is entirely inscribed within the original circle? | [
"\\( \\ln(2) \\)"
] | [
"Three points are chosen randomly in a circle. What is the probability that the circumcircle of the triangle formed by these points is entirely inscribed within the original circle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-873f8e47d026dac032384f092f5cacf9ebf54fbaf2c0ddd634f9585fde6b6aeb | Three points are chosen uniformly at random on a circle. What is the probability that no two of these points form an obtuse triangle with the circle's center? | [
"\\frac{3}{16}"
] | [
"Three points are chosen uniformly at random on a circle. What is the probability that no two of these points form an obtuse triangle with the circle's center?"
] | [
"MATH_training_set"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 6
} | 20 | 0 | 0 |
omr-ws-10cf058caf1a02f16c341cfa09340a15a69bb8ce920c86e5e4378ac8f6dcecfe | Three points are located at the vertices of an equilateral triangle with side length $a$. They all start moving simultaneously with constant velocity $v$, with the first point heading continually for the second, the second for the third, and the third for the first. How long will it take for the points to converge? | [
"\\(\\frac{a}{v}\\)"
] | [
"Three points are located at the vertices of an equilateral triangle with side length $a$. They all start moving simultaneously with constant velocity $v$, with the first point heading continually for the second, the second for the third, and the third for the first. How long will it take for the points to converge... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b3c0f5484a0f314df44e2529f9732fbd19661d4ef4a9c8ca57c1459a9bba942e | Three points are picked at random along the edges of a unit square. Assuming that the three points all lie on different edges, find the probability that the area of the triangle defined by the three points is larger than \(\frac{1}{4}\). | [
"\\dfrac{1}{2}"
] | [
"Three points are picked at random along the edges of a unit square. Assuming that the three points all lie on different edges, find the probability that the area of the triangle defined by the three points is larger than \\(\\frac{1}{4}\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 4
} | 13 | 0 | 0 |
omr-ws-5bb21b860775d8164cffaa18f06084f9d61bfbf9c779d614255ec721809e3549 | Three points are picked on a circle of radius 1. What is the probability that the sum of the pairwise distances between these points is less than 2? | [
"0"
] | [
"Three points are picked on a circle of radius 1. What is the probability that the sum of the pairwise distances between these points is less than 2?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-4ee62cfdc614885c33449588b257dd6203b2d65b79f6fba9589b71b2ebdc8e4d | Three points are picked on the perimeter of each of three concentric squares with sides of length $1$, $2$, and $3$, respectively, such that no two points are on the perimeter of the same square. What is the maximal area of the triangle formed by these points? | [
"$\\frac{9}{4}$"
] | [
"Three points are picked on the perimeter of each of three concentric squares with sides of length $1$, $2$, and $3$, respectively, such that no two points are on the perimeter of the same square. What is the maximal area of the triangle formed by these points?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 8
} | 23 | 0 | 0 |
omr-ws-52f48ed2db05229a96288d17139c01d2e290f439c1f85455f3a0326438668a44 | Three points are picked on the perimeter of each of three concentric squares with sides of length $1$, $2$, and $3$, respectively, such that no two points are on the perimeter of the same square. What is the maximal perimeter of the triangle formed by these points? | [
"\\(\\frac{\\sqrt{10}+5\\sqrt{2}}{2}+\\sqrt{5}\\)"
] | [
"Three points are picked on the perimeter of each of three concentric squares with sides of length $1$, $2$, and $3$, respectively, such that no two points are on the perimeter of the same square. What is the maximal perimeter of the triangle formed by these points?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 8,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-53107a4af654077bf22197910885596e276570f2d2d6614341482d3807c4dbf5 | Three points are picked on the perimeter of each of three concentric squares with sides of length $1$, $2$, and $3$, respectively, such that no two points are on the perimeter of the same square. What is the minimum perimeter of the triangle formed by these points? | [
"2"
] | [
"Three points are picked on the perimeter of each of three concentric squares with sides of length $1$, $2$, and $3$, respectively, such that no two points are on the perimeter of the same square. What is the minimum perimeter of the triangle formed by these points?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 5
} | 16 | 0 | 0 |
omr-ws-2aa4e6afb81bc1237fc827e5b5462c9deb46e27a77ea827534d2d436f2e65a5e | Three points are placed randomly and independently on a circle. Find the probability that all three pairwise distances between the points are less than the radius of the circle. | [
"\\(\\frac{1}{12}\\)"
] | [
"Three points are placed randomly and independently on a circle. Find the probability that all three pairwise distances between the points are less than the radius of the circle."
