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-932e7128d521c79facaf4876a33a323477bd44d07b0ad279aa8328724a8ffeb5 | A right circular cone has a radius of 3 cm and a height of 9 cm. A plane cuts the cone parallel to the element and divides the base into two semicircles. Find the resulting cross-sectional area. | [
"6"
] | [
"A right circular cone has a radius of 3 cm and a height of 9 cm. A plane cuts the cone parallel to the element and divides the base into two semicircles. Find the resulting cross-sectional area."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-8d6151b09aad293f1c1ef8cdd3340d6fbac9787df4e7f28b3dee5c8fdcfc5adf | A right circular cone has a radius of 6 cm and a height of 9 cm. A cutting plane is passed obliquely, cutting all the elements of the cone such that the topmost portion and the bottommost portion of the oblique elliptical upper base are 7 cm and 2 cm from the circular base, respectively. Calculate the volume of the por... | [
"\\(\\frac{1340}{27} \\pi\\)"
] | [
"A right circular cone has a radius of 6 cm and a height of 9 cm. A cutting plane is passed obliquely, cutting all the elements of the cone such that the topmost portion and the bottommost portion of the oblique elliptical upper base are 7 cm and 2 cm from the circular base, respectively. Calculate the volume of th... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-5f27577fdb5eaa8956b5f5b22395b4dba9d9f2338686d391579b0819ef28dee1 | A right circular cone has a volume of $12\pi$ cubic centimeters. The height of the cone is 4 cm. How many centimeters is the circumference of the base of the cone, in terms of $\pi$? | [
"6\\pi"
] | [
"A right circular cone has a volume of $12\\pi$ cubic centimeters. The height of the cone is 4 cm. How many centimeters is the circumference of the base of the cone, in terms of $\\pi$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1a30059abf73425a1c91f9c6f172e4feb8af10c41f092e27ce9c0c4bfe7ab909 | A right circular cone has base radius $r$ and height $h$. The cone lies on its side on a flat table. As the cone rolls on the surface of the table without slipping, the point where the cone's base meets the table traces a circular arc centered at the point where the vertex touches the table. The cone first returns to i... | [
"14"
] | [
"A right circular cone has base radius $r$ and height $h$. The cone lies on its side on a flat table. As the cone rolls on the surface of the table without slipping, the point where the cone's base meets the table traces a circular arc centered at the point where the vertex touches the table. The cone first returns... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8c485fd0919255c400db60d56a21e4261e509da90389e5f0f9baea00fca857b7 | A right circular cone has diameter $6$ and altitude $9$. Find the volume of the largest (in volume) rectangular box with a square base that fits inside the cone, with its square base flat against the base of the cone. | [
"24"
] | [
"A right circular cone has diameter $6$ and altitude $9$. Find the volume of the largest (in volume) rectangular box with a square base that fits inside the cone, with its square base flat against the base of the cone."
] | [
"aops_c4_high_school_math"
] | [
"cot",
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"cot": 16,
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omr-ws-194c703e9fcc9ceb90ccad1aa392119bb8b011b38681dfbd34780e0dd91b0c48 | A right circular cone has height $4$ cm and base radius $3$ cm. A cube of side length $s$ cm is completely contained inside the cone, and one of its faces lies on the base of the cone. What is the maximum possible value of $s$? | [
"\\( 36 - 24\\sqrt{2} \\)"
] | [
"A right circular cone has height $4$ cm and base radius $3$ cm. A cube of side length $s$ cm is completely contained inside the cone, and one of its faces lies on the base of the cone. What is the maximum possible value of $s$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
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] | {
"cot": 12,
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"tir": 29
} | 46 | 0 | 0 |
omr-ws-f7cda5b48a06236bcdda78ded6fdb87806d1a281395845abd71b0bee625dbb6a | A right circular cone has the radius of the base $r$ and an angle $\alpha $ in the axial section. Two identical balls of radius $R$ touch each other, the lateral surface of the cone (externally) and the plane of the base of the cone. Find the area of the triangle whose vertices are the centers of the balls and the cent... | [
"$R\\left[r+R\\cot\\left(\\frac{\\pi }{4}+\\frac{\\alpha }{4}\\right)\\right]$"
] | [
"A right circular cone has the radius of the base $r$ and an angle $\\alpha $ in the axial section. Two identical balls of radius $R$ touch each other, the lateral surface of the cone (externally) and the plane of the base of the cone. Find the area of the triangle whose vertices are the centers of the balls and th... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-338db513ed375b0ff2379414c97709e96e8a8e58d45a82b1ef00879fc1b2dbc6 | A right circular cone is 4 inches tall with a base radius of 3 inches. The entire surface, including the base, is painted. A plane parallel to the base divides the cone into a smaller cone $C$ and a frustum $F$. The ratio between the areas of the painted surfaces of $C$ and $F$ and the ratio between the volumes of $C$ ... | [
"512"
] | [
"A right circular cone is 4 inches tall with a base radius of 3 inches. The entire surface, including the base, is painted. A plane parallel to the base divides the cone into a smaller cone $C$ and a frustum $F$. The ratio between the areas of the painted surfaces of $C$ and $F$ and the ratio between the volumes of... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 11
} | 26 | 0 | 0 |
omr-ws-51ed0696d5ccc74c0f0c4fb385e2b43742e2947f3139b46a8741321d3b1f313a | A right circular cone is completely contained in a sphere. Let \( p \) be the maximum possible ratio of the volume of the cone to the volume of the sphere, and let \( q \) be the maximum possible ratio of the lateral surface area of the cone to the surface area of the sphere. What is the value of \( \frac{p^7}{q^{16}} ... | [
"864"
] | [
"A right circular cone is completely contained in a sphere. Let \\( p \\) be the maximum possible ratio of the volume of the cone to the volume of the sphere, and let \\( q \\) be the maximum possible ratio of the lateral surface area of the cone to the surface area of the sphere. What is the value of \\( \\frac{p^... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-635fbe47731855cd9bdd7f2e7b824df76a6c4865182b511cd3a06d0f8c61fbc5 | A right circular cone is inscribed in a sphere of radius $R$. Find the lateral surface area of the cone if the altitude is $h$. | [
"$\\pi h\\sqrt {2R(2R-h)}$"
] | [
"A right circular cone is inscribed in a sphere of radius $R$. Find the lateral surface area of the cone if the altitude is $h$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9b8c53b1d7973ca5ac92d302d5368fcb940c9d9729dad99b1df7fd28e4278243 | A right circular cone is inscribed in a sphere such that the circumference of its base lies on the interior of the sphere’s surface, as does its apex. The cone has a base radius of 5 and a height of 10. If the radius of the sphere can be expressed as \(\frac{m}{n}\), where \(m\) and \(n\) are relatively prime positive ... | [
"29"
] | [
"A right circular cone is inscribed in a sphere such that the circumference of its base lies on the interior of the sphere’s surface, as does its apex. The cone has a base radius of 5 and a height of 10. If the radius of the sphere can be expressed as \\(\\frac{m}{n}\\), where \\(m\\) and \\(n\\) are relatively pri... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-f576910dff3eeb96f84fbfb7e9cdb5b6b54fc7156948143eac02c655385edcec | A right circular cone is sliced into three parts by two planes parallel to its base, dividing its total height into three equal parts. If these planes divide the volume of the cone into three parts with ratio \( p : q : r \) (from top to bottom, respectively), where the greatest common divisor of \( p, q, \) and \( r \... | [
