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-a8dc9e9de842b8e9f65a24a84054f62c5bc700371395157f27954341bc806337 | Two of the roots for \( x \) of a polynomial equation with integral coefficients are \( \sqrt{5 + \sqrt{3}} \) and \( \sqrt{7 - \sqrt{2}} \). If the equation is \( 209x^4 + x^a - 24x^k - px^2 + m = 0 \), find the value of \( a + k + p + m \). | [
"-18"
] | [
"Two of the roots for \\( x \\) of a polynomial equation with integral coefficients are \\( \\sqrt{5 + \\sqrt{3}} \\) and \\( \\sqrt{7 - \\sqrt{2}} \\). If the equation is \\( 209x^4 + x^a - 24x^k - px^2 + m = 0 \\), find the value of \\( a + k + p + m \\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8d9a4d73b2029006b89eb4bb803f7c826c5ccb48d55684fb55d08db44c38e1c5 | Two of the roots of \[ax^3 + (a + 2b) x^2 + (b - 3a) x + (8 - a) = 0\]are $-2$ and 3. Find the third root. | [
"\\frac{4}{3}"
] | [
"Two of the roots of\n\\[ax^3 + (a + 2b) x^2 + (b - 3a) x + (8 - a) = 0\\]are $-2$ and 3. Find the third root."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-17806fc46660aba42141b67ee16c7c66bfaffc1dca0350e21180099614a68c05 | Two of the roots of the equation \[ax^3+bx^2+cx+d=0\]are $3$ and $-2.$ Given that $a \neq 0,$ compute $\frac{b+c}{a}.$ | [
"-7"
] | [
"Two of the roots of the equation \\[ax^3+bx^2+cx+d=0\\]are $3$ and $-2.$ Given that $a \\neq 0,$ compute $\\frac{b+c}{a}.$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4344ccc73d8743b90b4102167acc9755314edc934badef3629d9a126f5a8ec77 | Two of the side lengths of a non-degenerate triangle are 3 and 5. What is the sum of all possible integer values of the third side length? | [
"25"
] | [
"Two of the side lengths of a non-degenerate triangle are 3 and 5. What is the sum of all possible integer values of the third side length?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-3177891ef349b79c1b3bc5e9cbef9fb564a6207556891116f7912bd1918d2f4e | Two of the solutions to $ax^3 + bx^2 + cx + d = 0$ are $-\frac{2}{3}$ and $4.$ If $a + b = -11$ and $c + d = -34,$ find the third solution to the cubic equation. | [
"\\(-\\frac{1}{2}\\)"
] | [
"Two of the solutions to $ax^3 + bx^2 + cx + d = 0$ are $-\\frac{2}{3}$ and $4.$ If $a + b = -11$ and $c + d = -34,$ find the third solution to the cubic equation."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-adc296813729cbde4aed42196ab615276c08659ab60d35f12988ea7470e77c46 | Two of the squares of a $7 \times 7$ checkerboard are painted yellow, and the rest are painted green. Two color schemes are considered equivalent if one can be obtained from the other by applying a rotation in the plane of the board. How many inequivalent color schemes are possible? | [
"300"
] | [
"Two of the squares of a $7 \\times 7$ checkerboard are painted yellow, and the rest are painted green. Two color schemes are considered equivalent if one can be obtained from the other by applying a rotation in the plane of the board. How many inequivalent color schemes are possible?"
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 40
} | 71 | 0 | 0 |
omr-ws-caa0419c974bee329e6e971fcc6ede2cf47f66a17b23d8e092e6f1a17be85798 | Two of the squares of a 8x8 checkerboard are painted yellow, and the rest are painted green. Two color schemes are considered equivalent if one can be obtained from the other by applying a rotation in the plane of the board. How many inequivalent color schemes are possible? | [
"624"
] | [
"Two of the squares of a 8x8 checkerboard are painted yellow, and the rest are painted green. Two color schemes are considered equivalent if one can be obtained from the other by applying a rotation in the plane of the board. How many inequivalent color schemes are possible?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-96de950cd9e9fcdfe778c1e76b721af26e4291648dafbfce96f24759f8b2ba8c | Two of the squares of a checkerboard are painted yellow, and the rest are painted green. Two color schemes are considered equivalent if one can be obtained from the other by applying a rotation in the plane of the board. How many inequivalent color schemes are possible? | [
"528"
] | [
"Two of the squares of a checkerboard are painted yellow, and the rest are painted green. Two color schemes are considered equivalent if one can be obtained from the other by applying a rotation in the plane of the board. How many inequivalent color schemes are possible?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-46e2ce2a1ccf96410bf38cf7ddb87681b87ab01fc5c6ce8dfa4c0df440ba5dea | Two of the squares on a $7 \times 7$ board are painted white, and the rest are painted blue. Two colorings are considered equal if one can be obtained from the other by applying a rotation to the board. Determine the number of different colorings that exist. | [
"300"
] | [
"Two of the squares on a $7 \\times 7$ board are painted white, and the rest are painted blue. Two colorings are considered equal if one can be obtained from the other by applying a rotation to the board. Determine the number of different colorings that exist."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-670cee1173531ffdd9699cd5e4e7e5685ca89e6ba947ebebbdd950f7ac553ebc | Two of the vertices of a regular hexagon have coordinates $(8, 0)$ and $(16, 4)$. Let $V$ be the maximum possible volume of the solid formed by rotating this hexagon about the $y$-axis. Compute $\lfloor V \rfloor$. | [
"20195"
] | [
"Two of the vertices of a regular hexagon have coordinates $(8, 0)$ and $(16, 4)$. Let $V$ be the maximum possible volume of the solid formed by rotating this hexagon about the $y$-axis. Compute $\\lfloor V \\rfloor$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 5
} | 14 | 0 | 0 |
omr-ws-3c8b34170af07ecf55df5fdd5127de3f45a58d9eae9a2c2aaf12a263244df201 | Two of the vertices of a regular octahedron are to be chosen at random. What is the probability that they will be the endpoints of an edge of the octahedron? Express your answer as a common fraction. [asy] size(150); pair A, B, C, D, E, F; A=(1,1); B=(-1,-1); C=(0,6); D=(0,-6); E=(6, 0); F=(-6,0); draw(C--F--D--E--C--B... | [
"\\frac{4}{5}"
] | [
"Two of the vertices of a regular octahedron are to be chosen at random. What is the probability that they will be the endpoints of an edge of the octahedron? Express your answer as a common fraction. [asy]\nsize(150);\npair A, B, C, D, E, F;\nA=(1,1);\nB=(-1,-1);\nC=(0,6);\nD=(0,-6);\nE=(6, 0);\nF=(-6,0);\ndraw(C-... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8ece6e83160ec7a567ec806f621763c6223f32ad9e4852ea0e80620e8814938e | Two opposite vertices $M$ and $M'$ were chosen from a cube, and flat sections $ABC$ and $A'B'C'$ cut off two triangular pyramids $MABC$ and $M'A'B'C'$, forming an octahedron. Given that the three distances between the pairs of lines $(AB, A'B')$, $(BC, B'C')$, and $(AC, A'C')$ are pairwise different, calculate the volu... | [
