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-e6a0b22d19d27a4c2ed286b0f85395d56c1d3c6f7246a55f029bb5b3c0f1f06e | A sphere with radius 3 is inscribed in a frustrum, and the frustrum has a slant height of 10. What is the volume of the frustrum? | [
"182\\pi"
] | [
"A sphere with radius 3 is inscribed in a frustrum, and the frustrum has a slant height of 10. What is the volume of the frustrum?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-5a24753cbe746c5723a5b0d1cda076de561d4fbec4001fc5b968aaff988cb1a8 | A sphere with radius 3 is inscribed in a frustum with a slant height of 10. What is the volume of the frustum? (A frustum is a non-conical region formed by cutting a cone with a plane parallel to its base.) | [
"182\\pi"
] | [
"A sphere with radius 3 is inscribed in a frustum with a slant height of 10. What is the volume of the frustum? (A frustum is a non-conical region formed by cutting a cone with a plane parallel to its base.)"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0664b6eb2a01d82deb1052e0fb4fbd15f4a7ef2c89288bc167a33c2da8ebae05 | A sphere with radius 30 and center \( O_1 \) and a sphere with radius 20 with center \( O_2 \) intersect, forming a circle of intersection. Given that \( A \) and \( B \) are a diameter of the circle of intersection, and \( \angle AO_1B \) is \( 60^\circ \), find the volume of the entire figure. | [
"\\dfrac{\\pi}{3} \\left(70000 + 30375\\sqrt{3} + 5125\\sqrt{7}\\right)"
] | [
"A sphere with radius 30 and center \\( O_1 \\) and a sphere with radius 20 with center \\( O_2 \\) intersect, forming a circle of intersection. Given that \\( A \\) and \\( B \\) are a diameter of the circle of intersection, and \\( \\angle AO_1B \\) is \\( 60^\\circ \\), find the volume of the entire figure."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 6
} | 20 | 0 | 0 |
omr-ws-6d2b98d2a22420f882db28a45598f6483ca17d27c1bf6c88a8f346e956045eef | A sphere with radius \( R = 1 \) is cut by \( 2n \) planes into disks \( D_k \). These disks are placed at distances \( d_i > 0 \) from the center of the sphere, where \( i = \overline{1, 2n} \). If \( \sum d_i = n \), find the minimum number of disks that a line passing through the center of the sphere must intersect. | [
"n"
] | [
"A sphere with radius \\( R = 1 \\) is cut by \\( 2n \\) planes into disks \\( D_k \\). These disks are placed at distances \\( d_i > 0 \\) from the center of the sphere, where \\( i = \\overline{1, 2n} \\). If \\( \\sum d_i = n \\), find the minimum number of disks that a line passing through the center of the sph... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-abd1877d33ba6a00bbeab00c7d2991b57a976e9c7f31dc1fae724c31d4bb0f27 | A sphere with radius \( R \) is cut by two parallel planes such that the center of the sphere is outside the region determined by these planes. Let \( S_1 \) and \( S_2 \) be the areas of the intersections, and \( d \) the distance between these planes. Find the area of the intersection of the sphere with a plane paral... | [
"\\dfrac{S_1 + S_2}{2} + \\dfrac{\\pi d^2}{4}"
] | [
"A sphere with radius \\( R \\) is cut by two parallel planes such that the center of the sphere is outside the region determined by these planes. Let \\( S_1 \\) and \\( S_2 \\) be the areas of the intersections, and \\( d \\) the distance between these planes. Find the area of the intersection of the sphere with ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-9ef927e33222e2a052aa7b7a37fe1971b980169857d5e0bf6e9d070294c9efa6 | A sphere with radius \( r_0 = 10^{-18} \) meters is circumscribed by a cube, which is then circumscribed by another sphere with radius \( r_1 \). This process of circumscribing a cube and then a sphere is repeated, resulting in a sequence of radii \( r_n \). What is the smallest value of \( n \) such that the radius \(... | [
"188"
] | [
"A sphere with radius \\( r_0 = 10^{-18} \\) meters is circumscribed by a cube, which is then circumscribed by another sphere with radius \\( r_1 \\). This process of circumscribing a cube and then a sphere is repeated, resulting in a sequence of radii \\( r_n \\). What is the smallest value of \\( n \\) such that ... | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-b50ac78d722b1f05822ae94e9c085414a32902d8a08a605da04428c08ba53c4c | A spherical ball is being inflated at a rate of 20 cubic centimetres per second. Find the rate of increase of its radius when its surface area is $100\pi$ square centimetres. | [
"\\dfrac{1}{5\\pi}"
] | [
"A spherical ball is being inflated at a rate of 20 cubic centimetres per second. Find the rate of increase of its radius when its surface area is $100\\pi$ square centimetres."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-738f20187bd016dac74d310ab39b1f7b06925241fed39a6ef043509583828b53 | A spherical ball is floating in a bath such that more than half of it is above the water level. The lowest point of the ball is 7 cm below the water level, and the points of contact between the surface of the water and the ball form a circle with a diameter of 30 cm. Find the diameter of the ball, correct to three sign... | [
"39.143"
] | [
"A spherical ball is floating in a bath such that more than half of it is above the water level. The lowest point of the ball is 7 cm below the water level, and the points of contact between the surface of the water and the ball form a circle with a diameter of 30 cm. Find the diameter of the ball, correct to three... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-61c25a58cf70757f31fcbea8379719e24f8080e67e90f502421d1e4e69dbcebf | A spherical ball of volatile material evaporates at a rate proportional to its surface area. If the initial radius is $r_0$ and at time $t = 1$, the radius is $r_0/2$, find the time at which the ball disappears completely. | [
"2"
] | [
"A spherical ball of volatile material evaporates at a rate proportional to its surface area. If the initial radius is $r_0$ and at time $t = 1$, the radius is $r_0/2$, find the time at which the ball disappears completely."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-5b903da462691c4c206afcebd243512c1f77e29e336d8f353bd676dfc79553fd | A spherical balloon is being inflated at a rate of 600 cm³/sec. When the radius of the balloon is 15 cm, what is the rate of increase of the radius? | [
"\\(\\frac{2}{3\\pi}\\) cm/sec"
] | [
"A spherical balloon is being inflated at a rate of 600 cm³/sec. When the radius of the balloon is 15 cm, what is the rate of increase of the radius?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 28
} | 29 | 0 | 0 |
omr-ws-521bb38c22ff90982d7f1f0b083dfa25ea332a9cf9f0a5340ac77dc7dc32d371 | A spherical balloon is being inflated at a rate of 600 cm³/sec. When the radius of the balloon is 15 cm, what is the rate of increase of the surface area? | [
"80 cm²/sec"
] | [
"A spherical balloon is being inflated at a rate of 600 cm³/sec. When the radius of the balloon is 15 cm, what is the rate of increase of the surface area?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-6a68ac78f85b444432700702eebfe02306d08d4ce675dbc740d6fec984465a09 | A spherical cow is being pulled out of a deep well. The bottom of the well is $100$ feet down, and the cow and the water on top of the cow weigh $200$ pounds. The cow is being hauled up at a constant rate with a chain which weighs $2$ pounds per foot. The water on the cow drips off at $\frac{1}{2}$ pound per foot as it... | [
"27500"
] | [