"13"
] | [
"A right circular cone is sliced into three parts by two planes parallel to its base, dividing its total height into three equal parts. If these planes divide the volume of the cone into three parts with ratio \\( p : q : r \\) (from top to bottom, respectively), where the greatest common divisor of \\( p, q, \\) a... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7a75fdbb1d802baa7f332732837af0bbb424b4836dacad24c158330f519cf763 | A right circular cone of height 2 is cut into two pieces by a plane parallel to its base. The lateral areas of the two pieces are equal. What is the distance from the cone's vertex to the plane? | [
"\\(\\sqrt{2}\\)"
] | [
"A right circular cone of height 2 is cut into two pieces by a plane parallel to its base. The lateral areas of the two pieces are equal. What is the distance from the cone's vertex to the plane?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-3b1d6e8c028be1af2f43ae396d230b018ec529e954c05b5b1a772cadaec9506e | A right circular cone sits on a table, pointing up. The cross-section triangle, perpendicular to the base, has a vertex angle of 60 degrees. The diameter of the cone's base is $12\sqrt{3}$ inches. A sphere is placed inside the cone so that it is tangent to the sides of the cone and sits on the table. What is the volume... | [
"288\\pi"
] | [
"A right circular cone sits on a table, pointing up. The cross-section triangle, perpendicular to the base, has a vertex angle of 60 degrees. The diameter of the cone's base is $12\\sqrt{3}$ inches. A sphere is placed inside the cone so that it is tangent to the sides of the cone and sits on the table. What is the ... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7847a9d3b63aaeccb08ccf9e950463f4bf4e7042b32da40f023370ed501832f2 | A right circular cone with apex angle $\alpha$ is cut by a plane inclined at an acute angle $\theta$ with the axis of the cone, generating an elliptical section. Find the eccentricity of the elliptical section. | [
"\\(\\frac{\\cos \\theta}{\\cos \\frac{\\alpha}{2}}\\)"
] | [
"A right circular cone with apex angle $\\alpha$ is cut by a plane inclined at an acute angle $\\theta$ with the axis of the cone, generating an elliptical section. Find the eccentricity of the elliptical section."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 6,
"tir": 7
} | 15 | 0 | 0 |
omr-ws-d8bc82ecd76390b8c4d7e237a41821551d777222c59b0f0dfbb612ff99f55759 | A right circular cone with base radius $r$ has its vertex $V$ at a distance of $1$ from every point on the perimeter of the base. A plane section of the cone forms an ellipse with lowest point $L$ and highest point $H$. Two routes from $L$ to $H$ are marked: $R_1$ along the semi-perimeter of the ellipse and $R_2$ as th... | [
"r > \\dfrac{1}{2}"
] | [
"A right circular cone with base radius $r$ has its vertex $V$ at a distance of $1$ from every point on the perimeter of the base. A plane section of the cone forms an ellipse with lowest point $L$ and highest point $H$. Two routes from $L$ to $H$ are marked: $R_1$ along the semi-perimeter of the ellipse and $R_2$ ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 4
} | 17 | 0 | 0 |
omr-ws-895e998669a9436a2d8abff7fa27ec9afb97c5d6eb138668f873ae5df21a88ea | A right circular cylinder has a volume of 500 cubic centimeters. The top and bottom are made of material that costs 6 cents per square centimeter, while the sides are made of material that costs 4 cents per square centimeter. Express the total cost \( C \) of the material as a function of the radius \( r \) of the cyli... | [
"\\( C = 12 \\pi r^2 + \\frac{4000}{r} \\)"
] | [
"A right circular cylinder has a volume of 500 cubic centimeters. The top and bottom are made of material that costs 6 cents per square centimeter, while the sides are made of material that costs 4 cents per square centimeter. Express the total cost \\( C \\) of the material as a function of the radius \\( r \\) of... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 10
} | 25 | 0 | 0 |
omr-ws-03bfeffb0401274748c49bf7f7d985d7e30af6b22ded44b7152c6ee2edbd75aa | A right circular cylinder has the same volume as a rectangular box with square bases. If both figures have the same lateral area, compute the ratio of the diameter of the circular base to the side of the square base. | [
"1:1"
] | [
"A right circular cylinder has the same volume as a rectangular box with square bases. If both figures have the same lateral area, compute the ratio of the diameter of the circular base to the side of the square base."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e8f37796f6177b2e20ac2344767b3e38bc2f8fe9e75719cd7e5b4ce1dbe6c87f | A right circular cylinder is circumscribed about a sphere of radius $R$. A point $C$ lies in the interior of the cylinder 3 on its axis and is at the distance $\frac{3}{4}R$ from the lower base. A plane $P$ is drawn through that point, which has only one common point with the circle of the lower base. A right circular ... | [
"$\\frac{48}{125}\\pi R^3$"
] | [
"A right circular cylinder is circumscribed about a sphere of radius $R$. A point $C$ lies in the interior of the cylinder 3 on its axis and is at the distance $\\frac{3}{4}R$ from the lower base. A plane $P$ is drawn through that point, which has only one common point with the circle of the lower base. A right cir... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9ba917f2ee88d1880c71f3f0e509733b914e21f8601d0afe58c6bcbd9d398558 | A right circular cylinder is inscribed in a sphere of radius \(r\). Find the largest possible surface area of such a cylinder. | [
"(1 + \\sqrt{5})\\pi r^2"
] | [
"A right circular cylinder is inscribed in a sphere of radius \\(r\\). Find the largest possible surface area of such a cylinder."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cd8daa904a07efc2f60031e0c221ddc3595020e010dc123cb01af289d692d7f3 | A right circular cylinder is inscribed in a sphere. How many times does the volume of the sphere exceed that of the cylinder if it is known that the ratio of the radius of the sphere to the radius of the base of the cylinder is half the ratio of the surface of the sphere to the lateral surface of the cylinder? | [
"$16:9$"
] | [
"A right circular cylinder is inscribed in a sphere. How many times does the volume of the sphere exceed that of the cylinder if it is known that the ratio of the radius of the sphere to the radius of the base of the cylinder is half the ratio of the surface of the sphere to the lateral surface of the cylinder?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2c87626a4a681c9a398eabe9da7fe8d83a41adb2a0f206493442853d3dde8640 | A right circular cylinder with a base diameter of $d=6\text{ cm}$ is cut by a plane, resulting in an ellipse with an eccentricity of $e=\frac{4}{5}$. Calculate the semi-axes of the ellipse. | [
"\\(a = 5 \\text{ cm}, b = 3 \\text{ cm}\\)"
] | [
"A right circular cylinder with a base diameter of $d=6\\text{ cm}$ is cut by a plane, resulting in an ellipse with an eccentricity of $e=\\frac{4}{5}$. Calculate the semi-axes of the ellipse."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 27
} | 40 | 0 | 0 |
omr-ws-09292d43a381142fbbe5e0d34ae37409dd2fef4ead0012125537b6b64f2899be | A right circular cylinder with its diameter equal to its height is inscribed in a right circular cone. The cone has a diameter of 10 and an altitude of 12, and the axes of the cylinder and cone coincide. Find the radius of the cylinder. | [
"\\(\\frac{30}{11}\\)",
"\\dfrac{30}{11}"
] | [
"A right circular cylinder with its diameter equal to its height is inscribed in a right circular cone. The cone has a diameter of 10 and an altitude of 12, and the axes of the cylinder and cone coincide. Find the radius of the cylinder."
