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-94296c321aaa102a0c72a81fc13c1991709b494295a1a931bd17a8903e9ec5b5 | Three pairwise perpendicular rays passing through the fixed point $P$ inside a given sphere intersect the sphere at points $A, B, C$. Find the geometric locus of the projections of $P$ on the $ABC$ plane. | [
"\\text{The geometric locus is the sphere centered at } P \\text{ with radius } \\sqrt{\\dfrac{R^2 - |OP|^2}{5}}"
] | [
"Three pairwise perpendicular rays passing through the fixed point $P$ inside a given sphere intersect the sphere at points $A, B, C$. Find the geometric locus of the projections of $P$ on the $ABC$ plane."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c8b5629ee55fcfec8986c254d8b46e98c14f618263d1147f99d8b4a14449ac2f | Three pairwise relatively prime positive numbers greater than 1 have a product of 2520. How many distinct ways can these numbers be chosen? | [
"14"
] | [
"Three pairwise relatively prime positive numbers greater than 1 have a product of 2520. How many distinct ways can these numbers be chosen?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-eeb7c56de1a69aaf6198d779090d383302e861fae1cb2087137cf402eb93c0bf | Three parallel chords of a circle have lengths 2, 3, and 4 and subtend angles \( x \), \( y \), and \( x + y \) at the center (where \( x + y < 180^\circ \)). Find \( \cos x \). | [
"\\(\\frac{17}{32}\\)"
] | [
"Three parallel chords of a circle have lengths 2, 3, and 4 and subtend angles \\( x \\), \\( y \\), and \\( x + y \\) at the center (where \\( x + y < 180^\\circ \\)). Find \\( \\cos x \\)."
] | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-1d1062c07949412e8d200490f17a596b4d52d9fbb59e63b69e9ad4df4aaf1a7c | Three parallel chords of a circle have lengths 2, 3, and 4 and subtend angles \(x\), \(y\), and \(x+y\) at the center (with \(x + y < 180^\circ\)). Find \(\cos x\). | [
"\\dfrac{17}{32}"
] | [
"Three parallel chords of a circle have lengths 2, 3, and 4 and subtend angles \\(x\\), \\(y\\), and \\(x+y\\) at the center (with \\(x + y < 180^\\circ\\)). Find \\(\\cos x\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2c7a5cc85d303b17f94e8c6849dc9c7b1f9ad23fb26f2a01ebf11a234b6ec2c3 | Three parallel chords of circle O have lengths 20 cm, 16 cm, and 12 cm. Find, to the nearest tenth of a centimeter, the length of chord XY, where Y is the intersection of the second chord from the top with the side of the circle, and X is the intersection of the third chord from the top with the same side of the circle... | [
"2.8"
] | [
"Three parallel chords of circle O have lengths 20 cm, 16 cm, and 12 cm. Find, to the nearest tenth of a centimeter, the length of chord XY, where Y is the intersection of the second chord from the top with the side of the circle, and X is the intersection of the third chord from the top with the same side of the c... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 5
} | 6 | 0 | 0 |
omr-ws-1fb51fdac4958166f24ab239ff26fd6290fb6f8b9b36819f4961b38206a1dd49 | Three parallel cuts are made through a sphere of radius 2 such that each of the four pieces has a width of 1 unit. What is the total surface area of the four pieces? | [
"\\( 36 \\pi \\)"
] | [
"Three parallel cuts are made through a sphere of radius 2 such that each of the four pieces has a width of 1 unit. What is the total surface area of the four pieces?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-11317ba80ab8fb3b1d40285c37d89d05c555acea08571385a843cc6bf6a4358c | Three parallel lines $AD, BE, CF$ are drawn through the vertices of triangle $ABC$, meeting the opposite sides in $D, E, F$ respectively. The points $P, Q, R$ divide $AD, BE, CF$ respectively in the same ratio $k : 1$ and $P, Q, R$ are collinear. Find the value of $k$. | [
"2"
] | [
"Three parallel lines $AD, BE, CF$ are drawn through the vertices of triangle $ABC$, meeting the opposite sides in $D, E, F$ respectively. The points $P, Q, R$ divide $AD, BE, CF$ respectively in the same ratio $k : 1$ and $P, Q, R$ are collinear. Find the value of $k$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-8759f40bb5b9009c0796bf92859823504f9823affc4db56610edce839a885c1c | Three parallel lines $\ell_a, \ell_b, \ell_c$ pass through the vertices of triangle $ABC$. A line $a$ is the reflection of altitude $AH_a$ about $\ell_a$. Lines $b$ and $c$ are defined similarly. If the point of concurrency of lines $a, b, c$ is denoted by $P$, and the coordinates of $A, B, C$ are $(0,0)$, $(1,0)$, and... | [
"\\left( p, \\dfrac{p(p - 1)}{q} \\right)"
] | [
"Three parallel lines $\\ell_a, \\ell_b, \\ell_c$ pass through the vertices of triangle $ABC$. A line $a$ is the reflection of altitude $AH_a$ about $\\ell_a$. Lines $b$ and $c$ are defined similarly. If the point of concurrency of lines $a, b, c$ is denoted by $P$, and the coordinates of $A, B, C$ are $(0,0)$, $(1... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 2
} | 6 | 0 | 0 |
omr-ws-f0b70691eb9e7cc864e4847a9d595fb2b73aff8ce61adefddfab5a6d04980946 | Three parallel lines \( d_a \), \( d_b \), \( d_c \) pass through the vertices \( A \), \( B \), and \( C \) of triangle \( ABC \). The reflections of \( d_a \), \( d_b \), and \( d_c \) in \( BC \), \( CA \), and \( AB \) respectively form triangle \( XYZ \). Find the locus of the incenters of such triangles \( XYZ \)... | [
"I"
] | [
"Three parallel lines \\( d_a \\), \\( d_b \\), \\( d_c \\) pass through the vertices \\( A \\), \\( B \\), and \\( C \\) of triangle \\( ABC \\). The reflections of \\( d_a \\), \\( d_b \\), and \\( d_c \\) in \\( BC \\), \\( CA \\), and \\( AB \\) respectively form triangle \\( XYZ \\). Find the locus of the ince... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 4
} | 12 | 0 | 0 |
omr-ws-60c0c679f31da6d9ac00270aecd30c3b7bd0fd002279de133dce7d59118e88e5 | Three parallel lines \( l \), \( m \), and \( n \) are drawn through the vertices of triangle \( ABC \). Points \( X \), \( Y \), and \( Z \) are arbitrary points on \( l \), \( m \), and \( n \) respectively. \( G \) and \( G_1 \) are the centroids of triangles \( ABC \) and \( XYZ \) respectively. If the distance bet... | [
"5"
] | [
"Three parallel lines \\( l \\), \\( m \\), and \\( n \\) are drawn through the vertices of triangle \\( ABC \\). Points \\( X \\), \\( Y \\), and \\( Z \\) are arbitrary points on \\( l \\), \\( m \\), and \\( n \\) respectively. \\( G \\) and \\( G_1 \\) are the centroids of triangles \\( ABC \\) and \\( XYZ \\) ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-b9338ab2f5512c6df4386791508324b04bb4b36f343b15599a46ac0b1db082ef | Three parallel lines \( l, m, n \) are given with \( m \) between \( l \) and \( n \). The distances between the lines are \( l \) and \( m \) are 3 units, \( m \) and \( n \) are 1 unit, and \( l \) and \( n \) are 4 units. If the vertices of an equilateral triangle lie on these lines, with one vertex on each line, fi... | [
"169"
] | [
"Three parallel lines \\( l, m, n \\) are given with \\( m \\) between \\( l \\) and \\( n \\). The distances between the lines are \\( l \\) and \\( m \\) are 3 units, \\( m \\) and \\( n \\) are 1 unit, and \\( l \\) and \\( n \\) are 4 units. If the vertices of an equilateral triangle lie on these lines, with on... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 7
} | 55 | 0 | 0 |
omr-ws-5f81f207b2db55919483610cd26dd64c58a85fd5893e1626e076b5dd710463ed | Three parallel lines \( l_1 \), \( l_2 \), and \( l_3 \) are drawn through the vertices \( A \), \( B \), and \( C \) of a square \( ABCD \). If the distance between \( l_1 \) and \( l_2 \) is 7 and between \( l_2 \) and \( l_3 \) is 12, find the area of square \( ABCD \). | [
"193"
] | [
"Three parallel lines \\( l_1 \\), \\( l_2 \\), and \\( l_3 \\) are drawn through the vertices \\( A \\), \\( B \\), and \\( C \\) of a square \\( ABCD \\). If the distance between \\( l_1 \\) and \\( l_2 \\) is 7 and between \\( l_2 \\) and \\( l_3 \\) is 12, find the area of square \\( ABCD \\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-20e6ed682ca4a0fb13050e84cf811b6cd9782079750179a7f00cdaae6a200e72 | Three parallel lines \(L_1, L_2, L_3\) are drawn in the plane such that the perpendicular distance between \(L_1\) and \(L_2\) is 3 and the perpendicular distance between \(L_2\) and \(L_3\) is also 3. A square \(ABCD\) is constructed such that \(A\) lies on \(L_1\), \(B\) lies on \(L_3\), and \(C\) lies on \(L_2\). Fi... | [
"45"
] | [
"Three parallel lines \\(L_1, L_2, L_3\\) are drawn in the plane such that the perpendicular distance between \\(L_1\\) and \\(L_2\\) is 3 and the perpendicular distance between \\(L_2\\) and \\(L_3\\) is also 3. A square \\(ABCD\\) is constructed such that \\(A\\) lies on \\(L_1\\), \\(B\\) lies on \\(L_3\\), and ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7b91a79eb813831b24bb972b0ba1d1737df265cb08e3ff22b263415c1af94d4e | Three parallel lines \(L_1\), \(L_2\), and \(L_3\) are drawn through the vertices \(A\), \(B\), and \(C\) of a square \(ABCD\). If the distance between \(L_1\) and \(L_2\) is 7 and the distance between \(L_2\) and \(L_3\) is 12, then what is the area of square \(ABCD\)? | [
"193"
] | [
"Three parallel lines \\(L_1\\), \\(L_2\\), and \\(L_3\\) are drawn through the vertices \\(A\\), \\(B\\), and \\(C\\) of a square \\(ABCD\\). If the distance between \\(L_1\\) and \\(L_2\\) is 7 and the distance between \\(L_2\\) and \\(L_3\\) is 12, then what is the area of square \\(ABCD\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-d2639e27131d32f608ab2cc141ceae7f8cef70083b9b3aa2e12f02359f2435a3 | Three parallel lines \(d_a, d_b, d_c\) pass through the vertices \(A, B, C\) of triangle \(ABC\). The reflections of \(d_a, d_b, d_c\) in \(BC, CA, AB\) respectively form triangle \(XYZ\). Find the locus of the incenters of such triangles \(XYZ\). | [
"The locus of the incenters of such triangles \\(XYZ\\) is the circle centered at \\(O\\) with radius \\(2R\\), where \\(R\\) is the circumradius of \\(\\triangle ABC\\)."