"\\dfrac{s^3}{3}"
] | [
"Two opposite vertices $M$ and $M'$ were chosen from a cube, and flat sections $ABC$ and $A'B'C'$ cut off two triangular pyramids $MABC$ and $M'A'B'C'$, forming an octahedron. Given that the three distances between the pairs of lines $(AB, A'B')$, $(BC, B'C')$, and $(AC, A'C')$ are pairwise different, calculate the... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2375aaf8cc2f8d83e3fdcfbff319de0872c84245972dfa72d38d8889a5b63c49 | Two orthogonal circles intersect at points A and B. For an arbitrary point C on the smaller arc AB, let AC and BC intersect one of the circles at points D and E, respectively. If the radius of the circle containing points A, B, D, and E is 10, find the length of ED. | [
"20"
] | [
"Two orthogonal circles intersect at points A and B. For an arbitrary point C on the smaller arc AB, let AC and BC intersect one of the circles at points D and E, respectively. If the radius of the circle containing points A, B, D, and E is 10, find the length of ED."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-588f9dec38f03e26df666a37553f46d0be381d1131fda8320fb91e2f37fc19c8 | Two orthogonal circles intersect at points A and B. For an arbitrary point C on the smaller arc AB, let AC and BC intersect one of the circles at points D and E, respectively. Let EA and DB intersect at point F. If the radius of the smaller circle is 3 and the radius of the larger circle is 5, find the length of segmen... | [
"3\\sqrt{2}"
] | [
"Two orthogonal circles intersect at points A and B. For an arbitrary point C on the smaller arc AB, let AC and BC intersect one of the circles at points D and E, respectively. Let EA and DB intersect at point F. If the radius of the smaller circle is 3 and the radius of the larger circle is 5, find the length of s... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8b68a64bf7547516070d769d6f868ee310ceb48c77229af20f8ed644f0f5237d | Two overlapping circles A and B have the same area $\frac{x^2}{4}$. If one-sixth of circle A is rotated 90 degrees around point C and fits exactly over a one-sixth section of circle B, find the area of the figure bounded by arc DC, arc CE, and the straight line DE. | [
"$\\frac{x^2}{8\\pi^2}$"
] | [
"Two overlapping circles A and B have the same area $\\frac{x^2}{4}$. If one-sixth of circle A is rotated 90 degrees around point C and fits exactly over a one-sixth section of circle B, find the area of the figure bounded by arc DC, arc CE, and the straight line DE."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c1e87352f8c27bca778bccea6a653800f5f9d6687561093cd3715ef158dedaf6 | Two overlapping rhombi \( P \) and \( Q \) each with side length 20 are drawn such that the three marked points are collinear. If \( P \) and \( Q \) have areas 160 and \( m\sqrt{n} \), respectively, where \( n \) is square-free, find \( m + n \). | [
"203"
] | [
"Two overlapping rhombi \\( P \\) and \\( Q \\) each with side length 20 are drawn such that the three marked points are collinear. If \\( P \\) and \\( Q \\) have areas 160 and \\( m\\sqrt{n} \\), respectively, where \\( n \\) is square-free, find \\( m + n \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 3
} | 8 | 0 | 0 |
omr-ws-b3203b56d4b74eda3cae8565527d42b0e6c94ec4553a59fe97ed32d866728fcc | Two overlapping squares form a hexagon with an area \( A \). The segments have lengths 8, 6, and 4. Compute \( A \). | [
"236"
] | [
"Two overlapping squares form a hexagon with an area \\( A \\). The segments have lengths 8, 6, and 4. Compute \\( A \\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6fc2a19b68b81f5015d9287ce649070351d12914714964887892f8284ae40a0f | Two parabolas $C_1$ and $C_2$ on a plane have a common focus $F$. Their directrix lines are $\ell_1$ and $\ell_2$, respectively. Let $\ell_1$ intersect $C_2$ at points $A$ and $B$, and $\ell_2$ intersect $C_2$ at points $C$ and $D$. If the coordinates of $A$, $B$, $C$, and $D$ are such that the circle passing through t... | [
"100"
] | [
"Two parabolas $C_1$ and $C_2$ on a plane have a common focus $F$. Their directrix lines are $\\ell_1$ and $\\ell_2$, respectively. Let $\\ell_1$ intersect $C_2$ at points $A$ and $B$, and $\\ell_2$ intersect $C_2$ at points $C$ and $D$. If the coordinates of $A$, $B$, $C$, and $D$ are such that the circle passing ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f03b90f3870b9bb19b331275c07cfa564903883542a6b5a528db711e0b402974 | Two parabolas are the graphs of the equations $y=2x^2-10x-10$ and $y=x^2-4x+6$. Find all points where they intersect. List the points in order of increasing $x$-coordinate, separated by semicolons. | [
"(8,38)"
] | [
"Two parabolas are the graphs of the equations $y=2x^2-10x-10$ and $y=x^2-4x+6$. Find all points where they intersect. List the points in order of increasing $x$-coordinate, separated by semicolons."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-01b9fc53f80e069147f19eb75b78ea35d36ce151a1f9a2a2e691176f6264e729 | Two parabolas are the graphs of the equations $y=2x^2-7x+1$ and $y=8x^2+5x+1$. Give all points where they intersect. List the points in order of increasing $x$-coordinate, separated by semicolons. | [
"(0, 1)"
] | [
"Two parabolas are the graphs of the equations $y=2x^2-7x+1$ and $y=8x^2+5x+1$. Give all points where they intersect. List the points in order of increasing $x$-coordinate, separated by semicolons."
] | [
"MATH_training_set"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-603a83b39cd816aac8abd1a5619b72b21e5cb3e650e5853cb0f4ccbbfb2d3df8 | Two parabolas are the graphs of the equations $y=3x^2+4x-5$ and $y=x^2+11$. Give all points where they intersect. List the points in order of increasing $x$-coordinate, separated by semicolons. | [
"(-4, 27);(2, 15)"
] | [
"Two parabolas are the graphs of the equations $y=3x^2+4x-5$ and $y=x^2+11$. Give all points where they intersect. List the points in order of increasing $x$-coordinate, separated by semicolons."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ea1343ef47312576a91221207fd7ed91bf064ced87da4139f04823a295b9daa5 | Two parabolas have equations \( y = x^2 + ax + b \) and \( y = x^2 + cx + d \), where \( a \), \( b \), \( c \), and \( d \) are integers chosen independently by rolling a fair six-sided die. What is the probability that the parabolas have at least one point in common? | [
"\\(\\frac{31}{36}\\)"
] | [
"Two parabolas have equations \\( y = x^2 + ax + b \\) and \\( y = x^2 + cx + d \\), where \\( a \\), \\( b \\), \\( c \\), and \\( d \\) are integers chosen independently by rolling a fair six-sided die. What is the probability that the parabolas have at least one point in common?"
] | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-ea3ba10cb08cc111bcac17f4f9035958d46aa562de4fe45c2b95b5c8b6103fd9 | Two parabolas have the same focus at the point (3, -28). Their directrices are the $x$-axis and the $y$-axis, respectively. Compute the slope of their common chord. | [
"-1"
] | [
"Two parabolas have the same focus at the point (3, -28). Their directrices are the $x$-axis and the $y$-axis, respectively. Compute the slope of their common chord."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-159c56ec2b2671c220b7cf1bf4aaa828ff0e1f27bab3999a42852b9dea574767 | Two parabolas have the same focus at the point (3, -28). Their directrices are the x-axis and the y-axis, respectively. Compute the slope of the common chord. | [
"-1"
] | [
"Two parabolas have the same focus at the point (3, -28). Their directrices are the x-axis and the y-axis, respectively. Compute the slope of the common chord."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-d1ba65638fdc4e09b4bb20db51c3175ab1f983cb33ac6119e3c3983f47b126e2 | Two parabolas have the same focus at the point (3, -28). Their directrices are the x-axis and the y-axis, respectively. Compute the slope of their common chord. | [
"-1"
] | [
"Two parabolas have the same focus at the point (3, -28). Their directrices are the x-axis and the y-axis, respectively. Compute the slope of their common chord."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-88dcc03530b99dc243d8e1f74d0cab8dce98e4a48c6cfbb89f93ffd98aaeda97 | Two parabolas have the same tangent at their vertices and are located in different half-planes with respect to this tangent. Construct the other common tangent between these two parabolas. | [
"\\text{The other common tangent is the line connecting the points on each parabola horizontally aligned with the other parabola's vertex.}"
] | [
"Two parabolas have the same tangent at their vertices and are located in different half-planes with respect to this tangent. Construct the other common tangent between these two parabolas."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-73193c9472a8d6aed3edb7492abafdd30c8ae3343bff4d781a49e5b2b6931c13 | Two parabolas on a plane have perpendicular axes of symmetry and a common focus $O$. Let the two parabolas intersect at points $A$ and $B$. A common tangent line $\ell$ intersects $AB$ at point $C$. If the coordinates of $O$ are $(0,0)$, the coordinates of $A$ are $(3,4)$, and the coordinates of $B$ are $(-3,-4)$, find... | [
"25"
] | [
"Two parabolas on a plane have perpendicular axes of symmetry and a common focus $O$. Let the two parabolas intersect at points $A$ and $B$. A common tangent line $\\ell$ intersects $AB$ at point $C$. If the coordinates of $O$ are $(0,0)$, the coordinates of $A$ are $(3,4)$, and the coordinates of $B$ are $(-3,-4)$... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-b1f1b16cdabeb70434743947fe55979597e593dc1be716bd7c4963670f6ceefc | Two parabolas on the plane have perpendicular lines of symmetry and intersect at four distinct points. If the coordinates of these points are $(x_1, y_1)$, $(x_2, y_2)$, $(x_3, y_3)$, and $(x_4, y_4)$, find the radius of the circle passing through these four points given that the equations of the parabolas are $y = x^2... | [
"\\dfrac{\\sqrt{2}}{2}"
] | [
"Two parabolas on the plane have perpendicular lines of symmetry and intersect at four distinct points. If the coordinates of these points are $(x_1, y_1)$, $(x_2, y_2)$, $(x_3, y_3)$, and $(x_4, y_4)$, find the radius of the circle passing through these four points given that the equations of the parabolas are $y ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-18251e9bbab98ca412d6a4f31eadc0514917a3413be722d60a7e14a57b7ba07e | Two parallel chords \(AB\) and \(CD\) are drawn in a circle with center \(O\). Circles with diameters \(AB\) and \(CD\) intersect at point \(P\). If the radius of the original circle is \(r\) and the distance between the chords \(AB\) and \(CD\) is \(d\), find the length of the segment \(OP\) in terms of \(r\) and \(d\... | [
"\\sqrt{r^2 - \\dfrac{d^2}{2}}"
] | [
"Two parallel chords \\(AB\\) and \\(CD\\) are drawn in a circle with center \\(O\\). Circles with diameters \\(AB\\) and \\(CD\\) intersect at point \\(P\\). If the radius of the original circle is \\(r\\) and the distance between the chords \\(AB\\) and \\(CD\\) is \\(d\\), find the length of the segment \\(OP\\)... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-1a57a32b56eaffe5475b881837311a0e9c1f1cc3c2ba053fb5eeb83677480767 | Two parallel chords in a circle have lengths 10 and 14, and the distance between them is 6. Find the length of the chord parallel to these chords and midway between them. Let the length be \(\sqrt{a}\). Find \(a\). | [
"184"
] | [
"Two parallel chords in a circle have lengths 10 and 14, and the distance between them is 6. Find the length of the chord parallel to these chords and midway between them. Let the length be \\(\\sqrt{a}\\). Find \\(a\\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-37b888036d24e601018db0ec9c480dfd2507552fc0345e09830ae91b8e69f0cc | Two parallel chords in a circle have lengths 10 and 14, and the distance between them is 6. The chord parallel to these chords and midway between them is of length \(\sqrt{a}\), where \(a\) is: | [
"184"
] | [
"Two parallel chords in a circle have lengths 10 and 14, and the distance between them is 6. The chord parallel to these chords and midway between them is of length \\(\\sqrt{a}\\), where \\(a\\) is:"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-d2f54bbc4d88a87b6797c071fd2c9ee635c6c63c4dd32d639e4f90a019149faf | Two parallel chords in a circle have lengths 10 and 14, and the distance between them is 6. The chord parallel to these chords and midway between them is of length \(\sqrt{a}\). Find \(a\). | [
"184"
] | [
"Two parallel chords in a circle have lengths 10 and 14, and the distance between them is 6. The chord parallel to these chords and midway between them is of length \\(\\sqrt{a}\\). Find \\(a\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-67c7ece06b63268570473d964fdad9eb38172bd6a878f047c576aa54a84f6510 | Two parallel chords in a circle have lengths 10 and 14, and the distance between them is 6. What is the length of the chord parallel to these chords and midway between them? | [
"\\( 2\\sqrt{46} \\)"
] | [
"Two parallel chords in a circle have lengths 10 and 14, and the distance between them is 6. What is the length of the chord parallel to these chords and midway between them?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-a0fcddfe4a443c12d212af6ee767cf672cac9db2e20a1e45d78032ed56847838 | Two parallel chords in a circle have lengths 40 and 56, and the distance between them is 24. The chord parallel to these chords and midway between them is of length \(\sqrt{a}\) where \(a\) is ... | [
"2944"
] | [
"Two parallel chords in a circle have lengths 40 and 56, and the distance between them is 24. The chord parallel to these chords and midway between them is of length \\(\\sqrt{a}\\) where \\(a\\) is ..."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-dc22ff961ef1e4ef5b3fd3e106e55f8b526bbe7b2d59df3ccadfd43cd5113df0 | Two parallel chords in a circle have lengths 40 and 56, and the distance between them is 24. The chord parallel to these chords and midway between them is of length \(\sqrt{a}\), where \(a\) is ... | [
"2944"
] | [
"Two parallel chords in a circle have lengths 40 and 56, and the distance between them is 24. The chord parallel to these chords and midway between them is of length \\(\\sqrt{a}\\), where \\(a\\) is ..."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-b31ad9b1ac6981bdd872b9a4c2417886260ff0b729927577dacb96e3543d4b2b | Two parallel chords in a circle have lengths 6 and 8. The distance between them is 1. What is the diameter of the circle? | [
"10"
] | [
"Two parallel chords in a circle have lengths 6 and 8. The distance between them is 1. What is the diameter of the circle?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1e4a457bc58abd0904e24a3757cf412c281f3665bcda3d7e7ba382b1e79753c0 | Two parallel chords in a circle have lengths 6 and 8. The distance between them is 1. What is the diameter of this circle? | [
"10"
] | [
"Two parallel chords in a circle have lengths 6 and 8. The distance between them is 1. What is the diameter of this circle?"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 31
} | 31 | 0 | 0 |
omr-ws-530449b52aad67dc15b13c68a5d78eecaad9bb1ff3d069c4d546c4f6427fd2ba | Two parallel chords of a circle have lengths 10 and 14. The distance between them is 6 units. The chord parallel to these chords and halfway between them has length \(\sqrt{a}\). Find \(a\). | [
"46"
] | [