] | [
"aops_c5_contests_amp_programs"
] | [
"genselect"
] | {
"genselect": 8
} | 8 | 0 | 0 |
omr-ws-93d0c194215bc0cb642e7a786d7360d3f4f5bb9a2ac726a0e9960636c6109fc0 | Three points are randomly chosen along the perimeter of a unit square so that at least two of them are on different sides. A fourth point is randomly chosen independently of the first three points within the interior of the square. The points are randomly split into two pairs and line segments are drawn connecting the ... | [
"77"
] | [
"Three points are randomly chosen along the perimeter of a unit square so that at least two of them are on different sides. A fourth point is randomly chosen independently of the first three points within the interior of the square. The points are randomly split into two pairs and line segments are drawn connecting... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 4
} | 10 | 0 | 0 |
omr-ws-e1b1291057581fba720653cfddcea055158ab1a407de920de2ef234e02ae0378 | Three points are randomly chosen inside a circle. What is the probability that there exists a line passing through the center of the circle such that all three points are on one side of the line? | [
"\\(\\frac{3}{4}\\)"
] | [
"Three points are randomly chosen inside a circle. What is the probability that there exists a line passing through the center of the circle such that all three points are on one side of the line?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-f35542a631ac2bd5b7578eff3ce555686accb4a5d7dd4eacd74959957bed58fd | Three points are randomly located on a circle. What is the probability that the shortest distance between each pair of points is less than or equal to the radius of the circle? | [
"\\(\\frac{1}{12}\\)",
"\\dfrac{1}{12}"
] | [
"Three points are randomly located on a circle. What is the probability that the shortest distance between each pair of points is less than or equal to the radius of the circle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 20,
"genselect": 15,
"tir": 14
} | 49 | 0 | 0 |
omr-ws-85e486089daeae7d0b4df2df2bcfca979c34a4bfeb3045a1eba462b3ff691453 | Three points are randomly selected on an ellipse with a major axis length of 9 and a minor axis length of 5 to form a triangle. What is the probability that the perimeter of the triangle is greater than 10? | [
"1"
] | [
"Three points are randomly selected on an ellipse with a major axis length of 9 and a minor axis length of 5 to form a triangle. What is the probability that the perimeter of the triangle is greater than 10?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-388f4294644a8ed0c4119c6d5f0bc534cdcf1ac21e47d1bf4006754b71940f9f | Three points are randomly selected within a square of side length 12. What is the probability that the three points all lie within a circle of radius 3? The circle is not predefined. | [
"\\dfrac{\\pi^3}{256}"
] | [
"Three points are randomly selected within a square of side length 12. What is the probability that the three points all lie within a circle of radius 3? The circle is not predefined."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 6
} | 11 | 0 | 0 |
omr-ws-10ac503284485ba58824c9e96f6baa4de18022d8429390daa47a0dacb601cea5 | Three points are selected at random from the circumference of a circle. If the probability that these three points determine an acute triangle can be expressed as $\dfrac{m}{n}$, find $m+n$. | [
"005"
] | [
"Three points are selected at random from the circumference of a circle. If the probability that these three points determine an acute triangle can be expressed as $\\dfrac{m}{n}$, find $m+n$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-499d579bbe14550b32b0c69030a22748ef4418fe55507babbe37cf56bd44fae9 | Three points are selected inside a square ABCD with side length 12. What is the probability that all three points lie inside a circle of radius 3? | [
"\\left( \\dfrac{\\pi}{16} \\right)^3"
] | [
"Three points are selected inside a square ABCD with side length 12. What is the probability that all three points lie inside a circle of radius 3?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0d82c7526f52779a8061499e7dd4ce90d3959866ffef94ea5d0aa8d997188fd7 | Three points are selected inside a square of side length 12. What is the probability that all three points lie within a circle of radius 3? (The circle can be centered anywhere within the square.) | [
"\\left( \\dfrac{\\pi}{16} \\right)^3"
] | [
"Three points are selected inside a square of side length 12. What is the probability that all three points lie within a circle of radius 3? (The circle can be centered anywhere within the square.)"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 4
} | 17 | 0 | 0 |
omr-ws-5271b69e3870344d01df7bb6b6f264c8dd1da0a1fc0f74d874a0475c9d5403e3 | Three points are selected randomly on the circumference of a circle. What is the probability that the triangle formed by these three points contains the center of the circle? | [
"\\dfrac{1}{4}"
] | [
"Three points are selected randomly on the circumference of a circle. What is the probability that the triangle formed by these three points contains the center of the circle?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 32
} | 32 | 0 | 0 |
omr-ws-10335c6885321af66136280426a7687797ca7f38e8dcdb75ab56f716c3800ae5 | Three points are taken at random on the surface of a given triangle. Find the probability that the triangle formed by joining these points is acute. | [
"\\dfrac{1}{4}"
] | [
"Three points are taken at random on the surface of a given triangle. Find the probability that the triangle formed by joining these points is acute."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-0afec80a2b83271bfe74042e91803dfa6be767daf63ab2d6f0cf13d067a28c14 | Three points are uniformly and independently chosen inside a given circle. What is the probability that their circumcircle lies entirely within the given circle? | [
"\\dfrac{1}{4}"
] | [
"Three points are uniformly and independently chosen inside a given circle. What is the probability that their circumcircle lies entirely within the given circle?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-9aa20d699db97573777c5425b8e61bbcbf6efa5b21a99a9763805ef68706f425 | Three points move in a circle with constant velocity so that the sum of the squares of all three distances between them remains constant. If not all points move at the same speed, find the number of distinct possible configurations of these points where two points are always diametrically opposite to each other. | [
"3"
] | [
"Three points move in a circle with constant velocity so that the sum of the squares of all three distances between them remains constant. If not all points move at the same speed, find the number of distinct possible configurations of these points where two points are always diametrically opposite to each other."
... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0bfe7db0d33889a087a8b9fc5f4a5b2f122beaec510c34f54b7f3937c93c8d9d | Three points move in a circle with constant velocity. If the sum of the squares of all three pairwise distances between them remains constant, and the circumference of the circle is $2\pi$, find the ratio of the speed of the fastest point to the speed of the slowest point. | [
"2"
] | [
"Three points move in a circle with constant velocity. If the sum of the squares of all three pairwise distances between them remains constant, and the circumference of the circle is $2\\pi$, find the ratio of the speed of the fastest point to the speed of the slowest point."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 6
} | 11 | 0 | 0 |
omr-ws-746547adb2bcd3e391bc0aed508c4fc95992a108fbaadc339df4d7a432826ccd | Three points move in a straight line at constant speeds $v_1$, $v_2$, and $v_3$. If the sum of the squares of all three pairwise distances between them remains constant, find the value of $\frac{v_1^2 + v_2^2 + v_3^2}{v_1v_2 + v_2v_3 + v_3v_1}$. | [
"1"
] | [
"Three points move in a straight line at constant speeds $v_1$, $v_2$, and $v_3$. If the sum of the squares of all three pairwise distances between them remains constant, find the value of $\\frac{v_1^2 + v_2^2 + v_3^2}{v_1v_2 + v_2v_3 + v_3v_1}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-af5cc4d3789c0622d2406090c1ecf7a5e37b5234d99fc3db1d45055d5a6f1987 | Three points were marked on the border of the square, connected by segments, and cut through them. As a result, the square broke up into 4 triangles. What is the largest number of these triangles that can be equal? | [
"3"
] | [
"Three points were marked on the border of the square, connected by segments, and cut through them. As a result, the square broke up into 4 triangles. What is the largest number of these triangles that can be equal?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4ba8785cd8aa4db2298cfebd96bcaf93b27895a2c9f083120579807bbfa3778c | Three points with $y$-coordinates $1$, $2$, and $3$ respectively form an isosceles triangle with a $120^\circ$ angle. What are all possible areas of this triangle? | [
"\\dfrac{7\\sqrt{3}}{3}"
] | [
"Three points with $y$-coordinates $1$, $2$, and $3$ respectively form an isosceles triangle with a $120^\\circ$ angle. What are all possible areas of this triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d9e2fbdc3ecc727d27f0925a53dd989cebf360c1fd6c32204de32e05f13d40ad | Three points, $A$, $B$, and $C$, are located so that $\angle ABC=60^\circ$. A car starts from point $A$ the moment a train leaves point $B$. The car travels towards $B$ at 80 km/h and the train travels towards point $C$ at 50 km/h. How long after they started will the car and the train be closest if $| AB | = 200$ km? | [
"$1\\frac{27}{43}$ h"
] | [
"Three points, $A$, $B$, and $C$, are located so that $\\angle ABC=60^\\circ$. A car starts from point $A$ the moment a train leaves point $B$. The car travels towards $B$ at 80 km/h and the train travels towards point $C$ at 50 km/h. How long after they started will the car and the train be closest if $| AB | = 20... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-73d54f27ac9b637c9755e33225844707b5579ff14c66b03f47a338ab7e938b68 | Three points, (9,12), (10,5), and (0,9), lie on Circle P. An element \( k \) is selected at random from the set \(\{-11, -7, 2\}\) and an element \( w \) is selected at random from the set \(\{-1, 6, 7, 10\}\). Find the probability that the point \((k, w)\) does not lie in the interior of Circle P. Express your answer ... | [
"1"
] | [
"Three points, (9,12), (10,5), and (0,9), lie on Circle P. An element \\( k \\) is selected at random from the set \\(\\{-11, -7, 2\\}\\) and an element \\( w \\) is selected at random from the set \\(\\{-1, 6, 7, 10\\}\\). Find the probability that the point \\((k, w)\\) does not lie in the interior of Circle P. E... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f6b6afa06c9e94259826e2cdeb4fadd56065844b23471321d80c7094f2907f6b | Three points, namely the circumcenter, incenter, and centroid, are non-collinear. The triangle formed by joining these points is similar to the original triangle. Find the angles of the original triangle. | [
"30^\\circ"
] | [
"Three points, namely the circumcenter, incenter, and centroid, are non-collinear. The triangle formed by joining these points is similar to the original triangle. Find the angles of the original triangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-76c98886df10f42e30faba1fd54f4b41272ffd9424a0bbfef6d7b5f6e1b759e9 | Three positive factors of $20^{10}$ are chosen at random. Find the probability that their product is $20^{10}$. | [
