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-7c6214d8d01684f2cb55f9d741a1ddbd28ad63eef0ad0e7f1d18e68289fc805f | A cylinder has a radius of 3 cm and a height of 8 cm. What is the longest segment, in centimeters, that would fit inside the cylinder? | [
"10"
] | [
"A cylinder has a radius of 3 cm and a height of 8 cm. What is the longest segment, in centimeters, that would fit inside the cylinder?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7c9e74aa1a2aed1f75697789d590d86443fedc80022eada36464d0c42522b2ca | A cylinder has radius $6$ and height $12$. An equilateral triangle is inscribed in both bases of the cylinder, and a triangular prism is removed from the cylinder by drilling perpendicular to the bases through the image of the equilateral triangles. If the surface area of the new figure can be expressed as $a\pi + b\sq... | [
"217"
] | [
"A cylinder has radius $6$ and height $12$. An equilateral triangle is inscribed in both bases of the cylinder, and a triangular prism is removed from the cylinder by drilling perpendicular to the bases through the image of the equilateral triangles. If the surface area of the new figure can be expressed as $a\\pi ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-993dbf28d17ed0908e3e00b4289727c13e41b4c60e41c42b86dc714592b4b22d | A cylinder is 10 inches tall and has a volume of 785 cubic inches. What is the radius of the cylinder? | [
"\\( r = \\sqrt{\\frac{785}{10\\pi}} \\)"
] | [
"A cylinder is 10 inches tall and has a volume of 785 cubic inches. What is the radius of the cylinder?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 2,
"tir": 6
} | 17 | 0 | 0 |
omr-ws-82462ecfaf2ed90258ac8fa117d9c2727ca2d7563fad1f0bcd5fc854f5d47cfe | A cylinder is inscribed in a cone with height 10 and base radius 5. If \( r \) is the radius of the cylinder and \( h \) is the height of the cylinder, find the values of \( r \) and \( h \) that maximize the volume of the cylinder. | [
"\\( r = \\frac{10}{3}, h = \\frac{10}{3} \\)"
] | [
"A cylinder is inscribed in a cone with height 10 and base radius 5. If \\( r \\) is the radius of the cylinder and \\( h \\) is the height of the cylinder, find the values of \\( r \\) and \\( h \\) that maximize the volume of the cylinder."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 6,
"tir": 29
} | 42 | 0 | 0 |
omr-ws-1bdbb7b15e7d74405285033585f15861d24ca713f2c70e93a797e646c1e6b611 | A cylinder is inscribed in a cone with radius \( R \) and height \( H \). The height of the cylinder is perpendicular to the height of the cone. Find the dimensions of the cylinder that maximize its volume. | [
"\\frac{2R}{3}"
] | [
"A cylinder is inscribed in a cone with radius \\( R \\) and height \\( H \\). The height of the cylinder is perpendicular to the height of the cone. Find the dimensions of the cylinder that maximize its volume."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-54f8c6ab1aab32a931bd1eb4ad926e2b23dad6669af4e17bb8edd9ed830173ff | A cylinder is inscribed in a cone, the altitude of the cylinder being equal to the radius of the base of the cone. Find the angle between the axis of the cone and its element if the ratio of the total surface area of the cylinder to the area of the base of the cone is $3:2$. | [
"\\(\\alpha = \\arctan\\frac{1}{2}\\)"
] | [
"A cylinder is inscribed in a cone, the altitude of the cylinder being equal to the radius of the base of the cone. Find the angle between the axis of the cone and its element if the ratio of the total surface area of the cylinder to the area of the base of the cone is $3:2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b117d83db93fce227ed232e9c9e60fdd25e74bae01a846237d1ca870dc39b142 | A cylinder is inscribed in a sphere of radius 1. a. Express the volume \( V \) of the cylinder as a function of the base radius \( r \). b. Find the radius and height of the cylinder having the maximum volume. | [
"\\dfrac{2\\sqrt{3}}{3}"
] | [
"A cylinder is inscribed in a sphere of radius 1.\na. Express the volume \\( V \\) of the cylinder as a function of the base radius \\( r \\).\nb. Find the radius and height of the cylinder having the maximum volume."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-39def7a0a2321588b399fbc6990334509d152019eb6bfd58a3d5e4beea36cb25 | A cylinder is inscribed in a sphere. The diameter of the cylinder is 13 cm, and its height is 13 cm. Find the volume of the sphere. | [
"\\(\\frac{2197 \\pi \\sqrt{2}}{3}\\)"
] | [
"A cylinder is inscribed in a sphere. The diameter of the cylinder is 13 cm, and its height is 13 cm. Find the volume of the sphere."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-96ac52be792540004dbff5e44f5c7d1dcdc84054eddc952bf1ad1376e6cdde3a | A cylinder is inscribed within a sphere of radius 10 such that its volume is $\frac{1}{2} + \frac{1}{250}$ times the volume of the sphere. Find the sum of all possible heights for the cylinder. | [
"4 + 4\\sqrt{22}"
] | [
"A cylinder is inscribed within a sphere of radius 10 such that its volume is $\\frac{1}{2} + \\frac{1}{250}$ times the volume of the sphere. Find the sum of all possible heights for the cylinder."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d3242a1c1463bee7ecf0be089c6117746374bcd762604b2d05489b2110ecb486 | A cylinder is inside a cube, which is inside a sphere, which is inside a cube, which is inside a sphere. If the smaller sphere has radius $x,$ what is the ratio of the volumes of the bigger sphere and the cylinder? | [
"18:1"
] | [
"A cylinder is inside a cube, which is inside a sphere, which is inside a cube, which is inside a sphere. If the smaller sphere has radius $x,$ what is the ratio of the volumes of the bigger sphere and the cylinder?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-b9964f25a2c3ee3fdd6cfce2e9878b4f889fce9b72c77c9fdacf2b37f0f6f7d2 | A cylinder is obtained by revolving a rectangle about the $x$-axis, where the base of the rectangle lies on the $x$-axis and the entire rectangle is between the curve $f(x) = \frac{x}{(x^2 + 1)}$ and the $x$-axis. Find the maximum possible volume of the cylinder. | [
"\\(\\frac{\\pi}{4}\\)"
] | [
"A cylinder is obtained by revolving a rectangle about the $x$-axis, where the base of the rectangle lies on the $x$-axis and the entire rectangle is between the curve $f(x) = \\frac{x}{(x^2 + 1)}$ and the $x$-axis. Find the maximum possible volume of the cylinder."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 26
} | 40 | 0 | 0 |
omr-ws-730836607851bd12483d64483db77f19a2e4dcdb45a81e18baf1dbb23a0ec5cd | A cylinder is obtained by rotating a rectangle around the x-axis, where the base of the rectangle is supported and its upper vertices are on the curve \( y = \frac{x}{x^2 + 1} \). What is the maximum volume that such a cylinder can have? | [
"\\(\\frac{\\pi}{4}\\)"
] | [
"A cylinder is obtained by rotating a rectangle around the x-axis, where the base of the rectangle is supported and its upper vertices are on the curve \\( y = \\frac{x}{x^2 + 1} \\). What is the maximum volume that such a cylinder can have?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 6,
"tir": 20
} | 41 | 0 | 0 |
omr-ws-3cbc4466aae15f55873ccd2efa4c862772aff6f5a6d9d918816b872557969fc4 | A cylinder is tangent to the inside of a sphere with radius 10. The cross-section of the sphere is a circle with radius 2. What is the surface area of the part of the cylinder contained inside the sphere? | [
"32\\pi\\sqrt{6}"
] | [
"A cylinder is tangent to the inside of a sphere with radius 10. The cross-section of the sphere is a circle with radius 2. What is the surface area of the part of the cylinder contained inside the sphere?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 3
} | 15 | 0 | 0 |
omr-ws-9e2d6351f8ae6d344bc8dc00e61de256cbd0bcbb5a01aeddaae4e07109c6f51e | A cylinder of maximum surface area is inscribed within a sphere of radius 10. What is the surface area of this cylinder? | [
"100\\pi(1 + \\sqrt{5})"
] | [