"Two parallel chords of a circle have lengths 10 and 14. The distance between them is 6 units. The chord parallel to these chords and halfway between them has length \\(\\sqrt{a}\\). Find \\(a\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ef9d1315109ca057822ecf5caa3fafaf88b64e5fc9df068d15929fca25661837 | Two parallel chords of a circle with radius $2$ are at a distance $\sqrt{3}+1$ apart. If the chords subtend angles of $\frac{\pi}{k}$ and $\frac{2\pi}{k}$ at the center, find $\lfloor{k}\rfloor$. | [
"3"
] | [
"Two parallel chords of a circle with radius $2$ are at a distance $\\sqrt{3}+1$ apart. If the chords subtend angles of $\\frac{\\pi}{k}$ and $\\frac{2\\pi}{k}$ at the center, find $\\lfloor{k}\\rfloor$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 48
} | 48 | 0 | 0 |
omr-ws-9219f06fe3fd54326af4d931006ce80591146fbec05c3766200f7c6f4cbd4c0f | Two parallel lines \( l_1 \) and \( l_2 \) lie on a plane, distance \( d \) apart. On \( l_1 \) there are an infinite number of points \( A_1, A_2, A_3, \cdots \), in that order, with \( A_nA_{n+1} = 2 \) for all \( n \). On \( l_2 \) there are an infinite number of points \( B_1, B_2, B_3, \cdots \), in that order and... | [
"\\(\\pi - \\tan^{-1}(1/d)\\)"
] | [
"Two parallel lines \\( l_1 \\) and \\( l_2 \\) lie on a plane, distance \\( d \\) apart. On \\( l_1 \\) there are an infinite number of points \\( A_1, A_2, A_3, \\cdots \\), in that order, with \\( A_nA_{n+1} = 2 \\) for all \\( n \\). On \\( l_2 \\) there are an infinite number of points \\( B_1, B_2, B_3, \\cdo... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 2,
"tir": 7
} | 15 | 0 | 0 |
omr-ws-ac239194041da753c017f9ba09725e0932f364b0260a43f79494839c7243d3ca | Two parallel lines and a circle between them are given. To fix ideas, let the lines be $x = -5$, $x = 4$, and let the circle be $x^2 + y^2 = 9$. Denote by $A = (-5, a)$ a point on the line $x = -5$. Find the range of values of $a$ such that there exists a point $B$ on the line $x = 4$ with the property that $AB$ inters... | [
"\\( a \\in ]-\\infty, \\frac{13\\sqrt{35}}{35}[\\)"
] | [
"Two parallel lines and a circle between them are given. To fix ideas, let the lines be $x = -5$, $x = 4$, and let the circle be $x^2 + y^2 = 9$. Denote by $A = (-5, a)$ a point on the line $x = -5$. Find the range of values of $a$ such that there exists a point $B$ on the line $x = 4$ with the property that $AB$ i... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-607c9ccd748465df6036155adbd29c3c94e4a9091608179ad9f2e002485f95b4 | Two parallel lines are intersected by a diagonal line that starts from one parallel line, turns in the opposite direction, and then meets the second parallel line. The first part of the diagonal line makes an obtuse angle of \(4x\) with the first parallel line. The angle made by the turn in the opposite direction is \(... | [
"30"
] | [
"Two parallel lines are intersected by a diagonal line that starts from one parallel line, turns in the opposite direction, and then meets the second parallel line. The first part of the diagonal line makes an obtuse angle of \\(4x\\) with the first parallel line. The angle made by the turn in the opposite directio... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 5,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-69677609d54054949601164ba7cd5d206400c912c05d86c07605faa4723e9458 | Two parallel lines cut a rhombus \(ABCD\) at points \(M\) (on \(AB\)), \(N\) (on \(BC\)), \(L\) (on \(CD\)), and \(K\) (on \(AD\)). The distance between \(MN\) and \(LK\) is identical to the distance between \(AB\) and \(CD\). Let \(R_1\) and \(R_2\) be the radii of the circumcircles of \(\triangle BMN\) and \(\triangl... | [
"\\dfrac{s}{\\sin \\theta}"
] | [
"Two parallel lines cut a rhombus \\(ABCD\\) at points \\(M\\) (on \\(AB\\)), \\(N\\) (on \\(BC\\)), \\(L\\) (on \\(CD\\)), and \\(K\\) (on \\(AD\\)). The distance between \\(MN\\) and \\(LK\\) is identical to the distance between \\(AB\\) and \\(CD\\). Let \\(R_1\\) and \\(R_2\\) be the radii of the circumcircles ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-cf5978656a86208aaea216c6197ea96b1dd3b11ac6f82988a5de8d0625d4db07 | Two parallel lines divide a circle into parts of areas $1$, $2$, and $3$. How far apart are the lines? | [
"1"
] | [
"Two parallel lines divide a circle into parts of areas $1$, $2$, and $3$. How far apart are the lines?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 7
} | 16 | 0 | 0 |
omr-ws-e2f9bf8d6108f8be14b83be7061841ca5e9ad0ee41fb9bd72cc3a8508b869dcc | Two parallel lines intersect the $x$-axis cutting off a line segment of length $3$. The same line also cuts off a segment of the $y$-axis of length $4$. What is the perpendicular distance between the lines? | [
"\\dfrac{12}{5}"
] | [
"Two parallel lines intersect the $x$-axis cutting off a line segment of length $3$. The same line also cuts off a segment of the $y$-axis of length $4$. What is the perpendicular distance between the lines?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-113157bd27242ccb6624954e03535cc2115a237e4190e3f83255c15d72be0b74 | Two parallel lines pass through the points $(0,1)$ and $(-1,0)$. Two other lines are drawn through $(1,0)$ and $(0,0)$, each perpendicular to the first two. The two sets of lines intersect in four points that are the vertices of a square. Find all possible equations for the first two lines. | [
"\\( y = 1 \\) and \\( y = 2x + 1 \\)"
] | [
"Two parallel lines pass through the points $(0,1)$ and $(-1,0)$. Two other lines are drawn through $(1,0)$ and $(0,0)$, each perpendicular to the first two. The two sets of lines intersect in four points that are the vertices of a square. Find all possible equations for the first two lines."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bea6925cbd8be5ecb53f36fba5516d5cc10e68f85cc1dc82b3bb9a44d5ae7d23 | Two parallel lines pass through the points (0, 1) and (-1, 0). Two other lines are drawn through (1, 0) and (0, 0), each perpendicular to the first two. The two sets of lines intersect in four points that are the vertices of a square. Find all possible equations for the first two lines. | [
"\\( y = 1 \\) and \\( y = 0 \\) or \\( y = 1 + 2x \\) and \\( y = 2x + 2 \\)"
] | [
"Two parallel lines pass through the points (0, 1) and (-1, 0). Two other lines are drawn through (1, 0) and (0, 0), each perpendicular to the first two. The two sets of lines intersect in four points that are the vertices of a square. Find all possible equations for the first two lines."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 12
} | 12 | 0 | 0 |
omr-ws-d16fbec3b9c9c57f580c339315c15b16ac78bcef3af3667ae8682cd46fe6a5c9 | Two particles move along the edges of an equilateral triangle \( \Delta ABC \) in the direction \( A \rightarrow B \rightarrow C \rightarrow A \), starting simultaneously and moving at the same speed. One particle starts at \( A \), and the other starts at the midpoint of \( BC \). The midpoint of the line segment join... | [
"\\(\\frac{1}{4}\\)"
] | [
"Two particles move along the edges of an equilateral triangle \\( \\Delta ABC \\) in the direction \\( A \\rightarrow B \\rightarrow C \\rightarrow A \\), starting simultaneously and moving at the same speed. One particle starts at \\( A \\), and the other starts at the midpoint of \\( BC \\). The midpoint of the ... | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8b94ce46f5b5a85ed635d9ad5110533c7301f42714ed2af911672d4ef0233cfc | Two particles move along the edges of an equilateral triangle \( \triangle ABC \) in the direction \( A \rightarrow B \rightarrow C \rightarrow A \), starting simultaneously and moving at the same speed. One particle starts at \( A \), and the other starts at the midpoint of \( \overline{BC} \). The midpoint of the lin... | [
"\\(\\frac{1}{16}\\)"
] | [
"Two particles move along the edges of an equilateral triangle \\( \\triangle ABC \\) in the direction \\( A \\rightarrow B \\rightarrow C \\rightarrow A \\), starting simultaneously and moving at the same speed. One particle starts at \\( A \\), and the other starts at the midpoint of \\( \\overline{BC} \\). The m... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 10,
"tir": 13
} | 23 | 0 | 0 |
omr-ws-7bddcf8caf1ddcb3437b2f062b21624c76fc3f58e2ce789f662bce39902fa542 | Two particles start at the point (0,0,0) in a three-dimensional coordinate system and each take n steps to return to (0,0,0). Each step moves a particle to an adjacent point. What is the maximum number of times these two particles can meet at the same point at the same time? | [
"\\( n+1 \\) (including the starting point) or \\( n \\) (excluding the starting point). \n\nSince the problem does not specify whether to include the starting point or not, both answers are valid based on the forum discussion."