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 31
} | 63 | 0 | 0 |
omr-ws-c43a4e2290d1ccbbacb19581bb5ac4cc13b21ca6b2274e650b8fe9591f4506ac | A right circular cylinder with its diameter equal to its height is inscribed in a right circular cone. The cone has diameter 10 and altitude 12, and the axes of the cylinder and cone coincide. Find the radius of the cylinder. Express your answer as a common fraction. | [
"\\frac{30}{11}"
] | [
"A right circular cylinder with its diameter equal to its height is inscribed in a right circular cone. The cone has diameter 10 and altitude 12, and the axes of the cylinder and cone coincide. Find the radius of the cylinder. Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f4a9e867f46b07e57e34806ea74de98625f8df2db3b23b7fd59d3066c6f64b29 | A right circular cylinder with no top has a surface area of \(3\pi\) square feet. What height \(h\) and base radius \(r\) will maximize the volume of the cylinder? (Hint: The area of the curved side of the cylinder is \(2\pi rh\)). | [
"\\( r = 1 \\), \\( h = 1 \\)"
] | [
"A right circular cylinder with no top has a surface area of \\(3\\pi\\) square feet. What height \\(h\\) and base radius \\(r\\) will maximize the volume of the cylinder? (Hint: The area of the curved side of the cylinder is \\(2\\pi rh\\))."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 28
} | 41 | 0 | 0 |
omr-ws-f2022648bb2080ddffe720626845429099eca5c9e27474ea9a2b8127f607a912 | A right circular cylinder with radius 2 is inscribed in a hemisphere with radius 5 so that its bases are parallel to the base of the hemisphere. What is the height of this cylinder? | [
"\\sqrt{21}"
] | [
"A right circular cylinder with radius 2 is inscribed in a hemisphere with radius 5 so that its bases are parallel to the base of the hemisphere. What is the height of this cylinder?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-26cf21fae8ab3be846951ffec1e7b6eb5569c59f9b7379bfeab3846a911497fa | A right circular tank (pointy end down) is being filled with liquid at a constant rate of 2 m³/s. The tank has a height of 12 m and its circular face (at the top) has a radius of 5 m. At what rate is the radius of the upper surface of the liquid increasing when the tank is half full? Given formula: \( V = \frac{3}{\pi}... | [
"\\(\\frac{5}{6\\pi \\cdot 62.5^{2/3}}\\)"
] | [
"A right circular tank (pointy end down) is being filled with liquid at a constant rate of 2 m³/s. The tank has a height of 12 m and its circular face (at the top) has a radius of 5 m. At what rate is the radius of the upper surface of the liquid increasing when the tank is half full? Given formula: \\( V = \\frac{... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 5,
"tir": 21
} | 33 | 0 | 0 |
omr-ws-7f8590c6cfc96e90511cbeac8e272e844e934217561eb92e1a420ac9bb5cac33 | A right cone has a base with a circumference of $16\pi$ inches and a height of 30 inches. The height of this cone is reduced while the circumference stays the same. The volume of the shorter cone is $192\pi$ cubic inches. What is the ratio of the shorter height to the original height? Express your answer as a common fr... | [
"\\frac{3}{10}"
] | [
"A right cone has a base with a circumference of $16\\pi$ inches and a height of 30 inches. The height of this cone is reduced while the circumference stays the same. The volume of the shorter cone is $192\\pi$ cubic inches. What is the ratio of the shorter height to the original height? Express your answer as a co... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e7ab9770b626377cac3e70baf9a9b88f71a4788b405ea9c5decd53f85037abe7 | A right cylinder is inscribed in a right circular cone with a radius of 3 cm and a height of 5 cm. Find the dimensions of the cylinder that has the maximum volume. | [
"\\( r = 2 \\) cm, \\( h = \\frac{5}{3} \\) cm"
] | [
"A right cylinder is inscribed in a right circular cone with a radius of 3 cm and a height of 5 cm. Find the dimensions of the cylinder that has the maximum volume."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 7,
"tir": 27
} | 34 | 0 | 0 |
omr-ws-fe79f9556af2fce70a78ac725bd94fde0e523b171c3a99c1811a84abf69fd74d | A right cylinder with a base radius of 3 units is inscribed in a sphere of radius 5 units. The total volume, in cubic units, of the space inside the sphere and outside the cylinder is $W\pi$. Find $W$, as a common fraction. | [
"\\frac{284}{3}"
] | [
"A right cylinder with a base radius of 3 units is inscribed in a sphere of radius 5 units. The total volume, in cubic units, of the space inside the sphere and outside the cylinder is $W\\pi$. Find $W$, as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9389659984b76fbe7621ae69903aaabbfbc73236ec5655bb83d6c2c9847719f4 | A right cylinder with a height of 5 inches has a radius of 2 inches. What is the area of the curved surface of the cylinder, in square inches? Express your answer in terms of $\pi$. | [
"20\\pi"
] | [
"A right cylinder with a height of 5 inches has a radius of 2 inches. What is the area of the curved surface of the cylinder, in square inches? Express your answer in terms of $\\pi$."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fa6f5d8cb1aebf308936bb80ddf9087fad95acb668d68389d3e5239b9e7f41fa | A right cylindrical oil tank is $15$ feet tall and its circular bases have diameters of $4$ feet each. When the tank is lying flat on its side (not on one of the circular ends), the oil inside is $3$ feet deep. How deep, in feet, would the oil have been if the tank had been standing upright on one of its bases? Express... | [
"12.1"
] | [
"A right cylindrical oil tank is $15$ feet tall and its circular bases have diameters of $4$ feet each. When the tank is lying flat on its side (not on one of the circular ends), the oil inside is $3$ feet deep. How deep, in feet, would the oil have been if the tank had been standing upright on one of its bases? Ex... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6e3c6500217618f21465fa3ab7528e16cf99b4f52a53b8936556c58d89904385 | A right hexagonal prism has bases $ABCDEF$ and $A'B'C'D'E'F'$, with edges $AA'$, $BB'$, $CC'$, $DD'$, $EE'$, $FF'$ each perpendicular to the hexagons. The height of the prism is $5$ and the side length of each hexagon is $6$. The plane $P$ passes through points $A$, $C'$, and $E$. The area of the portion of $P$ contain... | [
"323"
] | [
"A right hexagonal prism has bases $ABCDEF$ and $A'B'C'D'E'F'$, with edges $AA'$, $BB'$, $CC'$, $DD'$, $EE'$, $FF'$ each perpendicular to the hexagons. The height of the prism is $5$ and the side length of each hexagon is $6$. The plane $P$ passes through points $A$, $C'$, and $E$. The area of the portion of $P$ co... | [
"aops_c4_high_school_math"
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"cot"
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"cot": 13
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omr-ws-13a157e269d7d0d53fa348480e27e4466956caf0dacea6276d1881fb2a9c07ed | A right hexagonal prism has height $2$. The bases are regular hexagons with side length $1$. Any $3$ of the $12$ vertices determine a triangle. Find the number of these triangles that are isosceles (including equilateral triangles). | [
"052",
"40",
"52"
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"A right hexagonal prism has height $2$. The bases are regular hexagons with side length $1$. Any $3$ of the $12$ vertices determine a triangle. Find the number of these triangles that are isosceles (including equilateral triangles)."
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omr-ws-61ca6a4eaba2611409c44eb7b957403597823399bfcf4ae9ef09441f3c24b0f7 | A right hexagonal prism has height 2. The bases are regular hexagons with side length 1. Any 3 of the 12 vertices determine a triangle. Find the number of these triangles that are isosceles (including equilateral triangles). | [
"52"
] | [
"A right hexagonal prism has height 2. The bases are regular hexagons with side length 1. Any 3 of the 12 vertices determine a triangle. Find the number of these triangles that are isosceles (including equilateral triangles)."
] | [
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omr-ws-5bde058c51e5531fd23865aa491ca93df35829f0e86e236b624df01576662ae1 | A right hexagonal pyramid has a base edge-to-side edge ratio of \( \frac{2}{3} \). Determine the cosine of the angle between the adjacent side walls of this pyramid. | [
"\\(\\frac{-11}{16}\\)"
] | [
"A right hexagonal pyramid has a base edge-to-side edge ratio of \\( \\frac{2}{3} \\). Determine the cosine of the angle between the adjacent side walls of this pyramid."