"A spherical cow is being pulled out of a deep well. The bottom of the well is $100$ feet down, and the cow and the water on top of the cow weigh $200$ pounds. The cow is being hauled up at a constant rate with a chain which weighs $2$ pounds per foot. The water on the cow drips off at $\\frac{1}{2}$ pound per foot... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-4ea5858438d2d9a3eadfd7428b33e24ae10d60d7b587dfb5382bb28daaf0010b | A spherical glass globe with a radius of 12 cm is inscribed in a cylinder. Latitude and longitude lines are painted on the globe. A light source at the center of the globe projects an image of the lines onto the cylinder. Find the area on the cylinder of the image of the region bounded by the lines of 0 and 45 degrees ... | [
"24\\pi"
] | [
"A spherical glass globe with a radius of 12 cm is inscribed in a cylinder. Latitude and longitude lines are painted on the globe. A light source at the center of the globe projects an image of the lines onto the cylinder. Find the area on the cylinder of the image of the region bounded by the lines of 0 and 45 deg... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-93763bae319f9392fa13453f1ca835b4a69351a2a2d95a08ab195d7222932d91 | A spherical head of Swiss cheese has holes in the form of non-overlapping balls with radii $1, 2.7, \ldots, 2.7^n$ mm. Find the smallest possible radius of the head of Swiss cheese. | [
"\\(3.7 \\times 2.7^{n-1}\\)"
] | [
"A spherical head of Swiss cheese has holes in the form of non-overlapping balls with radii $1, 2.7, \\ldots, 2.7^n$ mm. Find the smallest possible radius of the head of Swiss cheese."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-b494cf7f3a0182e8bc52b8b587dbdcfa33d4ea73b0bfc2ddc331480cdadec7e0 | A spherical meteor is submerged in the water on a minor planet with a radius of 30 kilometers, causing the water level to rise by 1 centimeter. What is the radius of the meteor in meters? | [
"300 meters"
] | [
"A spherical meteor is submerged in the water on a minor planet with a radius of 30 kilometers, causing the water level to rise by 1 centimeter. What is the radius of the meteor in meters?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-b7a60791b0fe9a95ef3822be36c291d7207531a1af287782e685f2911988c6b6 | A spherical planet with center at $(0,0,0)$ and radius 20 has a temperature function $T(x,y,z) = (x+y)^2 + (y-z)^2$ at any point $(x,y,z)$ on its surface. What is the average temperature of the surface of this planet? | [
"478.539"
] | [
"A spherical planet with center at $(0,0,0)$ and radius 20 has a temperature function $T(x,y,z) = (x+y)^2 + (y-z)^2$ at any point $(x,y,z)$ on its surface. What is the average temperature of the surface of this planet?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-10f77b54466b52d7cd1ef5896e386945e25586da57d083d657f6cfffb13e5f7f | A spherical planet with center at (0,0,0) and radius 20 has a temperature function given by \( T = (x+y)^2 + (y-z)^2 \) at any point \((x,y,z)\) on its surface. What is the average temperature of the surface of this planet? | [
"\\dfrac{1600}{3}"
] | [
"A spherical planet with center at (0,0,0) and radius 20 has a temperature function given by \\( T = (x+y)^2 + (y-z)^2 \\) at any point \\((x,y,z)\\) on its surface. What is the average temperature of the surface of this planet?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-002247d6b29f0e7595012dc97f6f4dbf25b20bdaf6c4bdd040e6267049ee62d6 | A spherical planet with center at \((0,0,0)\) and radius 20 has a temperature function \(T(x,y,z) = (x+y)^2 + (y-z)^2\) at any point \((x,y,z)\) on its surface. What is the average temperature of the surface? | [
"\\(\\frac{1600}{3}\\)"
] | [
"A spherical planet with center at \\((0,0,0)\\) and radius 20 has a temperature function \\(T(x,y,z) = (x+y)^2 + (y-z)^2\\) at any point \\((x,y,z)\\) on its surface. What is the average temperature of the surface?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 27
} | 75 | 0 | 0 |
omr-ws-34a674eab217d69502302edbbfc1347b05a445c40f1b992c4e7ddc70354fe162 | A spherical planet with center at \((0,0,0)\) and radius \(20\) has a temperature function \(T(x,y,z) = (x+y)^2 + (y-z)^2\) at any point \((x,y,z)\) on its surface. What is the average temperature of the surface of this planet? | [
"400"
] | [
"A spherical planet with center at \\((0,0,0)\\) and radius \\(20\\) has a temperature function \\(T(x,y,z) = (x+y)^2 + (y-z)^2\\) at any point \\((x,y,z)\\) on its surface. What is the average temperature of the surface of this planet?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-45b39369032ea7dc8184e3530b6a0d9730652ffef1431d2b9ad011aa225cea06 | A spherical rock with a radius of 4 decimetres and a cylindrical log with a diameter of 4 decimetres are used to create leverage with a pole. The points of tangency of the rock and the log with the ground are 9 decimetres apart. How long is the pole, assuming the thickness of the pole is negligible? | [
"17"
] | [
"A spherical rock with a radius of 4 decimetres and a cylindrical log with a diameter of 4 decimetres are used to create leverage with a pole. The points of tangency of the rock and the log with the ground are 9 decimetres apart. How long is the pole, assuming the thickness of the pole is negligible?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-30edcf9bb7e801efadfe0fe621db2ad36b40e389fd5b3a1dcbbf6c77d88b9e04 | A spherical scoop of ice cream with diameter $2$ inches is centered uniformly at random inside the cube $0 \le x, y, z \le 1$. The sphere is then cut through the planes $x = n$, $y = n$, and $z = n$ for every integer $n$. What is the expected number of pieces the ice cream is cut into? | [
"27"
] | [
"A spherical scoop of ice cream with diameter $2$ inches is centered uniformly at random inside the cube $0 \\le x, y, z \\le 1$. The sphere is then cut through the planes $x = n$, $y = n$, and $z = n$ for every integer $n$. What is the expected number of pieces the ice cream is cut into?"
] | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 6
} | 6 | 0 | 0 |
omr-ws-8ac2a5edc50fcb811161ef41a63e8423feab9f0cb29f0c45af386aece4216058 | A spherical scoop of vanilla ice cream with radius of 2 inches is dropped onto the surface of a dish of hot chocolate sauce. As it melts, the ice cream spreads out uniformly forming a cylindrical region 8 inches in radius. Assuming the density of the ice cream remains constant, how many inches deep is the melted ice cr... | [
"\\frac{1}{6}"
] | [
"A spherical scoop of vanilla ice cream with radius of 2 inches is dropped onto the surface of a dish of hot chocolate sauce. As it melts, the ice cream spreads out uniformly forming a cylindrical region 8 inches in radius. Assuming the density of the ice cream remains constant, how many inches deep is the melted i... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c320be1f7db3528b4dfb708549f9fe30436ec58666793a3c83573368451daa07 | A spherical soap bubble lands on a horizontal wet surface and forms a hemisphere of the same volume. Given the radius of the hemisphere is $3\sqrt[3]{2}$ cm, find the radius of the original bubble. | [
"3"
] | [
"A spherical soap bubble lands on a horizontal wet surface and forms a hemisphere of the same volume. Given the radius of the hemisphere is $3\\sqrt[3]{2}$ cm, find the radius of the original bubble."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b774c321721a1c0f52be719eda1802972fa573cc06761252f5f2cb4d99fda5b6 | A spherical tank initially contains 81.637 gallons of water. For the next 6 minutes, water flows out of the tank at the rate of \( 9\sin{\sqrt{t+1}} \) gallons per minute. How many gallons of water are in the tank at the end of the 6 minutes? | [
"36.453"
] | [
"A spherical tank initially contains 81.637 gallons of water. For the next 6 minutes, water flows out of the tank at the rate of \\( 9\\sin{\\sqrt{t+1}} \\) gallons per minute. How many gallons of water are in the tank at the end of the 6 minutes?"