] | [
"Two particles start at the point (0,0,0) in a three-dimensional coordinate system and each take n steps to return to (0,0,0). Each step moves a particle to an adjacent point. What is the maximum number of times these two particles can meet at the same point at the same time?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-736d274fbea61b762ab51bf3a082ecf47c5801ae4d93dcb9c07b0fd5811872a3 | Two particles start simultaneously from vertices B and C of a right triangle ABC (with the right angle at C) and move towards vertex A at constant speeds, arriving at A at the same time. The time of travel is given by \((AB/AC)^{3/2}\), and the length of BC is equal to the sum of the constant speeds of the particles. D... | [
"\\sqrt{2 + \\sqrt{5}}"
] | [
"Two particles start simultaneously from vertices B and C of a right triangle ABC (with the right angle at C) and move towards vertex A at constant speeds, arriving at A at the same time. The time of travel is given by \\((AB/AC)^{3/2}\\), and the length of BC is equal to the sum of the constant speeds of the parti... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-fffce744fa0762169a166d5ac935a795703057be1ffdc7f84ae6498a359033bc | Two paths cross a park. One is a semicircle, and the other is a straight line forming the diameter of the semicircle. The length of the straight path is 8 miles. Person A walks the semicircular path starting at noon at a speed of $\pi$ miles per hour. Person B walks the straight path starting at 1 PM at a speed of $\fr... | [
"4 PM"
] | [
"Two paths cross a park. One is a semicircle, and the other is a straight line forming the diameter of the semicircle. The length of the straight path is 8 miles. Person A walks the semicircular path starting at noon at a speed of $\\pi$ miles per hour. Person B walks the straight path starting at 1 PM at a speed o... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 8,
"tir": 12
} | 24 | 0 | 0 |
omr-ws-a4c42198369b3b7cb8241cb52d19a09af04b42d247077996bf3b3b859870d150 | Two paths cross a park. One is a semicircle, and the other is a straight line forming the diameter of the semicircle. The length of the straight path is 8 miles. Person A walks the semicircular path starting at noon at a speed of $\pi$ miles per hour. Person B walks the straight path starting at 1 PM at a speed of $\fr... | [
"2:03:55 PM",
"4 \\text{ PM}"
] | [
"Two paths cross a park. One is a semicircle, and the other is a straight line forming the diameter of the semicircle. The length of the straight path is 8 miles. Person A walks the semicircular path starting at noon at a speed of $\\pi$ miles per hour. Person B walks the straight path starting at 1 PM at a speed o... | [
"aops_c6_high_school_olympiads",
"aops_c7_college_math",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 9,
"tir": 15
} | 29 | 0 | 0 |
omr-ws-7ffeb1e8cfdae22e8301c8d29e5c1bc76bd4f613cb091b0d4d459c2a0fc87117 | Two pedestrians standing 10 feet and 40 feet away from the base of a billboard structure have equivalent views (angles) of the billboard. If the billboard is 9 feet tall, what is the height \(h\) above the ground of the billboard? | [
"16"
] | [
"Two pedestrians standing 10 feet and 40 feet away from the base of a billboard structure have equivalent views (angles) of the billboard. If the billboard is 9 feet tall, what is the height \\(h\\) above the ground of the billboard?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-59677ae1aa6be5d612c3fddeb3a06079eb385449e3b0063d6e57643deaa4dc4f | Two pedestrians start simultaneously from points $A$ and $B$ which are 27 km apart and move along straight line $AB$. If they move in opposite directions, they meet in 3 hours, while walking in the same direction, one catches up with the other in 9 hours. Find the speed of each pedestrian. | [
"6 km/h and 3 km/h"
] | [
"Two pedestrians start simultaneously from points $A$ and $B$ which are 27 km apart and move along straight line $AB$. If they move in opposite directions, they meet in 3 hours, while walking in the same direction, one catches up with the other in 9 hours. Find the speed of each pedestrian."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6cdcc29218eb0ae70cfa2cebf9860f69332a942a0e6877a47490594db66294ff | Two pedestrians started simultaneously from points $A$ and $B$ to meet each other. When the first pedestrian covered half the path, there remained 24 km for the second pedestrian to complete his walk. When the second pedestrian covered half the path, the first was at a distance of 15 km from the finish. How many km wil... | [
"8 km"
] | [
"Two pedestrians started simultaneously from points $A$ and $B$ to meet each other. When the first pedestrian covered half the path, there remained 24 km for the second pedestrian to complete his walk. When the second pedestrian covered half the path, the first was at a distance of 15 km from the finish. How many k... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-37d0090b3bb2db7a03cf97377379a932083529b47774f13b9f0fac38ccc34e7b | Two pedestrians started simultaneously, the first from $A$ to $B$ and the second from $B$ to $A$. When the distance between them became one-sixth of the initial distance, a cyclist started from $B$ to $A$. The first pedestrian met him the moment the second pedestrian covered 4/9 of the distance from $B$ to $A$. The cyc... | [
"$1:2$, $1:3$"
] | [
"Two pedestrians started simultaneously, the first from $A$ to $B$ and the second from $B$ to $A$. When the distance between them became one-sixth of the initial distance, a cyclist started from $B$ to $A$. The first pedestrian met him the moment the second pedestrian covered 4/9 of the distance from $B$ to $A$. T... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2496b8720f0d4c9831a2fe3dd50bddd803c86e5fb80eef5f0e8a21d841324c56 | Two pedestrians, facing each other on opposite sides of a road, start walking at the same time. The road is \( d \) miles long, and the two pedestrians meet after \( t \) minutes. What is the distance that each pedestrian had passed until the two of them met, if one pedestrian requires \( n \) minutes more to pass each... | [
"\\dfrac{d n + 2 t - \\sqrt{d^2 n^2 + 4 t^2}}{2 n}"
] | [
"Two pedestrians, facing each other on opposite sides of a road, start walking at the same time. The road is \\( d \\) miles long, and the two pedestrians meet after \\( t \\) minutes. What is the distance that each pedestrian had passed until the two of them met, if one pedestrian requires \\( n \\) minutes more t... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5e0c9068adcdf836ef17368e2d0efec509db4f7e7c1b53b782efc3671496dc6e | Two pentagons are given in the plane such that neither of them has a vertex on any line that contains a side of the other. Find the maximum number of intersection points their sides can form. | [
"20"
] | [
"Two pentagons are given in the plane such that neither of them has a vertex on any line that contains a side of the other. Find the maximum number of intersection points their sides can form."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 26,
"genselect": 7,
"tir": 1
} | 34 | 0 | 0 |
omr-ws-ff404c53cc17926fb0e8e65917fdb6f88456590b68a800b70330a7f0fc79c91c | Two people $A$ and $B$ play a game where they alternately remove numbers from the set $\{0, 1, 2, 3, \ldots, 1024\}$. The sequence of removals is $512$, $256$, $128$, $64$, $32$, $16$, $8$, $4$, $2$, $1$ numbers, with $A$ starting by removing $512$ numbers. After all removals, two numbers $a$ and $b$ remain ($a < b$), ... | [
"32"
] | [
"Two people $A$ and $B$ play a game where they alternately remove numbers from the set $\\{0, 1, 2, 3, \\ldots, 1024\\}$. The sequence of removals is $512$, $256$, $128$, $64$, $32$, $16$, $8$, $4$, $2$, $1$ numbers, with $A$ starting by removing $512$ numbers. After all removals, two numbers $a$ and $b$ remain ($a... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 4
} | 8 | 0 | 0 |
omr-ws-248a259ded83b2a2c626e90fcd20348bf19b8b905c41f66d5e48ad3398be7d45 | Two people $A$ and $B$ start from the same place at the same time to travel around a circular track of length $100$ m in opposite directions. First, $B$ goes more slowly than $A$ until they meet. Then, by doubling his rate, $B$ next meets $A$ at the starting point. Let $d$ m be the distance travelled by $B$ before he m... | [
"41"
] | [
"Two people $A$ and $B$ start from the same place at the same time to travel around a circular track of length $100$ m in opposite directions. First, $B$ goes more slowly than $A$ until they meet. Then, by doubling his rate, $B$ next meets $A$ at the starting point. Let $d$ m be the distance travelled by $B$ before... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 3
} | 25 | 0 | 0 |
omr-ws-38290af5d95dc8a237a6765bff4942635d0fac2f722cb287366ea8d423d63209 | Two people (labeled A and B) play a game where they take turns counting positive integers. Let \( m \) and \( n \) be relatively prime integers. If a person says an integer that is a multiple of \( m \), they write down \( m \), and similarly for \( n \). If an integer is a multiple of both \( m \) and \( n \), they wr... | [
