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-39c30e9169cd80b1bf39f7b8a3f679e29346d19f989de3562e0d61f109547955 | Two sides of an isosceles triangle are 22 and 61. Compute the area of the triangle. | [
"330"
] | [
"Two sides of an isosceles triangle are 22 and 61. Compute the area of the triangle."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-99521516ab4b1ed9ade129374ab14944d4b96cc769d0bee589739c804ac578ec | Two silvered mirrors each pass $1/4$ of the light striking them head-on and reflect $3/4$ of it. They are set parallel to each other, and a beam of light is directed head-on at the pair. What fraction of the original light eventually gets through to the other side of both mirrors? | [
"\\dfrac{1}{7}"
] | [
"Two silvered mirrors each pass $1/4$ of the light striking them head-on and reflect $3/4$ of it. They are set parallel to each other, and a beam of light is directed head-on at the pair. What fraction of the original light eventually gets through to the other side of both mirrors?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e430cc5a6c33998f356b80dd1a8c75cc3b9c65929e0d6526f68ac8e2f06c8304 | Two similar coins are tossed 1000 times, out of which 300 times 2 heads and 200 times 2 tails occur. What is the probability that exactly one head occurs? | [
"\\(\\frac{1}{2}\\)"
] | [
"Two similar coins are tossed 1000 times, out of which 300 times 2 heads and 200 times 2 tails occur. What is the probability that exactly one head occurs?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b88d78fe0250939420abd4bc48899ef5c82bb1e83d011f28d3983042a9af308e | Two similar right triangles have areas of 6 square inches and 150 square inches. The length of the hypotenuse of the smaller triangle is 5 inches. What is the sum of the lengths of the legs of the larger triangle? | [
"35"
] | [
"Two similar right triangles have areas of 6 square inches and 150 square inches. The length of the hypotenuse of the smaller triangle is 5 inches. What is the sum of the lengths of the legs of the larger triangle?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-61feae04276196589278305ad1f4b841d0bed631f00b7cda45aa07142524e9cc | Two similar triangles $A_1B_1C_1$ and $A_2B_2C_2$ are given, and their orientation coincides. Points $A_3, B_3, C_3$ are the midpoints of segments $A_1A_2$, $B_1B_2$, $C_1C_2$, respectively. If the ratio of similarity between $A_1B_1C_1$ and $A_2B_2C_2$ is $k$, find the ratio of similarity between triangle $A_3B_3C_3$ ... | [
"\\dfrac{1 + k}{2}"
] | [
"Two similar triangles $A_1B_1C_1$ and $A_2B_2C_2$ are given, and their orientation coincides. Points $A_3, B_3, C_3$ are the midpoints of segments $A_1A_2$, $B_1B_2$, $C_1C_2$, respectively. If the ratio of similarity between $A_1B_1C_1$ and $A_2B_2C_2$ is $k$, find the ratio of similarity between triangle $A_3B_3... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-23bab683730d7e8380dbd9e7d24cdcab7e99d64592a0d88ae4a307306f56d4e1 | Two similar triangles are each cut into two smaller triangles. One part from the first triangle is similar to one part from the second triangle. Let the similarity ratio between the original triangles be $k$. If the area of the part from the first triangle that is similar to the part from the second triangle is $A$, fi... | [
"A(k^2 - 1)"
] | [
"Two similar triangles are each cut into two smaller triangles. One part from the first triangle is similar to one part from the second triangle. Let the similarity ratio between the original triangles be $k$. If the area of the part from the first triangle that is similar to the part from the second triangle is $A... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-bde69b79de9f1013c860fd51905e4dabcbb05943bd086bdc88a8c5607e42868e | Two similar triangles with integral sides have two of their sides the same. The third sides differ by 387. What are the lengths of the sides? | [
"\\(125, 200, 320\\) and \\(200, 320, 512\\)"
] | [
"Two similar triangles with integral sides have two of their sides the same. The third sides differ by 387. What are the lengths of the sides?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 6
} | 6 | 0 | 0 |
omr-ws-398386a223990c70102f57d84c9b61faf1353f21243119d80151479dd66501d0 | Two sisters ascend 40-step escalators moving at the same speed. The older sister can only take 10 steps on the crowded "up" escalator, while the younger sister runs up the empty "down" escalator unimpeded, arriving at the top at the same time as her sister. How many steps does the younger sister take? | [
"70"
] | [
"Two sisters ascend 40-step escalators moving at the same speed. The older sister can only take 10 steps on the crowded \"up\" escalator, while the younger sister runs up the empty \"down\" escalator unimpeded, arriving at the top at the same time as her sister. How many steps does the younger sister take?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a58916300650e85d3e7770dabeb4208fbc72778ad6e20c83246bd10725432e02 | Two sisters ascend 40-step escalators moving at the same speed. The older sister can only take 10 steps up the crowded "up" escalator, while the younger sister runs up the empty "down" escalator unimpeded, arriving at the top at the same time as her sister. How many steps does the younger sister take? | [