] | [
"Three parallel lines \\(d_a, d_b, d_c\\) pass through the vertices \\(A, B, C\\) of triangle \\(ABC\\). The reflections of \\(d_a, d_b, d_c\\) in \\(BC, CA, AB\\) respectively form triangle \\(XYZ\\). Find the locus of the incenters of such triangles \\(XYZ\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fd3a1f92f76aa834b3120951337458cfabe81e72654c018568e0b293c59a4e85 | Three parallel lines are such that one passes through each vertex of triangle \(ABC\). The line through \(A\) meets \(BC\) (extended if necessary) at \(X\), and the lines through \(B\) and \(C\) meet \(CA\) and \(AB\) (extended if necessary) at \(Y\) and \(Z\) respectively. If the area of triangle \(ABC\) is 12 square ... | [
"24"
] | [
"Three parallel lines are such that one passes through each vertex of triangle \\(ABC\\). The line through \\(A\\) meets \\(BC\\) (extended if necessary) at \\(X\\), and the lines through \\(B\\) and \\(C\\) meet \\(CA\\) and \\(AB\\) (extended if necessary) at \\(Y\\) and \\(Z\\) respectively. If the area of trian... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a37a71816673ba6749f5ed05d9b400b7b87e065040c49835ece08d7d9ab3b8e9 | Three parallel lines are such that one passes through each vertex of triangle \(ABC\). The line through \(A\) meets \(BC\) (extended if necessary) at \(X\), the line through \(B\) meets \(CA\) (extended if necessary) at \(Y\), and the line through \(C\) meets \(AB\) (extended if necessary) at \(Z\). If the area of tria... | [
"24"
] | [
"Three parallel lines are such that one passes through each vertex of triangle \\(ABC\\). The line through \\(A\\) meets \\(BC\\) (extended if necessary) at \\(X\\), the line through \\(B\\) meets \\(CA\\) (extended if necessary) at \\(Y\\), and the line through \\(C\\) meets \\(AB\\) (extended if necessary) at \\(... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-89d77fc318513aacfe596e3bd4756cc1c6928f621fb130fae84e7bf24ef14c04 | Three parallel lines are such that one passes through each vertex of triangle \(ABC\). The line through \(A\) meets \(BC\) at \(X\) (extended if necessary), and the lines through \(B\) and \(C\) meet \(CA\) and \(AB\) (extended if necessary) at \(Y\) and \(Z\) respectively. If the area of triangle \(ABC\) is 15 square ... | [
"30"
] | [
"Three parallel lines are such that one passes through each vertex of triangle \\(ABC\\). The line through \\(A\\) meets \\(BC\\) at \\(X\\) (extended if necessary), and the lines through \\(B\\) and \\(C\\) meet \\(CA\\) and \\(AB\\) (extended if necessary) at \\(Y\\) and \\(Z\\) respectively. If the area of trian... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-266b0909ee7660d71977377ddc53736330a0d6d22f62a2c895eda3e5dff1a03c | Three parallel lines drawn through the vertices of a triangle \(ABC\) intersect a straight line \(d\) at points \(a\), \(b\), and \(c\) respectively. The parallel through \(b\) to \(AB\) intersects the parallel through \(c\) to \(AC\) at \(A'\). Similarly, find points \(B'\) and \(C'\). If the coordinates of \(A\), \(B... | [
"1"
] | [
"Three parallel lines drawn through the vertices of a triangle \\(ABC\\) intersect a straight line \\(d\\) at points \\(a\\), \\(b\\), and \\(c\\) respectively. The parallel through \\(b\\) to \\(AB\\) intersects the parallel through \\(c\\) to \\(AC\\) at \\(A'\\). Similarly, find points \\(B'\\) and \\(C'\\). If ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 2
} | 6 | 0 | 0 |
omr-ws-73ab755d64a6a041e276cdf0187a26938cd85739f1d40aa2f382324a8b021d16 | Three parallel lines passing through the vertices \( A, B \), and \( C \) of triangle \( ABC \) meet its circumcircle again at points \( A_1, B_1 \), and \( C_1 \) respectively. Points \( A_2, B_2 \), and \( C_2 \) are the reflections of points \( A_1, B_1 \), and \( C_1 \) in \( BC, CA \), and \( AB \) respectively. L... | [
"26"
] | [
"Three parallel lines passing through the vertices \\( A, B \\), and \\( C \\) of triangle \\( ABC \\) meet its circumcircle again at points \\( A_1, B_1 \\), and \\( C_1 \\) respectively. Points \\( A_2, B_2 \\), and \\( C_2 \\) are the reflections of points \\( A_1, B_1 \\), and \\( C_1 \\) in \\( BC, CA \\), and... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-696f6ea8a0f992fd18bcc23bca2db8f47533677c579dbd1ff100de4db21a5bd8 | Three particles are moving counterclockwise around a circle, starting from the same point. Their speeds are in the ratio of $1:2:3$. What is the maximum possible area formed by these three points? | [
"\\dfrac{3\\sqrt{3}}{4}"
] | [
"Three particles are moving counterclockwise around a circle, starting from the same point. Their speeds are in the ratio of $1:2:3$. What is the maximum possible area formed by these three points?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 2
} | 16 | 0 | 0 |
omr-ws-0277817932d7a838def06745e0725257f5d93b2210f1e24aa86b1709d215ce20 | Three passengers will arrive on three different flights, with the time of arrival between any two flights being independent and exponentially distributed with parameter \( b \). Find the expected time of the second arrival. | [
"\\(\\frac{5}{6b}\\)"
] | [
"Three passengers will arrive on three different flights, with the time of arrival between any two flights being independent and exponentially distributed with parameter \\( b \\). Find the expected time of the second arrival."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 12
} | 26 | 0 | 0 |
omr-ws-12fc99290e77a51189a8598a4e863f8c971914b747e4361ab3e6c8fd5b8274a3 | Three peas form a unit equilateral triangle on a flat table. Sebastian moves one of the peas a distance \( d \) along the table to form a right triangle. Determine the minimum possible value of \( d \). | [
"\\(\\frac{\\sqrt{3} - 1}{2}\\)"
] | [
"Three peas form a unit equilateral triangle on a flat table. Sebastian moves one of the peas a distance \\( d \\) along the table to form a right triangle. Determine the minimum possible value of \\( d \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-01aa236a5777d202674ef2eb98dd80270a71ca30a05f2826a442e6896fe5eb00 | Three pebbles are placed on the number line at -2020, 2020, and ?, where ? is an integer between -2020 and 2020. Players take turns moving either the leftmost or the rightmost pebble to an integer between the other two pebbles. The game ends when the pebbles occupy three consecutive integers. Odette wins if their sum i... | [
"2020"
] | [
"Three pebbles are placed on the number line at -2020, 2020, and ?, where ? is an integer between -2020 and 2020. Players take turns moving either the leftmost or the rightmost pebble to an integer between the other two pebbles. The game ends when the pebbles occupy three consecutive integers. Odette wins if their ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-87b91c9ff5761911072199a6a2b821c63a1a70b38a30f291b8dd747d08b95ced | Three pebbles are placed on the number line at -2020, 2020, and ?, where ? is an integer between -2020 and 2020. Players take turns moving either the leftmost or the rightmost pebble to an integer between the other two pebbles. The game ends when the pebbles occupy three consecutive integers. Odette wins if their sum i... | [
"2020"
] | [
"Three pebbles are placed on the number line at -2020, 2020, and ?, where ? is an integer between -2020 and 2020. Players take turns moving either the leftmost or the rightmost pebble to an integer between the other two pebbles. The game ends when the pebbles occupy three consecutive integers. Odette wins if their ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 1