"\\dfrac{2}{1617}"
] | [
"Three positive factors of $20^{10}$ are chosen at random. Find the probability that their product is $20^{10}$."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-6709c5325fa6c670c415299e2c78aeb7d1a3646f4cd7695bfbfdefffd76f6ed6 | Three positive integers $a$, $b$, and $c$ satisfy $a\cdot b\cdot c=8!$ and $a<b<c$. What is the smallest possible value of $c-a$? | [
"4"
] | [
"Three positive integers $a$, $b$, and $c$ satisfy $a\\cdot b\\cdot c=8!$ and $a<b<c$. What is the smallest possible value of $c-a$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ab816838c0a80a386706fe23535f2a9a0df39908fffed1e198c9f6a914ecf07d | Three positive integers $a, b,$ and $c$ are chosen so that the graph of the function $f(x) = \sqrt{ax - b} - c$ has exactly one point of intersection with its inverse function. What is the smallest possible value of $a + b + c$? | [
"10"
] | [
"Three positive integers $a, b,$ and $c$ are chosen so that the graph of the function $f(x) = \\sqrt{ax - b} - c$ has exactly one point of intersection with its inverse function. What is the smallest possible value of $a + b + c$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-2194b9fb0c8d7df8203ac8b01db63e693c9c10c8cda7ec2052d284fade9e6cd7 | Three positive integers $a,$ $b,$ and $x$ form an O'Hara triple $(a,b,x)$ if $\sqrt{a}+\sqrt{b}=x.$ For example, $(1,4,3)$ is an O'Hara triple because $\sqrt{1}+\sqrt{4}=3.$ If $(36,25,x)$ is an O'Hara triple, determine the value of $x.$ | [
"11"
] | [
"Three positive integers $a,$ $b,$ and $x$ form an O'Hara triple $(a,b,x)$ if $\\sqrt{a}+\\sqrt{b}=x.$ For example, $(1,4,3)$ is an O'Hara triple because $\\sqrt{1}+\\sqrt{4}=3.$\n\nIf $(36,25,x)$ is an O'Hara triple, determine the value of $x.$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-16979038fb25a5b197b756eb034d6e109a0c6cf06849d72a03dd1abb75599c20 | Three positive integers \(a, b, c\) are written on the board. Every minute, Elbek replaces the numbers \(a, b, c\) with \(a + \gcd(b, c)\), \(b + \gcd(a, c)\), and \(c + \gcd(a, b)\). After 5 minutes, let \(d\) be the greatest common divisor of the three numbers on the board. Find the smallest possible value of \(d\). | [
"2"
] | [
"Three positive integers \\(a, b, c\\) are written on the board. Every minute, Elbek replaces the numbers \\(a, b, c\\) with \\(a + \\gcd(b, c)\\), \\(b + \\gcd(a, c)\\), and \\(c + \\gcd(a, b)\\). After 5 minutes, let \\(d\\) be the greatest common divisor of the three numbers on the board. Find the smallest possi... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-60cdea06f4927d1120b3dfadc0d6cfafc58e1d4a829094c2f88663bbbf66f0b6 | Three positive integers \(a, b, c\) satisfy the equations \(ac + b + c = bc + a + 66\) and \(a + b + c = 32\). Find the value of \(abc\). | [
"798"
] | [
"Three positive integers \\(a, b, c\\) satisfy the equations \\(ac + b + c = bc + a + 66\\) and \\(a + b + c = 32\\). Find the value of \\(abc\\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 2,
"tir": 2
} | 13 | 0 | 0 |
omr-ws-2e5244117d7f7c8d53c104bdaf1493e060dab81e4333ef91da67f35ad07af89e | Three positive integers \(a, b, c\) with \(a > c\) satisfy the equations: \[ ac + b + c = bc + a + 66 \] \[ a + b + c = 32 \] Find the value of \(a\). | [
"19"
] | [
"Three positive integers \\(a, b, c\\) with \\(a > c\\) satisfy the equations:\n\\[ ac + b + c = bc + a + 66 \\]\n\\[ a + b + c = 32 \\]\nFind the value of \\(a\\)."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-996cd824c5630f4fb7b62bba6e23e2f059f2a603b502a20852b7d60e8b1fa1f0 | Three positive integers \(a\), \(b\), and \(c\) are such that \(0 < a < b < c\) and their differences \(b - a\), \(c - a\), and \(c - b\) are squares of some integer. What is the smallest value that \(c\) can take? | [
"26"
] | [
"Three positive integers \\(a\\), \\(b\\), and \\(c\\) are such that \\(0 < a < b < c\\) and their differences \\(b - a\\), \\(c - a\\), and \\(c - b\\) are squares of some integer. What is the smallest value that \\(c\\) can take?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-0cb7e83b713a55019ad512fbbaff93a4231d4de39d3b3a20cbd1a700155a9a0e | Three positive integers \(a\), \(b\), and \(c\) satisfy the equation \[4^a \cdot 5^b \cdot 6^c = 8^8 \cdot 9^9 \cdot 10^{10}.\] Determine the sum \(a + b + c\). | [
"36"
] | [
"Three positive integers \\(a\\), \\(b\\), and \\(c\\) satisfy the equation\n\n\\[4^a \\cdot 5^b \\cdot 6^c = 8^8 \\cdot 9^9 \\cdot 10^{10}.\\]\n\nDetermine the sum \\(a + b + c\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-90cc86114ba87bc1f4d3b602ebf73dddd81cd7bdd99eb178ab7e16f276906877 | Three positive integers are each greater than $1$, have a product of $ 27000 $, and are pairwise relatively prime. What is their sum? | [