"70 steps"
] | [
"Two sisters ascend 40-step escalators moving at the same speed. The older sister can only take 10 steps up the crowded \"up\" escalator, while the younger sister runs up the empty \"down\" escalator unimpeded, arriving at the top at the same time as her sister. How many steps does the younger sister take?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-6970fca5c9afed5ee41d5f2398beac48ed467661f2611f5de5e14a70a1fd96be | Two sisters go up a 40-step escalator. The older sister rides the escalator and takes 10 steps while the escalator moves her up. The younger sister runs up an empty down escalator and arrives at the top at the same time as her sister. How many steps does the younger sister take? | [
"70"
] | [
"Two sisters go up a 40-step escalator. The older sister rides the escalator and takes 10 steps while the escalator moves her up. The younger sister runs up an empty down escalator and arrives at the top at the same time as her sister. How many steps does the younger sister take?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-82462821b609527477e7c6702361718046284fc5774af399b2825637c0bb2430 | Two six-sided dice and one seven-sided die are rolled. What is the probability that the sum of the dice is eighteen? | [
"\\(\\frac{1}{84}\\)"
] | [
"Two six-sided dice and one seven-sided die are rolled. What is the probability that the sum of the dice is eighteen?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-f02eeb2f19b70b8041bb4405ca514f949679eac6e33243d9f66fe78023efe3e3 | Two six-sided dice are rolled. One die has the $4$ replaced by $3$, and the other die has the $3$ replaced by $4$. What is the probability that the sum of the numbers rolled is an odd number? | [
"\\(\\frac{5}{9}\\)"
] | [
"Two six-sided dice are rolled. One die has the $4$ replaced by $3$, and the other die has the $3$ replaced by $4$. What is the probability that the sum of the numbers rolled is an odd number?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-3561fc6e1c13a59b1d4edfa65db9dbc271186650ee42ddd22b584a4d47bc6e46 | Two skaters have to run a ring $S$ km long. When the winner finished, the other skater had to run one more complete circle. Find the length of the race-course if the winner, who completed every full circle $a$ s quicker than the other skater, ran the distance in $t$ min. | [
"$\\frac{-a+\\sqrt {a^2+240at}}{120t}S$"
] | [
"Two skaters have to run a ring $S$ km long. When the winner finished, the other skater had to run one more complete circle. Find the length of the race-course if the winner, who completed every full circle $a$ s quicker than the other skater, ran the distance in $t$ min."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-46ecf27a2001033ab2fb0744f465575a505947cb81d056d42bb667eb4a27edf8 | Two skaters, Allie and Billie, are at points $A$ and $B$, respectively, on a flat, frozen lake. The distance between $A$ and $B$ is $100$ meters. Allie leaves $A$ and skates at a speed of $8$ meters per second on a straight line that makes a $60^\circ$ angle with $AB$. At the same time Allie leaves $A$, Billie leaves $... | [
"160"
] | [
"Two skaters, Allie and Billie, are at points $A$ and $B$, respectively, on a flat, frozen lake. The distance between $A$ and $B$ is $100$ meters. Allie leaves $A$ and skates at a speed of $8$ meters per second on a straight line that makes a $60^\\circ$ angle with $AB$. At the same time Allie leaves $A$, Billie le... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 16
} | 31 | 0 | 0 |
omr-ws-d396ba60ad971c6d12d06aa712e83de4c024e29a8339b8b3bef1085bc8702801 | Two skaters, Allie and Billie, are at points $A$ and $B$, respectively, on a flat, frozen lake. The distance between $A$ and $B$ is $100$ meters. Allie leaves $A$ and skates at a speed of $8$ meters per second on a straight line that makes a $60^\circ$ angle with $AB$. At the same time Allie leaves $A$, Billie leaves $... | [
"160"
] | [
"Two skaters, Allie and Billie, are at points $A$ and $B$, respectively, on a flat, frozen lake. The distance between $A$ and $B$ is $100$ meters. Allie leaves $A$ and skates at a speed of $8$ meters per second on a straight line that makes a $60^\\circ$ angle with $AB$. At the same time Allie leaves $A$, Billi... | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 17
} | 32 | 0 | 0 |
omr-ws-273297fa04641cc2d6da035d796bc86a6e2a94cb2d6a4836c1bff7ae0c6c0e31 | Two skaters, Allie and Billie, are at points A and B, respectively, on a flat, frozen lake. The distance between A and B is 100 meters. Allie skates at a speed of 8 meters per second on a straight line that makes a 60-degree angle with AB. Billie skates at a speed of 7 meters per second and follows the straight path th... | [