] | [
"aops_c7_college_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 5
} | 6 | 0 | 0 |
omr-ws-42ee41c41e5ea21365515f17f610a47a57d518c6dbb002d670647de8774a490c | A spherical water tank of radius 4 feet is filled with water up to six feet from the bottom. How much work is required to pump all the water over the upper rim of the tank? (weight density of water = 62.4 lb/ft^3) | [
"\\( 20241.6 \\pi \\)"
] | [
"A spherical water tank of radius 4 feet is filled with water up to six feet from the bottom. How much work is required to pump all the water over the upper rim of the tank? (weight density of water = 62.4 lb/ft^3)"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-d440f21834dcdb9835ccf5a0ae14317adbeaf43be7d2cc506ac58009dc9fc088 | A spider and a fly are on diametrically opposite vertices of a regular hexagon. The spider can walk freely along the edges and at each vertex, it randomly chooses between two adjacent edges with equal probability. On average, how many edge lengths will the spider walk before reaching the fly? | [
"9"
] | [
"A spider and a fly are on diametrically opposite vertices of a regular hexagon. The spider can walk freely along the edges and at each vertex, it randomly chooses between two adjacent edges with equal probability. On average, how many edge lengths will the spider walk before reaching the fly?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 28,
"tir": 4
} | 32 | 0 | 0 |
omr-ws-446d61ff21ec691a5d0f0e2e6bf85a01a262dbb7224136cc984e28ae03a0a5fd | A spider and a fly are on the surface of a cube with edge length $1$. The fly positions itself at the point $(0,0,1)$ and the spider at $(x,y,z)$. Determine the maximum possible shortest path length the fly can take to reach the spider, walking along the surface of the cube, and find the coordinates $(x,y,z)$ that achi... | [
"2"
] | [
"A spider and a fly are on the surface of a cube with edge length $1$. The fly positions itself at the point $(0,0,1)$ and the spider at $(x,y,z)$. Determine the maximum possible shortest path length the fly can take to reach the spider, walking along the surface of the cube, and find the coordinates $(x,y,z)$ that... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6ea9da0dcc22406796474f4ec0e87f097217dae293016f4458540e0c991d245a | A spider and a fly are sitting on a cube with side length 1. The fly is at a vertex of the cube, and the spider is at another vertex. Let $d$ be the shortest path distance along the surface of the cube from the fly to the spider. If the spider is at vertex $(0,0,0)$, find the coordinates of the vertex where the fly sho... | [
"(1,1,1)"
] | [
"A spider and a fly are sitting on a cube with side length 1. The fly is at a vertex of the cube, and the spider is at another vertex. Let $d$ be the shortest path distance along the surface of the cube from the fly to the spider. If the spider is at vertex $(0,0,0)$, find the coordinates of the vertex where the fl... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-13f73e63a3db6364ee8d5f91a6b953a5bdcee3e2526e36b785dd02f7ad01e4a2 | A spider built a web on the unit circle. The web is a planar graph with straight edges inside the circle, bounded by the circumference of the circle. Each vertex of the graph lying on the circle belongs to a unique edge, which goes perpendicularly inward to the circle. For each vertex of the graph inside the circle, th... | [
"0"
] | [
"A spider built a web on the unit circle. The web is a planar graph with straight edges inside the circle, bounded by the circumference of the circle. Each vertex of the graph lying on the circle belongs to a unique edge, which goes perpendicularly inward to the circle. For each vertex of the graph inside the circl... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d066388052aa6b748df4a14710990d8f9855ea55716584c37dd6655436101d8b | A spider eats 3 flies a day. Given that 4 flies have already attempted to pass today, what is the probability that the next fly will live? | [
"1"
] | [
"A spider eats 3 flies a day. Given that 4 flies have already attempted to pass today, what is the probability that the next fly will live?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-6299e6fd1af41689b77079f6476822b481786a64c8c1f1da6bc8efed96d3ecc8 | A spider has 8 feet, 8 different shoes, and 8 different socks. Find the number of ways in which the spider can put on the 8 socks and the 8 shoes, given that a shoe can only be put on after the corresponding sock is already on. | [
"\\(\\frac{16!}{2^8}\\)"
] | [
"A spider has 8 feet, 8 different shoes, and 8 different socks. Find the number of ways in which the spider can put on the 8 socks and the 8 shoes, given that a shoe can only be put on after the corresponding sock is already on."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 6,
"tir": 11
} | 31 | 0 | 0 |
omr-ws-ea5c4688f64d829c363c7e72803b78f1826cbfcaddd0511d32ac40cd299462bf | A spider has 8 legs and needs to wear 8 socks and 8 shoes. How many ways can the spider put them on, given that a sock must be put on before the shoe on each leg? | [
"\\(\\frac{16!}{2^8}\\)"
] | [
"A spider has 8 legs and needs to wear 8 socks and 8 shoes. How many ways can the spider put them on, given that a sock must be put on before the shoe on each leg?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 31
} | 46 | 0 | 0 |
omr-ws-9206268daca933077d400ddc063a818860c3b65d89f1803dba346769d3f67f8a | A spider has one sock and one shoe for each of its eight legs. In how many different orders can the spider put on its socks and shoes, assuming that, on each leg, the sock must be put on before the shoe? | [
"\\(\\frac{16!}{2^8}\\)"
] | [
"A spider has one sock and one shoe for each of its eight legs. In how many different orders can the spider put on its socks and shoes, assuming that, on each leg, the sock must be put on before the shoe?"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 63
} | 95 | 0 | 0 |
omr-ws-1ed5a69286bf807b28b19c4c9d7ce7ac195d69135ca9d70656472e1e3529366a | A spider is at the center of a hexagonal web with side length 4, which is tiled by equilateral triangles of side length 1. Each minute, the spider moves a unit distance along one thread. What is the probability, in lowest terms, that after four minutes the spider is either at the edge of the web or in the center? If th... | [
"41"
] | [
"A spider is at the center of a hexagonal web with side length 4, which is tiled by equilateral triangles of side length 1. Each minute, the spider moves a unit distance along one thread. What is the probability, in lowest terms, that after four minutes the spider is either at the edge of the web or in the center? ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 11
} | 12 | 0 | 0 |
omr-ws-d177eeca3137178fcdc39394adc6404350471c3aba3fcbdb382265e046171d20 | A spider is on the rim of an empty conical cup with a diameter of 36 cm and a depth of 24 cm. The spider spies a fly one third of the way around the rim. What is the shortest path the spider can take to reach the fly? | [
"\\( 15\\sqrt{10-2\\sqrt{5}} \\)"
] | [
"A spider is on the rim of an empty conical cup with a diameter of 36 cm and a depth of 24 cm. The spider spies a fly one third of the way around the rim. What is the shortest path the spider can take to reach the fly?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 8,
"tir": 21
} | 41 | 0 | 0 |
omr-ws-199cc6e550595bc84f962cc367f75de7e007acc77670bc9c5c792874f7e2fffc | A spider is sitting on the end wall of a room that is $4$ m wide, $6$ m long, and $5$ m high. The spider is $1$ m from the ceiling and $2$ m from each of the side walls. The fly is at the opposite end of the room, $1$ m from the floor and $2$ m from each side wall. Find the length of the shortest route the spider can t... | [
"\\(\\sqrt{109}\\)"
] | [
"A spider is sitting on the end wall of a room that is $4$ m wide, $6$ m long, and $5$ m high. The spider is $1$ m from the ceiling and $2$ m from each of the side walls. The fly is at the opposite end of the room, $1$ m from the floor and $2$ m from each side wall. Find the length of the shortest route the spider ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 1
} | 3 | 0 | 0 |
omr-ws-f4d087f76b963edee69532b923e32a12d851460ec5941a37aa76ab4206ad2e32 | A spider lives at one corner of a cube with sides of length one. The spider roams along the surface but never goes more than 2 units away from its nest. Describe the region on the cube where a fly could land safely. | [
"\\text{Three quarter-circles with radius } 2 - \\sqrt{3} \\text{ on each face opposite the spider's corner}"
] | [
"A spider lives at one corner of a cube with sides of length one. The spider roams along the surface but never goes more than 2 units away from its nest. Describe the region on the cube where a fly could land safely."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 4
} | 5 | 0 | 0 |
omr-ws-38f59b6b77a535ba4f3a31b2e3f295b465515da337ea859cffad69cbf8e8067a | A spider must eat 3 flies a day to survive and stops after catching 3 flies. The spider has an even chance of catching each fly that passes by. Given that 5 flies have already passed by today (with some being caught and some surviving), what is the probability that the next fly will survive? | [
"\\(\\frac{3}{4}\\)"
] | [
"A spider must eat 3 flies a day to survive and stops after catching 3 flies. The spider has an even chance of catching each fly that passes by. Given that 5 flies have already passed by today (with some being caught and some surviving), what is the probability that the next fly will survive?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 11,
"tir": 13
} | 24 | 0 | 0 |
omr-ws-72769d123f95346cd4106654fa64627b31f02bbe91f8893b9faadab13e7a6961 | A spider named Johny has 12 feet, with numbers corresponding to the distances of each foot from his head. For Johny to take a nap, he must hold at least one foot down on the web, and the total distances of his left feet on the web from his head must equal the total distances of his right feet on the web from his head. ... | [
"235"
] | [
"A spider named Johny has 12 feet, with numbers corresponding to the distances of each foot from his head. For Johny to take a nap, he must hold at least one foot down on the web, and the total distances of his left feet on the web from his head must equal the total distances of his right feet on the web from his h... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 5
} | 15 | 0 | 0 |
omr-ws-3eda9c0ffd2e780d8c9f21c710a43b2322d2b9196d20806f6f753ccd62a7bb27 | A spider named KM starts at $(0,0)$ on the Cartesian plane. From any point $(x,y)$, KM can move up one to $(x,y+1)$ or, if $y>x$, KM can move to $(y,x)$. How many paths can KM take to reach $(10,10)$? | [