} | 6 | 0 | 0 |
omr-ws-c3a35b8284b1effd0e5e3f995cb8421a768ccc9e1e145cfd80eb09d2c83f9e89 | Three pebbles are placed on the number line at -2020, 2020, and an integer ?, where ? is between -2020 and 2020. Players take turns moving either the leftmost or the rightmost pebble to an integer between the other two pebbles. The game ends when the pebbles occupy three consecutive integers. Odette wins if their sum i... | [
"0"
] | [
"Three pebbles are placed on the number line at -2020, 2020, and an integer ?, where ? is between -2020 and 2020. Players take turns moving either the leftmost or the rightmost pebble to an integer between the other two pebbles. The game ends when the pebbles occupy three consecutive integers. Odette wins if their ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bbbf2396c9e5f0b210353f2f9744e5651a4889a67bd0cf7fb39b299836a7931c | Three pebbles are placed on the number line at -2020, 2020, and an integer ?, where ? is between -2020 and 2020. Players take turns moving either the leftmost or the rightmost pebble to an integer between the other two pebbles. The game ends when the pebbles occupy three consecutive integers. Odette wins if their sum i... | [
"0"
] | [
"Three pebbles are placed on the number line at -2020, 2020, and an integer ?, where ? is between -2020 and 2020. Players take turns moving either the leftmost or the rightmost pebble to an integer between the other two pebbles. The game ends when the pebbles occupy three consecutive integers. Odette wins if their ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d9e923ad99557b2b0eee0a7a97a0646d945e13ecb51f711b401c6041f9c39340 | Three pedestrians started simultaneously, each of them having his own route, t hours later it remained for the second pedestrian to walk half as much again as the first pedestrian had covered, and it remained for the first pedestrian to walk thrice as far as the third pedestrian had covered, $2t$ hours after he started... | [
"$\\frac{13}{4}t,\\ \\frac{11}{5}t$"
] | [
"Three pedestrians started simultaneously, each of them having his own route, t hours later it remained for the second pedestrian to walk half as much again as the first pedestrian had covered, and it remained for the first pedestrian to walk thrice as far as the third pedestrian had covered, $2t$ hours after he st... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-da798e750dedd6f837ca39625f7bfbfafe2807e00432e05159d0802805400185 | Three people $A$, $B$, and $C$ play Rock-Paper-Scissors until one of them wins exclusively. If a player loses, they do not play in the next round. What is the probability of having a winner in the second round? | [
"\\(\\frac{1}{3}\\)"
] | [
"Three people $A$, $B$, and $C$ play Rock-Paper-Scissors until one of them wins exclusively. If a player loses, they do not play in the next round. What is the probability of having a winner in the second round?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 10
} | 24 | 0 | 0 |
omr-ws-c5454693c9941916965067e4e1371086221d6587da669edbb738aabbc0d6db8a | Three people $A$, $B$, and $C$ play Rock-Paper-Scissors. Each player has an equal probability of winning. What is the probability of $A$ and $B$ being the winners? | [
"\\(\\frac{1}{9}\\)"
] | [
"Three people $A$, $B$, and $C$ play Rock-Paper-Scissors. Each player has an equal probability of winning. What is the probability of $A$ and $B$ being the winners?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 25
} | 39 | 0 | 0 |
omr-ws-bfcbdf97226b17c60112fdc7785d5c67b146aef3933c5257b95df87d8f0a21b3 | Three people $A$, $B$, and $C$ play Rock-Paper-Scissors. Each player has an equal probability of winning. What is the probability of having a draw? | [
"\\(\\frac{1}{3}\\)"
] | [
"Three people $A$, $B$, and $C$ play Rock-Paper-Scissors. Each player has an equal probability of winning. What is the probability of having a draw?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 26
} | 48 | 0 | 0 |
omr-ws-71877c2fe990409832ee6b024bc7b9b9d4939ffa08f9d81699fc090d2e26faf2 | Three people $A$, $B$, and $C$ play Rock-Paper-Scissors. Each player has an equal probability of winning. What is the probability of having only one winner? | [
"\\(\\frac{1}{3}\\)"
] | [
"Three people $A$, $B$, and $C$ play Rock-Paper-Scissors. Each player has an equal probability of winning. What is the probability of having only one winner?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 25
} | 40 | 0 | 0 |
omr-ws-6ba874abb65f5ac27485c71e4d767f2a55ac4d44ff2ef13315ff3789bbf9f528 | Three people (A, B, and C) each choose a real number between $0$ and $1$, denoted as $a$, $b$, and $c$, respectively. A random number $n$ is then selected between $0$ and $1$. The person whose chosen number is closest to $n$ wins. What is the best strategy for each player? Who is most likely to win? | [
"\\text{The players choosing } \\frac{1}{4} \\text{ and } \\frac{3}{4}"
] | [
"Three people (A, B, and C) each choose a real number between $0$ and $1$, denoted as $a$, $b$, and $c$, respectively. A random number $n$ is then selected between $0$ and $1$. The person whose chosen number is closest to $n$ wins. What is the best strategy for each player? Who is most likely to win?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fbc75ad2e0451b09cd66c35acede8453f8a19fac493e89b3460611b9b2634083 | Three people are each given a hat that is either red or blue. Each person can see the hats of the other two but not their own. They must simultaneously guess the color of their own hat. They win if at least one person guesses correctly. What is the best strategy to maximize their probability of winning, which should be... | [
"2/3"
] | [
"Three people are each given a hat that is either red or blue. Each person can see the hats of the other two but not their own. They must simultaneously guess the color of their own hat. They win if at least one person guesses correctly. What is the best strategy to maximize their probability of winning, which shou... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-91aa476b77961c692a0da9ed4212d10597f3dcd52413c7dd6407d1654cec0b57 | Three people are each given a hat that is either red, blue, or green. Each person can see the hats of the other two but not their own. They must simultaneously guess the color of their own hat. They win if at least one person guesses correctly. What is the best strategy to maximize their probability of winning, which s... | [
"The strategy guarantees a win with a probability greater than \\( \\frac{1}{3} \\)."
] | [
"Three people are each given a hat that is either red, blue, or green. Each person can see the hats of the other two but not their own. They must simultaneously guess the color of their own hat. They win if at least one person guesses correctly. What is the best strategy to maximize their probability of winning, wh... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 2,
"tir": 7
} | 19 | 0 | 0 |
omr-ws-a490917b3d4e4ed364549fe20941a93fe98ba1ae2fd7d1d68bb5fc7160d33586 | Three people are in a room, each wearing a hat with a number either 1, 2, or 3. They can see the numbers on the others' hats but not their own. They can meet beforehand to decide a strategy but cannot communicate once in the room. Find a strategy such that at least one person can correctly guess their number. | [
"Each person guesses their number such that the total sum modulo 3 matches their assigned residue (0, 1, or 2), ensuring at least one correct guess."