"160"
] | [
"Two skaters, Allie and Billie, are at points A and B, respectively, on a flat, frozen lake. The distance between A and B is 100 meters. Allie skates at a speed of 8 meters per second on a straight line that makes a 60-degree angle with AB. Billie skates at a speed of 7 meters per second and follows the straight pa... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-209ef4ab73ceb8403b302325a8137409d657d643c5231da1e6652d1c10fef6b4 | Two skiers started from point $A$ in the same direction, the second skier starting six minutes after the first and overtook the first skier 2 km from the start. Having covered 5 km in all, the second skier returned back and met the first skier 4 km from the start. Find the speed of the second skier. | [
"10 km/h"
] | [
"Two skiers started from point $A$ in the same direction, the second skier starting six minutes after the first and overtook the first skier 2 km from the start. Having covered 5 km in all, the second skier returned back and met the first skier 4 km from the start. Find the speed of the second skier."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bff4bba48eac0d39c94c2393fee24fabfbe5c7a8600b3267a9467b7736fc7ed4 | Two small boats, A and B, move with constant speed in straight and perpendicular paths. At a certain moment, boat A is 4.0 km from point P, and boat B is 3.0 km from point P, where the paths intersect. What is the minimum distance, measured in kilometers, between boats A and B? | [
"\\( 2\\sqrt{5} \\) km"
] | [
"Two small boats, A and B, move with constant speed in straight and perpendicular paths. At a certain moment, boat A is 4.0 km from point P, and boat B is 3.0 km from point P, where the paths intersect. What is the minimum distance, measured in kilometers, between boats A and B?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 2
} | 13 | 0 | 0 |
omr-ws-07f36f5de7450eab5116654deb06358197f0f25828eeb05fa3325041e59d6bd5 | Two small squares are chosen at random on a chessboard. What is the probability that the two squares share a common side? | [
"$\\frac{1}{18}$"
] | [
"Two small squares are chosen at random on a chessboard. What is the probability that the two squares share a common side?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-9af5f4ff729e545e2f0ab24c67625c45c3724eacc545f8ee2a5338bf1628ef57 | Two soccer players run a drill on a \( 100 \) foot by \( 300 \) foot rectangular soccer field. The two players start on two different corners of the rectangle separated by \( 100 \) feet, then run parallel along the long edges of the field, passing a soccer ball back and forth between them. Assume that the ball travels... | [
"500"
] | [
"Two soccer players run a drill on a \\( 100 \\) foot by \\( 300 \\) foot rectangular soccer field. The two players start on two different corners of the rectangle separated by \\( 100 \\) feet, then run parallel along the long edges of the field, passing a soccer ball back and forth between them. Assume that the b... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4b9a7844e4aa394093f9e8509b429b00ba5f7cc5a0292ccd88343af7d5dd5d06 | Two soccer players, Matt and John, each have a probability \( p \) of scoring on a penalty. They each take 5 penalties. Determine \( p \) such that the probability that Matt wins the game is equal to the probability that the game ends in a draw. | [
"\\( \\frac{1}{2} \\)"
] | [
"Two soccer players, Matt and John, each have a probability \\( p \\) of scoring on a penalty. They each take 5 penalties. Determine \\( p \\) such that the probability that Matt wins the game is equal to the probability that the game ends in a draw."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-c6e2803991aa333374606c7180e7dcdba9dd7fdd8aa437f1754d4258d0a750d3 | Two solutions of \[x^4 - 3x^3 + 5x^2 - 27x - 36 = 0\]are pure imaginary. Enter these solutions, separated by commas. | [
"3i,-3i"
] | [
"Two solutions of\n\\[x^4 - 3x^3 + 5x^2 - 27x - 36 = 0\\]are pure imaginary. Enter these solutions, separated by commas."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-50fa21361d65eee4513a3e9b1e68e7757025afab2c15be65d25319f7e54f5645 | Two solutions, the first containing 800 grams and the second, 600 grams of anhydrous sulphuric acid, are mixed to produce 10 kg of a new solution of sulphuric acid. Determine the weights of the first and second solutions in the mixture, if it is known that the content of anhydrous sulphuric acid in the first solution i... | [
"4 kg and 6 kg"
] | [
"Two solutions, the first containing 800 grams and the second, 600 grams of anhydrous sulphuric acid, are mixed to produce 10 kg of a new solution of sulphuric acid. Determine the weights of the first and second solutions in the mixture, if it is known that the content of anhydrous sulphuric acid in the first solut... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bcc46ec2072ab07627e0004a763d842412e121a76110b6d9d0b8846ecd1d0593 | Two spheres $P_1$ and $P_2$, each of radius $a$, externally touch all five spheres $K_1, K_2, K_3, K_4, K_5$ (each with center at a vertex of a regular pentagon $A_1A_2A_3A_4A_5$ and radius $a$). If $P_1$ and $P_2$ have a common point, find the distance between the centers of $P_1$ and $P_2$. | [