"512"
] | [
"A spider named KM starts at $(0,0)$ on the Cartesian plane. From any point $(x,y)$, KM can move up one to $(x,y+1)$ or, if $y>x$, KM can move to $(y,x)$. How many paths can KM take to reach $(10,10)$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 1,
"tir": 13
} | 19 | 0 | 0 |
omr-ws-59b0f3b6d15556d6d4b6381c7f83cdad53816ac6a68d5a0d71a5999c413996f9 | A spider spins a web starting at point (1,0) and proceeds counterclockwise to (0, 1/2), (-1/4,0), (0, -1/8), and so forth forever, always ending a strand at the boundary between quadrants and each time cutting the distance to the origin by half. What is the sum of the lengths of all these strands? | [
"\\(\\frac{\\sqrt{5}}{2}\\)"
] | [
"A spider spins a web starting at point (1,0) and proceeds counterclockwise to (0, 1/2), (-1/4,0), (0, -1/8), and so forth forever, always ending a strand at the boundary between quadrants and each time cutting the distance to the origin by half. What is the sum of the lengths of all these strands?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-42d027d023c49b492e0065ce257bef1c6b52bbf36b3b97b193f31636ba2baced | A spiral staircase turns $270^\circ$ as it rises 10 feet. The radius of the staircase is 3 feet. What is the number of feet in the length of the handrail? Express your answer as a decimal to the nearest tenth. | [
"17.3"
] | [
"A spiral staircase turns $270^\\circ$ as it rises 10 feet. The radius of the staircase is 3 feet. What is the number of feet in the length of the handrail? Express your answer as a decimal to the nearest tenth."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a08c87c8e552950d44466c7d9ec54590903a232ed0d559fd37b45324ed1b6a75 | A spiral staircase with a radius of 3 feet turns 270 degrees as it rises 10 feet. What is the length of the outer handrail, which is 3 feet away from the axis of the staircase? | [
"\\(\\frac{\\sqrt{81\\pi^2 + 400}}{2}\\)"
] | [
"A spiral staircase with a radius of 3 feet turns 270 degrees as it rises 10 feet. What is the length of the outer handrail, which is 3 feet away from the axis of the staircase?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 8,
"tir": 1
} | 21 | 0 | 0 |
omr-ws-c13566c688f3004fcbd7c540d5ec83d48b4892d5f6ff523f35247bbdfabb3ae6 | A spiral staircase with a radius of 3 feet turns 270 degrees as it rises 10 feet. What is the length of the outer handrail? | [
"\\(\\frac{\\sqrt{81\\pi^2+400}}2\\) feet"
] | [
"A spiral staircase with a radius of 3 feet turns 270 degrees as it rises 10 feet. What is the length of the outer handrail?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 11
} | 33 | 0 | 0 |
omr-ws-e08c8178b6625da6f3ab6c6faceae10e6749ab12e01668b3733b7dbbc9eb4295 | A sporting goods manufacturer makes a $5.00 profit on each soccer ball and a $4.00 profit on each volleyball. The cutting process requires 2 hours to make 75 soccer balls and 3 hours to make 60 volleyballs. The sewing process requires 3 hours to make 75 soccer balls and 2 hours to make 60 volleyballs. The cutting depar... | [
"\\$55050"
] | [
"A sporting goods manufacturer makes a $5.00 profit on each soccer ball and a $4.00 profit on each volleyball. The cutting process requires 2 hours to make 75 soccer balls and 3 hours to make 60 volleyballs. The sewing process requires 3 hours to make 75 soccer balls and 2 hours to make 60 volleyballs. The cutting ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7932bfefe7b7c99ac27e5decb58ba5c505c84a66d6760dd3faac659e2ce4b9f6 | A sports conference has 14 teams divided into two divisions of 7 teams each. Each team must play every other team in its own division twice and every team in the other division once. How many games are in a complete season for the conference? | [
"133"
] | [
"A sports conference has 14 teams divided into two divisions of 7 teams each. Each team must play every other team in its own division twice and every team in the other division once. How many games are in a complete season for the conference?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-74a292a45238d8b109740fb74d6ddbb99b0f41d13f1ce367fdfdc21b5bbdcbc4 | A sports conference has 14 teams in two divisions of 7. How many games are in a complete season for the conference if each team must play every other team in its own division twice and every team in the other division once? | [
"133"
] | [
"A sports conference has 14 teams in two divisions of 7. How many games are in a complete season for the conference if each team must play every other team in its own division twice and every team in the other division once?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-33e2eb2b95b8a2d17560dade03a50e0e9c9018b0140c2598e77a38be597ea76c | A sports meet is organized for 4 days. If each day half of the remaining medals plus one more medal is given away, find the total number of medals. | [
"30"
] | [
"A sports meet is organized for 4 days. If each day half of the remaining medals plus one more medal is given away, find the total number of medals."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-a7e476b981a9dce72b4282510c317e626a5c758639a3c4784f84ec585fcbec9c | A sports plane flew along a diamond-shaped route in windy weather. It took $a$, $b$, and $c$ hours to fly through the first three sides of the rhombus, respectively. How long did it take to cover the fourth side of the rhombus? (The speed of the aircraft is a vector equal to the sum of two vectors: the aircraft’s own s... | [
"\\dfrac{ac}{b}"
] | [
"A sports plane flew along a diamond-shaped route in windy weather. It took $a$, $b$, and $c$ hours to fly through the first three sides of the rhombus, respectively. How long did it take to cover the fourth side of the rhombus? (The speed of the aircraft is a vector equal to the sum of two vectors: the aircraft’s ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-ffa8879a918e39201f52ca894dfd8392a4d645218b78af87ff2a2229accebea4 | A sportsman climbing a mountain reaches an altitude of 800 m by the expiry of the first hour. Each subsequent hour he ascends to a height by 25 m less than during the preceding hour. How many hours does it take the sportsman to reach an altitude of 5700 m? | [
"8 hours"
] | [
"A sportsman climbing a mountain reaches an altitude of 800 m by the expiry of the first hour. Each subsequent hour he ascends to a height by 25 m less than during the preceding hour. How many hours does it take the sportsman to reach an altitude of 5700 m?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a1fa29c0824ae0eb114037c809130eb12c019ad5670dfc91dca21788b521cd19 | A spotlight beam on a small island has a length of $1$ km in the fog (a segment of $1$ km long) and rotates at a speed of $1$ rpm. The boat wants to land troops on the island. What should be the minimum speed of the boat to be able to do this? | [
"60"
] | [
"A spotlight beam on a small island has a length of $1$ km in the fog (a segment of $1$ km long) and rotates at a speed of $1$ rpm. The boat wants to land troops on the island. What should be the minimum speed of the boat to be able to do this?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-5943a946e265e05d1a6a804a42cf29e1a74afdb1075c1731896f478d1f024932 | A spotlight is 30 meters away from a very large wall and casts a beam of light in the shape of a right circular cone. The illuminated area on the wall is a circular disk with an area of $300\pi$ square meters. When the spotlight is swiveled so that the center of the illuminated area lies on the perimeter of the new ill... | [
"1201"
] | [
"A spotlight is 30 meters away from a very large wall and casts a beam of light in the shape of a right circular cone. The illuminated area on the wall is a circular disk with an area of $300\\pi$ square meters. When the spotlight is swiveled so that the center of the illuminated area lies on the perimeter of the n... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c92f0fe5985512709033aa52a333eb75768a1b987a54938c7ce8e2c62ec839d4 | A spotlight on the ground shines on a wall 12 m away. If a 2 m tall man walks from the spotlight straight towards the building at a speed of 1.6 m/s, how fast is the length of his shadow on the building decreasing at the moment when he is 4 m from the building? | [
"-0.6 m/sec"
] | [
"A spotlight on the ground shines on a wall 12 m away. If a 2 m tall man walks from the spotlight straight towards the building at a speed of 1.6 m/s, how fast is the length of his shadow on the building decreasing at the moment when he is 4 m from the building?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 18
} | 33 | 0 | 0 |
omr-ws-fd05cff29fdbb5ff7cfb2efe0912b9cf77f3150302ffe90fae05339e708afb90 | A spotlight on the ground shines on the outside wall of a parking garage 12 meters away. If a 2-meter tall man walks toward the garage at a speed of 0.75 meters per second, how fast is the height of the man's shadow on the garage wall decreasing when he is 4 meters from the building? | [
"\\(-1.125\\)"
] | [
"A spotlight on the ground shines on the outside wall of a parking garage 12 meters away. If a 2-meter tall man walks toward the garage at a speed of 0.75 meters per second, how fast is the height of the man's shadow on the garage wall decreasing when he is 4 meters from the building?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-7e52f123e91642c013042db5b51376becff6a98db58fc35c476f08f604bc66ce | A spotted lanternfly is at point $(0, 0, 0)$ and wants to land on an unassuming CMU student at point $(2, 3, 4)$. It can move one unit at a time in either the $+x$, $+y$, or $+z$ directions. However, there is another student waiting at $(1, 2, 3)$ who will stomp on the lanternfly if it passes through that point. How ma... | [
"900"
] | [
"A spotted lanternfly is at point $(0, 0, 0)$ and wants to land on an unassuming CMU student at point $(2, 3, 4)$. It can move one unit at a time in either the $+x$, $+y$, or $+z$ directions. However, there is another student waiting at $(1, 2, 3)$ who will stomp on the lanternfly if it passes through that point. H... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 22
} | 37 | 0 | 0 |
omr-ws-82d9f332dca2f6e929752d00730dd2bd04c47448f4091e18a992d48edbd95f2e | A sprinkler distributes water in a circular pattern, supplying water to a depth of \( e^{-r} \) feet per hour at a distance of \( r \) feet from the sprinkler. A. What is the total amount of water supplied per hour inside a circle of radius 18 feet? B. What is the total amount of water that goes through the sprinkler p... | [
"\\( 2\\left(1-e^{-18}\\right)\\pi \\) for part A and \\( 2\\pi \\) for part B."