] | [
"Three people are in a room, each wearing a hat with a number either 1, 2, or 3. They can see the numbers on the others' hats but not their own. They can meet beforehand to decide a strategy but cannot communicate once in the room. Find a strategy such that at least one person can correctly guess their number."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-fc11c6b24e2b8a63ec2c939393d087bd4a4e237c6605e92117338afac8dde046 | Three people are on a ballot, and there are 10 voters. Each voter sees the previous voter’s ballot and does not want to vote for the same person as the voter before him. How many ways are there for there to be a winner with a total of 5 votes? | [
"540"
] | [
"Three people are on a ballot, and there are 10 voters. Each voter sees the previous voter’s ballot and does not want to vote for the same person as the voter before him. How many ways are there for there to be a winner with a total of 5 votes?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 5
} | 11 | 0 | 0 |
omr-ws-dd69863d00e3e2a873e2d4492c29dd5da63dd320405a1fb6a034eb7e804313a4 | Three people are playing a game where they take balls from an urn with $n$ balls. The players can take $1$, $3$, or $5$ balls from it, in turn. Find all $n$ such that the first player can always manage to take the last ball, assuming the second and third players are cooperating to not let the first player win. | [
"n \\equiv 1, 3, 5 \\pmod{7}"
] | [
"Three people are playing a game where they take balls from an urn with $n$ balls. The players can take $1$, $3$, or $5$ balls from it, in turn. Find all $n$ such that the first player can always manage to take the last ball, assuming the second and third players are cooperating to not let the first player win."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-0f8e83eb0a6e13ddeec82eb3e350d8ce69a71e73ff28c4a6112299f0c240ba4c | Three people are racing up a stairway. Person $A$ leaps five steps with each stride, Person $B$ leaps six steps with each stride, and Person $C$ leaps seven steps with each stride. At one point, they all land simultaneously on the same step, with Person $A$ 21 steps from the top, Person $B$ seven steps from the top, an... | [
"211"
] | [
"Three people are racing up a stairway. Person $A$ leaps five steps with each stride, Person $B$ leaps six steps with each stride, and Person $C$ leaps seven steps with each stride. At one point, they all land simultaneously on the same step, with Person $A$ 21 steps from the top, Person $B$ seven steps from the to... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f852fa98122d11734b1ca4ef55a3065ff8d23e075c945e0604ed54b8403eefb3 | Three people are standing at points (0,0), (3,0), and (0,3) in the plane. Only one person can move at a time, and that person can move any distance in a line parallel to the line joining the other two people. Can the three people end up at the points (-1,1), (-1,3), and (2,5)? | [
"\\text{No}"
] | [
"Three people are standing at points (0,0), (3,0), and (0,3) in the plane. Only one person can move at a time, and that person can move any distance in a line parallel to the line joining the other two people. Can the three people end up at the points (-1,1), (-1,3), and (2,5)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f4ac0c7b01772a969c24001b92be0130923a1c1c62144f387f29a5fe1f410551 | Three people are standing in a row and need to move two paces forward. They can each move one step at a time, forward or back, but they cannot move behind the starting line or beyond the finishing line. A person cannot move if there is someone to their right, nor can they move if that move will result in someone being ... | [
"26"
] | [
"Three people are standing in a row and need to move two paces forward. They can each move one step at a time, forward or back, but they cannot move behind the starting line or beyond the finishing line. A person cannot move if there is someone to their right, nor can they move if that move will result in someone b... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7d22121375268e363db733afbae03fbb1866e4f03fd59870bebaf82f64484ec2 | Three people are to arrive at a certain location at some randomly chosen time between 1:00 PM and 2:00 PM. Each person will wait for 20 minutes before leaving. What is the probability that they will all meet? | [
"$\\frac{299}{324}$"
] | [
"Three people are to arrive at a certain location at some randomly chosen time between 1:00 PM and 2:00 PM. Each person will wait for 20 minutes before leaving. What is the probability that they will all meet?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d6536e78f2aa43e54b87bc15cff32b7bb783dc553af994d6da3a68129ba95f19 | Three people are to arrive at a certain location at some randomly chosen time between 1:00 PM and 2:00 PM. The first person will wait for 15 minutes, the second for 20 minutes, and the third for 30 minutes before leaving. What is the probability that they will all meet? | [
"\\dfrac{1435}{5184}"
] | [
"Three people are to arrive at a certain location at some randomly chosen time between 1:00 PM and 2:00 PM. The first person will wait for 15 minutes, the second for 20 minutes, and the third for 30 minutes before leaving. What is the probability that they will all meet?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 7
} | 19 | 0 | 0 |
omr-ws-6c356539e5746a5abb05e27f1060f7a64856d9dd2ef5ecc761a58277550dc59a | Three people at the base of a long stairway begin a race up the stairs. Person A leaps five steps with each stride (landing on steps $5$, $10$, $15$, etc.). Person B leaps a little more slowly but covers six steps with each stride. Person C leaps seven steps with each stride. A picture taken near the end of the race sh... | [
"211"
] | [
"Three people at the base of a long stairway begin a race up the stairs. Person A leaps five steps with each stride (landing on steps $5$, $10$, $15$, etc.). Person B leaps a little more slowly but covers six steps with each stride. Person C leaps seven steps with each stride. A picture taken near the end of the ra... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-dc5d6223622cfbccdb59db4d23f7687c3579517533192a55111ffc5194331bd2 | Three people go to bed at three random times, \(a < b < c\), between 10 pm and 11 pm. What is the probability that \(|a - b| \le 5\) minutes and \(|b - c| \le 5\) minutes? | [
"\\(\\frac{11}{288}\\)"
] | [
"Three people go to bed at three random times, \\(a < b < c\\), between 10 pm and 11 pm. What is the probability that \\(|a - b| \\le 5\\) minutes and \\(|b - c| \\le 5\\) minutes?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-223a8fb4e4fdec60bc19824e2833bc663466a71f16633ddd8548f2c554b8282a | Three people named $A$, $B$, and $C$ sit in a circle and take turns playing a game starting with $A$. Together they count all the consecutive integers starting at $1$ and ending at $12$. For each turn, a player may say either one or two numbers. Whoever says $12$ wins. Assuming each person employs optimal play, compute... | [
"0"
] | [
"Three people named $A$, $B$, and $C$ sit in a circle and take turns playing a game starting with $A$. Together they count all the consecutive integers starting at $1$ and ending at $12$. For each turn, a player may say either one or two numbers. Whoever says $12$ wins. Assuming each person employs optimal play, co... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-fcb4cc93a258a92d597be1ab86b17c9b5ca49ac24d888184a163497d11058c51 | Three people play a game with a total of 24 counters where the result is always that one person loses and two people win. The loser must then double the number of counters that each of the other players has at that time. At the end of three games, each player has lost one game and each person has 8 counters. At the beg... | [
"13"
] | [
"Three people play a game with a total of 24 counters where the result is always that one person loses and two people win. The loser must then double the number of counters that each of the other players has at that time. At the end of three games, each player has lost one game and each person has 8 counters. At th... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-6faf900b38dd1ed11889930824f6cedcfd68ce4ad81ff0d2ca5958ed28ea66d6 | Three people play a game with a total of 24 counters where the result is always that one person loses and two people win. The loser must then double the number of counters that each of the other players has at that time. At the end of three games, each player has lost one game and each person has 8 counters. At the beg... | [
"13"
] | [
"Three people play a game with a total of 24 counters where the result is always that one person loses and two people win. The loser must then double the number of counters that each of the other players has at that time. At the end of three games, each player has lost one game and each person has 8 counters. At th... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 6
} | 21 | 0 | 0 |
omr-ws-d11740a93db4cc5742439c50f5f7fc8a52549f7ea57ba3edc7f46f1c410723be | Three people sold their chickens at the market. The first person had 10 chickens, the second had 16, and the third had 26. They sold some chickens at a higher price in the morning and the rest at a lower price in the afternoon. Each person collected a total of 35 monies by the end of the day. Determine the price of eac... | [
"\\( a = 3.75 \\), \\( b = 1.25 \\)"
] | [
"Three people sold their chickens at the market. The first person had 10 chickens, the second had 16, and the third had 26. They sold some chickens at a higher price in the morning and the rest at a lower price in the afternoon. Each person collected a total of 35 monies by the end of the day. Determine the price o... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 17