"2a"
] | [
"Two spheres $P_1$ and $P_2$, each of radius $a$, externally touch all five spheres $K_1, K_2, K_3, K_4, K_5$ (each with center at a vertex of a regular pentagon $A_1A_2A_3A_4A_5$ and radius $a$). If $P_1$ and $P_2$ have a common point, find the distance between the centers of $P_1$ and $P_2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3f6f1baf9b5f6e45f9cc5eafaeb6398e31e292ee2ccb1759e3141ac65892e894 | Two spheres are externally tangent at point $P$. Segments $AB$ and $CD$ touch the spheres with $A$ and $C$ lying on the first sphere and $B$ and $D$ on the second. Let $M$ and $N$ be the projections of the midpoints of segments $AC$ and $BD$ onto the line connecting the centers of the spheres. If the radii of the spher... | [
"0"
] | [
"Two spheres are externally tangent at point $P$. Segments $AB$ and $CD$ touch the spheres with $A$ and $C$ lying on the first sphere and $B$ and $D$ on the second. Let $M$ and $N$ be the projections of the midpoints of segments $AC$ and $BD$ onto the line connecting the centers of the spheres. If the radii of the ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-88dac1cf8807313323762fe04351e29d5ce2d45158b8acb6b426e21c80b72283 | Two spheres are inscribed in two cubes with side lengths 10 and 30. In the larger sphere, another tetrahedron is inscribed. What is the ratio of the volume of the tetrahedron to the volume of the sphere with radius 10? | [
"\\(\\frac{81 \\sqrt{3}}{2 \\pi}\\)"
] | [
"Two spheres are inscribed in two cubes with side lengths 10 and 30. In the larger sphere, another tetrahedron is inscribed. What is the ratio of the volume of the tetrahedron to the volume of the sphere with radius 10?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-34074515fafe7c101d6f523c7f8fa807bf54671700fc4cc4bbf03072d740dbaf | Two spheres both have radii $1$, and each of the spheres pass through the center of the other. Find the volume of the region inside the two spheres. | [
"\\(\\frac{5\\pi}{12}\\)"
] | [
"Two spheres both have radii $1$, and each of the spheres pass through the center of the other. Find the volume of the region inside the two spheres."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-fe365de17a0359c8e11ee3e3387d5b8c9c7a9b558dee4f8c8cba3d5dc888ddec | Two spheres each with radii $1$ are placed in space such that their centers are $1$ unit apart. Find the volume of the region where the two spheres intersect. | [
"\\(\\frac{5\\pi}{12}\\)"
] | [
"Two spheres each with radii $1$ are placed in space such that their centers are $1$ unit apart. Find the volume of the region where the two spheres intersect."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-b59c91940f06b1c6f395c3f35c1ac3d45810d7f4aa87399e4a04eb9275f7dadd | Two spheres each with radii $1$ are placed in space such that their centers are $\sqrt{2}$ units apart. Find the volume of the region where the two spheres intersect. | [
"\\dfrac{(8 - 5\\sqrt{2})\\pi}{6}"
] | [
"Two spheres each with radii $1$ are placed in space such that their centers are $\\sqrt{2}$ units apart. Find the volume of the region where the two spheres intersect."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8669d426c60063be6df2e40d5648e92a9f39892605381917e3dc217d97104943 | Two spheres each with radii $1$ are placed in space such that their centers are $\sqrt{3}$ units apart. Find the volume of the region where the two spheres intersect. | [
"\\dfrac{(16 - 9\\sqrt{3})\\pi}{12}"
] | [
"Two spheres each with radii $1$ are placed in space such that their centers are $\\sqrt{3}$ units apart. Find the volume of the region where the two spheres intersect."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-22bb1de9d6ca51cfef17d6b9a8110a5019ab8e79cd1af47e44d6bb8420d6c47d | Two spheres of radii $r$ and $R$ lie on a plane without meeting each other. The distance between the centers of the spheres is $\rho $. Find the minimal possible radius of the sphere which would lie on that plane and touch the given spheres. | [
"$\\frac{\\rho ^2-(R-r)^2}{4\\left(\\sqrt R+\\sqrt r\\right)^2}$"
] | [
"Two spheres of radii $r$ and $R$ lie on a plane without meeting each other. The distance between the centers of the spheres is $\\rho $. Find the minimal possible radius of the sphere which would lie on that plane and touch the given spheres."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-161d3025b6add4a3cce0f75225ca49c4f81727219bdeeeb5ca9ef0d4f710dafc | Two spheres of radii \(a\) and \(b\) are tangent to each other and to a plane at different points. Find the radius of the largest sphere that can pass between the two spheres and the plane. | [
"\\(\\frac{ab}{a + b}\\)"
] | [