] | [
"A sprinkler distributes water in a circular pattern, supplying water to a depth of \\( e^{-r} \\) feet per hour at a distance of \\( r \\) feet from the sprinkler. \n\nA. What is the total amount of water supplied per hour inside a circle of radius 18 feet?\n\nB. What is the total amount of water that goes through... | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 6
} | 6 | 0 | 0 |
omr-ws-d2ab9ea103f292f53bfe520989556c190f7b1beefff25be9796ba68c78a22692 | A squad of $4n$ men form groups of 4. Find the probability that two specified men are next to each other in the same group. | [
"\\(\\frac{1}{2n}\\)"
] | [
"A squad of $4n$ men form groups of 4. Find the probability that two specified men are next to each other in the same group."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b968aae980b3d133875ca5fba81d47e38e3803c7cf4c9fcbaf8a269cbceb9f09 | A squadron of $10$ ships is located on a $10 \times 10$ square checkered field. Each ship lies on a $1 \times 2$ rectangle with sides along the grid lines (no two ships share the same rectangle). Determine if it is possible to hit at least one ship with $30$ shots. | [
"\\text{No}"
] | [
"A squadron of $10$ ships is located on a $10 \\times 10$ square checkered field. Each ship lies on a $1 \\times 2$ rectangle with sides along the grid lines (no two ships share the same rectangle). Determine if it is possible to hit at least one ship with $30$ shots."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f516a6992911c427b0a8c25b0a6c182fdbcf81ae7c89e3c70a419037957c4b65 | A squadron of 10 powerful destroyers and 20 small boats is positioned in a straight line, equally spaced. Two torpedo boats, each with 10 torpedoes, aim to protect an island. The first torpedo boat can shoot at any 10 successive boats, while the second can shoot at any 10 boats that are next to each other. Both boats s... | [
"3"
] | [
"A squadron of 10 powerful destroyers and 20 small boats is positioned in a straight line, equally spaced. Two torpedo boats, each with 10 torpedoes, aim to protect an island. The first torpedo boat can shoot at any 10 successive boats, while the second can shoot at any 10 boats that are next to each other. Both bo... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 31,
"genselect": 5
} | 36 | 0 | 0 |
omr-ws-51896b5e081edc153a2223fb4653a30e298e59123e7b030c7c2fbdebf81354a7 | A square $5 \times 5$ (consisting of 25 cells) is to be cut along the grid lines into several figures such that the perimeter of each figure (calculated in cells) is equal to $P$. For which values of $P < 25$ is this possible? | [
"8, 10, 12, 14, 16"
] | [
"A square $5 \\times 5$ (consisting of 25 cells) is to be cut along the grid lines into several figures such that the perimeter of each figure (calculated in cells) is equal to $P$. For which values of $P < 25$ is this possible?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c91fc1a7fd3355d540e9fc5843302c664f9acbe3fbf81406916c7563a5335ec6 | A square $666 \times 666$ has been covered using tiles $5 \times 1$. A single $1 \times 1$ square remains uncovered. Determine the number of possible positions for the uncovered $1 \times 1$ square. | [
"17689"
] | [
"A square $666 \\times 666$ has been covered using tiles $5 \\times 1$. A single $1 \\times 1$ square remains uncovered. Determine the number of possible positions for the uncovered $1 \\times 1$ square."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-926543dd0b9bfb4fc714b6696f840e12e36b84d94426e618ddd5c5f6a9d49994 | A square $A BCD$ is inscribed into a circle $x^2+y^2=169$. Find the coordinates of the vertices $B$, $C$, and $D$ if $A (5, -12)$. | [
"$B(12,5),\\ C(-5,12),\\ D(-12,-5)$"
] | [
"A square $A BCD$ is inscribed into a circle $x^2+y^2=169$. Find the coordinates of the vertices $B$, $C$, and $D$ if $A (5, -12)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4bc5943b8b3164fb3fa53d766fbfb3574b7c5bc6f174529f5fc9b0f58a9103e1 | A square $ABCD$ has a side length of $\sqrt{3} + 1$. Point $P$ is on $\overline{AC}$ such that $AP = \sqrt{2}$. The square region bounded by $ABCD$ is rotated $90^\circ$ counterclockwise with center $P$, sweeping out a region whose area is $\frac{1}{c} (a \pi + b)$, where $a$, $b$, and $c$ are positive integers and $\t... | [
"19"
] | [
"A square $ABCD$ has a side length of $\\sqrt{3} + 1$. Point $P$ is on $\\overline{AC}$ such that $AP = \\sqrt{2}$. The square region bounded by $ABCD$ is rotated $90^\\circ$ counterclockwise with center $P$, sweeping out a region whose area is $\\frac{1}{c} (a \\pi + b)$, where $a$, $b$, and $c$ are positive integ... | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0cb4eaa8bb56ee679312a4ec27b76059975e300f7fd49de67c54524b369f5a31 | A square $ABCD$ has side length $1$. A circle passes through the vertices of the square. Let $P, Q, R, S$ be the midpoints of the arcs which are symmetrical to the arcs $AB$, $BC$, $CD$, $DA$ when reflected on sides $AB$, $BC$, $CD$, $DA$, respectively. The area of square $PQRS$ is $a+b\sqrt{2}$, where $a$ and $b$ are ... | [
"1"
] | [
"A square $ABCD$ has side length $1$. A circle passes through the vertices of the square. Let $P, Q, R, S$ be the midpoints of the arcs which are symmetrical to the arcs $AB$, $BC$, $CD$, $DA$ when reflected on sides $AB$, $BC$, $CD$, $DA$, respectively. The area of square $PQRS$ is $a+b\\sqrt{2}$, where $a$ and $b... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 5
} | 14 | 0 | 0 |
omr-ws-0d1b15f0301b88cd869b44a71c481f4770dcae8e0d1141d0e4cd6886869b2ae3 | A square $ABCD$ is built inside a circle $S$ with chord $AB$. Point $E$ is on $S$ such that $CE = CD$. The extension of segment $ED$ intersects $S$ at point $F$. Given $DE = 2$ and the distance from $C$ to $DE$ is $5$, calculate $12x$ where $x$ is the length of segment $DF$. | [
"\\(\\frac{15912}{289}\\) or \\(\\frac{18768}{289}\\)"
] | [
"A square $ABCD$ is built inside a circle $S$ with chord $AB$. Point $E$ is on $S$ such that $CE = CD$. The extension of segment $ED$ intersects $S$ at point $F$. Given $DE = 2$ and the distance from $C$ to $DE$ is $5$, calculate $12x$ where $x$ is the length of segment $DF$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e7f3f1a6689529774aeb57f68cc4bdf51004b8ce237707710370aac0af153cdb | A square $ABCD$ is divided into $100^2$ equal small squares by lines parallel to its sides. A king of chess starts from $A$ and can move only rightward, leftward, upwards, or downwards, but not crosswise. There are $100$ pawns on the chessboard, and the king wants to capture them. What is the maximum number of movement... | [
"2000"
] | [
"A square $ABCD$ is divided into $100^2$ equal small squares by lines parallel to its sides. A king of chess starts from $A$ and can move only rightward, leftward, upwards, or downwards, but not crosswise. There are $100$ pawns on the chessboard, and the king wants to capture them. What is the maximum number of mov... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a2acdf9776cb192a3dff8717a77ad1f3d9f9c73ab73d9b98a65870a2ae78cc11 | A square $ABCD$ is divided into $n^2$ equal small squares by lines parallel to its sides. A spider starts from $A$ and moves only rightward or upwards, trying to reach $C$. Each "movement" of the spider consists of $k$ steps rightward and $m$ steps upwards or $m$ steps rightward and $k$ steps upwards. The spider comple... | [
"\\(\\sum_a \\binom{\\ell}{a}\\binom{\\ell(m-k)}{a(m-k)} \\cdot \\binom{m+k}{m}^l\\)"
] | [
"A square $ABCD$ is divided into $n^2$ equal small squares by lines parallel to its sides. A spider starts from $A$ and moves only rightward or upwards, trying to reach $C$. Each \"movement\" of the spider consists of $k$ steps rightward and $m$ steps upwards or $m$ steps rightward and $k$ steps upwards. The spider... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3688089928b0f824bdc4cc760a2605282d26ea6deee7466c0e74a73190d324a8 | A square $ABCD$ is inscribed in a circle $k$. A point $P$ is given on the shorter arc $AB$ of $k$. Let $X=\overline{PD} \cap \overline{AB}$, $Y=\overline{PC} \cap \overline{BD}$. If the side length of the square is $s$, find the length of segment $\overline{XY}$ in terms of $s$. | [