] | [
"aops_c4_high_school_math"
] | [
"cot",
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omr-ws-2d587c59e4b27297c6ea61ab0652a25c093e9885f69dc0a28dd54e2cb2827771 | A right parallelepiped has sides of lengths 10 cm, 10 cm, and 20 cm. A ladybug is placed on one of the vertices and wants to crawl to the opposite vertex at a constant speed. What is the shortest distance the ladybug must crawl to reach the opposite vertex? | [
"\\(20\\sqrt{2}\\)"
] | [
"A right parallelepiped has sides of lengths 10 cm, 10 cm, and 20 cm. A ladybug is placed on one of the vertices and wants to crawl to the opposite vertex at a constant speed. What is the shortest distance the ladybug must crawl to reach the opposite vertex?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
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"cot": 15,
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} | 52 | 0 | 0 |
omr-ws-286e19c3aeb2c2fa43151739cd6762f205a418e70451639c7e72d1fd4923efbb | A right parallelepiped, whose base is a rhombus with the side $a$ and acute angle $\alpha $ is cut with a plane passing through the vertex of the angle $\alpha $, the section yielding a rhombus with the acute angle $\frac{\alpha }{2}$. Determine the area of this section. | [
"$S_{sec}=2a^2\\sin^2 \\frac{\\alpha }{2}\\cot\\frac{\\alpha }{4}$"
] | [
"A right parallelepiped, whose base is a rhombus with the side $a$ and acute angle $\\alpha $ is cut with a plane passing through the vertex of the angle $\\alpha $, the section yielding a rhombus with the acute angle $\\frac{\\alpha }{2}$. Determine the area of this section."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e988f26167e549ece429aa9a649841ce2631a761468ba3144c944f3a02952eba | A right prism has a square base. Consider the set of all angles formed by a diagonal of a face and a diagonal of the prism. If one such angle has a tangent of 2, find the sum of the tangents of all the angles in the set. | [
"\\(\\frac{56}{3}\\)"
] | [
"A right prism has a square base. Consider the set of all angles formed by a diagonal of a face and a diagonal of the prism. If one such angle has a tangent of 2, find the sum of the tangents of all the angles in the set."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
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"additional_problems": 1
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omr-ws-ba03a3f2b348795e292b1b123150d48ad3df721994c8f8f39ca89265a7aa670f | A right prism has an equilateral triangle as its base. The plane drawn through one of its sides at an angle a to the base cuts off a triangular pyramid of volume $v$ from the prism. Find the area of the section. | [
"$\\sqrt 3\\, \\frac{\\sqrt [3]{v^2\\cot^2 \\alpha }}{\\cos \\alpha }$"
] | [
"A right prism has an equilateral triangle as its base. The plane drawn through one of its sides at an angle a to the base cuts off a triangular pyramid of volume $v$ from the prism. Find the area of the section."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
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"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f64aaafaff0576a48f2c6181ca9b64ff8066665969d2b1cf55543aea0d2f2aad | A right prism with height $h$ has bases that are regular hexagons with sides of length $12$. A vertex $A$ of the prism and its three adjacent vertices are the vertices of a triangular pyramid. The dihedral angle (the angle between the two planes) formed by the face of the pyramid that lies in a base of the prism and th... | [
"108"
] | [
"A right prism with height $h$ has bases that are regular hexagons with sides of length $12$. A vertex $A$ of the prism and its three adjacent vertices are the vertices of a triangular pyramid. The dihedral angle (the angle between the two planes) formed by the face of the pyramid that lies in a base of the prism a... | [
"MATH_training_set"
] | [
"cot",
"genselect",
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] | {
"cot": 15,
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} | 29 | 0 | 0 |
omr-ws-a4f62b34e6eb589d854a432238fd132c54b78b2788346c5a92e989d07d91d98b | A right prism with height \( h \) has bases that are regular hexagons with sides of length 12. A vertex \( A \) of the prism and its three adjacent vertices form a triangular pyramid. The dihedral angle between the face of the pyramid that lies in a base of the prism and the face of the pyramid that does not contain \(... | [
"108"
] | [
"A right prism with height \\( h \\) has bases that are regular hexagons with sides of length 12. A vertex \\( A \\) of the prism and its three adjacent vertices form a triangular pyramid. The dihedral angle between the face of the pyramid that lies in a base of the prism and the face of the pyramid that does not c... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-c83f4e7b449686c4bd7741abc96e80e4a2f1b8ff11e3930cfde6699456405ea9 | A right pyramid has a regular octagon $ABCDEFGH$ with side length $1$ as its base and apex $V$. Segments $\overline{AV}$ and $\overline{DV}$ are perpendicular. What is the square of the height of the pyramid? | [
"\\(\\frac{1+\\sqrt2}{2}\\)"
] | [
"A right pyramid has a regular octagon $ABCDEFGH$ with side length $1$ as its base and apex $V$. Segments $\\overline{AV}$ and $\\overline{DV}$ are perpendicular. What is the square of the height of the pyramid?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-639cbb45f74feb5d7d6b1cd12ba952f69eb0a8838b10af993fe96b79ea65e92f | A right pyramid has a square base with perimeter 24 inches. Its apex is 9 inches from each of the other vertices. What is the height of the pyramid from its peak to the center of its square base, in inches? | [
"3\\sqrt{7}"
] | [
"A right pyramid has a square base with perimeter 24 inches. Its apex is 9 inches from each of the other vertices. What is the height of the pyramid from its peak to the center of its square base, in inches?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-eb06b457e1d651f3d6d553f65d22677908c26e232cdf51ffdaa7df07c7a97f7b | A right pyramid has a square base with side length 10 cm. Its peak is 12 cm above the center of its base. What is the total surface area of the pyramid, in square centimeters? | [
"360"
] | [
"A right pyramid has a square base with side length 10 cm. Its peak is 12 cm above the center of its base. What is the total surface area of the pyramid, in square centimeters?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c7df0a0fbdc735367ccd8f10471d7ad3365cb97f00bd3561d2586ba0d6a7bfcc | A right pyramid with a square base has total surface area 432 square units. The area of each triangular face is half the area of the square face. What is the volume of the pyramid in cubic units? | [
"288\\sqrt{3}"
] | [
"A right pyramid with a square base has total surface area 432 square units. The area of each triangular face is half the area of the square face. What is the volume of the pyramid in cubic units?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b1f64d4578ac35549e847fb594f259b669e19c6b2add5bf89734ba5e93d18d79 | A right rectangular prism $ABCDEFGH$ has edge lengths measuring 12, 18, and 20 units long, such that $AB = CD = ED = GH = 12$ units, $AH = BG = CF = EF = 18$ units, and the remaining parallel edges each measure 20 units. The midpoints of $AH$ and $DE$ are points $M$ and $N$, respectively. A plane containing the points ... | [
"21"
] | [
"A right rectangular prism $ABCDEFGH$ has edge lengths measuring 12, 18, and 20 units long, such that $AB = CD = ED = GH = 12$ units, $AH = BG = CF = EF = 18$ units, and the remaining parallel edges each measure 20 units. The midpoints of $AH$ and $DE$ are points $M$ and $N$, respectively. A plane containing the po... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-9f9c8531a9bcf1dfd362a2d5c60ead889907b5298a065460e3fb65fc26683657 | A right rectangular prism $P$ (i.e., a rectangular parallelpiped) has sides of integral length $a, b, c,$ with $a\le b\le c.$ A plane parallel to one of the faces of $P$ cuts $P$ into two prisms, one of which is similar to $P,$ and both of which have nonzero volume. Given that $b=1995,$ for how many ordered triples $(a... | [