} | 21 | 0 | 0 |
omr-ws-19e6c2439af1e8236d850cf1ec049008d68889e9a676fd1ef399e692c0695b64 | Three people take the AMC and each has a probability \( p \) of getting any given problem correct. If all 25 of their answers match up, what is the probability that they all got everything correct? | [
"\\(\\left(\\dfrac{p^3}{p^3 + \\frac{(1 - p)^3}{16}}\\right)^{25}\\)"
] | [
"Three people take the AMC and each has a probability \\( p \\) of getting any given problem correct. If all 25 of their answers match up, what is the probability that they all got everything correct?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 8,
"tir": 2
} | 22 | 0 | 0 |
omr-ws-39db07c31bc53ab7c7ddd1737d97196753d6cce2daa606cf8ab47d75ff4712f6 | Three people toss a coin to determine who pays for a meal. If all coins show heads or all show tails, they toss again. Otherwise, the person with the different coin pays. Determine the number of all possible outcomes if the game ends on: a. The first toss. b. The second toss. c. The third toss. d. The tenth toss. | [
"a. 2, b. 4, c. 8, d. 1024"
] | [
"Three people toss a coin to determine who pays for a meal. If all coins show heads or all show tails, they toss again. Otherwise, the person with the different coin pays. Determine the number of all possible outcomes if the game ends on:\na. The first toss.\nb. The second toss.\nc. The third toss.\nd. The tenth to... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c9789f6e43a075c488cd876d83506f4a4141f51a5ab6efa883c905c082b199cc | Three people with one double-seater motorcycle simultaneously headed from city A to city B. How should they act so that the time for which the last of them will arrive is the smallest? Determine this time. The pedestrian speed is 5 km/h, the motorcycle speed is 45 km/h, and the distance from A to B is 60 kilometers. | [
"12"
] | [
"Three people with one double-seater motorcycle simultaneously headed from city A to city B. How should they act so that the time for which the last of them will arrive is the smallest? Determine this time. The pedestrian speed is 5 km/h, the motorcycle speed is 45 km/h, and the distance from A to B is 60 kilometer... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-62cdbb56313e4e203251bce9cba8ad8fedc997a03bd20c161ccd2654dbd0942d | Three people, $A$, $B$, and $C$, run on a beach line at constant velocities. When someone reaches an extreme of the beach, they turn back and run in the opposite direction. If $A$ meets $B$ six times and $A$ meets $C$ eight times before they all meet again at the starting point, how many times did $B$ and $C$ meet? | [
"14"
] | [
"Three people, $A$, $B$, and $C$, run on a beach line at constant velocities. When someone reaches an extreme of the beach, they turn back and run in the opposite direction. If $A$ meets $B$ six times and $A$ meets $C$ eight times before they all meet again at the starting point, how many times did $B$ and $C$ meet... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-e202a18200bb4f8f5b29e089874bfb2784d4e97f985f740a5a9678c532c7da4e | Three people, $A$, $B$, and $C$, run on a straight beach at constant velocities. They start and finish at the same point. Not counting the start and finish, $A$ meets $B$ six times and $A$ meets $C$ eight times. How many times do $B$ and $C$ meet? | [
"13"
] | [
"Three people, $A$, $B$, and $C$, run on a straight beach at constant velocities. They start and finish at the same point. Not counting the start and finish, $A$ meets $B$ six times and $A$ meets $C$ eight times. How many times do $B$ and $C$ meet?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4929d812dd9d642614064ee329c6e5bc74e4cb355ca0681b283861693867415d | Three people, A, B, and C are running a race. The probability that A finishes before B is 60%. The probability that B finishes before C is 90%. What is the probability that C finishes before A? | [
"\\(\\frac{23}{71}\\)"
] | [
"Three people, A, B, and C are running a race. The probability that A finishes before B is 60%. The probability that B finishes before C is 90%. What is the probability that C finishes before A?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fca16feb0332f285065eb6e2e131ff702927fcc9f50a14a31b401e4b7cb30338 | Three people, A, B, and C, are given three unknown positive integers \( x \), \( y \), and \( z \). It is known that two of these numbers sum to 25, and the product of two of these numbers equals 120. A, B, and C each know one of the numbers and initially state that they do not know the values of the other two numbers.... | [
"\\( x = 6 \\), \\( y = 5 \\), \\( z = 20 \\)"
] | [
"Three people, A, B, and C, are given three unknown positive integers \\( x \\), \\( y \\), and \\( z \\). It is known that two of these numbers sum to 25, and the product of two of these numbers equals 120. A, B, and C each know one of the numbers and initially state that they do not know the values of the other t... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-05b5dd4cd3f212b03e356577c02b0446a91e6008fb5ecb0c19551ec4c90eb7c2 | Three people, A, B, and C, have speeds of 4 m/hr, 7 m/hr, and 14 m/hr, respectively, and are running on a circular track with a perimeter of 12 meters. When will they all meet at the same point on the track? | [
"12 hours"
] | [
"Three people, A, B, and C, have speeds of 4 m/hr, 7 m/hr, and 14 m/hr, respectively, and are running on a circular track with a perimeter of 12 meters. When will they all meet at the same point on the track?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-de57386a99c58941c952af6103df47f5245834a9eed36c8454a4ef678c026c3a | Three people, Annabel, Keith, and Sylvia, each have a hat labeled with a whole number greater than 0. The greatest number among the three is the sum of the other two. Annabel and Keith cannot determine their own numbers, but Sylvia deduces that her number is 25. What are the numbers on Annabel's and Keith's hats? | [
"Annabel: 10, Keith: 15"
] | [
"Three people, Annabel, Keith, and Sylvia, each have a hat labeled with a whole number greater than 0. The greatest number among the three is the sum of the other two. Annabel and Keith cannot determine their own numbers, but Sylvia deduces that her number is 25. What are the numbers on Annabel's and Keith's hats?"... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-7597e935107d661d0ce37056926c29a5d0d48b50beb5faf53521eea12fd5d578 | Three people—A, B, and C—sit in a circle and successively toss a fair coin in the order A, B, C, A, B, C, etc. The winner is the first person to throw a "head." What is the probability that person A wins this game? | [
"\\(\\frac{4}{7}\\)"
] | [
"Three people—A, B, and C—sit in a circle and successively toss a fair coin in the order A, B, C, A, B, C, etc. The winner is the first person to throw a \"head.\" What is the probability that person A wins this game?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-9fb17fd9606055a762abe3b75439351163c7ccdd128fa3d82c29c0c58b5d0711 | Three persons $A, B, C$ play a game where a subset with $k$ elements of the set $\{1, 2, 3, ..., 1986\}$ is selected randomly. The winner is $A, B, C$, according to whether the sum of the elements of the selected subset is congruent to $0, 1,$ or $2$ modulo $3$. Find all values of $k$ for which $A, B, C$ have equal cha... | [
"k \\not\\equiv 0 \\pmod{3}"
] | [
"Three persons $A, B, C$ play a game where a subset with $k$ elements of the set $\\{1, 2, 3, ..., 1986\\}$ is selected randomly. The winner is $A, B, C$, according to whether the sum of the elements of the selected subset is congruent to $0, 1,$ or $2$ modulo $3$. Find all values of $k$ for which $A, B, C$ have eq... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-759997e9e0a978f9833585804082ce9ac14c0a008291868ea20f60b90770ff02 | Three persons P, Q, and R of the same mass travel with speed \( u \) along the sides of an equilateral triangle of side length \( d \) such that each person always faces the other. After how much time will they meet each other? | [
"\\( \\frac{2d}{3u} \\)"
] | [
"Three persons P, Q, and R of the same mass travel with speed \\( u \\) along the sides of an equilateral triangle of side length \\( d \\) such that each person always faces the other. After how much time will they meet each other?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 17
} | 32 | 0 | 0 |
omr-ws-7c40dc8434eb0e583237c2fba3e63647c325eb20c741e537e88ed452264a8d50 | Three persons, A, B, and C, whose probabilities of speaking the truth are $\frac{1}{2}$, $\frac{2}{3}$, and $\frac{3}{4}$ respectively, assert that a ball drawn from a bag containing 5 balls of different colors (one of which is white) is white. What is the probability that their assertion is true? | [
"\\(\\frac{3}{5}\\)"
] | [
"Three persons, A, B, and C, whose probabilities of speaking the truth are $\\frac{1}{2}$, $\\frac{2}{3}$, and $\\frac{3}{4}$ respectively, assert that a ball drawn from a bag containing 5 balls of different colors (one of which is white) is white. What is the probability that their assertion is true?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 2,
"tir": 30
} | 47 | 0 | 0 |
omr-ws-717d0bfb74b9254e481f51c8e2304d4481bdcf3a0cd2158e68fc6076251a1ead | Three piles of stones are given, initially containing 2000, 4000, and 4899 stones respectively. Ali and Baba play a game, taking turns, with Ali starting first. In one move, a player can choose two piles and transfer some stones from one pile to the other, provided that at the end of the move, the pile from which the s... | [
"Ali has a winning strategy."