"s(\\sqrt{2} - 1)"
] | [
"A square $ABCD$ is inscribed in a circle $k$. A point $P$ is given on the shorter arc $AB$ of $k$. Let $X=\\overline{PD} \\cap \\overline{AB}$, $Y=\\overline{PC} \\cap \\overline{BD}$. If the side length of the square is $s$, find the length of segment $\\overline{XY}$ in terms of $s$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-7e1c229f79c2077d03ff0abebd70d1923b4698d87a99ee9bee4718029c33b8a4 | A square $ABCD$ is inscribed in a circle $k$. A square $PQRS$ is circumscribed to the same circle $k$ such that the vertices of the circumscribed square lie on the extensions of the sides of the inscribed square. Determine the size of the angle $PSA$ formed by the sides of the squares. | [
"\\( 15^\\circ \\)"
] | [
"A square $ABCD$ is inscribed in a circle $k$. A square $PQRS$ is circumscribed to the same circle $k$ such that the vertices of the circumscribed square lie on the extensions of the sides of the inscribed square. Determine the size of the angle $PSA$ formed by the sides of the squares."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 18,
"genselect": 5,
"tir": 5
} | 28 | 0 | 0 |
omr-ws-ea4f0bdba8de40272e081a5b652fc552e32a31272d3772823ee40402ba288b80 | A square $ABCD$ is inscribed in a circle. Point $M$ lies on arc $BC$. Line $AM$ intersects $BD$ at point $P$, and line $DM$ intersects $AC$ at point $Q$. If the side length of the square is $s$, find the value of $\frac{[APQD]}{s^2}$. | [
"\\dfrac{1}{2}"
] | [
"A square $ABCD$ is inscribed in a circle. Point $M$ lies on arc $BC$. Line $AM$ intersects $BD$ at point $P$, and line $DM$ intersects $AC$ at point $Q$. If the side length of the square is $s$, find the value of $\\frac{[APQD]}{s^2}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-b3bb744db7d114514439bf4a32cad6f8047e5337458c01da55df3fe43d792de2 | A square $ABCD$ is inscribed into a circle. Point $M$ lies on arc $BC$, $AM$ meets $BD$ at point $P$, and $DM$ meets $AC$ at point $Q$. If the side length of the square is $s$, find the area of quadrilateral $APQD$ in terms of $s$. | [
"\\dfrac{s^2}{2}"
] | [
"A square $ABCD$ is inscribed into a circle. Point $M$ lies on arc $BC$, $AM$ meets $BD$ at point $P$, and $DM$ meets $AC$ at point $Q$. If the side length of the square is $s$, find the area of quadrilateral $APQD$ in terms of $s$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-f0b9d9ac9f022080558f415b2b2924813602309d91becc203bc18e23e2f6310b | A square $ABCD$ is partitioned into rectangles (with sides parallel to $ABCD$) such that no point is a vertex of 4 rectangles. Let $N$ be the total number of vertices created by the partition, including the vertices of the square. Determine the minimum number of vertices that must be colored black if the vertices are t... | [
""
] | [
"A square $ABCD$ is partitioned into rectangles (with sides parallel to $ABCD$) such that no point is a vertex of 4 rectangles. Let $N$ be the total number of vertices created by the partition, including the vertices of the square. Determine the minimum number of vertices that must be colored black if the vertices ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 1 |
omr-ws-723bc732fb54c094c72ff3507069a5b14e1c557d7dfb383d53a6e7c0fe847789 | A square $ABCD$ of side length $2a$ is given on a plane $\Pi$. Let $S$ be a point on the ray $Ax$ perpendicular to $\Pi$ such that $AS = 2a$. $(a)$ Let $M \in BC$ and $N \in CD$ be two variable points. Find the positions of $M, N$ such that $BM + DN \ge \frac{3}{2}$, planes $SAM$ and $SMN$ are perpendicular, and $BM \c... | [
"0"
] | [
"A square $ABCD$ of side length $2a$ is given on a plane $\\Pi$. Let $S$ be a point on the ray $Ax$ perpendicular to $\\Pi$ such that $AS = 2a$. \n$(a)$ Let $M \\in BC$ and $N \\in CD$ be two variable points. Find the positions of $M, N$ such that $BM + DN \\ge \\frac{3}{2}$, planes $SAM$ and $SMN$ are perpendicula... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 1
} | 8 | 0 | 0 |
omr-ws-1b2d20a55507b9dd3442a7e02c3c1f216ae62fa16ce8aa9a8f4bc73deb3a6309 | A square $ABCD$ with side length $2023$ is divided into $2023$ unit squares. Each unit square is colored either blue or red such that two different cells that are symmetrical about the line $AC$ have different colors. Let $S$ be the set of different triples $(m, n, p)$ (regardless of order); $m,n,p \in \{ 1, 2, 3, \dot... | [
"\\dfrac{2023 \\times 2022 \\times 2021}{6}"
] | [
"A square $ABCD$ with side length $2023$ is divided into $2023$ unit squares. Each unit square is colored either blue or red such that two different cells that are symmetrical about the line $AC$ have different colors. Let $S$ be the set of different triples $(m, n, p)$ (regardless of order); $m,n,p \\in \\{ 1, 2, ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-782249eeff2c244a0e0b93696f1f68435a91ab821f566f63dac8921ee89a1f6a | A square $ABDE$ is constructed on the hypotenuse $AB$ of a right triangle $ABC$ in the half-plane to which triangle $ABC$ does not belong. Find the distance from vertex $C$ to the center of the square, given that $BC = a$ and $AC = b$. | [
"\\dfrac{(a + b)\\sqrt{2}}{2}"
] | [
"A square $ABDE$ is constructed on the hypotenuse $AB$ of a right triangle $ABC$ in the half-plane to which triangle $ABC$ does not belong. Find the distance from vertex $C$ to the center of the square, given that $BC = a$ and $AC = b$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 5
} | 19 | 0 | 0 |
omr-ws-700cc0eaa6f8e8a8b4ae0bf3d70b125f04c458258932f2b2a9e8c8a6c18c56f1 | A square $ACEG$ of side length $a$ is rotated by $45^\circ$ around its center $S$ and then displaced by a distance $h$ perpendicular to its plane, resulting in the square $BDFH$. The points $A, C, E, G$ are mapped to $B, D, F, H$ respectively. Determine the distance between the intersection point $X$ of the planes $(AB... | [
"h(3 + 2\\sqrt{2})"
] | [
"A square $ACEG$ of side length $a$ is rotated by $45^\\circ$ around its center $S$ and then displaced by a distance $h$ perpendicular to its plane, resulting in the square $BDFH$. The points $A, C, E, G$ are mapped to $B, D, F, H$ respectively. Determine the distance between the intersection point $X$ of the plane... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 7
} | 11 | 0 | 0 |
omr-ws-04db6248a5ad5ddb59a338fbdb663f6f1a7c0695b0460219e79939d392e05974 | A square $n \times n$ is given on the coordinate plane with vertices at integer points, the sides of which are parallel to the coordinate axes. How many different rectangles can be specified, the sides of which are parallel to the axes, and all the vertices are integer points belonging to the given square? | [
"\\(\\left( \\frac{(n+1)n}{2} \\right)^2\\)"
] | [
"A square $n \\times n$ is given on the coordinate plane with vertices at integer points, the sides of which are parallel to the coordinate axes. How many different rectangles can be specified, the sides of which are parallel to the axes, and all the vertices are integer points belonging to the given square?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 11
} | 26 | 0 | 0 |
omr-ws-fc0fd46e7d106e138b8deafa9ef424ba6d9cebb753ec61c099fd99a5f508bd05 | A square $n \times n$ is given on the coordinate plane with vertices at integer points, the sides of which are parallel to the coordinate axes. How many different rectangles can be specified, the sides of which are parallel to the axes, and all vertices are integer points belonging to the given square, so that each poi... | [
"\\left\\lfloor \\dfrac{n+1}{2} \\right\\rfloor^2"
] | [