"160"
] | [
"Three positive integers are each greater than $1$, have a product of $ 27000 $, and are pairwise relatively prime. What is their sum?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f150a8f4887d2b946cb3125fbf1e74b343f1e94a62457a237c246d678f6f1d1d | Three positive integers between $1$ and $100$, inclusive, are randomly chosen. What is the probability that their sum will be odd? | [
"\\(\\frac{1}{2}\\)"
] | [
"Three positive integers between $1$ and $100$, inclusive, are randomly chosen. What is the probability that their sum will be odd?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 32
} | 47 | 0 | 0 |
omr-ws-f32e00a270320b3c020d713f624784449b08b3eb159db767fe66a5b414f74c06 | Three positive integers form an arithmetic progression. If the smallest number is increased by one, the arithmetic progression becomes a geometric progression. If the largest number is increased by two, the arithmetic progression also becomes a geometric progression. Compute the smallest number of the original arithmet... | [
"8"
] | [
"Three positive integers form an arithmetic progression. If the smallest number is increased by one, the arithmetic progression becomes a geometric progression. If the largest number is increased by two, the arithmetic progression also becomes a geometric progression. Compute the smallest number of the original ari... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-dfca785117e863b649ba12bf4789a1506f1cbac1c25a72e628c04802eb925de2 | Three positive integers have a product of 2450 and their sum is the age of mathematician $M$. Initially, $M$ cannot determine the three integers. After being told that each of the three integers is smaller than the age of logician $L$, $M$ can determine the integers. Find the three integers and the ages of $M$ and $L$.... | [
"The three integers are \\(7, 10, 35\\). The age of \\(M\\) is \\(52\\) and the age of \\(L\\) is \\(50\\)."
] | [
"Three positive integers have a product of 2450 and their sum is the age of mathematician $M$. Initially, $M$ cannot determine the three integers. After being told that each of the three integers is smaller than the age of logician $L$, $M$ can determine the integers. Find the three integers and the ages of $M$ and... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bd06fb299ecdf24e39a67260bcadc085c12c9381a7070b5f0158281042e83296 | Three positive integers have a sum of 72 and are in the ratio 1:3:4. What is the least of these three integers? | [
"9"
] | [
"Three positive integers have a sum of 72 and are in the ratio 1:3:4. What is the least of these three integers?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4e344a397f8321104f5194dc74d5bd218fcdfed13b78c9f72bb1a84b42ba3b4f | Three positive integers have an arithmetic mean of 26 and median of 27. If the median is 5 less than the largest number, what is the smallest number? | [
"19"
] | [
"Three positive integers have an arithmetic mean of 26 and median of 27. If the median is 5 less than the largest number, what is the smallest number?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e4f0ed9e63667df758eb7ed3b63602629052f33494bd5bde89b83de7ab5ac7c4 | Three positive integers sum to $16$. What is the least possible value of the sum of their squares? | [
"86"
] | [
"Three positive integers sum to $16$. What is the least possible value of the sum of their squares?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 3
} | 18 | 0 | 0 |
omr-ws-2b44f86bb3da5aeb1a323c199e21df0534688c223849682b361eeaafd9d90e16 | Three positive numbers $a$, $b$, and $c$ are such that the sum of any one of them with the sum of squares of the remaining two numbers is the same. If $a \leq b \leq c$, find the value of $\frac{c}{a}$. | [
"2"
] | [
"Three positive numbers $a$, $b$, and $c$ are such that the sum of any one of them with the sum of squares of the remaining two numbers is the same. If $a \\leq b \\leq c$, find the value of $\\frac{c}{a}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 8
} | 11 | 0 | 0 |
omr-ws-690a68dcf68caffc8cc9ff5730264749cbfad38d716daa516858243eee65fa0e | Three positive numbers \(a, b,\) and \(c\) form an arithmetic progression. If \(a\) is increased by 1, the sequence becomes a geometric progression. If \(c\) is increased by 2, the sequence also becomes a geometric progression. What is the value of \(b\)? | [
"12"
] | [
"Three positive numbers \\(a, b,\\) and \\(c\\) form an arithmetic progression. If \\(a\\) is increased by 1, the sequence becomes a geometric progression. If \\(c\\) is increased by 2, the sequence also becomes a geometric progression. What is the value of \\(b\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1d837a9c8746d7b1101507f5617a2745c568e2bd4ac12e7bd17483ae5cb5e6e7 | Three positive rational numbers \(a, b, c\) are written on the board. Every minute, the numbers \(a, b, c\) are replaced with \[ a + \frac{1}{bc}, \quad b + \frac{1}{ca}, \quad c + \frac{1}{ab} \] respectively. After 10 minutes, let \(S\) be the sum of the three numbers on the board. If the initial numbers are \(a = \f... | [