"A cylinder of maximum surface area is inscribed within a sphere of radius 10. What is the surface area of this cylinder?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-81436ccf93da10d1018110388ec0d0d7defeabec9adad43f8abed81c2ab21b85 | A cylinder of maximum volume is inscribed within a sphere of radius 10. What is the surface area of this cylinder? | [
"\\( 100\\pi (\\sqrt{5} + 1) \\)"
] | [
"A cylinder of maximum volume is inscribed within a sphere of radius 10. What is the surface area of this cylinder?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0840456fd409868b2a02209ec6128158362b0ee3a16756665b271096fc016abe | A cylinder of radius $1$ passes through a sphere of radius $3$ such that the center axis of the cylinder passes through the center of the sphere. Find the volume $V$ cut out of the sphere. | [
"\\( 4\\pi \\left( 9 - \\frac{16\\sqrt{2}}{3} \\right) \\)"
] | [
"A cylinder of radius $1$ passes through a sphere of radius $3$ such that the center axis of the cylinder passes through the center of the sphere. Find the volume $V$ cut out of the sphere."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 7,
"tir": 21
} | 43 | 0 | 0 |
omr-ws-5c82dca6fce3a7a155be77c623fc0817dddb54675be52be82fbb27a33f77f439 | A cylinder with a diameter of 27 and a height of 30 contains two lead spheres with radii 6 and 9, with the larger sphere sitting on the bottom of the cylinder. Water is poured into the cylinder so that it just covers both spheres. Find the volume of water required. | [
"\\dfrac{8415}{2} \\pi"
] | [
"A cylinder with a diameter of 27 and a height of 30 contains two lead spheres with radii 6 and 9, with the larger sphere sitting on the bottom of the cylinder. Water is poured into the cylinder so that it just covers both spheres. Find the volume of water required."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-80c395a16b24dbbafdadbe9bd28805e30a7937a5148890480333abaa34891c57 | A cylinder with a diameter of 74 mm and a length of 100 mm is inclined at an angle of 72 degrees. An horizontal plane, located 50 mm from the bottom of the cylinder, intersects the cylinder. Compute the length and width of the ellipse formed by this intersection. | [
"Width = 74 mm, Length = 30.9 mm"
] | [
"A cylinder with a diameter of 74 mm and a length of 100 mm is inclined at an angle of 72 degrees. An horizontal plane, located 50 mm from the bottom of the cylinder, intersects the cylinder. Compute the length and width of the ellipse formed by this intersection."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e7486d9d9d3f65333867cc6fab8abee78eb51539229eed34768c9f19ab15d8f8 | A cylinder with a height of 10 cm and a radius of 6 cm is rotating at 1 revolution per minute. A paintbrush 4 cm wide starts at the top and moves vertically down the cylinder at 1 cm per minute. Compute the time in minutes it takes for the paintbrush to paint three-fifths of the lateral surface area of the cylinder. | [
"3"
] | [
"A cylinder with a height of 10 cm and a radius of 6 cm is rotating at 1 revolution per minute. A paintbrush 4 cm wide starts at the top and moves vertically down the cylinder at 1 cm per minute. Compute the time in minutes it takes for the paintbrush to paint three-fifths of the lateral surface area of the cylinde... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 3
} | 5 | 0 | 0 |
omr-ws-51ea810657300b48dbcb8b826dfdd0846a4e6333b1c65b5901272e57e17c5184 | A cylinder with height \( H \) and radius \( R \) is filled with water. A hole with radius \( r \) is drilled into the cylinder at a distance \( h \) above the ground. Water pours out of the hole and reaches a distance \( x \) along the ground. Express \( x \) in terms of \( H \), \( R \), \( r \), and \( h \). | [
"\\( x = 2 \\sqrt{h (H - h)} \\)"
] | [
"A cylinder with height \\( H \\) and radius \\( R \\) is filled with water. A hole with radius \\( r \\) is drilled into the cylinder at a distance \\( h \\) above the ground. Water pours out of the hole and reaches a distance \\( x \\) along the ground. Express \\( x \\) in terms of \\( H \\), \\( R \\), \\( r \\... | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 9
} | 9 | 0 | 0 |
omr-ws-222019b84b16a0f1bbb75fc2b506e78fe8a5b1c307c9044a35f356eb40ac4212 | A cylinder with radius $24$ and height $18$ has points $X$ and $Y$ on its top face such that the minor arc $XY$ subtends an angle of $120^\circ$. A plane slices the cylinder through $X$, $Y$, and the center of the cylinder. The area of the shape the plane cuts through is expressed as $G \sqrt{B} + F\pi$, where $G$, $B$... | [
"603"
] | [
"A cylinder with radius $24$ and height $18$ has points $X$ and $Y$ on its top face such that the minor arc $XY$ subtends an angle of $120^\\circ$. A plane slices the cylinder through $X$, $Y$, and the center of the cylinder. The area of the shape the plane cuts through is expressed as $G \\sqrt{B} + F\\pi$, where ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e1950582baaaecbeaae3c39394bcc9cfedd9b70a5c33eb0994bbb2d170e2f6d8 | A cylinder with radius $5$ and height $1$ is rolling on the floor. Inside the cylinder, there is water that maintains a constant height of $\frac{15}{2}$. What is the volume of the water? | [
"\\(\\frac{50\\pi}{3}+\\frac{25\\sqrt{3}}{4}\\)"
] | [
"A cylinder with radius $5$ and height $1$ is rolling on the floor. Inside the cylinder, there is water that maintains a constant height of $\\frac{15}{2}$. What is the volume of the water?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-d7366b4c1e468eb4f4e6ccb38d5a406aa4425da85e8e364774f8417f3adf1994 | A cylindrical barrel 50cm high with a radius of 18cm is filled with water up to 1cm from the brim. A solid sphere with radius 9cm is placed into the water so that it is half submerged. How much water spills out? | [
"\\( 162\\pi \\) cubic centimeters or approximately \\( 508.938 \\) mL."
] | [
"A cylindrical barrel 50cm high with a radius of 18cm is filled with water up to 1cm from the brim. A solid sphere with radius 9cm is placed into the water so that it is half submerged. How much water spills out?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-2fc9f3a63d3bbe7c22208c8dab610ad24d72a14f5a03a84d0aef1adf1c31072f | A cylindrical barrel with radius $4$ feet and height $10$ feet is full of water. A solid cube with side length $8$ feet is set into the barrel so that the diagonal of the cube is vertical. The volume of water thus displaced is $v$ cubic feet. Find $v^2$. | [
"384"
] | [
"A cylindrical barrel with radius $4$ feet and height $10$ feet is full of water. A solid cube with side length $8$ feet is set into the barrel so that the diagonal of the cube is vertical. The volume of water thus displaced is $v$ cubic feet. Find $v^2$."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 6
} | 8 | 0 | 0 |
omr-ws-9bca14887ad83d31116a23e7fd6f27de9bb5c724b33efa78ba2d651c2f972bc4 | A cylindrical barrel with radius $4$ feet and height $10$ feet is full of water. A solid cube with side length $8$ feet is set into the barrel so that the diagonal of the cube is vertical. The volume of water thus displaced is $v$ cubic feet. Find $v^2$. [asy] import three; import solids; size(5cm); currentprojection=o... | [
"384"
] | [
"A cylindrical barrel with radius $4$ feet and height $10$ feet is full of water. A solid cube with side length $8$ feet is set into the barrel so that the diagonal of the cube is vertical. The volume of water thus displaced is $v$ cubic feet. Find $v^2$.\n[asy] import three; import solids; size(5cm); currentprojec... | [
"MATH_training_set"
] | [
"additional_problems",
"cot",
"genselect"
] | {
"additional_problems": 1,
"cot": 1,
"genselect": 2
} | 4 | 0 | 0 |
omr-ws-da49ba744bf9703652bc9cdca819dd2244b7427b3575e93918d309d7c7f62acf | A cylindrical bearing with radius \( r = 2 \) cm and length \( l = 6 \) cm is immersed into a brass solution, resulting in a brass coating with thickness varying from 0.02 cm to 0.06 cm. Find an approximation for the volume of the coated bearing. | [
"77.15 < Volume < 78.916"
] | [
"A cylindrical bearing with radius \\( r = 2 \\) cm and length \\( l = 6 \\) cm is immersed into a brass solution, resulting in a brass coating with thickness varying from 0.02 cm to 0.06 cm. Find an approximation for the volume of the coated bearing."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 19