"A square $n \\times n$ is given on the coordinate plane with vertices at integer points, the sides of which are parallel to the coordinate axes. How many different rectangles can be specified, the sides of which are parallel to the axes, and all vertices are integer points belonging to the given square, so that ea... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-aaa985f885c96b4cd0201f97902ec27dc4fa851e46a95b1356bb8a28fca439f6 | A square $n \times n$ is given on the coordinate plane with vertices at integer points, the sides of which are parallel to the coordinate axes. How many different right triangles can be specified, the vertices of which are integer points belonging to the given square? | [
"2n^3(n + 1)"
] | [
"A square $n \\times n$ is given on the coordinate plane with vertices at integer points, the sides of which are parallel to the coordinate axes. How many different right triangles can be specified, the vertices of which are integer points belonging to the given square?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 6
} | 7 | 0 | 0 |
omr-ws-3e066e03245beb17a17b802015e37dcb960c0932be2f95fa5f06f11080038071 | A square $n \times n$ is given on the coordinate plane with vertices at integer points, the sides of which are parallel to the coordinate axes. How many different squares can be specified, the sides of which are parallel to the axes, and all vertices are integer points belonging to the given square? | [
"\\(\\frac{n(n+1)(2n+1)}{6}\\)"
] | [
"A square $n \\times n$ is given on the coordinate plane with vertices at integer points, the sides of which are parallel to the coordinate axes. How many different squares can be specified, the sides of which are parallel to the axes, and all vertices are integer points belonging to the given square?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-c2ef2341148363abf6a522d2ba6740cd48c33fd3729a04bb7ea7e6fb15d31ac7 | A square $n \times n$ is given on the coordinate plane with vertices at integer points, the sides of which are parallel to the coordinate axes. How many unequal squares can be specified, the sides of which are parallel to the axes, and all vertices are integer points belonging to the given square? | [
"\\( n \\)"
] | [
"A square $n \\times n$ is given on the coordinate plane with vertices at integer points, the sides of which are parallel to the coordinate axes. How many unequal squares can be specified, the sides of which are parallel to the axes, and all vertices are integer points belonging to the given square?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 7
} | 10 | 0 | 0 |
omr-ws-2b14a786ed7fe58a5169b2e7e0dc6660f30cf62b760669c594d2b07008f43b05 | A square $n \times n$ table is filled with real numbers. For any given row and column, the sum of their $2n - 1$ numbers is greater than or equal to a real number $X$. What is the minimal possible value of the sum of all numbers in the table? | [
"\\(\\frac{n^2 X}{2n - 1}\\)"
] | [
"A square $n \\times n$ table is filled with real numbers. For any given row and column, the sum of their $2n - 1$ numbers is greater than or equal to a real number $X$. What is the minimal possible value of the sum of all numbers in the table?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 3
} | 18 | 0 | 0 |
omr-ws-a2570ff3072fc2ba424ff4c3649ef6e29865df5eeed56889b33055c06d98675d | A square 2 by 2 dartboard is situated in the xy-plane with its center at the origin and its sides parallel to the coordinate axes. Darts that are thrown land randomly on the dartboard. Find the probability that a dart will land at a point closer to the point (0,1) than the bottom edge of the board. | [
"\\(\\frac{11}{24}\\)"
] | [
"A square 2 by 2 dartboard is situated in the xy-plane with its center at the origin and its sides parallel to the coordinate axes. Darts that are thrown land randomly on the dartboard. Find the probability that a dart will land at a point closer to the point (0,1) than the bottom edge of the board."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-1dc7fd87a40bbfea3443975bc91ea013199e50c3a5686b131ec64a5110f66ad3 | A square ABCD has each side divided into 9 equal parts. Connecting the division points with straight lines according to the direction of the diagonal AC, a hatching is formed. Calculate the sum of the lengths of the 17 segments thus obtained. | [
"\\( 9s\\sqrt{2} \\)"
] | [
"A square ABCD has each side divided into 9 equal parts. Connecting the division points with straight lines according to the direction of the diagonal AC, a hatching is formed. Calculate the sum of the lengths of the 17 segments thus obtained."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 3,
"tir": 21
} | 28 | 0 | 0 |
omr-ws-6ff705451563d75294da4f6d4834dd15e287b40f3b29401ef0a1f3979f59859c | A square ABCD has side length 37453 cm. A point P is randomly chosen inside the square. What is the probability that triangle APB is acute? Express your answer to the nearest whole percent. | [
"61%"
] | [
"A square ABCD has side length 37453 cm. A point P is randomly chosen inside the square. What is the probability that triangle APB is acute? Express your answer to the nearest whole percent."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-cb8ad7454ff730293fe3678f6151025e7ba09d82b5b7aeef6517201462c9287b | A square ABCD is folded along a straight line such that point B coincides with the midpoint of side DC. The fold divides side BC into two segments of lengths \(a\) and \(b\) with \(a \leq b\). What is the ratio \(b/a\)? | [
"\\(\\frac{5}{3}\\)"
] | [
"A square ABCD is folded along a straight line such that point B coincides with the midpoint of side DC. The fold divides side BC into two segments of lengths \\(a\\) and \\(b\\) with \\(a \\leq b\\). What is the ratio \\(b/a\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-7fdad82f82e39d7acdd1fab8abd542d8079a336477fd9d4adebe2fc865d7a1d3 | A square HOMF has side length OM = 5 cm. A triangle ABC has H as the intersection of the altitudes, O as the center of the circumscribed circle, M as the midpoint of BC, and F as the foot of the altitude from A. What is the area of triangle ABC in square centimeters? | [
"150"
] | [
"A square HOMF has side length OM = 5 cm. A triangle ABC has H as the intersection of the altitudes, O as the center of the circumscribed circle, M as the midpoint of BC, and F as the foot of the altitude from A. What is the area of triangle ABC in square centimeters?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 40,
"genselect": 8,
"tir": 1
} | 49 | 0 | 0 |
omr-ws-be5dcc08d78065b4eee597f21774ef22fecd4c5c0c9b7fce72b7117996f15a7f | A square \( 2 \) by \( 2 \) dartboard is situated in the \( xy \)-plane with its center at the origin and its sides parallel to the coordinate axes. Darts that are thrown land randomly on the dartboard. Find the probability that a dart will land at a point of the dartboard that is nearer to the point \( (0, 1) \) than ... | [
"\\dfrac{11}{24}"
] | [
"A square \\( 2 \\) by \\( 2 \\) dartboard is situated in the \\( xy \\)-plane with its center at the origin and its sides parallel to the coordinate axes. Darts that are thrown land randomly on the dartboard. Find the probability that a dart will land at a point of the dartboard that is nearer to the point \\( (0,... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-61d44261b55904cccb473e88fef828f0a62096214308ca7616671cee0fd341a8 | A square \( ABCD \) is inscribed in a regular hexagon \( PQRSTU \) of side length 7 cm such that \( AB \) is parallel to \( QR \) and \( DC \) is parallel to \( TU \). Find the area of square \( ABCD \). | [
"588 - 294\\sqrt{3}"
] | [
"A square \\( ABCD \\) is inscribed in a regular hexagon \\( PQRSTU \\) of side length 7 cm such that \\( AB \\) is parallel to \\( QR \\) and \\( DC \\) is parallel to \\( TU \\). Find the area of square \\( ABCD \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ecd256fded9bc7f09f1bcd1b49ef51c2f60882f9376869f47691ccd0df240d4c | A square \( Q \) is divided into smaller squares by lines parallel to its sides. The boundary of \( Q \) is oriented such that moving in the chosen direction, the internal points of \( Q \) always stay on one side. For each of the other edges of the subdivision, a direction is specified such that for each vertex inside... | [