"no_valid_answer_found"
] | [
"Three positive rational numbers \\(a, b, c\\) are written on the board. Every minute, the numbers \\(a, b, c\\) are replaced with \n\\[ a + \\frac{1}{bc}, \\quad b + \\frac{1}{ca}, \\quad c + \\frac{1}{ab} \\]\nrespectively. After 10 minutes, let \\(S\\) be the sum of the three numbers on the board. If the initial... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cb6ba07b4d434364d9e37f87a10fbbe160bee570ccbf8e4e41f5413e15bd5079 | Three positive real numbers \( x, y, z \) satisfy the equations: \[ x^2 + y^2 = 3^2 \] \[ y^2 + yz + z^2 = 4^2 \] \[ x^2 + \sqrt{3}xz + z^2 = 5^2 \] Find the value of \( 2xy + xz + \sqrt{3}yz \). | [
"24"
] | [
"Three positive real numbers \\( x, y, z \\) satisfy the equations:\n\\[ x^2 + y^2 = 3^2 \\]\n\\[ y^2 + yz + z^2 = 4^2 \\]\n\\[ x^2 + \\sqrt{3}xz + z^2 = 5^2 \\]\nFind the value of \\( 2xy + xz + \\sqrt{3}yz \\)."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 12,
"tir": 21
} | 33 | 0 | 0 |
omr-ws-eef764cadf4dec4abe4b3476dfbcfdd4473b0e864b4c86ffb6515312535b0b39 | Three positive real numbers \(a, b, c\) satisfy \(a^b = 343\), \(b^c = 10\), and \(a^c = 7\). Find \(b^b\). | [
"1000"
] | [
"Three positive real numbers \\(a, b, c\\) satisfy \\(a^b = 343\\), \\(b^c = 10\\), and \\(a^c = 7\\). Find \\(b^b\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-da79f0954b49330c15310c30d405f3627dd52de064c1a59f9a6455f5642f4e88 | Three positive reals \( x, y, z \) satisfy the equations: \[ x^2 + y^2 = 3^2, \] \[ y^2 + yz + z^2 = 4^2, \] \[ x^2 + \sqrt{3}xz + z^2 = 5^2. \] Find the value of \( 2xy + xz + \sqrt{3}yz \). | [
"24"
] | [
"Three positive reals \\( x, y, z \\) satisfy the equations:\n\\[ x^2 + y^2 = 3^2, \\]\n\\[ y^2 + yz + z^2 = 4^2, \\]\n\\[ x^2 + \\sqrt{3}xz + z^2 = 5^2. \\]\nFind the value of \\( 2xy + xz + \\sqrt{3}yz \\)."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 10,
"tir": 25
} | 35 | 0 | 0 |
omr-ws-98355c264e9474a6471ed14222f5b31425aa4db787adeef06fb1ac0458312ba6 | Three prisoners, Mark, Luke, and Thomas, are in a cell. Two of them will be shot, but each prisoner does not know if they will be shot or not. Mark asks the guard if either Luke or Thomas will be shot, and the guard says Luke will be shot. Let $p$ be the probability that Mark will be shot given this information. Calcul... | [
"2"
] | [
"Three prisoners, Mark, Luke, and Thomas, are in a cell. Two of them will be shot, but each prisoner does not know if they will be shot or not. Mark asks the guard if either Luke or Thomas will be shot, and the guard says Luke will be shot. Let $p$ be the probability that Mark will be shot given this information. C... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e679ba4928f4ba24707585d5ba8d478a7d22d2e051f47345a1cb2b2e60ff1472 | Three professors and ten students are to sit in a row of 13 chairs. The professors take their seats first, before the students enter the room. If the professors agree among themselves that each will sit between two students, how many ways can they choose their seats? | [
"504"
] | [
"Three professors and ten students are to sit in a row of 13 chairs. The professors take their seats first, before the students enter the room. If the professors agree among themselves that each will sit between two students, how many ways can they choose their seats?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 45,
"genselect": 8,
"tir": 3
} | 56 | 0 | 0 |
omr-ws-b1f725c18b6b7fbe29328d3e580b3510f18b0bed89b96a624591ea9859b7e992 | Three pumps simultaneously began pumping out the water each from its own reservoir. When the third pump emptied an $\alpha $th of the volume of its reservoir $(\alpha < 1/2)$, the second pump had to pump out as much water as the first pump had drained; when the third pump had to pump out $(1 -\alpha )$th of the volume,... | [
"Assume that $T_i$ $(i = 1, 2, 3)$ is the time needed for the $i$th pump to pump out the water from its reservoir. Then $T_1>T_3>T_2$, and $T_1:T_3:T_2=\\left(1+\\sqrt \\alpha \\right):1:\\left[\\alpha \\left(1+\\sqrt \\alpha \\right)\\right]$, if $\\alpha <\\frac{3-\\sqrt 5}{2}$, $T_1LT_3:T_2=\\frac{1-\\alpha }{\\... | [