"Two spheres of radii \\(a\\) and \\(b\\) are tangent to each other and to a plane at different points. Find the radius of the largest sphere that can pass between the two spheres and the plane."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-437273871e8e4c9b2285d9c6f00c64e4cd3164a7ceb4a62c35532ec214928038 | Two spheres touch 10 units above the ground. The smaller sphere has a radius of 8. What is the radius of the bigger sphere? | [
"\\(\\frac{40}{3}\\)"
] | [
"Two spheres touch 10 units above the ground. The smaller sphere has a radius of 8. What is the radius of the bigger sphere?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 25
} | 73 | 0 | 0 |
omr-ws-d7f7e57e8a07edae07cc9e8a12f8a3feed69e5230d4bf8fedc8594e3c490b6b0 | Two spheres touch a plane $P$ at points $A$ and $B$ and lie on different sides of the plane. The distance between the centers of the sphere is equal to 10. A third sphere touches the two given spheres and its center $O$ lies in the plane $P$. It is known that $|AO|=|OB|=2\sqrt {10}$, $|AB|=8$. Find the radius of the th... | [
"4"
] | [
"Two spheres touch a plane $P$ at points $A$ and $B$ and lie on different sides of the plane. The distance between the centers of the sphere is equal to 10. A third sphere touches the two given spheres and its center $O$ lies in the plane $P$. It is known that $|AO|=|OB|=2\\sqrt {10}$, $|AB|=8$. Find the radius of ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fdedbb67cb0b35e4e680732e49bb985545dd06a96fe6eded76f71688b7c98607 | Two spheres with different radii are externally tangent. The straight line that joins the centers is the axis of a right circular cone that touches both spheres externally. There are $n$ spheres, all tangent externally to the first two spheres, all tangent to the cone, and each tangent externally to two others, forming... | [
"5"
] | [
"Two spheres with different radii are externally tangent. The straight line that joins the centers is the axis of a right circular cone that touches both spheres externally. There are $n$ spheres, all tangent externally to the first two spheres, all tangent to the cone, and each tangent externally to two others, fo... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-49dec7426d88e6fa64de9daa8499c75463f321af5cb5c9902f97014cae8eda38 | Two spheres with radii $36$ and one sphere with radius $13$ are each externally tangent to the other two spheres and to two different planes $\mathcal{P}$ and $\mathcal{Q}$. The intersection of planes $\mathcal{P}$ and $\mathcal{Q}$ is the line $\ell$. The distance from line $\ell$ to the point where the sphere with ra... | [
"335"
] | [
"Two spheres with radii $36$ and one sphere with radius $13$ are each externally tangent to the other two spheres and to two different planes $\\mathcal{P}$ and $\\mathcal{Q}$. The intersection of planes $\\mathcal{P}$ and $\\mathcal{Q}$ is the line $\\ell$. The distance from line $\\ell$ to the point where the sph... | [
"MATH_training_set",
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 78,
"genselect": 15,
"tir": 20
} | 113 | 0 | 0 |
omr-ws-7e257ec023a4f5adba84b672a546655020e13f1ae434489b6b670f85769a4e1a | Two spheres with radii 30 cm and 40 cm intersect, with their centers 50 cm apart. Compute the area of the circle formed by their intersection. | [
"\\( 576\\pi \\)"
] | [
"Two spheres with radii 30 cm and 40 cm intersect, with their centers 50 cm apart. Compute the area of the circle formed by their intersection."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-d6d62bd20ff732d8165bb6ea693bc1088730585311c3a21832ae2fa98f33aa9b | Two spheres with radii 36 and one sphere with radius 13 are each externally tangent to the other two spheres and to two different planes \(\mathcal{P}\) and \(\mathcal{Q}\). The intersection of planes \(\mathcal{P}\) and \(\mathcal{Q}\) is the line \(\ell\). The distance from line \(\ell\) to the point where the sphere... | [
"335"
] | [
"Two spheres with radii 36 and one sphere with radius 13 are each externally tangent to the other two spheres and to two different planes \\(\\mathcal{P}\\) and \\(\\mathcal{Q}\\). The intersection of planes \\(\\mathcal{P}\\) and \\(\\mathcal{Q}\\) is the line \\(\\ell\\). The distance from line \\(\\ell\\) to the... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 7
} | 19 | 0 | 0 |
omr-ws-8a44d9a11dae62280f712be59348262bd20a5053591ed283eb288f032019abe0 | Two spheres, each with radius 1, are tangent to each other's centers. What is the area of the region that is common to both spheres? | [
"\\(\\frac{5\\pi}{12}\\)"
] | [
"Two spheres, each with radius 1, are tangent to each other's centers. What is the area of the region that is common to both spheres?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 1
} | 2 | 0 | 0 |
omr-ws-dc9fe1e305f682c7773283baff3a497fc823a5b83683cf5331183f2bab21a963 | Two spherical balls lie on the ground touching. If one of the balls has a radius of 8 inches, and the point of contact is 10 inches above the ground, what is the radius of the other ball? | [
"\\(\\frac{40}{3}\\)"
] | [