} | 19 | 0 | 0 |
omr-ws-6026a9da5f006b18d3179087dbd2bd8219c815ff9a69ca740d3d4713f5c291ca | A cylindrical bucket without a top has a circumference of 6 inches and a height of 4 inches. On the inside of the bucket, one inch from the top, is a drop of honey; on the opposite side of the bucket, one inch from the bottom, on the outside, is an ant. What is the shortest distance the ant could crawl to reach the hon... | [
"\\(\\sqrt{13}\\)"
] | [
"A cylindrical bucket without a top has a circumference of 6 inches and a height of 4 inches. On the inside of the bucket, one inch from the top, is a drop of honey; on the opposite side of the bucket, one inch from the bottom, on the outside, is an ant. What is the shortest distance the ant could crawl to reach th... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 3
} | 4 | 0 | 0 |
omr-ws-d227a146c9f110cf90b2c4c3728d03c6a84545af834631f54f98de82f315f27b | A cylindrical can with a radius of 6 inches is filled with 2 inches of water. When a smaller cylindrical can 4 inches tall is placed inside of the first with its bottom lying on the bottom of the first, the water in the first can rises a further inch to 3 inches. What is the radius of the smaller can? | [
"\\(2\\sqrt{3}\\)"
] | [
"A cylindrical can with a radius of 6 inches is filled with 2 inches of water. When a smaller cylindrical can 4 inches tall is placed inside of the first with its bottom lying on the bottom of the first, the water in the first can rises a further inch to 3 inches. What is the radius of the smaller can?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-233ca9d745b460e53b0a87239f7f611103e503793edf2c2c99e6b108637a7fa9 | A cylindrical container has a radius of 3 inches and a height of 5.5 inches. If racquetballs have a radius of 1 inch, what is the maximum number of racquetballs that will fit in the container? Assume none of the racquetballs are compressed. | [
"20"
] | [
"A cylindrical container has a radius of 3 inches and a height of 5.5 inches. If racquetballs have a radius of 1 inch, what is the maximum number of racquetballs that will fit in the container? Assume none of the racquetballs are compressed."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 6,
"tir": 1
} | 9 | 0 | 0 |
omr-ws-be9ca948f4adaef5c8a2dfdf2122c2abc965ffd8352f17ed70d1507e703574d7 | A cylindrical container has height $6 \text{ cm}$ and radius $4 \text{ cm}$. It rests on a circular hoop, also of radius $4 \text{ cm}$, fixed in a horizontal plane with its axis vertical and with each circular rim of the cylinder touching the hoop at two points. The cylinder is now moved so that each of its circular r... | [
"A circle of radius \\(4 \\text{ cm}\\) in a plane parallel to the hoop and at a height of \\(6 \\text{ cm}\\) above the hoop."
] | [
"A cylindrical container has height $6 \\text{ cm}$ and radius $4 \\text{ cm}$. It rests on a circular hoop, also of radius $4 \\text{ cm}$, fixed in a horizontal plane with its axis vertical and with each circular rim of the cylinder touching the hoop at two points. The cylinder is now moved so that each of its ci... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 1
} | 2 | 0 | 0 |
omr-ws-83d89023f89a1fedeab8d1d2f38db72f329174f65155a66a26739cfe7aaa1bf0 | A cylindrical container has height $6$ cm and radius $4$ cm. It rests on a circular hoop with radius $4$ cm, which is fixed in a horizontal plane. The cylinder's axis is horizontal, and each of its circular rims touches the hoop at two points. When the cylinder is moved while maintaining contact with the hoop at two po... | [
"The locus is a circle with radius \\(4\\) cm centered at the center of the hoop."
] | [
"A cylindrical container has height $6$ cm and radius $4$ cm. It rests on a circular hoop with radius $4$ cm, which is fixed in a horizontal plane. The cylinder's axis is horizontal, and each of its circular rims touches the hoop at two points. When the cylinder is moved while maintaining contact with the hoop at t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 4,
"tir": 1
} | 7 | 0 | 0 |
omr-ws-b5256edfb6c6771a9af0b5d62e9e21be1cde3a824ad7f11ebd5c2d260538300e | A cylindrical container is 10 units high and 4 units in diameter. It is partially filled with water and then tilted so that the water level reaches 9 units up the side at the highest point and 3 units up at the lowest point. Compute the volume of the water in the cylinder. | [
"\\( 24\\pi \\)"
] | [
"A cylindrical container is 10 units high and 4 units in diameter. It is partially filled with water and then tilted so that the water level reaches 9 units up the side at the highest point and 3 units up at the lowest point. Compute the volume of the water in the cylinder."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 24
} | 39 | 0 | 0 |
omr-ws-aa7a53bec22cdd423c1763de90660dc108f7153eec714844bbefed2993c6e1ac | A cylindrical glass beaker is 8 cm high and has a circumference of 12 cm. Inside the beaker, 3 cm from the edge, there is a tiny drop of honey. On the outer surface of the beaker, in the plane passing through the axis of the cylinder and the drop of honey, and located 1 cm from the base, there is a fly. What is the sho... | [
"\\(\\sqrt{193}\\)"
] | [
"A cylindrical glass beaker is 8 cm high and has a circumference of 12 cm. Inside the beaker, 3 cm from the edge, there is a tiny drop of honey. On the outer surface of the beaker, in the plane passing through the axis of the cylinder and the drop of honey, and located 1 cm from the base, there is a fly. What is th... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a4631bacbd154c9dcd47d15371560943687e12ec14d5d680d4e9fb389a6a08a0 | A cylindrical glass filled to the brim with water stands on a horizontal plane. The height of the glass is $2$ times the diameter of the base. At what angle must the glass be tilted from the vertical so that exactly $1/3$ of the water it contains pours out? | [
"\\(\\arctan\\frac{4}{3}\\)"
] | [
"A cylindrical glass filled to the brim with water stands on a horizontal plane. The height of the glass is $2$ times the diameter of the base. At what angle must the glass be tilted from the vertical so that exactly $1/3$ of the water it contains pours out?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 5,
"tir": 3
} | 13 | 0 | 0 |
omr-ws-57bbd7ca071fa4be09cacab641da14ed51a166d7140ceafd8ba56f57ec9b9761 | A cylindrical glass is half full of lemonade. The ratio of lemon juice to water in the lemonade is 1:11. If the glass is 6 inches tall and has a diameter of 2 inches, what is the volume of lemon juice in the glass? Express your answer as a decimal to the nearest hundredth. | [
".79"
] | [
"A cylindrical glass is half full of lemonade. The ratio of lemon juice to water in the lemonade is 1:11. If the glass is 6 inches tall and has a diameter of 2 inches, what is the volume of lemon juice in the glass? Express your answer as a decimal to the nearest hundredth."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a720323239c0050fd5ad0bfe761a30535eeb05e5a97cba55d9169c06158bfae8 | A cylindrical glass of radius \( r \) and height \( L \) is filled with water and then tilted until the water exactly covers half the base. Find the volume of water in the glass. | [
"\\( \\frac{2}{3} L r^2 \\)"
] | [
"A cylindrical glass of radius \\( r \\) and height \\( L \\) is filled with water and then tilted until the water exactly covers half the base. Find the volume of water in the glass."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 6,
"tir": 4
} | 21 | 0 | 0 |
omr-ws-75b492ce6925ac160a9febe2381b2a912ca19f78c537db565539166eb1925da4 | A cylindrical glass of radius \( r \) and height \( L \) is filled with water and then tilted until the water remaining in the glass exactly covers half the base. Use cross sections that are circular segments to determine the volume of water in the cylinder. | [
"\\(\\frac{2}{3}r^{2}L\\)"
] | [