] | [
"Three piles of stones are given, initially containing 2000, 4000, and 4899 stones respectively. Ali and Baba play a game, taking turns, with Ali starting first. In one move, a player can choose two piles and transfer some stones from one pile to the other, provided that at the end of the move, the pile from which ... | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 3,
"tir": 17
} | 32 | 0 | 0 |
omr-ws-bd24db3f28a1313f81012e8d1bb412e26307fa3ec8a9be7bdc420e7d93a4c110 | Three pirates find a chest of gold coins which are to be divided between them. Before dividing, one pirate secretly counts the number of coins and finds that there is one left over after dividing into three equal piles. Stereotypically, the pirate adds the coin to one pile and steals that pile. Later, the second pirate... | [
"25"
] | [
"Three pirates find a chest of gold coins which are to be divided between them. Before dividing, one pirate secretly counts the number of coins and finds that there is one left over after dividing into three equal piles. Stereotypically, the pirate adds the coin to one pile and steals that pile. Later, the second p... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ae6d6df2da30deda7580fe9ed0068a967aaaf266032806d8dcaea9cc716e8b23 | Three planets are orbiting a sun in coplanar orbits. One planet completes an orbit every 21 years, another every 33 years, and the third every 77 years. If all planets and the sun are currently aligned, how long will it be before they are aligned again? | [
"231"
] | [
"Three planets are orbiting a sun in coplanar orbits. One planet completes an orbit every 21 years, another every 33 years, and the third every 77 years. If all planets and the sun are currently aligned, how long will it be before they are aligned again?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 11
} | 26 | 0 | 0 |
omr-ws-0ec6ab7e730f4dfc3dd3c9319466a427a6112e9822130bf15ddfcc95af3ee7ac | Three planets orbit a star circularly in the same plane with periods of 60, 84, and 140 years. The three planets and the star are currently collinear. What is the fewest number of years from now that they will all be collinear again? | [
"105"
] | [
"Three planets orbit a star circularly in the same plane with periods of 60, 84, and 140 years. The three planets and the star are currently collinear. What is the fewest number of years from now that they will all be collinear again?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 9
} | 32 | 0 | 0 |
omr-ws-813c35da973f4446c18f3f735024b61250390959c9955d68d030be3beb89461e | Three planets orbit a star circularly in the same plane with periods of 60, 84, and 140 years. They are currently collinear with the star. What is the fewest number of years from now that they will all be collinear again? | [
"105"
] | [
"Three planets orbit a star circularly in the same plane with periods of 60, 84, and 140 years. They are currently collinear with the star. What is the fewest number of years from now that they will all be collinear again?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 44,
"genselect": 8,
"tir": 6
} | 58 | 0 | 0 |
omr-ws-cc010054323520d55e90231587dc63252cc13ecceb5394b58ebcb0f79eb10087 | Three planets orbit a star circularly in the same plane. Each moves in the same direction and moves at constant speed. Their periods are 60, 84, and 140 years. The three planets and the star are currently collinear. What is the fewest number of years from now that they will all be collinear again? | [
"105"
] | [
"Three planets orbit a star circularly in the same plane. Each moves in the same direction and moves at constant speed. Their periods are 60, 84, and 140 years. The three planets and the star are currently collinear. What is the fewest number of years from now that they will all be collinear again?"
] | [
"MATH_training_set"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 7,
"tir": 4
} | 26 | 0 | 0 |
omr-ws-47043eef11ad5b526fc324ea1d2ce269219e666ae7ff41730fd7f7f386819654 | Three planets revolve about a star in coplanar circular orbits with the star at the center. All planets revolve in the same direction, each at a constant speed, and the periods of their orbits are 60, 84, and 140 years. The positions of the star and all three planets are currently collinear. They will next be collinear... | [
"105"
] | [
"Three planets revolve about a star in coplanar circular orbits with the star at the center. All planets revolve in the same direction, each at a constant speed, and the periods of their orbits are 60, 84, and 140 years. The positions of the star and all three planets are currently collinear. They will next be coll... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 7,
"tir": 4
} | 23 | 0 | 0 |
omr-ws-78c7792fd2f9982ec474fcebd2fef0464de89a7abec1fdffa21927f1ef765ee4 | Three planets revolve about a star in coplanar circular orbits with the star at the center. The periods of their orbits are 60, 84, and 140 years, respectively. The positions of the star and all three planets are currently collinear. They will next be collinear after \( n \) years. Find \( n \). | [
"105"
] | [
"Three planets revolve about a star in coplanar circular orbits with the star at the center. The periods of their orbits are 60, 84, and 140 years, respectively. The positions of the star and all three planets are currently collinear. They will next be collinear after \\( n \\) years. Find \\( n \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 7,
"tir": 9
} | 26 | 0 | 0 |
omr-ws-706b6649bbb9966fa67201ee4ebd06af46ee1befe688e19f2f3a4e754ea9d509 | Three planets revolve about a star in coplanar circular orbits with the star at the center. The periods of their orbits are 60, 84, and 140 years. The positions of the star and all three planets are currently collinear. They will next be collinear after \( n \) years. Find \( n \). | [
"105"
] | [
"Three planets revolve about a star in coplanar circular orbits with the star at the center. The periods of their orbits are 60, 84, and 140 years. The positions of the star and all three planets are currently collinear. They will next be collinear after \\( n \\) years. Find \\( n \\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 9
} | 30 | 0 | 0 |
omr-ws-d7c12a7d635405a0e4b93507c827374b15de5b24b95cd9ca4dbeaca236371f6e | Three planets with coplanar, circular, and concentric orbits are shown. The radii of the three circles are $3, 4$, and $5$. Initially, the three planets are collinear. Every hour, the outermost planet moves one-sixth of its full orbit, the middle planet moves one-fourth of its full orbit, and the innermost planet moves... | [
"16"
] | [
"Three planets with coplanar, circular, and concentric orbits are shown. The radii of the three circles are $3, 4$, and $5$. Initially, the three planets are collinear. Every hour, the outermost planet moves one-sixth of its full orbit, the middle planet moves one-fourth of its full orbit, and the innermost planet ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-14dfdbd9235caaeba662614501d1a7a6f540656390439462724e39cda45c5814 | Three players $A$, $B$, and $C$ play games. In the first round, $A$ matches $B$. In the second and later rounds, the winner matches the rest. If one player wins two consecutive rounds, they become the champion. Matches are repeated until someone wins two consecutive rounds. The probability of every player winning a mat... | [
"\\dfrac{5}{14}"
] | [
"Three players $A$, $B$, and $C$ play games. In the first round, $A$ matches $B$. In the second and later rounds, the winner matches the rest. If one player wins two consecutive rounds, they become the champion. Matches are repeated until someone wins two consecutive rounds. The probability of every player winning ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-81af451e2670bba94244ca5fb0228ba1a90ab7adefe9732e3d05050f01d3188e | Three players $A$, $B$, and $C$ take turns taking stones from a pile of $N$ stones. They play in the order $A$, $B$, $C$, $A$, $B$, $C$, ... with $A$ starting the game. The player who takes the last stone loses. $B$ can take $1, 2, 3, 4,$ or $5$ stones on each move, while $A$ and $C$ can each take $1, 2,$ or $3$ stones... | [
"B \\text{ wins if } N \\equiv 1 \\pmod{9}; \\text{ otherwise, } A \\text{ and } C \\text{ win}"
] | [
"Three players $A$, $B$, and $C$ take turns taking stones from a pile of $N$ stones. They play in the order $A$, $B$, $C$, $A$, $B$, $C$, ... with $A$ starting the game. The player who takes the last stone loses. $B$ can take $1, 2, 3, 4,$ or $5$ stones on each move, while $A$ and $C$ can each take $1, 2,$ or $3$ s... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 2
} | 3 | 0 | 0 |
omr-ws-9a57702f3655c135656a311dc0acd1c794fa44b4bf5e919265425028fc718c42 | Three players $A$, $B$, and $C$ take turns to roll a die in the order $ABCABC...$. Calculate the probability that $A$ is the first to throw a 6, $B$ is the second, and $C$ is the third. Express your answer as a fraction in lowest terms and find the sum of the numerator and the denominator. | [