"Two spherical balls lie on the ground touching. If one of the balls has a radius of 8 inches, and the point of contact is 10 inches above the ground, what is the radius of the other ball?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-8d1c4acfef216d997538748b09ba1182604255d6bf399aed3036bbba5fd9f974 | Two spiders are positioned at one vertex of a cube, and a fly is positioned at the opposite vertex. The spiders and the fly can move along the edges of the cube at the same speed. The spiders move in harmony to capture the fly. Determine the minimum number of moves required for the spiders to guarantee capturing the fl... | [
"3"
] | [
"Two spiders are positioned at one vertex of a cube, and a fly is positioned at the opposite vertex. The spiders and the fly can move along the edges of the cube at the same speed. The spiders move in harmony to capture the fly. Determine the minimum number of moves required for the spiders to guarantee capturing t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 5
} | 10 | 0 | 0 |
omr-ws-69290881ca394016dc469c5082e1d003eb9622cff38f736bea82795a60f54ac0 | Two sportsmen run along the same closed track in a stadium. The speed of each runner is constant but the first sportsman makes a round 10 s quicker than the second. If they begin at the same start and run in the same direction, then they will meet again in 720 s. How much of the track does each runner cover per second? | [
"1/80, 1/90"
] | [
"Two sportsmen run along the same closed track in a stadium. The speed of each runner is constant but the first sportsman makes a round 10 s quicker than the second. If they begin at the same start and run in the same direction, then they will meet again in 720 s. How much of the track does each runner cover per se... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7e80af2f2085b1accb6dc4e4aa2110bd19b48bd9fbbd7ae9d1a05528aa3d5e72 | Two square carpets were brought to a large square hall. The side of one carpet is double the side of the other. When placed in opposite corners of the hall, they cover a common area of $4$ m$^2$. When placed in adjacent corners, they cover a common area of $14$ m$^2$. What is the size of the hall? | [
"19"
] | [
"Two square carpets were brought to a large square hall. The side of one carpet is double the side of the other. When placed in opposite corners of the hall, they cover a common area of $4$ m$^2$. When placed in adjacent corners, they cover a common area of $14$ m$^2$. What is the size of the hall?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-20564b860a252ab96f125b963aa30d4dfdd063a03a52e0e5898183533d12e2aa | Two square pieces of paper have a total area of 2 square units. These squares are to be placed inside a rectangle such that their sides are parallel to the sides of the rectangle. What is the least possible area of the rectangle that can enclose these two squares? | [
"2"
] | [
"Two square pieces of paper have a total area of 2 square units. These squares are to be placed inside a rectangle such that their sides are parallel to the sides of the rectangle. What is the least possible area of the rectangle that can enclose these two squares?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-6b21fe195358e9c8e21a04b3deb722ba18fd4d3e7bb4da3185aa61b9e19d39f2 | Two square sheets each have an area of 2003 and are divided into 2003 nonoverlapping polygons of unitary area. When the two sheets are overlaid, determine the minimum number of punctures required such that each of the 4006 polygons gets punctured at least once. | [
"2003"
] | [
"Two square sheets each have an area of 2003 and are divided into 2003 nonoverlapping polygons of unitary area. When the two sheets are overlaid, determine the minimum number of punctures required such that each of the 4006 polygons gets punctured at least once."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5d93f1df036a03005164e67889bd1871fbd4093b2fa0c033317d6848487f322a | Two square sheets of paper each have an area of 2013 and are divided into 2013 non-overlapping polygons, each with an area of 1. When one sheet is laid exactly on top of the other, determine the minimum number of punctures required such that each of the 4026 polygons gets punctured at least once. | [
"2013"
] | [
"Two square sheets of paper each have an area of 2013 and are divided into 2013 non-overlapping polygons, each with an area of 1. When one sheet is laid exactly on top of the other, determine the minimum number of punctures required such that each of the 4026 polygons gets punctured at least once."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9334a3ce7faa65c187ee000d336f1a26e621fb0dc1b5c6346b504f362fb53bf3 | Two squares $ABCD$ and $EFGH$ with side length $1$ lie in the plane such that point $F$ is outside square $ABCD$, point $B$ is inside segment $\overline{EF}$, and point $C$ is inside segment $\overline{FG}$. If $H$ can be positioned in three distinct regions relative to square $ABCD$ (inside, on the edge, or outside), ... | [