"40"
] | [
"A right rectangular prism $P$ (i.e., a rectangular parallelpiped) has sides of integral length $a, b, c,$ with $a\\le b\\le c.$ A plane parallel to one of the faces of $P$ cuts $P$ into two prisms, one of which is similar to $P,$ and both of which have nonzero volume. Given that $b=1995,$ for how many ordered trip... | [
"MATH_training_set"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 6
} | 13 | 0 | 0 |
omr-ws-a67219fc83db9469a6f290eba350071b5668208dad7f6335076d81eccebe9b0e | A right rectangular prism \( P \) has sides of integral length \( a, b, c \) with \( a \le b \le c \). A plane parallel to one of the faces of \( P \) cuts \( P \) into two prisms, one of which is similar to \( P \), and both of which have nonzero volume. Given that \( b = 1995 \), for how many ordered triples \( (a, b... | [
"41"
] | [
"A right rectangular prism \\( P \\) has sides of integral length \\( a, b, c \\) with \\( a \\le b \\le c \\). A plane parallel to one of the faces of \\( P \\) cuts \\( P \\) into two prisms, one of which is similar to \\( P \\), and both of which have nonzero volume. Given that \\( b = 1995 \\), for how many ord... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-53311a88f67346a97d85593868241cdbdb6f409e9d294e9cd9714a234dded4a6 | A right rectangular prism has edge lengths $\log_2 x$, $\log_3 x$, and $\log_4 x$. If the surface area and volume of the prism are numerically equal, what is the value of $x$? | [
"576"
] | [
"A right rectangular prism has edge lengths $\\log_2 x$, $\\log_3 x$, and $\\log_4 x$. If the surface area and volume of the prism are numerically equal, what is the value of $x$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-491e2d16ce05b802353eb20848bf05dab3823e45badc7ff5003c3b8c7078659a | A right rectangular prism has integer side lengths \(a\), \(b\), and \(c\). Given that \(\text{lcm}(a,b)=72\), \(\text{lcm}(a,c)=24\), and \(\text{lcm}(b,c)=18\), what is the sum of the minimum and maximum possible volumes of the prism? | [
"3024"
] | [
"A right rectangular prism has integer side lengths \\(a\\), \\(b\\), and \\(c\\). Given that \\(\\text{lcm}(a,b)=72\\), \\(\\text{lcm}(a,c)=24\\), and \\(\\text{lcm}(b,c)=18\\), what is the sum of the minimum and maximum possible volumes of the prism?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 15
} | 30 | 0 | 0 |
omr-ws-11b34ffb2bb14f78a601a4e2c29dcba280f25894f48dec869c17b47b061b74fc | A right rectangular prism whose surface area and volume are numerically equal has edge lengths $\log_2 x$, $\log_3 x$, and $\log_4 x$. What is $x$? | [
"576"
] | [
"A right rectangular prism whose surface area and volume are numerically equal has edge lengths $\\log_2 x$, $\\log_3 x$, and $\\log_4 x$. What is $x$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f35f8b62525f4ef071718e14e71b0fab31d7f2276abc76a7102302a2513870cd | A right square pyramid has base edges of length \(5\sqrt{2}\) meters and slant edges of length 13 meters. The pyramid is cut by a plane that is parallel to its base and 2 meters above its base. What is the volume, in cubic meters, of this new, smaller pyramid that is cut off by the plane? | [
"\\dfrac{3125}{27}"
] | [
"A right square pyramid has base edges of length \\(5\\sqrt{2}\\) meters and slant edges of length 13 meters. The pyramid is cut by a plane that is parallel to its base and 2 meters above its base. What is the volume, in cubic meters, of this new, smaller pyramid that is cut off by the plane?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f06589110bfaacd00f891b0969efc35041043d3f8eaa28f7014ab71bb40b636d | A right square pyramid with base edges of length $8\sqrt{2}$ units each and slant edges of length 10 units each is cut by a plane that is parallel to its base and 3 units above its base. What is the volume, in cubic units, of the new pyramid that is cut off by this plane? [asy] import three; size(2.5inch); currentproje... | [
"32"
] | [
"A right square pyramid with base edges of length $8\\sqrt{2}$ units each and slant edges of length 10 units each is cut by a plane that is parallel to its base and 3 units above its base. What is the volume, in cubic units, of the new pyramid that is cut off by this plane? [asy]\nimport three;\nsize(2.5inch);\ncur... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-20c776aeb2903c0501760c48f8fbae918c1d37e4a471969212d10fb0b22afed8 | A right square pyramid with volume $54$ has a base with side length $6$. The five vertices of the pyramid all lie on a sphere with radius $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | [
"21"
] | [
"A right square pyramid with volume $54$ has a base with side length $6$. The five vertices of the pyramid all lie on a sphere with radius $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-f8582ebf459fa063aa50cea581a0a2616fc762ebd29684517e9d79b129a4da7a | A right square pyramid with volume $54$ has a base with side length $6.$ The five vertices of the pyramid all lie on a sphere with radius $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | [
"021"
] | [
"A right square pyramid with volume $54$ has a base with side length $6.$ The five vertices of the pyramid all lie on a sphere with radius $\\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$."
] | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-225900355e78111a0edfb17af153376298b5d4f3dc3db9883108db8a1b26f4bd | A right trapezoid is constructed from tangents to a circle with radius 12. If the area of the trapezoid is 588, find the difference between the lengths of the parallel sides. | [
"7"
] | [
"A right trapezoid is constructed from tangents to a circle with radius 12. If the area of the trapezoid is 588, find the difference between the lengths of the parallel sides."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-5380173b5d7709175c0ba329b5747488a6b4e3c8f614c23b2d129e325c19f0df | A right triangle $ABC$ has hypotenuse $AB = 20$ inches. Its legs $BC = a$ inches and $AC = b$ inches. $D$ is the midpoint of $\overline{BC}$ and $E$ is the midpoint of $\overline{AC}$. When a new right triangle is formed with the lengths $AD$ and $BE$ as its legs, what is the length of its hypotenuse? | [
"10\\sqrt{5}"
] | [
"A right triangle $ABC$ has hypotenuse $AB = 20$ inches. Its legs $BC = a$ inches and $AC = b$ inches. $D$ is the midpoint of $\\overline{BC}$ and $E$ is the midpoint of $\\overline{AC}$. When a new right triangle is formed with the lengths $AD$ and $BE$ as its legs, what is the length of its hypotenuse?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f6dac9888cbc8a89cfdc8311fbe8416e5c92d3692fbecb382ba4cbfef6211302 | A right triangle $ABC$ with hypotenuse $AB$ is inscribed in a circle. Let $K$ be the midpoint of the arc $BC$ not containing $A$, $N$ the midpoint of side $AC$, and $M$ a point of intersection of ray $KN$ with the circle. Let $E$ be a point of intersection of tangents to the circle at points $A$ and $C$. If $\angle EMK... | [
"\\dfrac{\\sqrt{2}}{2}"
] | [
"A right triangle $ABC$ with hypotenuse $AB$ is inscribed in a circle. Let $K$ be the midpoint of the arc $BC$ not containing $A$, $N$ the midpoint of side $AC$, and $M$ a point of intersection of ray $KN$ with the circle. Let $E$ be a point of intersection of tangents to the circle at points $A$ and $C$. If $\\ang... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f49ab2010974b6cb6ac584e968bef569693025c5ed193bbd9150c02c4702e422 | A right triangle $A_1 A_2 A_3$ with side lengths $6,\,8,$ and $10$ on a plane $\mathcal P$ is given. Three spheres $S_1, S_2,$ and $S_3$ with centers $O_1, O_2,$ and $O_3,$ respectively, are located on the same side of the plane $\mathcal P$ such that $S_i$ is tangent to $\mathcal P$ at $A_i$ for $i = 1, 2, 3.$ Assume ... | [
"\\(\\sqrt{769}\\)"
] | [