"A cylindrical glass of radius \\( r \\) and height \\( L \\) is filled with water and then tilted until the water remaining in the glass exactly covers half the base. Use cross sections that are circular segments to determine the volume of water in the cylinder."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 6,
"tir": 9
} | 28 | 0 | 0 |
omr-ws-4feca83c9c8f6fadfe992ca68de754b38dec2cce12479d374e9ab654b01d7ad1 | A cylindrical hole is drilled through the center of a sphere of radius \( R \). Use the method of cylindrical shells to find the volume of the remaining solid, given that the solid is 6 cm high. | [
"\\( 36\\pi \\)"
] | [
"A cylindrical hole is drilled through the center of a sphere of radius \\( R \\). Use the method of cylindrical shells to find the volume of the remaining solid, given that the solid is 6 cm high."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 27
} | 41 | 0 | 0 |
omr-ws-b17ce59272574840f564a76a05e660407462c093a852c47ffb833ef38349b006 | A cylindrical hole of radius 1 is drilled along one of the diagonals of a cube with side length 3. Find the area of one of the six congruent faces of the resulting solid. (Express your answer in simplest radical form in terms of \(\pi\)). | [
"\\( 9 - \\frac{\\pi \\sqrt{3}}{3} \\)"
] | [
"A cylindrical hole of radius 1 is drilled along one of the diagonals of a cube with side length 3. Find the area of one of the six congruent faces of the resulting solid. (Express your answer in simplest radical form in terms of \\(\\pi\\))."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 1
} | 3 | 0 | 0 |
omr-ws-31171c17441139120782117de3cc3210100a7322e73f670f03ebe2f53a4b2455 | A cylindrical hole of radius 1 is drilled along one of the long diagonals of a cube of side length 3. Find the area of one of the six congruent faces of the resulting solid. | [
"\\( 9 - \\frac{\\pi \\sqrt{3}}{3} \\)"
] | [
"A cylindrical hole of radius 1 is drilled along one of the long diagonals of a cube of side length 3. Find the area of one of the six congruent faces of the resulting solid."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-b6f9a64e2ba87990caa8c1865c0332c759ab08213ef05c0b6a07b07ea4baac63 | A cylindrical log has a diameter of 12 inches. A wedge is cut from the log by making two planar cuts that go entirely through the log. The first cut is perpendicular to the axis of the cylinder, and the plane of the second cut forms a $45^\circ$ angle with the plane of the first cut. The intersection of these two plane... | [
"216"
] | [
"A cylindrical log has a diameter of 12 inches. A wedge is cut from the log by making two planar cuts that go entirely through the log. The first cut is perpendicular to the axis of the cylinder, and the plane of the second cut forms a $45^\\circ$ angle with the plane of the first cut. The intersection of these two... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 4,
"tir": 1
} | 18 | 0 | 0 |
omr-ws-d3bd4db26a6bc7976f1f0112ea2226f32233bf4b7e7346e21352df7e976e9522 | A cylindrical log has diameter $12$ inches. A wedge is cut from the log by making two planar cuts that go entirely through the log. The first is perpendicular to the axis of the cylinder, and the plane of the second cut forms a $45^\circ$ angle with the plane of the first cut. The intersection of these two planes has e... | [
"216"
] | [
"A cylindrical log has diameter $12$ inches. A wedge is cut from the log by making two planar cuts that go entirely through the log. The first is perpendicular to the axis of the cylinder, and the plane of the second cut forms a $45^\\circ$ angle with the plane of the first cut. The intersection of these two planes... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 29
} | 29 | 0 | 0 |
omr-ws-20d7f233a66c11d8a145175f567675c668be7e70ae78d5071bdf8be61c32f2e3 | A cylindrical mug with a diameter of 1 and a height of 2, filled to the top with water, is tilted at an angle of $45^\circ$ from the vertical. What fraction of the water spills out? | [
"\\(\\frac{1}{4}\\)"
] | [
"A cylindrical mug with a diameter of 1 and a height of 2, filled to the top with water, is tilted at an angle of $45^\\circ$ from the vertical. What fraction of the water spills out?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 7,
"tir": 5
} | 15 | 0 | 0 |
omr-ws-70c920c3af3cb6d256512b5a7b5eeb3f2e68de020601a245fd7115e20a6f7dff | A cylindrical oil tank, lying horizontally, has an interior length of 10 feet and an interior diameter of 6 feet. If the rectangular surface of the oil has an area of 40 square feet, what is the depth of the oil? The answer choices are: \[ \textbf{(A)}\ \sqrt{5} \qquad \textbf{(B)}\ 2\sqrt{5} \qquad \textbf{(C)}\ 3-\sq... | [
"either \\(3 - \\sqrt{5}\\) or \\(3 + \\sqrt{5}\\)"
] | [
"A cylindrical oil tank, lying horizontally, has an interior length of 10 feet and an interior diameter of 6 feet. If the rectangular surface of the oil has an area of 40 square feet, what is the depth of the oil?\n\n The answer choices are:\n \\[\n \\textbf{(A)}\\ \\sqrt{5} \\qquad\n \\textbf{(B)}\\ 2\\sqrt{5}... | [
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-4b916fb601dd1bc719dfba576005608f6ffff90b063c39f8085c433077e893c5 | A cylindrical open-topped can has a radius of 3 cm and a height of 10 cm. If the can is tilted at a 45-degree angle, what volume of water will it hold? | [
"\\( 63\\pi \\)"
] | [
"A cylindrical open-topped can has a radius of 3 cm and a height of 10 cm. If the can is tilted at a 45-degree angle, what volume of water will it hold?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 2,
"tir": 4
} | 14 | 0 | 0 |
omr-ws-9013c9d881f7b1237cae3563b8059f09d773cec317d115ad7f55d1c00b27ac48 | A cylindrical pipe with a piston is placed vertically into a tank of water so that there is a column of air $h$ meters high between the piston and the water (at the atmospheric pressure). The piston is then elevated $b$ meters above the water level in the tank. Compute the height of the column of water in the pipe if i... | [
"x=\\frac{1}{2}\\left[(b+c)-\\sqrt{(b-c)^2+4hc}\\right]"
] | [
"A cylindrical pipe with a piston is placed vertically into a tank of water so that there is a column of air $h$ meters high between the piston and the water (at the atmospheric pressure). The piston is then elevated $b$ meters above the water level in the tank. Compute the height of the column of water in the pipe... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-17ac73b5c21c90f31e8dde4f67fad2a3f5b2c0a53f530eba20b39ef8c928cd49 | A cylindrical tank 20 feet long and 8 feet in diameter is placed horizontally. When the tank is filled to a depth of 2 feet, how many cubic feet of liquid does it contain? | [
"\\(\\frac{320\\pi}{3} - 80\\sqrt{3}\\)"
] | [
"A cylindrical tank 20 feet long and 8 feet in diameter is placed horizontally. When the tank is filled to a depth of 2 feet, how many cubic feet of liquid does it contain?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 7
} | 30 | 0 | 0 |
omr-ws-39b6e95187de735b0cfe44ab4c9cbb928d3de61cc0c85d9410114dc813a5c88d | A cylindrical tank with a height of 3 units and a diameter of 2 units is partially filled with water to a height of \( h \) units. When the tank is turned horizontal, the maximum depth of the water is \(\frac{1}{2}\) units. Find the value of \( h \). | [
"\\( h = 1 - \\frac{3\\sqrt{3}}{4\\pi} \\)"
] | [
"A cylindrical tank with a height of 3 units and a diameter of 2 units is partially filled with water to a height of \\( h \\) units. When the tank is turned horizontal, the maximum depth of the water is \\(\\frac{1}{2}\\) units. Find the value of \\( h \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-0e4200529602d4654fbb293012ab299784f24aae1bd61479157e2f851fe5ba34 | A cylindrical tank with radius $4$ feet and height $9$ feet is lying on its side. The tank is filled with water to a depth of $2$ feet. What is the volume of the water, in cubic feet? | [
"\\( 48\\pi - 36\\sqrt{3} \\)"
] | [
"A cylindrical tank with radius $4$ feet and height $9$ feet is lying on its side. The tank is filled with water to a depth of $2$ feet. What is the volume of the water, in cubic feet?"
] | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-feb1121f8870d67c649729a55a84021b4d282216c478403d4c53d4b18df0e1cc | A cylindrical tank with radius 4 feet and height 9 feet is lying on its side. The tank is filled with water to a depth of 2 feet. What is the volume of the water in cubic feet? | [
"\\( 48\\pi - 36\\sqrt{3} \\)"
] | [
"A cylindrical tank with radius 4 feet and height 9 feet is lying on its side. The tank is filled with water to a depth of 2 feet. What is the volume of the water in cubic feet?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 23
} | 38 | 0 | 0 |
omr-ws-613c598bf3661c4a064f4995ba1ac6efc86409d575501a9dd73f7e02d9939dd2 | A cylindrical vertical tree trunk has a radius of $R$. A notch is cut in the trunk by two half-planes intersecting at an angle $\theta$ and each making an angle $\alpha$ and $\beta$ with the horizontal plane, respectively. Find the value of $\alpha + \beta$ that minimizes the volume of the notch, given that $\alpha + \... | [
"\\dfrac{\\pi}{2}"
] | [
"A cylindrical vertical tree trunk has a radius of $R$. A notch is cut in the trunk by two half-planes intersecting at an angle $\\theta$ and each making an angle $\\alpha$ and $\\beta$ with the horizontal plane, respectively. Find the value of $\\alpha + \\beta$ that minimizes the volume of the notch, given that $... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-2e8558f808f3a156dde73e64775452d585f539fb4c8ee10dacabaa7fd0200f35 | A cylindrical vertical tree trunk has a radius of $R$. A notch is cut in the trunk by two half-planes intersecting at an angle $\theta$ and each making an angle $\alpha$ and $\beta$ with the horizontal plane. If $\theta = 60^\circ$, find the value of $\alpha + \beta$ that minimizes the volume of the notch. Express your... | [
"60"
] | [
"A cylindrical vertical tree trunk has a radius of $R$. A notch is cut in the trunk by two half-planes intersecting at an angle $\\theta$ and each making an angle $\\alpha$ and $\\beta$ with the horizontal plane. If $\\theta = 60^\\circ$, find the value of $\\alpha + \\beta$ that minimizes the volume of the notch. ... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 2
} | 6 | 0 | 0 |
omr-ws-5a1966eafee27c8dc4537e55754c664daf6cde29fb03c6ea811d159a3efd16c2 | A cylindrical water pitcher is 13 inches tall and has a radius of 5 inches. A solid cylinder, half as tall as the pitcher and with a radius of 2.5 inches, is placed inside the pitcher. By what fraction is the amount of water the pitcher can hold reduced? | [
"\\(\\frac{1}{8}\\)"
] | [
"A cylindrical water pitcher is 13 inches tall and has a radius of 5 inches. A solid cylinder, half as tall as the pitcher and with a radius of 2.5 inches, is placed inside the pitcher. By what fraction is the amount of water the pitcher can hold reduced?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-bc8c5ddce75314356261e7d8a52971727c3391314277530ea5b7019c04c55de8 | A cylindrical water tank 12 feet high with a radius of 8 feet is located on a tower so that the bottom of the tank is 20 feet above ground level. How much work is done in filling the tank half full of water through a hole in the bottom of the tank, using water from ground level? | [
"32*58*6*pi*d*g"
] | [
"A cylindrical water tank 12 feet high with a radius of 8 feet is located on a tower so that the bottom of the tank is 20 feet above ground level. How much work is done in filling the tank half full of water through a hole in the bottom of the tank, using water from ground level?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ddc50a1abf76c383dfe56acacfd178f340caf4a79fdd46c4ba1e09ba8a3b7988 | A cylindrical water tank is $\frac{1}{5}$ full. If three liters were added, the tank would be $\frac{1}{4}$ full. How many liters does the tank hold when it is full? | [
"60"
] | [
"A cylindrical water tank is $\\frac{1}{5}$ full. If three liters were added, the tank would be $\\frac{1}{4}$ full. How many liters does the tank hold when it is full?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9bde05d97348b4476346d5284c8bfa8fecb2ca5f49acfe67dd0f0734b07ed535 | A cylindrical water tank with radius 2 feet and length 6 feet is filled with water to a depth of 3 feet when in a horizontal position. If the tank is turned upright, what is the depth of the water? | [
"\\(4 + \\frac{3\\sqrt{3}}{2\\pi}\\)"
] | [
"A cylindrical water tank with radius 2 feet and length 6 feet is filled with water to a depth of 3 feet when in a horizontal position. If the tank is turned upright, what is the depth of the water?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 18
} | 32 | 0 | 0 |
omr-ws-249178ace4cacb486c3cc8a6557fce454518f476593978c03c07c2ca9148564d | A dance with 2018 couples takes place in Havana. For the dance, 2018 distinct points labeled $0, 1, \ldots, 2017$ are marked in a circumference and each couple is placed on a different point. For $i \geq 1$, let $s_i = i \ (\textrm{mod}\ 2018)$ and $r_i = 2i \ (\textrm{mod}\ 2018)$. The dance begins at minute $0$. On t... | [
"505"
] | [
"A dance with 2018 couples takes place in Havana. For the dance, 2018 distinct points labeled $0, 1, \\ldots, 2017$ are marked in a circumference and each couple is placed on a different point. For $i \\geq 1$, let $s_i = i \\ (\\textrm{mod}\\ 2018)$ and $r_i = 2i \\ (\\textrm{mod}\\ 2018)$. The dance begins at min... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-6c0887e9a3b5fe4edf2a52d1b2f2be366530ddeff146244de6740e751d627691 | A dart board is a regular octagon divided into regions. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is the probability that the dart lands within the center square? | [
"\\(\\frac{\\sqrt{2}-1}{2}\\)"
] | [
"A dart board is a regular octagon divided into regions. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is the probability that the dart lands within the center square?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 22,
"genselect": 8,
"tir": 13
} | 43 | 0 | 0 |
omr-ws-9700a259ed2dd35a7abeb13be3364df4894750e3e2e11d9c606b9c76d7001f40 | A dart hits a square target board with uniform probability. What is the probability that the dart is closer to the center of the board than to any edge of the board? | [
"\\dfrac{4\\sqrt{2} - 5}{3}"
] | [
"A dart hits a square target board with uniform probability. What is the probability that the dart is closer to the center of the board than to any edge of the board?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-84138f075c61efc12aba18e194b6a0b6a80e592b1b268db75a82f1a85077c6ec | A dart is planted 12 cm from the center of a circular target with a radius of 20 cm. Another dart is launched and hits the target at a random point. What is the probability that the second dart is closer to the first dart than to the edge of the target? | [
"\\(\\frac{1}{5}\\)"
] | [
"A dart is planted 12 cm from the center of a circular target with a radius of 20 cm. Another dart is launched and hits the target at a random point. What is the probability that the second dart is closer to the first dart than to the edge of the target?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-d1ca96a0d1d4765417a8a425612c2cc9f714545271a7b88e38c89345473a2496 | A dart is thrown at a regular hexagon $BOATIS$ and lands at a uniformly random point inside the hexagon. The probability that the dart lands in the interior of exactly one of the quadrilaterals $BOAT$ or $OTIS$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m + n$. | [
"005"
] | [
"A dart is thrown at a regular hexagon $BOATIS$ and lands at a uniformly random point inside the hexagon. The probability that the dart lands in the interior of exactly one of the quadrilaterals $BOAT$ or $OTIS$ can be expressed as $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 5
} | 13 | 0 | 0 |
omr-ws-9393893c7dc8830f19cfb1e8f97072822ae3bbf12dcd4e37e6d14c3855bf0cc4 | A dart is thrown at a square target. Find the probability that the point hit is nearer to the center of the square than to any of its edges. | [
"\\(\\frac{4\\sqrt{2} - 5}{3}\\)",
"\\(\\frac{4\\sqrt{2}-5}{3}\\)",
"\\dfrac{4\\sqrt{2} - 5}{3}"
] | [
"A dart is thrown at a square target. Find the probability that the point hit is nearer to the center of the square than to any of its edges."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 37,
"genselect": 13,
"tir": 26
} | 76 | 0 | 0 |
omr-ws-5f77f5d97e491851fe83ac2fa3de8846f29082db3d38a5f94e476e3da078c9aa | A dartboard has an area of 143 cm², with a "double" ring of area 15 cm² and a "triple" ring of area 10 cm² (the rings do not overlap). If you throw three darts independently and at random, what is the probability that you get exactly one triple and not more than one double? | [
"\\(\\frac{523920}{2922507}\\)"
] | [
"A dartboard has an area of 143 cm², with a \"double\" ring of area 15 cm² and a \"triple\" ring of area 10 cm² (the rings do not overlap). If you throw three darts independently and at random, what is the probability that you get exactly one triple and not more than one double?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 19
} | 19 | 0 | 0 |
omr-ws-ef89be34711ef2d16bcb9d0c8c0f458964650094a83ae86a2007886922755d92 | A dartboard has three regions where one can score points. The three regions are worth 28, 35, and 53 points. You may throw any number of darts at the dartboard. What is the greatest score that cannot be achieved? | [