"7"
] | [
"Two squares $ABCD$ and $EFGH$ with side length $1$ lie in the plane such that point $F$ is outside square $ABCD$, point $B$ is inside segment $\\overline{EF}$, and point $C$ is inside segment $\\overline{FG}$. If $H$ can be positioned in three distinct regions relative to square $ABCD$ (inside, on the edge, or out... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-fd9f6a1b41cb9674716e71ba2598baddaedd02277181b73aa666bf774a1dbfc0 | Two squares $ABDE$ and $ACFG$ are constructed on $AB$ and $AC$ outside the triangle $ABC$. Points $P$ and $Q$ are on line $EG$ such that $BP$ and $CQ$ are perpendicular to $BC$. If $AB = 3$, $AC = 4$, and $\angle BAC = 90^\circ$, calculate the minimum value of $BP + CQ - BC - EG$. | [
"\\dfrac{5}{24}"
] | [
"Two squares $ABDE$ and $ACFG$ are constructed on $AB$ and $AC$ outside the triangle $ABC$. Points $P$ and $Q$ are on line $EG$ such that $BP$ and $CQ$ are perpendicular to $BC$. If $AB = 3$, $AC = 4$, and $\\angle BAC = 90^\\circ$, calculate the minimum value of $BP + CQ - BC - EG$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-083113f3ccd12b6b8db615e78dc02e60f4f8c1c17d5353f137829564c189a6a2 | Two squares $ACXE$ and $CBDY$ are constructed in the exterior of an acute-angled triangle $ABC$. Let $P$ be the intersection of the lines $AD$ and $BE$. If $H$ is the foot of the altitude from $C$ to $AB$, find the ratio $\frac{CP}{CH}$ when $AC = 3$ and $BC = 4$. | [
"no_valid_answer_found"
] | [
"Two squares $ACXE$ and $CBDY$ are constructed in the exterior of an acute-angled triangle $ABC$. Let $P$ be the intersection of the lines $AD$ and $BE$. If $H$ is the foot of the altitude from $C$ to $AB$, find the ratio $\\frac{CP}{CH}$ when $AC = 3$ and $BC = 4$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-57d721e65ee18d3038abd111afbc62706f2af05392a282b9842b475e26dfdf60 | Two squares (each with side $12$) are placed so that a corner of one lies at the center of the other. Find the area of quadrilateral $EJCK$ if $BJ = 4$. | [
"36"
] | [
"Two squares (each with side $12$) are placed so that a corner of one lies at the center of the other. Find the area of quadrilateral $EJCK$ if $BJ = 4$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 7
} | 13 | 0 | 0 |
omr-ws-56d5e24a47a1d7ad7308a50ab1b655acf9aa9e8232a1fd5e053ea0f4aeb9e25e | Two squares \(ABCD\) and \(CDEF\) are in perpendicular planes. If \(\ell\) is the side length of the squares, find the distance between the lines \(BD\) and \(CE\) in terms of \(\ell\). | [
"\\dfrac{\\ell \\sqrt{3}}{3}"
] | [
"Two squares \\(ABCD\\) and \\(CDEF\\) are in perpendicular planes. If \\(\\ell\\) is the side length of the squares, find the distance between the lines \\(BD\\) and \\(CE\\) in terms of \\(\\ell\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 6
} | 21 | 0 | 0 |
omr-ws-2dd0e3892fb49840befe23ba2bed86a9efc91c7ddc2f57ae204d235d27ab64af | Two squares are chosen at random from the small squares of an $8 \times 8$ chessboard. What is the probability that the two squares chosen have exactly one corner in common? | [
"$\\frac{7}{144}$"
] | [
"Two squares are chosen at random from the small squares of an $8 \\times 8$ chessboard. What is the probability that the two squares chosen have exactly one corner in common?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 31
} | 79 | 0 | 0 |
omr-ws-9990ff9e766a9049e58ab705dd0363d9f5bd2c8354b5237fa079c63e26379630 | Two squares are chosen at random on a chessboard. What is the probability that they have a side in common? Consider whether the sample space should be $\binom{64}{2}$ or $64 \times 63$. | [
"\\(\\frac{1}{9}\\)"
] | [
"Two squares are chosen at random on a chessboard. What is the probability that they have a side in common? Consider whether the sample space should be $\\binom{64}{2}$ or $64 \\times 63$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f30b6f85ff7f2c9abf2c780479fdc59fb9be837b948977fb65510298bf5c4185 | Two squares are inscribed in a right-angled triangle. One square is parallel to the hypotenuse and has an area of 441, while the other square is parallel to the legs and has an area of 440. Find the sum of the lengths of the two shorter sides of the triangle. | [
"462"
] | [
"Two squares are inscribed in a right-angled triangle. One square is parallel to the hypotenuse and has an area of 441, while the other square is parallel to the legs and has an area of 440. Find the sum of the lengths of the two shorter sides of the triangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-762e7cce2605bb19caa7e16e5d8a52ecc511e144ded7189b46bc2555e65e2ce1 | Two squares are inscribed in a right-angled triangle. The first square has an area of 441, and the second square has an area of 440. Find the sum of the two shorter sides of the triangle. | [
"462"
] | [
"Two squares are inscribed in a right-angled triangle. The first square has an area of 441, and the second square has an area of 440. Find the sum of the two shorter sides of the triangle."