"Three pumps simultaneously began pumping out the water each from its own reservoir. When the third pump emptied an $\\alpha $th of the volume of its reservoir $(\\alpha < 1/2)$, the second pump had to pump out as much water as the first pump had drained; when the third pump had to pump out $(1 -\\alpha )$th of th... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e3ff87edd276c35617192f67c1edcfa6c8ff7ddfaa9ab1702c0616d137fd5c96 | Three quadratic prisms, each with a base that is a regular 12-gon of the same size, are positioned such that their projections from above or from the sides form a plus sign. How many faces does the polyhedron formed by the intersection of all three prisms have? | [
"144"
] | [
"Three quadratic prisms, each with a base that is a regular 12-gon of the same size, are positioned such that their projections from above or from the sides form a plus sign. How many faces does the polyhedron formed by the intersection of all three prisms have?"
] | [
"aops_c6_high_school_olympiads",
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-96befc7b48c213710fe3bd4b707c6d800c8ee25717dab76a90cc28b476d8d471 | Three racers, $A$, $B$, and $C$, start simultaneously but from different points and race at constant speeds in the same direction along a circular highway. At the moment of the start racer $B$ was ahead of racer $A$ by 1/3 of the length of the highway, and racer $C$ was ahead of racer $B$ by the same distance. Racer $A... | [
"1/4 h"
] | [
"Three racers, $A$, $B$, and $C$, start simultaneously but from different points and race at constant speeds in the same direction along a circular highway. At the moment of the start racer $B$ was ahead of racer $A$ by 1/3 of the length of the highway, and racer $C$ was ahead of racer $B$ by the same distance. Rac... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-635c0fbe9979daf03d11c9a107eddc6f30221862fa499f760fbeafb977fdbff9 | Three racers, first $A$, then $B$ and then $C$, start with an interval of 1 min from the same point on a circular highway and race in the same direction at constant speeds. It takes each racer more than 2 min to make a circle. Having made three circles, racer $A$ overtakes racer $B$ for the first time at the start, and... | [
"3 min"
] | [
"Three racers, first $A$, then $B$ and then $C$, start with an interval of 1 min from the same point on a circular highway and race in the same direction at constant speeds. It takes each racer more than 2 min to make a circle. Having made three circles, racer $A$ overtakes racer $B$ for the first time at the star... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b6a447600482776bdda1b90fd9288bee4a29f473a7aad495e2fddc1241beb58a | Three radii are drawn at random in a circle, dividing it into three sectors. A point is chosen at random in each sector. Find the probability that the triangle formed by joining these three points is acute. | [
"\\dfrac{1}{4}"
] | [
"Three radii are drawn at random in a circle, dividing it into three sectors. A point is chosen at random in each sector. Find the probability that the triangle formed by joining these three points is acute."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-93ed43b6cf2dda3b1e944dcb33b27cea39889302e8e9fdee4ebc21d609133726 | Three random points are picked on the circumference of a circle. What is the probability that the triangle formed by these points will have the center of the circle inside the triangle? | [
"\\dfrac{1}{4}"
] | [
"Three random points are picked on the circumference of a circle. What is the probability that the triangle formed by these points will have the center of the circle inside the triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6022ef080892e05e9121624cdb963071de3ec25d72f5af209583028ab4c87d5b | Three random points are thrown on a circle. Find the probability that the triangle formed by these points contains the center of the circle. | [
"$\\frac{1}{4}$"
] | [
"Three random points are thrown on a circle. Find the probability that the triangle formed by these points contains the center of the circle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-d69326f019d9d84faba2074e663151047cc25d99b4b42d19087093b08a9d6783 | Three random points, A, B, and C are chosen on a unit circle centered at the origin. Three red circles are drawn using AB, BC, and CA as diameters. What is the probability that the origin is located inside at least one of these red circles? Express your answer as a common fraction. | [
"\\dfrac{13}{16}"
] | [
"Three random points, A, B, and C are chosen on a unit circle centered at the origin. Three red circles are drawn using AB, BC, and CA as diameters. What is the probability that the origin is located inside at least one of these red circles? Express your answer as a common fraction."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 4
} | 12 | 0 | 0 |