"311"
] | [
"A dartboard has three regions where one can score points. The three regions are worth 28, 35, and 53 points. You may throw any number of darts at the dartboard. What is the greatest score that cannot be achieved?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 5
} | 10 | 0 | 0 |
omr-ws-defbe61745b03a49d9efd4d25e299b27f4bc519bb8cf3e4a99f4c137624924fa | A dartboard is created by drawing 2022 concentric circles with radii from 1 to 2022. The point values of the 2022 sections created by the circles are proportional to the area of the section, with the center section having a point value of 1. What is the point value of the outermost section? | [
"4043"
] | [
"A dartboard is created by drawing 2022 concentric circles with radii from 1 to 2022. The point values of the 2022 sections created by the circles are proportional to the area of the section, with the center section having a point value of 1. What is the point value of the outermost section?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-0292636387e3b2ba3c9ff6867d074fe54df17464ec8db5702f5d4731c0d6a170 | A dartboard is in the shape of a regular hexagon. Three random vertices of the board are chosen and connected to form a triangle. Three darts are thrown at the dartboard. If a dart lands inside the triangle, a point is scored. What is the expected value of the number of points scored? | [
"0.9"
] | [
"A dartboard is in the shape of a regular hexagon. Three random vertices of the board are chosen and connected to form a triangle. Three darts are thrown at the dartboard. If a dart lands inside the triangle, a point is scored. What is the expected value of the number of points scored?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 31
} | 45 | 0 | 0 |
omr-ws-988d3d2fd27fd0d88c5e72c1e22b49ce9ea29b4e751076e5fc70053ef18535d9 | A dartboard is in the shape of a star formed by two equilateral triangles that trisect each other's sides. What is the probability that a dart hitting the board will land inside the hexagon formed by the intersection of the triangles? | [
"\\(\\frac{1}{2}\\)"
] | [
"A dartboard is in the shape of a star formed by two equilateral triangles that trisect each other's sides. What is the probability that a dart hitting the board will land inside the hexagon formed by the intersection of the triangles?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 1,
"tir": 14
} | 27 | 0 | 0 |
omr-ws-1c4096f9fb203997c621b6807ce42cdc1b00e0c1889720e15cdc7a6d8b44b50b | A dartboard is shaped as an equilateral triangle with a side length of 4 units. What is the probability that a randomly thrown dart, which hits the dartboard, lands more than 2 units away from any of the corners of the dartboard? | [
"1 - \\dfrac{\\pi}{2\\sqrt{3}}"
] | [
"A dartboard is shaped as an equilateral triangle with a side length of 4 units. What is the probability that a randomly thrown dart, which hits the dartboard, lands more than 2 units away from any of the corners of the dartboard?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7a2f8d8e4d5c15ff83c14ebff357287535945c838d6b59ff65d5b56e39f8568c | A dartboard is the region $B$ in the coordinate plane consisting of points $(x,y)$ such that $|x| + |y| \leq 8$. A target $T$ is the region where $(x^2 + y^2 - 25)^2 \leq 49$. A dart is thrown and lands at a random point in $B$. The probability that the dart lands in $T$ can be expressed as $\frac{m}{n} \cdot \pi$, whe... | [
"71"
] | [
"A dartboard is the region $B$ in the coordinate plane consisting of points $(x,y)$ such that $|x| + |y| \\leq 8$. A target $T$ is the region where $(x^2 + y^2 - 25)^2 \\leq 49$. A dart is thrown and lands at a random point in $B$. The probability that the dart lands in $T$ can be expressed as $\\frac{m}{n} \\cdot ... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 1
} | 24 | 0 | 0 |
omr-ws-d91f297f9d6230d0e7f14f7bbb49e18347ecdfc66c44719ec2faf9c38a3d2345 | A dartboard is the region B in the coordinate plane consisting of points $(x, y)$ such that $|x| + |y| \le 8$. A target T is the region where $(x^2 + y^2 - 25)^2 \le 49$. A dart is thrown at a random point in B. The probability that the dart lands in T can be expressed as $\frac{m}{n} \pi$, where $m$ and $n$ are relati... | [
"71"
] | [
"A dartboard is the region B in the coordinate plane consisting of points $(x, y)$ such that $|x| + |y| \\le 8$. A target T is the region where $(x^2 + y^2 - 25)^2 \\le 49$. A dart is thrown at a random point in B. The probability that the dart lands in T can be expressed as $\\frac{m}{n} \\pi$, where $m$ and $n$ a... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-75ef489669a8055caabf350b0bf87588a77936e9e9cf193054ea71678238beaa | A dartboard's center is 6 ft. from the floor. A woman standing 8 ft. away throws a dart from a height of 5 ft. above the ground at an angle $\cos^{-1}(4/5)$. What initial velocity is required to hit the bull's eye? | [
"\\(40\\) feet per second."
] | [
"A dartboard's center is 6 ft. from the floor. A woman standing 8 ft. away throws a dart from a height of 5 ft. above the ground at an angle $\\cos^{-1}(4/5)$. What initial velocity is required to hit the bull's eye?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-eaf52628d8649db43206e5e25657df1edc8c203a0f4d388f41b1ade3ccda5f83 | A data set consists of 6 (not distinct) positive integers: 1, 7, 5, 2, 5, and \(X\). The average (arithmetic mean) of the 6 numbers equals a value in the data set. What is the sum of all possible values of \(X\)? | [
"36"
] | [
"A data set consists of 6 (not distinct) positive integers: 1, 7, 5, 2, 5, and \\(X\\). The average (arithmetic mean) of the 6 numbers equals a value in the data set. What is the sum of all possible values of \\(X\\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-ab81c444f95a8709919bc930ac99bc1dfdce486af341698d2083792e3914036e | A data set consists of 6 (not distinct) positive integers: \(1\), \(7\), \(5\), \(2\), \(5\), and \(X\). The average (arithmetic mean) of the 6 numbers equals a value in the data set. What is the sum of all positive values of \(X\)? | [
"36"
] | [
"A data set consists of 6 (not distinct) positive integers: \\(1\\), \\(7\\), \\(5\\), \\(2\\), \\(5\\), and \\(X\\). The average (arithmetic mean) of the 6 numbers equals a value in the data set. What is the sum of all positive values of \\(X\\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-5f8fe57e2e20caccb70f2f81de7a9510cd12303b6e038bb4b28abe4cc750631d | A data set with four real numbers \( 2n-4 \), \( 2n-6 \), \( n^2-8 \), \( 3n^2-6 \) has an average of 0 and a median of \( \frac{9}{2} \). What is the largest number in the data set? | [
"21"
] | [
"A data set with four real numbers \\( 2n-4 \\), \\( 2n-6 \\), \\( n^2-8 \\), \\( 3n^2-6 \\) has an average of 0 and a median of \\( \\frac{9}{2} \\). What is the largest number in the data set?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 9
} | 24 | 0 | 0 |
omr-ws-3ab801bd1f0b818d85646fd46280c74019ef0b2323e664c8ffd2887c0529df99 | A date is called "cool" if the sum of the squares of the month and the day is equal to the square of the last 2 digits of the year. For example, 12/16/20 is a cool date. What is the last cool date before 2100? | [
"10/24/26"
] | [
"A date is called \"cool\" if the sum of the squares of the month and the day is equal to the square of the last 2 digits of the year. For example, 12/16/20 is a cool date. What is the last cool date before 2100?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-b3dbe417a910037c97ced6ac415792305edc2cdbc8a7cfe2837dcd8eda8f2133 | A day can be evenly divided into 86,400 periods of 1 second; 43,200 periods of each 2 seconds; or in many other ways. In total, how many ways are there to divide a day into $n$ periods of $m$ seconds, where $n$ and $m$ are positive integers? | [
"96"
] | [
"A day can be evenly divided into 86,400 periods of 1 second; 43,200 periods of each 2 seconds; or in many other ways. In total, how many ways are there to divide a day into $n$ periods of $m$ seconds, where $n$ and $m$ are positive integers?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7c0a121de95252f875cece50376501b145d67eed29f8d5fbb76aab5006f15ece | A daycare has babies that are one year, two years, and four years old. If the two-year-old babies are excluded, the average age of the remaining babies is $2.0$; if the four-year-old babies are excluded, the average age of the remaining babies is $1.9$; and if the one-year-old babies are excluded, the average age of th... | [
"\\(\\frac{40}{19}\\)"
] | [
"A daycare has babies that are one year, two years, and four years old. If the two-year-old babies are excluded, the average age of the remaining babies is $2.0$; if the four-year-old babies are excluded, the average age of the remaining babies is $1.9$; and if the one-year-old babies are excluded, the average age ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-e3e58ec5c5a933f460067600acdca3af8ec0ff130707c2f51a786ad4d78b943d | A dead body is found at 2 PM with a temperature of 29 degrees in a room where the temperature is 22 degrees. It is known that under these conditions, it takes 2 hours for the body's temperature to drop to 35 degrees. Determine the time of the murder. | [
"3:20 a.m."