"81"
] | [
"A square \\( Q \\) is divided into smaller squares by lines parallel to its sides. The boundary of \\( Q \\) is oriented such that moving in the chosen direction, the internal points of \\( Q \\) always stay on one side. For each of the other edges of the subdivision, a direction is specified such that for each ve... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7765b883ecf56a5e906e586483ce85a70056a0b6a7ec3064d1683d4d1d6daa07 | A square \( S_1 \) with side length \( 10^{2021} + 1 \) is drawn. For each integer \( n > 1 \), a square \( S_{n+1} \) is drawn with points that divide the sides of \( S_n \) into the ratio \( 1:10^{2021} \). Let \( A_n \) be the area of \( S_n \) and \( S = \sum_{n=1}^{\infty} A_n \). Compute the sum of the digits of ... | [
"12"
] | [
"A square \\( S_1 \\) with side length \\( 10^{2021} + 1 \\) is drawn. For each integer \\( n > 1 \\), a square \\( S_{n+1} \\) is drawn with points that divide the sides of \\( S_n \\) into the ratio \\( 1:10^{2021} \\). Let \\( A_n \\) be the area of \\( S_n \\) and \\( S = \\sum_{n=1}^{\\infty} A_n \\). Compute ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-4ef30b3663ebc437f8a66f263b97cb62eb69294183460cdf18ddab583bf3d804 | A square \( n \times n \) (\( n \geq 2 \)) is divided into \( n^2 \) unit squares, each colored either black or white. The condition is that the squares at the four corners of any rectangle (containing at least four squares) must have different colors. Find the maximum possible value of \( n \). | [
"4"
] | [
"A square \\( n \\times n \\) (\\( n \\geq 2 \\)) is divided into \\( n^2 \\) unit squares, each colored either black or white. The condition is that the squares at the four corners of any rectangle (containing at least four squares) must have different colors. Find the maximum possible value of \\( n \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-f531f559c6c2586664dd3941f06b161fde1d24927f8d6c441a6d55f72a0b4046 | A square \(ABCD\) has a point \(P\) on line segment \([AB]\) such that \(\frac{|BP|}{|AP|}=7\). The line segment \([CD]\) has a point \(Q\) such that \(\frac{|DQ|}{|CQ|}=7\). The perimeter formed by connecting the vertices \(A\), \(P\), \(Q\), and \(D\) is equal to 26 centimetres. What is the area of this figure? | [
"32 cm^2"
] | [
"A square \\(ABCD\\) has a point \\(P\\) on line segment \\([AB]\\) such that \\(\\frac{|BP|}{|AP|}=7\\). The line segment \\([CD]\\) has a point \\(Q\\) such that \\(\\frac{|DQ|}{|CQ|}=7\\). The perimeter formed by connecting the vertices \\(A\\), \\(P\\), \\(Q\\), and \\(D\\) is equal to 26 centimetres. What is t... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-82c5b0383f82230ccc27b160285f3caa19a3b9ec38bbaa737f0356815b502a23 | A square \(ABCD\) has a side length of 1. A circle passes through the vertices of the square. Let \(P\), \(Q\), \(R\), and \(S\) be the midpoints of the arcs which are symmetrical to the arcs \(AB\), \(BC\), \(CD\), and \(DA\) when reflected on sides \(AB\), \(BC\), \(CD\), and \(DA\), respectively. The area of square ... | [
"1"
] | [
"A square \\(ABCD\\) has a side length of 1. A circle passes through the vertices of the square. Let \\(P\\), \\(Q\\), \\(R\\), and \\(S\\) be the midpoints of the arcs which are symmetrical to the arcs \\(AB\\), \\(BC\\), \\(CD\\), and \\(DA\\) when reflected on sides \\(AB\\), \\(BC\\), \\(CD\\), and \\(DA\\), re... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 7
} | 19 | 0 | 0 |
omr-ws-87261bd05e67a493756d2d348ed86e5266c671c44ae3d1f48ada2de0e544cd33 | A square \(ABCD\) has side length 1 and center \(O\). A point \(P\), distinct from \(O\), is chosen in the interior of \(ABCD\). Let \(P_a\) lie on the ray \(AP\) such that \(AP_a \cdot AP = 1\). Define \(P_b\), \(P_c\), and \(P_d\) similarly. Suppose that \(P_aPP_c\) and \(P_bPP_d\) are non-degenerate triangles. Let \... | [
"1"
] | [
"A square \\(ABCD\\) has side length 1 and center \\(O\\). A point \\(P\\), distinct from \\(O\\), is chosen in the interior of \\(ABCD\\). Let \\(P_a\\) lie on the ray \\(AP\\) such that \\(AP_a \\cdot AP = 1\\). Define \\(P_b\\), \\(P_c\\), and \\(P_d\\) similarly. Suppose that \\(P_aPP_c\\) and \\(P_bPP_d\\) are... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-fa4946e4c08b7a0676b8b0a1a0ca8729ea25cb9a400da47b8f19831dd13bd5ce | A square \(ABCD\) has side length 1 and centre \(O\). A point \(P\), distinct from \(O\), is chosen in the interior of \(ABCD\). Let \(P_a\) lie on the ray \(AP\) such that \(AP_a \cdot AP = 1\). Define \(P_b\), \(P_c\), and \(P_d\) similarly. Suppose that \(P_aPP_c\) and \(P_bPP_d\) are non-degenerate triangles. Let \... | [
"no_valid_answer_found"
] | [
"A square \\(ABCD\\) has side length 1 and centre \\(O\\). A point \\(P\\), distinct from \\(O\\), is chosen in the interior of \\(ABCD\\). Let \\(P_a\\) lie on the ray \\(AP\\) such that \\(AP_a \\cdot AP = 1\\). Define \\(P_b\\), \\(P_c\\), and \\(P_d\\) similarly. Suppose that \\(P_aPP_c\\) and \\(P_bPP_d\\) are... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0cad76687857e364156ae0ea886c823d659ce47e2f787540d75867e885b9f80e | A square \(ABCD\) is inscribed in a circle. If \(M\) is a point on the shorter arc \(AB\), find the minimum value of the expression \(\frac{MC \cdot MD}{MA \cdot MB}\). | [
"3 + 2\\sqrt{2}"
] | [
"A square \\(ABCD\\) is inscribed in a circle. If \\(M\\) is a point on the shorter arc \\(AB\\), find the minimum value of the expression \\(\\frac{MC \\cdot MD}{MA \\cdot MB}\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-de2d867741bad7146dda5316bda3d3a255af7751b960b14a7903e15910dd6e01 | A square \(ABCD\) of side length 2 is given on a plane. The segment \(AB\) is moved continuously towards \(CD\) until \(A\) and \(C\) coincide with \(C\) and \(D\), respectively. Let \(S\) be the area of the region formed by the segment \(AB\) while moving. Find the maximum possible value of \(S\) such that \(S < \frac... | [
"2.00"
] | [
"A square \\(ABCD\\) of side length 2 is given on a plane. The segment \\(AB\\) is moved continuously towards \\(CD\\) until \\(A\\) and \\(C\\) coincide with \\(C\\) and \\(D\\), respectively. Let \\(S\\) be the area of the region formed by the segment \\(AB\\) while moving. Find the maximum possible value of \\(S... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 7
} | 10 | 0 | 0 |
omr-ws-e7c2ee4055d9ceff579d90e2e70dbd50970cdbaf3b8b822b93edbe68b345f523 | A square \(ABCD\) with sides of length 1 is divided into two congruent trapezoids and a pentagon, which have equal areas, by joining the center of the square with points \(E\), \(F\), and \(G\), where \(E\) is the midpoint of \(BC\), \(F\) and \(G\) are on \(AB\) and \(CD\) respectively, and \(AF < FB\), \(DG < GC\), a... | [
"\\(\\frac{5}{6}\\)"
] | [
"A square \\(ABCD\\) with sides of length 1 is divided into two congruent trapezoids and a pentagon, which have equal areas, by joining the center of the square with points \\(E\\), \\(F\\), and \\(G\\), where \\(E\\) is the midpoint of \\(BC\\), \\(F\\) and \\(G\\) are on \\(AB\\) and \\(CD\\) respectively, and \\... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 16
} | 37 | 0 | 0 |
omr-ws-1cae85559f526274c08066dc20c19c88143ccb54c76255f5146bb746d6cbb479 | A square and a circle intersect so that each side of the square contains a chord of the circle equal in length to the radius of the circle. What is the ratio of the area of the square to the area of the circle? Express your answer as a common fraction in terms of $\pi$. | [
"\\frac{3}{\\pi}"
] | [
"A square and a circle intersect so that each side of the square contains a chord of the circle equal in length to the radius of the circle. What is the ratio of the area of the square to the area of the circle? Express your answer as a common fraction in terms of $\\pi$."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
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