omr-ws-50bfa9c22d69b59ceaaf8561d8698d94176f145c127aad8b3a5f8dc98e02bd2b | Three random real numbers \(a, b, c\) are chosen between 0 and 10, inclusive. What is the probability that \(abc + ab + bc + ac + a + b + c < 4\)? | [
"\\dfrac{5(\\ln 5)^2 - 10\\ln 5 + 8}{2000}"
] | [
"Three random real numbers \\(a, b, c\\) are chosen between 0 and 10, inclusive. What is the probability that \\(abc + ab + bc + ac + a + b + c < 4\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5290116ec51b90b861647125fc7c90cc27582ac62c38b37f7327a738d1c35671 | Three random squares are chosen from a regular chessboard. Find the probability that they form the letter 'L'. | [
"\\(\\frac{6 \\times 49}{64 \\times 63 \\times 62}\\)"
] | [
"Three random squares are chosen from a regular chessboard. Find the probability that they form the letter 'L'."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-09bd76021e05f33967260d7a6442679c5fda642904159972af80164721ae48ce | Three rational fractions have denominators of $20$, $21$, and $30$ when expressed in lowest terms. What is the least possible denominator that the sum of the three fractions, expressed in lowest terms, could have? | [
"28"
] | [
"Three rational fractions have denominators of $20$, $21$, and $30$ when expressed in lowest terms. What is the least possible denominator that the sum of the three fractions, expressed in lowest terms, could have?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ea5c5dca83bc61f2e1cc015e493034f4bd96c77394ddf7942509d9ab32cd6acd | Three rational numbers are in geometric progression. If we add 8 to the second number, the progression becomes arithmetic. Now if we add 64 to the third number, the progression returns to be geometric. What are these numbers? | [
"\\( (4, 12, 36) \\) and \\( \\left(\\frac{4}{9}, \\frac{20}{9}, \\frac{100}{9}\\right) \\)"
] | [
"Three rational numbers are in geometric progression. If we add 8 to the second number, the progression becomes arithmetic. Now if we add 64 to the third number, the progression returns to be geometric. What are these numbers?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a54dca2ae92cbfd3b656eea51b69163025c099bc386e3e92b811c818cfd8ef67 | Three rational numbers have denominators $20$, $21$, and $30$ when written in lowest terms. What is the smallest possible denominator their sum could have? | [
"420"
] | [
"Three rational numbers have denominators $20$, $21$, and $30$ when written in lowest terms. What is the smallest possible denominator their sum could have?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-edf4d701197c559aa98bb1de64ad953df2d6a093c529ee80546c8b6b8f1f232a | Three rational positive numbers (none of which are integers) are written on a board. Every minute, the numbers $x, y, z$ on the board are erased, and the numbers $x+\frac{1}{yz}, y+\frac{1}{zx}, z+\frac{1}{xy}$ are written instead. After 5 minutes, the numbers on the board are $a, b, c$. Calculate the sum of all possib... | [
"\\dfrac{325}{12}"
] | [
"Three rational positive numbers (none of which are integers) are written on a board. Every minute, the numbers $x, y, z$ on the board are erased, and the numbers $x+\\frac{1}{yz}, y+\\frac{1}{zx}, z+\\frac{1}{xy}$ are written instead. After 5 minutes, the numbers on the board are $a, b, c$. Calculate the sum of al... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ddab6304b5cf6fa067af8a7df49f36167ba15d9c6a235368168dc288065cfd16 | Three rays $OC$, $OD$, and $OE$ were drawn inside the obtuse angle $\angle AOB$, with $OC \perp OA$. $OD$ is the bisector of the angle $\angle AOB$ and $OE$ is the bisector of the angle $\angle BOC$. Find the measure of the angle $\angle DOE$. | [
"45^\\circ"
] | [
"Three rays $OC$, $OD$, and $OE$ were drawn inside the obtuse angle $\\angle AOB$, with $OC \\perp OA$. $OD$ is the bisector of the angle $\\angle AOB$ and $OE$ is the bisector of the angle $\\angle BOC$. Find the measure of the angle $\\angle DOE$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-63369fb53cb12198ae5076ba57a612d97744a82aca4ae98a036503b0aa320464 | Three rays $\ell_1, \ell_2, \ell_3$ are drawn through a point $M$ lying outside the circle $S$ ($\ell_2$ lies between $\ell_1$ and $\ell_3$). $\ell_1$ intersects $S$ successively at points $A$ and $B$. The circles $\omega_1$ and $\omega_2$ with centers $O_1$ and $O_2$ lying on $\ell_2$ ($O_1$ lies between $O_2$ and $M$... | [
"1"
] | [
"Three rays $\\ell_1, \\ell_2, \\ell_3$ are drawn through a point $M$ lying outside the circle $S$ ($\\ell_2$ lies between $\\ell_1$ and $\\ell_3$). $\\ell_1$ intersects $S$ successively at points $A$ and $B$. The circles $\\omega_1$ and $\\omega_2$ with centers $O_1$ and $O_2$ lying on $\\ell_2$ ($O_1$ lies betwee... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
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