"\\( m + n \\)"
] | [
"Two people (labeled A and B) play a game where they take turns counting positive integers. Let \\( m \\) and \\( n \\) be relatively prime integers. If a person says an integer that is a multiple of \\( m \\), they write down \\( m \\), and similarly for \\( n \\). If an integer is a multiple of both \\( m \\) and... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 4
} | 5 | 0 | 0 |
omr-ws-a05c69821f4f981af0e447d13e8132e97996884d8a53bff3290226226668e890 | Two people agree to meet at a coffee shop. They each independently pick a random moment in time between 8 a.m. and 9 a.m. and show up exactly at their selected time. But they are very impatient and only stay for 10 minutes after they arrive. What is the probability they meet? Express the answer as a common fraction. | [
"\\(\\frac{11}{36}\\)"
] | [
"Two people agree to meet at a coffee shop. They each independently pick a random moment in time between 8 a.m. and 9 a.m. and show up exactly at their selected time. But they are very impatient and only stay for 10 minutes after they arrive. What is the probability they meet? Express the answer as a common fractio... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-f8f9b7d9c3be6793006702e163065b17401f8656e23b49924ec66e9fdc5cc604 | Two people agree to meet between 12:00 and 1:00. If one person arrives and will wait for 20 minutes, what is the probability that the second person will arrive within that 20-minute window? | [
"\\dfrac{5}{9}"
] | [
"Two people agree to meet between 12:00 and 1:00. If one person arrives and will wait for 20 minutes, what is the probability that the second person will arrive within that 20-minute window?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-3c62b4e24c3fde8077572c28822deb2659526e4f2b1e145a42bde579e116bec1 | Two people are in a best-of-11 match. Person A has a 5/7 chance of winning each game, and person B has a 2/7 chance of winning each game. Given that person A already won 1 game, what is the probability that person A will win the match? | [
"\\dfrac{38834375}{40353607}"
] | [
"Two people are in a best-of-11 match. Person A has a 5/7 chance of winning each game, and person B has a 2/7 chance of winning each game. Given that person A already won 1 game, what is the probability that person A will win the match?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-a6feb39537e2bdd75c0fd4231069402dbf6d82231c17825200806ec0f8ba00c0 | Two people are playing a card game with decks of 300 cards each. Each card is red, green, or blue, with red > green > blue > red in terms of strength. Each deck is shuffled and placed face down. Players reveal the top card of their decks simultaneously. If the cards are of different colors, the player with the stronger... | [
"\\(\\frac{1}{3}\\)"
] | [
"Two people are playing a card game with decks of 300 cards each. Each card is red, green, or blue, with red > green > blue > red in terms of strength. Each deck is shuffled and placed face down. Players reveal the top card of their decks simultaneously. If the cards are of different colors, the player with the str... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-faafdcb80854e3923b7306db182293eff2ed900aadf32252aa9e09da1cd8f724 | Two people are playing a game where the first person flips a fair coin twice. If the first person gets a head on the second flip, they continue flipping until they do not get a head, at which point they pass the coin to the second person. The second person follows the same rules. What is the expected number of flips un... | [
"16"
] | [
"Two people are playing a game where the first person flips a fair coin twice. If the first person gets a head on the second flip, they continue flipping until they do not get a head, at which point they pass the coin to the second person. The second person follows the same rules. What is the expected number of fli... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 5
} | 11 | 0 | 0 |
omr-ws-c3f686146f107b23bf29b4c65dec5df4d2ea280c2f67c2c45609f13a13b50893 | Two people are playing a game where the first person is given 5 cards from a deck that has 4 suits and cards numbered from 1 to 13. The first person gives 4 of these cards to the second person. How can the second person determine the 5th card? | [
"The second person uses the suit of one of the four cards (which must have a duplicate in the original five) and the order of the remaining three cards to encode the rank of the fifth card through modular arithmetic."
] | [
"Two people are playing a game where the first person is given 5 cards from a deck that has 4 suits and cards numbered from 1 to 13. The first person gives 4 of these cards to the second person. How can the second person determine the 5th card?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5c40bcced56c592300ab5236949cca982fe495e0b4e3467cf9c378a191c82c03 | Two people are playing a game where they start with the number 2006 and repeatedly subtract between 1 and 6 from it. The player who reduces the number to 0 wins. Player 1 takes the first turn. Which player should win? | [
"Player 1 wins."
] | [
"Two people are playing a game where they start with the number 2006 and repeatedly subtract between 1 and 6 from it. The player who reduces the number to 0 wins. Player 1 takes the first turn. Which player should win?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-9696b1ee29079c17fdc7a94727bcec67053407eaebec65094559166ade50fb7d | Two people are playing a game. Each person can either roll a die or flip a coin. If any person rolls a $6$, they win. If any person flips a head, they get to go again twice. What is the probability that the first person who plays wins? | [
"\\dfrac{\\sqrt{5} - 1}{2}"
] | [
"Two people are playing a game. Each person can either roll a die or flip a coin. If any person rolls a $6$, they win. If any person flips a head, they get to go again twice. What is the probability that the first person who plays wins?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-210a39e2b8c093505fb318a29cd9d12d37468a5dce076ab4bf66d4c5b80449c6 | Two people are playing with a standard deck of 52 cards, where Jack, Queen, and King are considered 11, 12, and 13 respectively. They shuffle the deck and each draw one card from the top of the deck, showing their cards to each other. The person with the higher number wins that round. They put the cards aside after eac... | [
"\\dfrac{1}{2^{25}}"
] | [
"Two people are playing with a standard deck of 52 cards, where Jack, Queen, and King are considered 11, 12, and 13 respectively. They shuffle the deck and each draw one card from the top of the deck, showing their cards to each other. The person with the higher number wins that round. They put the cards aside afte... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 4
} | 18 | 0 | 0 |
omr-ws-6ac0e31e8759dc257cca56ac447a67dcd9286ac4eede69c0840e185fc0184de0 | Two people are walking randomly on the number line. Each takes a step of length $1$ every minute, with equal probability of going right or left. After $K$ steps, find the probability that they are at the same place. | [
"\\(\\frac{\\binom{2K}{K}}{4^K}\\)"
] | [
"Two people are walking randomly on the number line. Each takes a step of length $1$ every minute, with equal probability of going right or left. After $K$ steps, find the probability that they are at the same place."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-1e3102c72e8384870fc4cf4f013e89cf3a657ff333e19a1654b44fd875899155 | Two people are walking randomly on the number line. Each takes a step of length $1$ every minute, with equal probability of going right or left. After $K$ steps, find the probability that they are two units apart. | [
"\\( 2 \\frac{\\binom{2k}{k-2}}{4^k} \\)"
] | [
"Two people are walking randomly on the number line. Each takes a step of length $1$ every minute, with equal probability of going right or left. After $K$ steps, find the probability that they are two units apart."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c5e6decf8561148a0b84bc6a3c757bf50c9b770151c9f04543e4f06a61501233 | Two people choose a random day to go shopping between November 1 and November 30. Given that one person stays for a random time between 30 and 90 minutes, and the other always stays for one hour, what is the probability that they meet during the period, given that they only shop once and both plan their shopping times ... | [
"\\dfrac{167}{11520}"
] | [
"Two people choose a random day to go shopping between November 1 and November 30. Given that one person stays for a random time between 30 and 90 minutes, and the other always stays for one hour, what is the probability that they meet during the period, given that they only shop once and both plan their shopping t... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-296d4f37c36f05f085e5091975dc9e9be0e42148c36a05b460951ca3114b7d1e | Two people divide four consecutive integers by their respective ages (both integers) and sum the remainders to get 40 in each case. What is the sum of their ages? | [