] | [
"A dead body is found at 2 PM with a temperature of 29 degrees in a room where the temperature is 22 degrees. It is known that under these conditions, it takes 2 hours for the body's temperature to drop to 35 degrees. Determine the time of the murder."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 7,
"tir": 23
} | 43 | 0 | 0 |
omr-ws-10103801065185375bc5316a52ba27511b72195beff266208347405409cc5fc6 | A dead body was found in a room with a constant temperature of 70°F. At the time of discovery, the core temperature of the body was 85°F. One hour later, the core temperature was 80°F. Assuming the time of death corresponds to \( t = 0 \) and the core temperature at that time was 98.6°F, how many hours elapsed before t... | [
"1.59 hours"
] | [
"A dead body was found in a room with a constant temperature of 70°F. At the time of discovery, the core temperature of the body was 85°F. One hour later, the core temperature was 80°F. Assuming the time of death corresponds to \\( t = 0 \\) and the core temperature at that time was 98.6°F, how many hours elapsed b... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-64cb0ef615a236bc25d731fa1c684f55129dd0e213246da6fd44c56617686f0a | A dead fly lies on a 1m by 1m square floor. A spider, also on the same floor, can move by selecting one of the corners of the floor and moving to the midpoint between that corner and its current position. After exactly 8 moves, the spider aims to be as close as possible to the fly. If the fly is located at the center o... | [
"0"
] | [
"A dead fly lies on a 1m by 1m square floor. A spider, also on the same floor, can move by selecting one of the corners of the floor and moving to the midpoint between that corner and its current position. After exactly 8 moves, the spider aims to be as close as possible to the fly. If the fly is located at the cen... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-b0c3ae69dd13ff3d17fd14218d8f1e1b9c83d2adcf758720b422f282fa71a831 | A dealer bought \( n \) radios for \( d \) dollars total (where \( d \) is a positive integer). If he sold two radios at half their cost and sold the rest at a profit of 8 dollars each, resulting in a total profit of 72 dollars, what is the least possible value of \( n \)? | [
"12"
] | [
"A dealer bought \\( n \\) radios for \\( d \\) dollars total (where \\( d \\) is a positive integer). If he sold two radios at half their cost and sold the rest at a profit of 8 dollars each, resulting in a total profit of 72 dollars, what is the least possible value of \\( n \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-5e43143cdca2178928396b9ab0f51c89cf5d11a960aa19705de1f54e6c64ab9b | A dealer has a deck of 1999 cards. He takes the top card off and "ducks" it, placing it on the bottom of the deck. He deals the second card onto the table. He ducks the third card, deals the fourth card, ducks the fifth card, deals the sixth card, and so forth, continuing until he has only one card left; he then ducks ... | [
"1"
] | [
"A dealer has a deck of 1999 cards. He takes the top card off and \"ducks\" it, placing it on the bottom of the deck. He deals the second card onto the table. He ducks the third card, deals the fourth card, ducks the fifth card, deals the sixth card, and so forth, continuing until he has only one card left; he then... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a79344060fdec98afbb496e2b6a3477becedaef5f06f293139a9b191eb7691ad | A dealer has a stock of 6 similar television sets, but one is withdrawn for repair. Given that the monthly demand for the television sets follows a Poisson distribution with a mean of 3.56, what is the expected number of television sets rented out in a month? | [
"3.56"
] | [
"A dealer has a stock of 6 similar television sets, but one is withdrawn for repair. Given that the monthly demand for the television sets follows a Poisson distribution with a mean of 3.56, what is the expected number of television sets rented out in a month?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3d717449bda06666afd54b80172120d3cd0a267bb0051feb409e5c030c7d2b8a | A dealer marks up the price of goods by 20%, then offers a 10% discount. To achieve a 50% profit, by what percentage must the price be increased from the discounted price? | [
"\\(\\frac{4}{9}\\cdot 100\\) or approximately \\(44.44\\%\\)"
] | [
"A dealer marks up the price of goods by 20%, then offers a 10% discount. To achieve a 50% profit, by what percentage must the price be increased from the discounted price?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f40e7eb208d8b98e67dd31034d568c9375c22cb6dbb44199dbc2195e4199aa9e | A dealer places a standard deck of 52 cards in a circle such that each card is adjacent to two other cards. If exactly two aces are adjacent (and not adjacent to more aces), you earn \$100. If exactly three aces are adjacent (and not adjacent to more aces), you earn \$1000. If all four aces are adjacent, you earn \$100... | [
"31.71"
] | [
"A dealer places a standard deck of 52 cards in a circle such that each card is adjacent to two other cards. If exactly two aces are adjacent (and not adjacent to more aces), you earn \\$100. If exactly three aces are adjacent (and not adjacent to more aces), you earn \\$1000. If all four aces are adjacent, you ear... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-19f25278aaef75e9fd5a13bdea3629575e6446578d8f6b3796365e0db5d1999b | A dealer sold 200 cars, and the data for some of those sales are recorded in this table. If the rest of the cars she sold were Hondas, how many Hondas did she sell? \begin{tabular}{ |c | c|} \hline \textbf{Type of Car} & \textbf{$\%$ of Total Cars Sold} \\ \hline Audi & $15\%$ \\ \hline Toyota & $22\%$ \\ \hline Acura ... | [
"70"
] | [
"A dealer sold 200 cars, and the data for some of those sales are recorded in this table. If the rest of the cars she sold were Hondas, how many Hondas did she sell?\n\n\\begin{tabular}{ |c | c|}\n\\hline \\textbf{Type of Car} & \\textbf{$\\%$ of Total Cars Sold} \\\\ \\hline\nAudi & $15\\%$ \\\\ \\hline\nToyota & ... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0c5a858ab77d12b405de5e3471c80173f724dccc08f5f0b864d22dba18590e6e | A dealer wishes to produce 300 gallons of oil worth $0.40 per quart. He has oil worth $0.36 per quart and oil worth $0.52 per quart. How many quarts of each type of oil should he mix to achieve this? | [
"900 quarts of $0.36 oil and 300 quarts of $0.52 oil."