"800227"
] | [
"Three players $A$, $B$, and $C$ take turns to roll a die in the order $ABCABC...$. Calculate the probability that $A$ is the first to throw a 6, $B$ is the second, and $C$ is the third. Express your answer as a fraction in lowest terms and find the sum of the numerator and the denominator."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 4
} | 9 | 0 | 0 |
omr-ws-ad8d19f0ca267b30e759d5bf53a27ad667745b7c363f39eabb7a861b2de71291 | Three players are given natural numbers. Each player can see the numbers of the other two players but not their own. It is known that the greatest number is the sum of the other two numbers. The first player is asked to state their number but cannot. The second player also cannot state their number. The third player th... | [
"10 and 15"
] | [
"Three players are given natural numbers. Each player can see the numbers of the other two players but not their own. It is known that the greatest number is the sum of the other two numbers. The first player is asked to state their number but cannot. The second player also cannot state their number. The third play... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-830001c27ea85ba8ccac92cb4c98989a1d49eacd46bda61175664117eb1a4895 | Three players each select a real number in the interval $(0,1)$ not picked by the others. Let the selections be $a_1, a_2, a_3$ respectively. Define $S_i = \left\{ x \in (0,1) \middle| |x-a_i| < \min_{j \neq i} |x-a_j| \right\}$, which is the set of $x \in (0,1)$ such that $x$ is strictly closer to $a_i$ than any of th... | [
"\\(\\frac{1}{4}\\)"
] | [
"Three players each select a real number in the interval $(0,1)$ not picked by the others. Let the selections be $a_1, a_2, a_3$ respectively. Define $S_i = \\left\\{ x \\in (0,1) \\middle| |x-a_i| < \\min_{j \\neq i} |x-a_j| \\right\\}$, which is the set of $x \\in (0,1)$ such that $x$ is strictly closer to $a_i$ ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 27,
"genselect": 1,
"tir": 11
} | 39 | 0 | 0 |
omr-ws-e3257e5d584ef414b42888bc0b2a27aaba702f3fd78b0d9b351b0e12c0ded4dd | Three players play a game of "rock-paper-scissors". In every round, each player simultaneously shows one of these shapes. Rock beats scissors, scissors beat paper, while paper beats rock. If in a round exactly two distinct shapes are shown (and thus one of them is shown twice), then 1 point is added to the score of the... | [
"23"
] | [
"Three players play a game of \"rock-paper-scissors\". In every round, each player simultaneously shows one of these shapes. Rock beats scissors, scissors beat paper, while paper beats rock. If in a round exactly two distinct shapes are shown (and thus one of them is shown twice), then 1 point is added to the score... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 8
} | 12 | 0 | 0 |
omr-ws-556a6d34ccdfa4f1aa6e31e9d0b643e1af4a2ed06b25422ab2e8d412fd39592f | Three players start with \(a\), \(b\), and \(c\) chips. Two players are picked at random, and one player gives a chip to the other. This process continues until one of the players runs out of chips. Find the expected number of turns until one of the players runs out of chips. | [
"\\(\\frac{3abc}{a+b+c}\\)"
] | [
"Three players start with \\(a\\), \\(b\\), and \\(c\\) chips. Two players are picked at random, and one player gives a chip to the other. This process continues until one of the players runs out of chips. Find the expected number of turns until one of the players runs out of chips."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 8
} | 12 | 0 | 0 |
omr-ws-8a30b0dacd11c0c27651613d104244fe78f74980851973e4fb46fb200a1af110 | Three players, $A$, $B$, and $C$, are playing chess with only one board. The first game is between $A$ and $B$, the next game is between $C$ and the winner of the first game, and the next game is between $A$ and the winner of the second game, and so on. A player is considered the winner of the series if they win two co... | [
"\\(\\frac{5}{14}\\)"
] | [
"Three players, $A$, $B$, and $C$, are playing chess with only one board. The first game is between $A$ and $B$, the next game is between $C$ and the winner of the first game, and the next game is between $A$ and the winner of the second game, and so on. A player is considered the winner of the series if they win t... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-5eeb809d86d90e245fa1e2bad45cea1a1f551f30a53c1cd5ee9d1915f14c3d0f | Three players, $A$, $B$, and $C$, are playing chess with only one board. The first game is between $A$ and $B$, the next game is between $C$ and the winner of the first game, and the next game is between $A$ and the winner of the second game, and so on. A player is considered the winner of the series if they win two co... | [
"\\(\\frac{2}{7}\\)"
] | [
"Three players, $A$, $B$, and $C$, are playing chess with only one board. The first game is between $A$ and $B$, the next game is between $C$ and the winner of the first game, and the next game is between $A$ and the winner of the second game, and so on. A player is considered the winner of the series if they win t... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 5,
"tir": 4
} | 14 | 0 | 0 |
omr-ws-c4faa0c3a3a90fd93bad8808dd8a113685766659071a5ace131e3ce036e6417a | Three players, $A$, $B$, and $C$, take turns rolling a fair six-sided die in the order $A$, $B$, $C$, $A$, $B$, $C$, and so on. Let $p$ be the probability that the first 6 to appear is thrown by $A$, the second 6 to appear is thrown by $B$, and the third 6 to appear is thrown by $C$. Calculate the value of $p \cdot 753... | [
"46656"
] | [
"Three players, $A$, $B$, and $C$, take turns rolling a fair six-sided die in the order $A$, $B$, $C$, $A$, $B$, $C$, and so on. Let $p$ be the probability that the first 6 to appear is thrown by $A$, the second 6 to appear is thrown by $B$, and the third 6 to appear is thrown by $C$. Calculate the value of $p \\cd... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-951c742e56f7b7c9671031c6d2e6bf1f7bf5955b1402906fc2cbf2826723d26a | Three players—Joey, Rithwik, and William—are playing a coin flipping game. Each turn, all three players flip a fair coin with equal probability of landing heads or tails. The player who flips a result different from the other two players gets a point. If all three players flip the same result, nobody gets a point. The ... | [
"8"
] | [
"Three players—Joey, Rithwik, and William—are playing a coin flipping game. Each turn, all three players flip a fair coin with equal probability of landing heads or tails. The player who flips a result different from the other two players gets a point. If all three players flip the same result, nobody gets a point.... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 5
} | 10 | 0 | 0 |
omr-ws-7a968fa8fb5a9a9b213d92072a94ed70b289c40489dc65a7945d98336efa3a51 | Three points $A$, $B$ and $C$ are connected by straight roads. Adjoining section $AB$ of the road is a square field with a side of $1/2 AB$. Adjoining $BC$ is a square with a side equal to $BC$, and adjoining $CA$ is a rectangular section of the woods length equal to $AC$ and of width 4 km. The wooded area is 20 square... | [
"The area of the forest is 40 km$^2$. Obtain the equation $AC=5+1/4BC^2+1/16AB^2$ from the statement of the problem; besides, for any three points $A$, $B$ and $C$ the inequality $AC\\le AB+BC$ is valid, whence $5+1/4BC^2+1/16AB^2\\le AB+BC$ or $(1/2BC-1)^2+(1/4AB-2)^2\\le 0,$ which is possible only for $AB=8$... | [
"Three points $A$, $B$ and $C$ are connected by straight roads. Adjoining section $AB$ of the road is a square field with a side of $1/2 AB$. Adjoining $BC$ is a square with a side equal to $BC$, and adjoining $CA$ is a rectangular section of the woods length equal to $AC$ and of width 4 km. The wooded area is 20 s... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9d451407f9d299a9039dd81e3cb4f44387a9fd6f528829b4eb027c72a43432bd | Three points $A$, $B$, $C$ are on a sphere with radius $10$. Find the maximum and minimum values of $\overrightarrow{AB} \cdot \overrightarrow{AC}$. No trigonometry or calculus is allowed. | [
"Maximum = 400, Minimum = -50"
] | [
"Three points $A$, $B$, $C$ are on a sphere with radius $10$. Find the maximum and minimum values of $\\overrightarrow{AB} \\cdot \\overrightarrow{AC}$. No trigonometry or calculus is allowed."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 6
} | 6 | 0 | 0 |
omr-ws-7816bb8ed48a7127823b263b775a17ffe4bd28e60763074d249418fb602f256c | Three points $A$, $B$, and $C$ are randomly chosen on a circle's circumference. What is the probability that $\angle ABC$ is acute? | [
"\\(\\frac{3}{4}\\)"
] | [
"Three points $A$, $B$, and $C$ are randomly chosen on a circle's circumference. What is the probability that $\\angle ABC$ is acute?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 7,
"tir": 10
} | 20 | 0 | 0 |