"59"
] | [
"Two people divide four consecutive integers by their respective ages (both integers) and sum the remainders to get 40 in each case. What is the sum of their ages?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-11a93965da5e981dd2368fcde91c95980f4f2f665eaa35cd29e5553890d347c8 | Two people divide four consecutive integers by their respective ages (both integers) and sum the remainders. In both cases, the sum of the remainders is 40. What is the sum of their ages? | [
"59"
] | [
"Two people divide four consecutive integers by their respective ages (both integers) and sum the remainders. In both cases, the sum of the remainders is 40. What is the sum of their ages?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 6,
"tir": 4
} | 23 | 0 | 0 |
omr-ws-fbfb3002f994d8fefb9acdb4e4f4a6f4c1271a653661b0d995aaa5d0b02c30cb | Two people each put in $10, and the winner receives the full $20. Person A believes that if two coins are flipped repeatedly, the number of times the outcome is 1 head and 1 tail should be at least twice as much as 2 tails or 2 heads. Person B disagrees and believes that the number of 2 tail outcomes should be more tha... | [
"-10"
] | [
"Two people each put in $10, and the winner receives the full $20. Person A believes that if two coins are flipped repeatedly, the number of times the outcome is 1 head and 1 tail should be at least twice as much as 2 tails or 2 heads. Person B disagrees and believes that the number of 2 tail outcomes should be mor... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b627a0b282ad673a8e85b12bc40e8641f52d9d042f7ccb13a625b30fe709d371 | Two people independently pick a whole number from 1 through 9. Whether one number is divided by the other or vice versa, the same remainder is obtained. What is this remainder? | [
"0"
] | [
"Two people independently pick a whole number from 1 through 9. Whether one number is divided by the other or vice versa, the same remainder is obtained. What is this remainder?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-087c35b9ffc1eeb29957370ef892f5c53deebd7bef617ee90af9672b71f8850f | Two people play a 'factorization' game with the number \(270000 = 2^4 \cdot 3^3 \cdot 5^4\). On each turn, a player takes a number \(N\) on the paper, deletes it, and writes two numbers \(X\) and \(Y\) such that \(N = XY\) and \(X\) and \(Y\) are not relatively prime. If the first player always plays optimally, calcula... | [
"46"
] | [
"Two people play a 'factorization' game with the number \\(270000 = 2^4 \\cdot 3^3 \\cdot 5^4\\). On each turn, a player takes a number \\(N\\) on the paper, deletes it, and writes two numbers \\(X\\) and \\(Y\\) such that \\(N = XY\\) and \\(X\\) and \\(Y\\) are not relatively prime. If the first player always pla... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b8ed4bdfdb3e9c44badde319be5651975bd52f8b4792d16b100d70e7d3787ea6 | Two people play a game in which they alternate turns and the probability of the player whose turn it is winning on turn \( n \) is \( \frac{n}{n+2} \). What is the probability that the second player to take a turn ends up winning the game? Note that the first player's first turn is turn 1, the second person's first tur... | [
"\\dfrac{1 - e^{-2}}{2}"
] | [
"Two people play a game in which they alternate turns and the probability of the player whose turn it is winning on turn \\( n \\) is \\( \\frac{n}{n+2} \\). What is the probability that the second player to take a turn ends up winning the game? Note that the first player's first turn is turn 1, the second person's... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 8
} | 18 | 0 | 0 |
omr-ws-81f068b37ba710df33beb5e220472e51b05ccefc0a80b0f046e6045de2054ec9 | Two people play a game in which they take turns rolling a single six-sided die. If someone rolls the number their opponent just rolled or any multiple of that number, they win. What is the probability that the first player wins the game on their second turn? | [
"\\( \\frac{59}{216} \\)"
] | [
"Two people play a game in which they take turns rolling a single six-sided die. If someone rolls the number their opponent just rolled or any multiple of that number, they win. What is the probability that the first player wins the game on their second turn?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 23
} | 38 | 0 | 0 |
omr-ws-9f9acbb83c55657d5799af063cc101115ec5d8baf8f70c406760d181bd61b447 | Two people play a game on a graph with 2022 points. Initially, there are no edges in the graph. They take turns and connect two non-neighbouring vertices with an edge. Whoever makes the graph connected loses. Which player has a winning strategy? | [
"First player (Alice) has a winning strategy."
] | [
"Two people play a game on a graph with 2022 points. Initially, there are no edges in the graph. They take turns and connect two non-neighbouring vertices with an edge. Whoever makes the graph connected loses. Which player has a winning strategy?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-eb44b38b1101d1125b8d50c3c14a0b9617e1204373ddc30ab3915b74fc10f49f | Two people play a game starting with the number $2004$. The first player subtracts any of its integer divisors from the number, and the result is passed to the opponent who must also subtract one of its integer divisors, and so on. The game is lost by the person who gets $0$ after completing their turn. Describe a stra... | [
"Always subtract an even divisor to keep the number even."
] | [
"Two people play a game starting with the number $2004$. The first player subtracts any of its integer divisors from the number, and the result is passed to the opponent who must also subtract one of its integer divisors, and so on. The game is lost by the person who gets $0$ after completing their turn. Describe a... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-a95a1ef23ff14cb91a0131c93ddb8d341ec4baf98160fe5ebd08fa027db7ddea | Two people play a game where the first person chooses \( n \) numbers and announces the sums of every pair of these numbers (some sums may be equal). Let \( g(n) \) be the smallest value of \( n \) for which the second person cannot uniquely determine the original \( n \) numbers. Find \( g(4) \). | [
"4"
] | [
"Two people play a game where the first person chooses \\( n \\) numbers and announces the sums of every pair of these numbers (some sums may be equal). Let \\( g(n) \\) be the smallest value of \\( n \\) for which the second person cannot uniquely determine the original \\( n \\) numbers. Find \\( g(4) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-c511bb6f4d1ed21d4f95f65f94fb4a4f288a8459a3e728f544a2e808cf6f1e78 | Two people play a game where they alternately take turns writing down successive terms of a sequence. The sequence begins $1, 2, 3$, and every term thereafter must be the sum of the term immediately preceding it and any term before that. The winner is the player who first writes down a number greater than or equal to $... | [
"\\text{First player}"
] | [
"Two people play a game where they alternately take turns writing down successive terms of a sequence. The sequence begins $1, 2, 3$, and every term thereafter must be the sum of the term immediately preceding it and any term before that. The winner is the player who first writes down a number greater than or equal... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-bae926cff44e57593e7ef015d1dad05c29c33cf161d255fb77db10d0980e614d | Two people play a game where they alternately take turns writing down successive terms of a sequence. The sequence begins $1, 2, 3$, and every term thereafter must be the sum of the term immediately preceding it and any term before that. The winner is the player who first writes down a number greater than or equal to $... | [
"Second player has a winning strategy."
] | [
"Two people play a game where they alternately take turns writing down successive terms of a sequence. The sequence begins $1, 2, 3$, and every term thereafter must be the sum of the term immediately preceding it and any term before that. The winner is the player who first writes down a number greater than or equal... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 2,
"tir": 11
} | 17 | 0 | 0 |
omr-ws-808b747806c27c89f7a1717fc08a5b7658b1b83454864a6bf054e82e456ef633 | Two people play a game where they alternately take turns writing down successive terms of a sequence. The sequence begins $1,2,3$, and every term thereafter must be the sum of the term immediately preceding it and any term before that. So, for instance, the first player begins play by writing $4=3+1$ or $5=3+2$; and th... | [
"The first player has a winning strategy."