"A right triangle $A_1 A_2 A_3$ with side lengths $6,\\,8,$ and $10$ on a plane $\\mathcal P$ is given. Three spheres $S_1, S_2,$ and $S_3$ with centers $O_1, O_2,$ and $O_3,$ respectively, are located on the same side of the plane $\\mathcal P$ such that $S_i$ is tangent to $\\mathcal P$ at $A_i$ for $i = 1, 2, 3.... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 28
} | 75 | 0 | 0 |
omr-ws-8e7756d917ce42913a2a74c4e1e9f5c10c477c0bb35317ba9a7029dec88c5444 | A right triangle $MNC$ is inscribed in a right isosceles triangle $A BC$ with $\angle B=90^\circ$ so that $\angle MNC=90^\circ$, point $N$ lies on the hypotenuse $AC$ and point $M$ on the side $AB$. In what ratio must the point $N$ divide the hypotenuse $AC$ for the area of $\triangle MNC$ to constitute 3/8 of the area... | [
"$\\frac{|NC|}{|AC|}=\\frac{3}{4}$"
] | [
"A right triangle $MNC$ is inscribed in a right isosceles triangle $A BC$ with $\\angle B=90^\\circ$ so that $\\angle MNC=90^\\circ$, point $N$ lies on the hypotenuse $AC$ and point $M$ on the side $AB$. In what ratio must the point $N$ divide the hypotenuse $AC$ for the area of $\\triangle MNC$ to constitute 3/8 o... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-251ace9d1f3d65c50dc6988ccef77b7c4704eda944bc0b5bf59086a6e0c1eef2 | A right triangle $OAB$ with $\angle A = 90^\circ$ lies inside another triangle with vertex $O$. The altitude from $A$ to the side of angle $O$ meets at point $M$. The distances from $M$ and $B$ to the second side of angle $O$ are $2$ and $1$ respectively. Find the length of $OA$. | [
"\\(\\sqrt{2}\\)"
] | [
"A right triangle $OAB$ with $\\angle A = 90^\\circ$ lies inside another triangle with vertex $O$. The altitude from $A$ to the side of angle $O$ meets at point $M$. The distances from $M$ and $B$ to the second side of angle $O$ are $2$ and $1$ respectively. Find the length of $OA$."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-405c56dc123b707d34f3cd3132498a36cd6d5b638782b9f8db73db49ce056de6 | A right triangle \(ABC\) with right angle at \(C\) is inscribed in circle \((O)\). Point \(M\) is an arbitrary point on circle \((O)\), different from \(A\), \(B\), and \(C\). Point \(N\) is the reflection of \(M\) in \(AB\), \(P\) is the perpendicular foot of \(N\) to \(AC\), and \(MP\) meets \((O)\) at a second point... | [
"\\dfrac{5}{2}"
] | [
"A right triangle \\(ABC\\) with right angle at \\(C\\) is inscribed in circle \\((O)\\). Point \\(M\\) is an arbitrary point on circle \\((O)\\), different from \\(A\\), \\(B\\), and \\(C\\). Point \\(N\\) is the reflection of \\(M\\) in \\(AB\\), \\(P\\) is the perpendicular foot of \\(N\\) to \\(AC\\), and \\(MP... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-8dfb470d0ea359b4e06d37aef99d0830fd3263a1cfae12b7db254a905ea7418e | A right triangle has a circumscribed circle of radius R and an inscribed circle of radius r. Find the area of the triangle in terms of R and r. | [
"r(r + 2R)"
] | [
"A right triangle has a circumscribed circle of radius R and an inscribed circle of radius r. Find the area of the triangle in terms of R and r."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1714825ea776fbc93cdce1061ccd55db7e3b6461bd555efcb51dfb70995d2aaa | A right triangle has a hypotenuse equal to the sum of the sine and cosine values of one of its acute angles. What is the maximum value of one of its legs? | [
"\\(\\frac{1+\\sqrt{2}}{2}\\)"
] | [
"A right triangle has a hypotenuse equal to the sum of the sine and cosine values of one of its acute angles. What is the maximum value of one of its legs?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-85fffbc71f5d47d40e5223ff55342fb756de988d639239de938923bf87a9fb03 | A right triangle has a hypotenuse of length 26. If one of the acute angles is decreasing at the rate of 10 degrees per second, how fast is the area of the triangle decreasing when the acute angle is 13 degrees? | [
"\\( 169 \\cos\\left(\\frac{77\\pi}{90}\\right) \\frac{\\pi}{9} \\)"
] | [
"A right triangle has a hypotenuse of length 26. If one of the acute angles is decreasing at the rate of 10 degrees per second, how fast is the area of the triangle decreasing when the acute angle is 13 degrees?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
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] | {
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} | 12 | 0 | 0 |
omr-ws-abc44b62a95dea0e6da8875de151c5d3d202a6b01775b1e162679a4d4dde91e9 | A right triangle has a perimeter of $2008$, and the area of a circle inscribed in the triangle is $100\pi^3$. Let $A$ be the area of the triangle. Compute $\lfloor A\rfloor$. | [
"31541"
] | [
"A right triangle has a perimeter of $2008$, and the area of a circle inscribed in the triangle is $100\\pi^3$. Let $A$ be the area of the triangle. Compute $\\lfloor A\\rfloor$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
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} | 39 | 0 | 0 |
omr-ws-d5f5768601df61be04ce294dc76011cf65906202a7fbf8c778cef6bb5abdb201 | A right triangle has a perimeter of 32 and an area of 20. What is the length of its hypotenuse? | [
"\\(\\frac{59}{4}\\)"
] | [
"A right triangle has a perimeter of 32 and an area of 20. What is the length of its hypotenuse?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-670292aefd0502617ba113cd1ccec132b8a3062946be157cf573543bce3dce64 | A right triangle has a perimeter of 60 and the product of its side lengths is 1740. What is the area of the triangle? | [
"30"
] | [
"A right triangle has a perimeter of 60 and the product of its side lengths is 1740. What is the area of the triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-d4717d7d73192f09b7836d98b5c0c671f47a72f90136b9eff0dfdc7973f401fd | A right triangle has a perimeter of length 7 and a hypotenuse of length 3. If $\theta$ is the largest non-right angle in the triangle, what is the value of $\cos{\theta}$? | [
"\\(\\frac{4+\\sqrt{2}}{6}\\)"
] | [
"A right triangle has a perimeter of length 7 and a hypotenuse of length 3. If $\\theta$ is the largest non-right angle in the triangle, what is the value of $\\cos{\\theta}$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3b95f89fa539c8d79396c303d77703e29032296fd67cce66ce9b9247c1796575 | A right triangle has a side length of 21 inches and a hypotenuse of 29 inches. A second triangle is similar to the first and has a hypotenuse of 87 inches. What is the length of the shortest side of the second triangle? | [
"60\\text{ inches}"
] | [
"A right triangle has a side length of 21 inches and a hypotenuse of 29 inches. A second triangle is similar to the first and has a hypotenuse of 87 inches. What is the length of the shortest side of the second triangle?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-53a89a7d8b2a3eac731429c4c2fc96d1245a528c6069542f07aa07e14dca5e69 | A right triangle has an area of 120 square units, and a leg length of 24 units. What is the perimeter of the triangle, in units? | [
"60"
] | [
"A right triangle has an area of 120 square units, and a leg length of 24 units. What is the perimeter of the triangle, in units?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8b35eb86426fb4ecceef5a1a0cc1248b9cadf3b67a13964d7f23835493d622f5 | A right triangle has an inscribed circle that cuts the hypotenuse into lengths of 7 and 11. What is the area of the triangle? | [
"77"
] | [
"A right triangle has an inscribed circle that cuts the hypotenuse into lengths of 7 and 11. What is the area of the triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-e900cb1902a6caa6a0bd3202cb134a4cf42a2db25c57e4b2b2fe5291f3f8099a | A right triangle has an inscribed circle, and another inscribed circle tangent to the hypotenuse, the second longest side, and the first circle, and a third circle tangent to the second longest side, the hypotenuse, and the second circle. What percent of the right triangle's area is covered by the three circles? | [
"70\\%"
] | [
"A right triangle has an inscribed circle, and another inscribed circle tangent to the hypotenuse, the second longest side, and the first circle, and a third circle tangent to the second longest side, the hypotenuse, and the second circle. What percent of the right triangle's area is covered by the three circles?"
... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-a12d4e4cbc498c399da147c4d7931f2455c173827e5b68aa85a129ee2291397f | A right triangle has area \( A \) and perimeter \( P \). Find the largest possible value for the positive constant \( k \) such that for every such triangle, \( P^2 \ge kA \). | [
"12 + 8\\sqrt{2}"
] | [
"A right triangle has area \\( A \\) and perimeter \\( P \\). Find the largest possible value for the positive constant \\( k \\) such that for every such triangle, \\( P^2 \\ge kA \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1417eb477994b2975a18be674194895133663dbfcdd4c10ab19c23bc02eb1462 | A right triangle has integer legs and hypotenuse \( 2009 \). What is its perimeter? | [
"4410"
] | [
"A right triangle has integer legs and hypotenuse \\( 2009 \\). What is its perimeter?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-30ea1c3e79afa21d9b5fae02c4ac9e700ff652f609dda3c6db3887d0668503ed | A right triangle has integer side lengths, and its longer leg is 1 less than its hypotenuse. Let $r$ be its inradius, and let $R$ be its circumradius. If $1 < |R - r| < 100$, find the sum of the possible values for the triangle's hypotenuse. | [
"669"
] | [
"A right triangle has integer side lengths, and its longer leg is 1 less than its hypotenuse. Let $r$ be its inradius, and let $R$ be its circumradius. If $1 < |R - r| < 100$, find the sum of the possible values for the triangle's hypotenuse."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-598ba1e64d3f8d872ed17528a3cbc41e93447ee25f589d4bb118f633b62389f3 | A right triangle has integer side lengths, and the sum of its area and the length of one of its legs equals $75$. Find the side lengths of the triangle. | [
"\\(8, 15, 17\\)"
] | [
"A right triangle has integer side lengths, and the sum of its area and the length of one of its legs equals $75$. Find the side lengths of the triangle."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 32
} | 36 | 0 | 0 |
omr-ws-823085c4c8f361e62fe74f2cfb8838d261bd466b2a75adad444fd88c0d7ecca0 | A right triangle has integer sides and hypotenuse 2022. Find its perimeter. | [
"4800"
] | [
"A right triangle has integer sides and hypotenuse 2022. Find its perimeter."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e128265485d8ba875c2f53993e669e6d815389b53aedf224680061d00933058a | A right triangle has legs $a$ and $b$ and hypotenuse $c.$ Find the largest possible value of \[\frac{a + b}{c}.\] | [
"\\sqrt{2}"
] | [
"A right triangle has legs $a$ and $b$ and hypotenuse $c.$ Find the largest possible value of\n\\[\\frac{a + b}{c}.\\]"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e4060b29017c9cdc52bba5fdcbe7608cfdf720aad247287d31b96e041377bbe3 | A right triangle has legs \(a\) and \(b\) and hypotenuse \(c\). Let \(a\) be the eccentricity of the ellipse with equation \(16x^2 + 9y^2 - 64x + 54y = -1\) and let \(b\) be the positive root of the curve \(y = x^3 + 2x^2 - x - 2\). Find the hypotenuse \(c\). | [
"\\(\\frac{\\sqrt{23}}{4}\\)"
] | [
"A right triangle has legs \\(a\\) and \\(b\\) and hypotenuse \\(c\\). Let \\(a\\) be the eccentricity of the ellipse with equation \\(16x^2 + 9y^2 - 64x + 54y = -1\\) and let \\(b\\) be the positive root of the curve \\(y = x^3 + 2x^2 - x - 2\\). Find the hypotenuse \\(c\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-86ff40826de10599abffaf44ff2f8bec98595db037a0dc6470ce733050352595 | A right triangle has legs measuring 20 inches and 21 inches. What is the length of the hypotenuse, in inches? | [
"29"
] | [
"A right triangle has legs measuring 20 inches and 21 inches. What is the length of the hypotenuse, in inches?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-97f7fab61c31f816ac151bbf4d641120b87700b39d9dcc16192a36ba9aac1c3a | A right triangle has legs of length \(x\) and \(y\). Given that \(2^x \cdot 2^y = 1024\) and \((2^x)^y = 16\), find the length of the hypotenuse. | [
"2\\sqrt{23}"
] | [
"A right triangle has legs of length \\(x\\) and \\(y\\). Given that \\(2^x \\cdot 2^y = 1024\\) and \\((2^x)^y = 16\\), find the length of the hypotenuse."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-0683cb403d9665257b9cc9ec561ebaa850cd6726e88d30aabccd018503bda5af | A right triangle has legs of lengths 13 and 15. The hypotenuse of the triangle is also the side of a square. Find (a) the perimeter of the square, and (b) the area of the square. | [
"Perimeter = \\( 4\\sqrt{394} \\), Area = \\( 394 \\)"
] | [
"A right triangle has legs of lengths 13 and 15. The hypotenuse of the triangle is also the side of a square. Find (a) the perimeter of the square, and (b) the area of the square."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 23
} | 23 | 0 | 0 |
omr-ws-b890680acf20832781689006ba0a2f42df2ca889bd4865b1f4393daab41c1b4e | A right triangle has one leg of length 10 and another of a random length between 0 and 10. What is the probability that the square units of area exceeds the units in the length of the hypotenuse? | [
"\\dfrac{12 - \\sqrt{6}}{12}"
] | [
"A right triangle has one leg of length 10 and another of a random length between 0 and 10. What is the probability that the square units of area exceeds the units in the length of the hypotenuse?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a1d4ca010b9f62289ce5e14f707b38677136aedcbc6f67ad2619d64a620235ef | A right triangle has one leg which equals 91 and hypotenuse 109. What is the length of the other leg? | [
"60"
] | [
"A right triangle has one leg which equals 91 and hypotenuse 109. What is the length of the other leg?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a7148aa26687757b324a3980e1f5edfda971b11843e86d0dc1d82ae5ac5454a3 | A right triangle has one side equal to $25.3$ and a perimeter of $68.4$. Find the length of the hypotenuse and the other leg. | [
"Hypotenuse = 29, Other leg = 14.1"
] | [
"A right triangle has one side equal to $25.3$ and a perimeter of $68.4$. Find the length of the hypotenuse and the other leg."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 21
} | 21 | 0 | 0 |
omr-ws-1e635c408f4bb04dfff0e700348a9b5e41e57192ef8a41f15d924916e2d089d2 | A right triangle has side lengths 3, 4, and 5 units. Another triangle has sides of length 4, \( x \), and \( y \) units and has the same area as the right triangle. If \( x \leq y \), what is the smallest possible value of \( y \)? Express your answer in simplest radical form. | [
"\\(\\sqrt{13}\\)"
] | [
"A right triangle has side lengths 3, 4, and 5 units. Another triangle has sides of length 4, \\( x \\), and \\( y \\) units and has the same area as the right triangle. If \\( x \\leq y \\), what is the smallest possible value of \\( y \\)? Express your answer in simplest radical form."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-a8cfe12fd8b1c191ee046b13c0830a39c162cf72a6eaaa2d2411c5d6aaec1135 | A right triangle has sides \(X = 50\) meters and \(Y = 60\) meters, with hypotenuse \(Z = \sqrt{6100} \approx 78\) meters. Side \(Z\) is falling against side \(X\) at a constant rate of \(8.333\) meters/minute, causing \(X\) to shrink. Simultaneously, side \(Z\) is sliding outwards along side \(Y\), causing \(Y\) to in... | [