omr-ws-99da36d6a389bf9de9a8d184345defa1e0f1c37917495904d0faafcae8335317 | Three points $A$, $B$, and $C$ are randomly chosen on the circumference of a circle. If $A$, $B$, and $C$ all lie on a semicircle, what is the probability that all angles of triangle $ABC$ are less than $120^\circ$? | [
"\\(\\frac{5}{9}\\)"
] | [
"Three points $A$, $B$, and $C$ are randomly chosen on the circumference of a circle. If $A$, $B$, and $C$ all lie on a semicircle, what is the probability that all angles of triangle $ABC$ are less than $120^\\circ$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d477ee08781c0c09fca03ab3f4724fa98cac3ce3b2cd9c7589efc427cbfb3287 | 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}\\)",
"\\dfrac{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_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 18
} | 49 | 0 | 0 |
omr-ws-5230e85da064c85a348658c9b0cee5434499c7cf6ffeea7fbdd640f4f1366ea3 | Three points $A$, $B$, and $C$ are such that $B \in ]AC[$. On the side of $AC$, three semicircles are drawn with diameters $[AB]$, $[BC]$, and $[AC]$. The common interior tangent at $B$ to the first two semicircles meets the third semicircle in $E$. Let $U$ and $V$ be the points of contact of the common exterior tangen... | [
"\\(\\frac{r_1 \\cdot r_2}{(r_1 + r_2)^2}\\)"
] | [
"Three points $A$, $B$, and $C$ are such that $B \\in ]AC[$. On the side of $AC$, three semicircles are drawn with diameters $[AB]$, $[BC]$, and $[AC]$. The common interior tangent at $B$ to the first two semicircles meets the third semicircle in $E$. Let $U$ and $V$ be the points of contact of the common exterior ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 4,
"tir": 7
} | 17 | 0 | 0 |
omr-ws-37802f8569ed6646faa98343a82f7a771b9de99df889cf6e54faf07c6912c484 | Three points $A$, $B$, and $C$ lie on a circle of radius $16$ in that order. The perpendicular from $A$ to $BC$ intersects the circle at point $L$, and the midpoint of the arc $BC$ not containing $A$ is point $M$. Points $D$ and $E$ are taken on $AB$ and $AC$ respectively such that $DB = EC = LM = 6$. It is given that ... | [
"186"
] | [
"Three points $A$, $B$, and $C$ lie on a circle of radius $16$ in that order. The perpendicular from $A$ to $BC$ intersects the circle at point $L$, and the midpoint of the arc $BC$ not containing $A$ is point $M$. Points $D$ and $E$ are taken on $AB$ and $AC$ respectively such that $DB = EC = LM = 6$. It is given ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 7,
"tir": 1
} | 9 | 0 | 0 |
omr-ws-ac81d2fda4fa7add0a584d31234af50920a0e0721e461277314715975448fcdb | Three points $A$, $B$, and $C$ lie on the same line such that $BC = 2AB$. Construct equilateral triangles $ABP$ and $BCQ$ with $P$ and $Q$ on the same half-plane. Let $M$ be the midpoint of $CQ$ and $N$ be the intersection point of $AM$ and $BQ$. Find $1000 \cdot \frac{QN}{NB}$. | [
"3000"
] | [
"Three points $A$, $B$, and $C$ lie on the same line such that $BC = 2AB$. Construct equilateral triangles $ABP$ and $BCQ$ with $P$ and $Q$ on the same half-plane. Let $M$ be the midpoint of $CQ$ and $N$ be the intersection point of $AM$ and $BQ$. Find $1000 \\cdot \\frac{QN}{NB}$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-e4d1868021d3254678746429471d5379d431245d03b691c28a093423c3db999b | 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 such that circle $O$ is the incircle of $\triangle APB$. Const... | [
"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 $T$. Let $P$ be the point such that circle $O$ is the incircle of $\\triangle APB$.... | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 21,
"genselect": 8
} | 29 | 0 | 0 |
omr-ws-b1a10413d8017daa841e83722fdcb2669253e2f6398362aeb41dda968f9653d0 | Three points $A, B, C$ are chosen at random (with uniform distribution) and independently inside a circle with center $O$. What is the probability $p$ that $C$ is inside the triangle $OAB$? | [
"\\(\\frac{4}{9\\pi^2}\\)"
] | [
"Three points $A, B, C$ are chosen at random (with uniform distribution) and independently inside a circle with center $O$. What is the probability $p$ that $C$ is inside the triangle $OAB$?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 31
} | 45 | 0 | 0 |
omr-ws-82b1b5afb74538da12703f71214b173c0c5ba682de1273398b6dfd3a5df014e9 | Three points $A, B, C$ are chosen at random (with uniform distribution) and independently inside a circle with center $O$. What is the probability $q$ that $O$ is inside the triangle $ABC$? | [
"\\(\\frac{1}{4}\\)"
] | [
"Three points $A, B, C$ are chosen at random (with uniform distribution) and independently inside a circle with center $O$. What is the probability $q$ that $O$ is inside the triangle $ABC$?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 27
} | 41 | 0 | 0 |
omr-ws-09ea8b2951347ff1ab07f2944f3c3d5f43982aa5ec3956c83e34749f9d62fb4e | Three points $A, B, C$ are chosen at random (with uniform distribution) and independently inside a circle with center $O$. What is the probability $r$ that the points $O, A, B, C$ are vertices of a convex $4$-gon? | [
"\\(\\frac{3}{4} - \\frac{4}{3\\pi^2}\\)"
] | [
"Three points $A, B, C$ are chosen at random (with uniform distribution) and independently inside a circle with center $O$. What is the probability $r$ that the points $O, A, B, C$ are vertices of a convex $4$-gon?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 2,
"tir": 11
} | 14 | 0 | 0 |
omr-ws-5bf47b915a99f320cd2f40551170b6dce2ee9f2e80146ae6fb5eb5b9392a6189 | Three points $A, B, C$ are chosen at random on a circle. The probability that there exists a point $P$ inside an equilateral triangle $A_1B_1C_1$ such that $PA_1 = BC$, $PB_1 = AC$, and $PC_1 = AB$ can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | [
"5"
] | [
"Three points $A, B, C$ are chosen at random on a circle. The probability that there exists a point $P$ inside an equilateral triangle $A_1B_1C_1$ such that $PA_1 = BC$, $PB_1 = AC$, and $PC_1 = AB$ can be expressed in the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$."
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 19
} | 19 | 0 | 0 |
omr-ws-a49e3fe402221a14cc9874b7c4701a9aee15ac9cf763d7e74151a9e5dd30df69 | Three points $A, B, C$ are chosen at random on a circle. The probability that there exists a point $P$ inside an equilateral triangle $A_1B_1C_1$ such that $PA_1=BC$, $PB_1=AC$, $PC_1=AB$ can be expressed in the form $\frac{m}{n}$, where $m, n$ are relatively prime positive integers. Find $m+n$. | [
"5"
] | [
"Three points $A, B, C$ are chosen at random on a circle. The probability that there exists a point $P$ inside an equilateral triangle $A_1B_1C_1$ such that $PA_1=BC$, $PB_1=AC$, $PC_1=AB$ can be expressed in the form $\\frac{m}{n}$, where $m, n$ are relatively prime positive integers. Find $m+n$."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-2845f1fd22d3e12a2ac7ff6340fc3c4b2c5f4a14e871fb10a61af458797a51f9 | Three points $A, B, C$ are chosen in circle $O$ with radius $10$ such that $ABC$ is acute. A circle $X$ in $O$ is mutually tangent to line $BC$ and the minor arc $BC$. Another circle $Y$ that is not $X$ in $O$ is drawn that fulfills the same conditions, and is drawn to intersect $X$ in two points, $P$ and $Q$. If $AB =... | [
"13"
] | [
"Three points $A, B, C$ are chosen in circle $O$ with radius $10$ such that $ABC$ is acute. A circle $X$ in $O$ is mutually tangent to line $BC$ and the minor arc $BC$. Another circle $Y$ that is not $X$ in $O$ is drawn that fulfills the same conditions, and is drawn to intersect $X$ in two points, $P$ and $Q$. If ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-bf9855edb799fadc7f7bdb428ea3aa3da34e70d797d56e2f28121cc34a219be5 | Three points $A, B, C$ are such that $B \in AC$. On one side of $AC$, draw the three semicircles with diameters $AB$, $BC$, and $CA$. The common interior tangent at $B$ to the first two semicircles meets the third semicircle at $E$. Let $U$ and $V$ be the points of contact of the common exterior tangent to the first tw... | [
"\\(\\frac{r_{1}r_{2}}{(r_{1}+r_{2})^2}\\)"
] | [
"Three points $A, B, C$ are such that $B \\in AC$. On one side of $AC$, draw the three semicircles with diameters $AB$, $BC$, and $CA$. The common interior tangent at $B$ to the first two semicircles meets the third semicircle at $E$. Let $U$ and $V$ be the points of contact of the common exterior tangent to the fi... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 27,
"tir": 3
} | 30 | 0 | 0 |
omr-ws-c38d81677048fb0779c0444923b70eecf153a1ac0477123738b2f16c63eb3c40 | Three points $A, B, C$ lie on the parabola $y = x^2$. Find the locus of the barycenter $G$ of triangle $ABC$, given that the area of the triangle is a constant value $S$. | [
"y = x^2 + \\dfrac{2}{3} S^{2/3}"
] | [
"Three points $A, B, C$ lie on the parabola $y = x^2$. Find the locus of the barycenter $G$ of triangle $ABC$, given that the area of the triangle is a constant value $S$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.