] | [
"A dealer wishes to produce 300 gallons of oil worth $0.40 per quart. He has oil worth $0.36 per quart and oil worth $0.52 per quart. How many quarts of each type of oil should he mix to achieve this?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 31
} | 45 | 0 | 0 |
omr-ws-e356339e8c2db33ab74722d70008638f287b2fab971abb456b75690a1bb3f485 | A debt of $12,000, which is due 10 years from now, is instead to be paid off by four payments: $3000 now, $2000 in 3 years, $2000 in 6 years, and a final payment at the end of 8 years. What would this final payment be if an interest rate of 5.5% compounded semi-annually is assumed? | [
"9039.04"
] | [
"A debt of $12,000, which is due 10 years from now, is instead to be paid off by four payments: $3000 now, $2000 in 3 years, $2000 in 6 years, and a final payment at the end of 8 years. What would this final payment be if an interest rate of 5.5% compounded semi-annually is assumed?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-ae949f8d7dbea2bd74b43d751dcdbaf08ba2f2c698a6ac3c4163b3a719a2cd94 | A decahedron $ABCDPQRSTUVW$ has two parallel bases: a square $ABCD$ and an octagon $PQRSTUVW$, with all angles of the octagon equal. The decahedron also has eight side faces: triangles $APQ, BRS, CTU, DVW$ and rectangles $DAPW$, $ABRQ$, $BCTS$, and $CDVU$. The cross-sectional area of the decahedron by the plane passing... | [
"\\(\\sqrt{13}\\)"
] | [
"A decahedron $ABCDPQRSTUVW$ has two parallel bases: a square $ABCD$ and an octagon $PQRSTUVW$, with all angles of the octagon equal. The decahedron also has eight side faces: triangles $APQ, BRS, CTU, DVW$ and rectangles $DAPW$, $ABRQ$, $BCTS$, and $CDVU$. The cross-sectional area of the decahedron by the plane pa... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c601764f07fe2ea724f73e162fefd5f1ba8ed7272fc92682cac073fa8b14db1a | A decimal representation of a number $A$ consists of 1001 sevens. Find the quotient and the remainder resulting from the division of $A$ by the number 1001. | [
"The quotient is equal to $\\underbrace{777000777000\\ldots 777000}_{\\substack{{\\rm the\\ combination\\ 777000}\\\\ {\\rm is\\ repeated\\ 166\\ times}}}77$. The remainder is equal to 700"
] | [
"A decimal representation of a number $A$ consists of 1001 sevens. Find the quotient and the remainder resulting from the division of $A$ by the number 1001."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b8b67b9ab438d1ee434535bfddc748fe7629d5383783f7f559c5e3ecd68dc2bf | A decision is made by tossing $2n + 1$ coins. Initially, all coins are fair. A mathematician replaces $n$ pairs of fair coins with $n$ pairs of biased coins, such that in each pair, the probability of obtaining heads on one coin is the same as the probability of obtaining tails on the other. Let $P(n)$ be the probabili... | [
"\\dfrac{1}{2}"
] | [
"A decision is made by tossing $2n + 1$ coins. Initially, all coins are fair. A mathematician replaces $n$ pairs of fair coins with $n$ pairs of biased coins, such that in each pair, the probability of obtaining heads on one coin is the same as the probability of obtaining tails on the other. Let $P(n)$ be the prob... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2df34ad305af234fcebff6b7ef18dcf1eefaa2150a754ea8565e81f937cc4c35 | A decision-making problem will be resolved by tossing $2n + 1$ coins. If heads come up in the majority, one option will be taken; if tails come up in the majority, the other option will be taken. Initially, all coins were fair. A mathematician replaced $n$ pairs of fair coins with $n$ pairs of biased coins, such that i... | [
"\\dfrac{1}{2}"
] | [
"A decision-making problem will be resolved by tossing $2n + 1$ coins. If heads come up in the majority, one option will be taken; if tails come up in the majority, the other option will be taken. Initially, all coins were fair. A mathematician replaced $n$ pairs of fair coins with $n$ pairs of biased coins, such t... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-953acf64aeeef8a65757abe56ad10b1ed878b5bb611834eb5dfbaf09832aaaba | A deck consists of 52 playing cards which is well shuffled. Draw 6 cards. Find the probability that among the 6 cards there will be a representative of all suits. | [
"\\(\\frac{9971}{78302}\\)"
] | [
"A deck consists of 52 playing cards which is well shuffled. Draw 6 cards. Find the probability that among the 6 cards there will be a representative of all suits."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 22
} | 22 | 0 | 0 |
omr-ws-23dbd54cd53e2dcda1d6731193ea20b1225aac66aecef3254fac8aca9c547c0e | A deck contains $n$ cards labeled $1, 2, 3, \ldots, n$. Starting with an arbitrary ordering of the cards, repeat the following operation: if the top card is labeled $k$, reverse the order of the top $k$ cards. Let $f(n)$ be the maximum number of operations required for the top card to become labeled $1$ for any initial... | [
"38"
] | [
"A deck contains $n$ cards labeled $1, 2, 3, \\ldots, n$. Starting with an arbitrary ordering of the cards, repeat the following operation: if the top card is labeled $k$, reverse the order of the top $k$ cards. Let $f(n)$ be the maximum number of operations required for the top card to become labeled $1$ for any i... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 6
} | 11 | 0 | 0 |
omr-ws-acbc22acad4a16cb089bc2c77f85c87e55cf3918aedb3232ee1dc79b04f2c7d2 | A deck contains 52 cards, with 20 being black and 32 being red. When 10 cards are randomly selected, calculate the exact probability $P$ of selecting an odd number of black cards. Express your answer as a fraction in lowest terms and find the sum of the numerator and the denominator of $P$. | [
"5944297621"
] | [
"A deck contains 52 cards, with 20 being black and 32 being red. When 10 cards are randomly selected, calculate the exact probability $P$ of selecting an odd number of black cards. Express your answer as a fraction in lowest terms and find the sum of the numerator and the denominator of $P$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-e06b46644d657ca6f592630f468ba252f5d27a0cf5ff958625692e31d2fc2318 | A deck of $2n$ cards numbered from $1$ to $2n$ is shuffled, and $n$ cards are dealt to players $A$ and $B$. Players alternately discard a card face up, starting with $A$. The game ends when the sum of the discards is first divisible by $2n + 1$, and the last person to discard wins. What is the probability that $A$ wins... | [
"0"
] | [
"A deck of $2n$ cards numbered from $1$ to $2n$ is shuffled, and $n$ cards are dealt to players $A$ and $B$. Players alternately discard a card face up, starting with $A$. The game ends when the sum of the discards is first divisible by $2n + 1$, and the last person to discard wins. What is the probability that $A$... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-327367c94501a0f015cba74363fc8416990de51b3aacafb26ba096c375a4d334 | A deck of $n \geq 3$ cards contains exactly one card of each positive integer $1, 2, 3, \ldots, n$. Three cards are chosen uniformly and randomly from the deck without replacement. Suppose $p$ is the probability that the sum of the numbers on the three cards chosen is even. If $p$ is greater than $50.1\%$, find the lar... | [
"39"
] | [
"A deck of $n \\geq 3$ cards contains exactly one card of each positive integer $1, 2, 3, \\ldots, n$. Three cards are chosen uniformly and randomly from the deck without replacement. Suppose $p$ is the probability that the sum of the numbers on the three cards chosen is even. If $p$ is greater than $50.1\\%$, find... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-112d4829a0ba603a8a47b91ae38f9940b442680dded311f0b0e3bc27bae78a5c | A deck of $n$ playing cards contains three aces. The cards are shuffled randomly, and then turned up one by one from the top until the second ace appears. Let $E(n)$ be the expected number of cards to be turned up. Calculate $E(100)$. | [
"50.5"
] | [
"A deck of $n$ playing cards contains three aces. The cards are shuffled randomly, and then turned up one by one from the top until the second ace appears. Let $E(n)$ be the expected number of cards to be turned up. Calculate $E(100)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ce550356b52488f5ac25253ff7fdebda1fed271a26bbfa260411d5b4007d3207 | A deck of $n$ playing cards, which contains three aces, is shuffled at random. The cards are then turned up one by one from the top until the second ace appears. Let $E(n)$ denote the expected number of cards to be turned up. Find $E(51)$. | [
"26"
] | [
"A deck of $n$ playing cards, which contains three aces, is shuffled at random. The cards are then turned up one by one from the top until the second ace appears. Let $E(n)$ denote the expected number of cards to be turned up. Find $E(51)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cdb51bbabc4cc718367564556a4079205d99ec923ac1859707be3bc559b2f805 | A deck of 100 cards numbered 1 through 100 is shuffled and turned over one card at a time. A "hit" occurs whenever card \( i \) is the \( i \)-th card to be turned over for \( i = 1, 2, \ldots, 100 \). Find the expectation and variance of the total number of hits. | [
"Expectation = 1, Variance = 1"
] | [
"A deck of 100 cards numbered 1 through 100 is shuffled and turned over one card at a time. A \"hit\" occurs whenever card \\( i \\) is the \\( i \\)-th card to be turned over for \\( i = 1, 2, \\ldots, 100 \\). Find the expectation and variance of the total number of hits."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 27
} | 27 | 0 | 0 |
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