] | [
"Two people play a game where they alternately take turns writing down successive terms of a sequence. The sequence begins $1,2,3$, and every term thereafter must be the sum of the term immediately preceding it and any term before that. So, for instance, the first player begins play by writing $4=3+1$ or $5=3+2$; a... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 7,
"tir": 9
} | 31 | 0 | 0 |
omr-ws-863707e603b50ef67e570c74bc083bf2fe39b85b96935e0b864ed7832013c83a | Two people play a game where they alternately take turns writing down successive terms of a sequence. The sequence begins \( 1, 2, 3 \), and every term thereafter must be the sum of the term immediately preceding it and any term before that. So, for instance, the first player begins play by writing \( 4 = 3 + 1 \) or \... | [
"\\text{First player}"
] | [
"Two people play a game where they alternately take turns writing down successive terms of a sequence. The sequence begins \\( 1, 2, 3 \\), and every term thereafter must be the sum of the term immediately preceding it and any term before that. So, for instance, the first player begins play by writing \\( 4 = 3 + 1... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-47244bee72f126c4f79af78862b603e790df718eea51315ad99aa405a0e5bc4e | Two people play a game with 2008 bags numbered from 1 to 2008, each containing 2008 frogs. On each turn, a player selects a bag and removes any number of frogs (at least one), leaving $x$ frogs in it ($x \geq 0$). After each turn, from each bag with a number higher than the selected one and having more than $x$ frogs, ... | [
"2007"
] | [
"Two people play a game with 2008 bags numbered from 1 to 2008, each containing 2008 frogs. On each turn, a player selects a bag and removes any number of frogs (at least one), leaving $x$ frogs in it ($x \\geq 0$). After each turn, from each bag with a number higher than the selected one and having more than $x$ f... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-557258fd9a19392fc0a794244905fd69bd34d1927fd254180e9505840d3e8032 | Two people play a game with 2010 coins. On each turn, a player can take 1, 2, 4, 8, ..., or \(2^n\) coins. The player who takes the last coin wins. Who is the winner? | [
"The first player is the winner."
] | [
"Two people play a game with 2010 coins. On each turn, a player can take 1, 2, 4, 8, ..., or \\(2^n\\) coins. The player who takes the last coin wins. Who is the winner?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3c46d72eeb6aed0dfecbbdef279b62e56c5d7a1135e23a8c683f50f70d02312e | Two people play a game with 2011 candies on the table. On each turn, a player can take 1, 2, or 6 candies. The player who picks the last candy wins. Assuming both players play optimally, who has the winning strategy? | [
"The first player has the winning strategy."
] | [
"Two people play a game with 2011 candies on the table. On each turn, a player can take 1, 2, or 6 candies. The player who picks the last candy wins. Assuming both players play optimally, who has the winning strategy?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-2227a17d3b95a1995a58dec05da048fe28317e1549413b0bc4ff6bf6692c40b7 | Two people play a game with 40 cards numbered from 1 to 10, each with 4 different signs. Each player starts with 20 cards. Each turn, a player either places a card on the table or removes cards from the table whose sum is 15. At the end of the game, one player has a 5 and a 3 in their hand, and there is a 9 on the tabl... | [
"8"
] | [
"Two people play a game with 40 cards numbered from 1 to 10, each with 4 different signs. Each player starts with 20 cards. Each turn, a player either places a card on the table or removes cards from the table whose sum is 15. At the end of the game, one player has a 5 and a 3 in their hand, and there is a 9 on the... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-87f57ef1ad512b46946fdfed56cd1c630643fbf50848be9efe654ec9ccb4a3e4 | Two people play a game with red and blue cards and two boxes. The first player distributes a number of red and blue cards between the two boxes. The second player then chooses one box and takes all the cards in that box, while doubling the number of cards of each color in the other box and giving them to the first play... | [
"a = 2b \\text{ or } b = 2a \\text{ or } a + b \\equiv 0 \\pmod{3}"
] | [
"Two people play a game with red and blue cards and two boxes. The first player distributes a number of red and blue cards between the two boxes. The second player then chooses one box and takes all the cards in that box, while doubling the number of cards of each color in the other box and giving them to the first... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 4
} | 13 | 0 | 0 |
omr-ws-37335d5f4f7517f65eb5f82ff9bb2c707fa8812f2c51351d6ed66c16d43f7087 | Two people play the following game. The first player arbitrarily arranges the numbers $1, 2, \ldots, 2007$ in a row. A move consists of choosing three adjacent numbers $a$, $b$, and $c$ in the row, and replacing them with the numbers $b$ and $a+c$. Beginning with the second player, the two players alternately make move... | [
"2006"
] | [
"Two people play the following game. The first player arbitrarily arranges the numbers $1, 2, \\ldots, 2007$ in a row. A move consists of choosing three adjacent numbers $a$, $b$, and $c$ in the row, and replacing them with the numbers $b$ and $a+c$. Beginning with the second player, the two players alternately mak... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 1
} | 6 | 0 | 0 |
omr-ws-6170cfedc163fd895fae8add96c11c263dea65506d60f6268a5675d412f4ee44 | Two people play the following game: there is a chip on the central node of a $10 \times 10$ checkered square. In one move, you can move a chip to any other node of the square, including on the border, but in such a way that the length of the move is greater than the length of the previous move made by the opponent. The... | [
"\\text{Second player}"
] | [
"Two people play the following game: there is a chip on the central node of a $10 \\times 10$ checkered square. In one move, you can move a chip to any other node of the square, including on the border, but in such a way that the length of the move is greater than the length of the previous move made by the opponen... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 8
} | 16 | 0 | 0 |
omr-ws-dcde1768f56e0debc05462ad6f566b4b23e5c8c7398ae13d88d11bc57a34acb0 | Two people randomly think of a real number between 0 and 100. What is the expected difference between their numbers? | [
"\\(\\frac{100}{3}\\)"
] | [
"Two people randomly think of a real number between 0 and 100. What is the expected difference between their numbers?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 32
} | 47 | 0 | 0 |
omr-ws-f75c1938df8dbf0c615184de5a914984b10489fa23743457a9e5009dd591d6f5 | Two people stand anywhere on a line segment of length 1. They can communicate and their goal is to find each other. What is the minimal sum of distances they need to walk to ensure they find each other every time? | [
"1"
] | [
"Two people stand anywhere on a line segment of length 1. They can communicate and their goal is to find each other. What is the minimal sum of distances they need to walk to ensure they find each other every time?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-ee6d9f67a8fcb193daa33a52d85a8be1f7556d31c66bbbbf764b5adc390cdc47 | Two people stand back to back next to the rails in a small railway station. As the head of the express train (which does not stop at the station) reaches them, they start to walk in opposite directions parallel to the rails. As the tail of the train reaches each of them, they stop, having walked 30 meters and 40 meters... | [
"240m"
] | [
"Two people stand back to back next to the rails in a small railway station. As the head of the express train (which does not stop at the station) reaches them, they start to walk in opposite directions parallel to the rails. As the tail of the train reaches each of them, they stop, having walked 30 meters and 40 m... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-34a49b1fcce06c5f26e0fb3ee179583acc0b1c43a36d1d8aac256820581cc56f | Two people take turns cutting up a rectangular $m \times n$ chocolate bar, where $m$ and $n$ are positive integers. You are allowed to cut only along a division between the squares and the cut can only be in a straight line. The last player who can break the chocolate wins. For which values of $m$ and $n$ does the firs... | [
"850"
] | [
"Two people take turns cutting up a rectangular $m \\times n$ chocolate bar, where $m$ and $n$ are positive integers. You are allowed to cut only along a division between the squares and the cut can only be in a straight line. The last player who can break the chocolate wins. For which values of $m$ and $n$ does th... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-ecbfcfe58bd0921f68f7a4509188ef1373cbf919b252af63427126b698fae74d | Two people take turns painting a square, a domino, or a corner of three cells on a \( 9 \times 10 \) board. You can only paint over previously unpainted cells. The one who cannot make a move loses. Who wins when played correctly? | [
"\\text{First player}"
] | [
"Two people take turns painting a square, a domino, or a corner of three cells on a \\( 9 \\times 10 \\) board. You can only paint over previously unpainted cells. The one who cannot make a move loses. Who wins when played correctly?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 6
} | 15 | 0 | 0 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.