"6.944 meters/minute"
] | [
"A right triangle has sides \\(X = 50\\) meters and \\(Y = 60\\) meters, with hypotenuse \\(Z = \\sqrt{6100} \\approx 78\\) meters. Side \\(Z\\) is falling against side \\(X\\) at a constant rate of \\(8.333\\) meters/minute, causing \\(X\\) to shrink. Simultaneously, side \\(Z\\) is sliding outwards along side \\(... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-fc21c10027f3ef3e3744b999fd9eb9f56d3b16c7c8b9a09d4290ee6196139a49 | A right triangle has sides of integer length and a perimeter of 2010 units. The length of the shortest side is equal to one-fifth of the sum of the lengths of the other two sides. What is the area of the triangle? | [
"134670"
] | [
"A right triangle has sides of integer length and a perimeter of 2010 units. The length of the shortest side is equal to one-fifth of the sum of the lengths of the other two sides. What is the area of the triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-d97ef093c71de28581022edb29d40057d4542971f2ccb0c8a1f498ce98995365 | A right triangle has sides with lengths \(4 + 2^{1/3}\) and \(3 + 7 \cdot 2^{1/3}\). What is the length of the hypotenuse? Write your answer in the form \(a + b \cdot c^{1/3}\), where \(a\), \(b\), and \(c\) are positive integers. | [
"\\(5 + 5 \\cdot 4^{1/3}\\)"
] | [
"A right triangle has sides with lengths \\(4 + 2^{1/3}\\) and \\(3 + 7 \\cdot 2^{1/3}\\). What is the length of the hypotenuse? Write your answer in the form \\(a + b \\cdot c^{1/3}\\), where \\(a\\), \\(b\\), and \\(c\\) are positive integers."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-7847bfd1685db80ba381d86910ff2f1ced4ced1e2ded91ff0b20de43ad0e459a | A right triangle has three integer sides, one of which is 2021 units. Find the minimum perimeter such a triangle could have. | [
"4230"
] | [
"A right triangle has three integer sides, one of which is 2021 units. Find the minimum perimeter such a triangle could have."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 9
} | 24 | 0 | 0 |
omr-ws-fc41a08bcbb91681480a5952996cdb2d8bf743e948fd7a744ce9663224e0ad1c | A right triangle has vertices at (2,-2,1), (2,-2,7), and (5,2,1). What is the length of the hypotenuse? | [
"\\(\\sqrt{61}\\)"
] | [
"A right triangle has vertices at (2,-2,1), (2,-2,7), and (5,2,1). What is the length of the hypotenuse?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-25fa5cfa96c1ea56c10e25048ef284ce4a4eacdbf5e420b932e7e78eaa074934 | A right triangle is bounded in a coordinate plane by the lines \( x = 0 \), \( y = 0 \), \( x = x_{100} \), and \( y = f(x) \), where \( f \) is a linear function with a negative slope and \( f(x_{100}) = 0 \). The lines \( x = x_1 \), \( x = x_2 \), \( \ldots \), \( x = x_{99} \) (\( x_1 < x_2 < \ldots < x_{100} \)) s... | [
"\\(100 + 30\\sqrt{11}\\)"
] | [
"A right triangle is bounded in a coordinate plane by the lines \\( x = 0 \\), \\( y = 0 \\), \\( x = x_{100} \\), and \\( y = f(x) \\), where \\( f \\) is a linear function with a negative slope and \\( f(x_{100}) = 0 \\). The lines \\( x = x_1 \\), \\( x = x_2 \\), \\( \\ldots \\), \\( x = x_{99} \\) (\\( x_1 < x... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 6,
"tir": 23
} | 42 | 0 | 0 |
omr-ws-270afd295d0407cf58161e96fae97ae999ef0bb44c19f4d3ece42431d29b0f89 | A right triangle is divided into two smaller triangles by drawing an altitude towards the hypotenuse. If the distance between the incenters of the two smaller triangles is 14, find the inradius of the original triangle. | [
"\\( 7\\sqrt{2} \\)"
] | [
"A right triangle is divided into two smaller triangles by drawing an altitude towards the hypotenuse. If the distance between the incenters of the two smaller triangles is 14, find the inradius of the original triangle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 7,
"tir": 8
} | 27 | 0 | 0 |
omr-ws-31333966099a046e2504f09de8ee6160220fe2e4333986872c723cba2ff6f9f1 | A right triangle is divided into two smaller triangles when an altitude is drawn towards the hypotenuse. If the distance between the incenters of the two smaller triangles is 14, find the inradius of the original triangle. | [
"\\( 7\\sqrt{2} \\)"
] | [
"A right triangle is divided into two smaller triangles when an altitude is drawn towards the hypotenuse. If the distance between the incenters of the two smaller triangles is 14, find the inradius of the original triangle."
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 26,
"genselect": 8,
"tir": 16
} | 50 | 0 | 0 |
omr-ws-c584f4cf2ba10d2dedfbcf4db473f7d84eb584753f51f565f0c460d29d3a435b | A right triangle is formed in the first quadrant by the x- and y-axes and a line through the point (1,2). Write the length of the hypotenuse as a function of x. | [
"\\(\\frac{x \\sqrt{x^2 - 2x + 5}}{x-1}\\)"
] | [
"A right triangle is formed in the first quadrant by the x- and y-axes and a line through the point (1,2). Write the length of the hypotenuse as a function of x."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 6,
"tir": 26
} | 45 | 0 | 0 |
omr-ws-3370a7341b038f7068fc53b5b0420251a8998dafdd99cf33921d48b366aa837b | A right triangle is inscribed in a rectangle \(ABCD\) such that one vertex of the triangle is at \(A\) and the other two vertices lie on the sides \(BC\) and \(CD\) respectively. Let \(T_1\), \(T_2\), and \(T_3\) be the areas of the three right triangles formed, where \(T_1\) is the area of the triangle with the hypote... | [
"1"
] | [
"A right triangle is inscribed in a rectangle \\(ABCD\\) such that one vertex of the triangle is at \\(A\\) and the other two vertices lie on the sides \\(BC\\) and \\(CD\\) respectively. Let \\(T_1\\), \\(T_2\\), and \\(T_3\\) be the areas of the three right triangles formed, where \\(T_1\\) is the area of the tri... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-ff29cb12eeb4acf069538779ccebb371e1eb456116158b5fa10e44e1b2b82ba8 | A right triangle is inscribed in the parabola \(y = x^2\) with vertices at \((a, a^2)\), \((b, b^2)\), and \((c, c^2)\), where \(a < b < c\). Let \(L\) be the length of the hypotenuse of this triangle. Find the minimum possible value of \(L^2\). | [
"4"
] | [
"A right triangle is inscribed in the parabola \\(y = x^2\\) with vertices at \\((a, a^2)\\), \\((b, b^2)\\), and \\((c, c^2)\\), where \\(a < b < c\\). Let \\(L\\) be the length of the hypotenuse of this triangle. Find the minimum possible value of \\(L^2\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-0964f074be3331e206a197b92cbd02400e446e73355afce14d00755f0af9a792 | A right triangle is inscribed in the parabola \(y = x^2\), with the hypotenuse parallel to the \(Ox\) axis. What can be the height of the triangle lowered to the hypotenuse? | [
"1"
] | [
"A right triangle is inscribed in the parabola \\(y = x^2\\), with the hypotenuse parallel to the \\(Ox\\) axis. What can be the height of the triangle lowered to the hypotenuse?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
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