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 24,
"genselect": 15,
"tir": 4
} | 43 | 0 | 0 |
omr-ws-584d449f7071697341200cff429a7160d062cdf2d2d2d62f9574b58c6a930c96 | Two squares are inscribed in a right-angled triangle. The first square has one vertex coinciding with the right angle, two sides on the perpendicular sides, and the opposite vertex on the hypotenuse. The second square has one side on the hypotenuse and two vertices on the perpendicular sides. The areas of the first and... | [
"462"
] | [
"Two squares are inscribed in a right-angled triangle. The first square has one vertex coinciding with the right angle, two sides on the perpendicular sides, and the opposite vertex on the hypotenuse. The second square has one side on the hypotenuse and two vertices on the perpendicular sides. The areas of the firs... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 8,
"tir": 19
} | 39 | 0 | 0 |
omr-ws-ed77412da226d50f1e57c8d5ef51b0cc3d5d4ba1b7e5dfa495cede08c7409d2b | Two squares are inscribed in rectangle ABCD. If the area of the first square $S_1$ is 440 square units and the area of the second square $S_2$ is 441 square units, what is the perimeter of rectangle ABCD? | [
"924"
] | [
"Two squares are inscribed in rectangle ABCD. If the area of the first square $S_1$ is 440 square units and the area of the second square $S_2$ is 441 square units, what is the perimeter of rectangle ABCD?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a77c7828823801d864970d718074d2e6fb695062ae425698a53898953495b15e | Two squares are overlapped. The smaller square has an area of $16$, and the grey triangle formed by the overlap has an area of $1$. What is the area of the larger square? | [
"18"
] | [
"Two squares are overlapped. The smaller square has an area of $16$, and the grey triangle formed by the overlap has an area of $1$. What is the area of the larger square?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-98800734e197e02c9b305155cc14faa2dc0b64562cf1906841e37c623df11d07 | Two squares are painted in blue and two squares are painted in red on a \( 3 \times 3 \) board in such a way that two squares with the same color are neither in the same row nor in the same column. In how many different ways can these four squares be painted? | [
"198"
] | [
"Two squares are painted in blue and two squares are painted in red on a \\( 3 \\times 3 \\) board in such a way that two squares with the same color are neither in the same row nor in the same column. In how many different ways can these four squares be painted?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 4
} | 15 | 0 | 0 |
omr-ws-25a13e06da496a3c4e5d56bb71a4e376933c2444d6db79f8ff2388f9269fc421 | Two squares are positioned so that the center of the first square is the vertex of the second square. If the side of the first square measures $12$ cm, what is the area common to the two squares? | [
"36"
] | [
"Two squares are positioned so that the center of the first square is the vertex of the second square. If the side of the first square measures $12$ cm, what is the area common to the two squares?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 5
} | 18 | 0 | 0 |
omr-ws-43673a2a2cbbf1bdeb787ca56fc04d2c2fea9a8c6fed36cd1cc3007edacab1fa | Two squares differ by 2016. Find their maximum possible sum. | [
"508034"
] | [
"Two squares differ by 2016. Find their maximum possible sum."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-b424b1af1291233e0366aa840965dfb2532ef9cfe3c7c4b9f29484aa83fe3faf | Two squares of a $7\times 7$ checkerboard are painted yellow, and the rest are painted green. Two color schemes are equivalent if one can be obtained from the other by applying a rotation in the plane board. How many inequivalent color schemes are possible? | [
"300"
] | [
"Two squares of a $7\\times 7$ checkerboard are painted yellow, and the rest are painted green. Two color schemes are equivalent if one can be obtained from the other by applying a rotation in the plane board. How many inequivalent color schemes are possible?"
] | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-889c486086ca298279cbf1ffb3e2d7a96ee42de1f154480c1c9f17ce8a5d01df | Two squares of a 7x7 checkerboard are painted yellow, and the rest are painted green. Two color schemes are considered equivalent if one can be obtained from the other by applying a rotation in the plane of the board. How many inequivalent color schemes are possible? | [
"300"
] | [
"Two squares of a 7x7 checkerboard are painted yellow, and the rest are painted green. Two color schemes are considered equivalent if one can be obtained from the other by applying a rotation in the plane of the board. How many inequivalent color schemes are possible?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-0609c0dde44f0a6c97691c82d86d968ecb865972400bca894b9d157d42f74d61 | Two squares of area 1 have sides parallel and their intersection is a rectangle of area at least $\frac{1}{2}$. Their union forms a nonconvex octagon. What is the maximum possible perimeter of this octagon? | [
"\\(8 - 2\\sqrt{2}\\)"
] | [
"Two squares of area 1 have sides parallel and their intersection is a rectangle of area at least $\\frac{1}{2}$. Their union forms a nonconvex octagon. What is the maximum possible perimeter of this octagon?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 12
} | 26 | 0 | 0 |
omr-ws-895e9038fe3acc7061ab4e480ee91850cb3f62efcfe4f8445ed1eccb26bad72a | Two squares of side length $2$ are glued together along their boundary so that the four vertices of the first square are glued to the midpoints of the four sides of the other square, and vice versa. This gluing results in a convex polyhedron. If the square of the volume of this polyhedron is written in simplest form as... | [
"651"
] | [
"Two squares of side length $2$ are glued together along their boundary so that the four vertices of the first square are glued to the midpoints of the four sides of the other square, and vice versa. This gluing results in a convex polyhedron. If the square of the volume of this polyhedron is written in simplest fo... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 8
} | 10 | 0 | 0 |
omr-ws-6d422dcbc8e8ae4921af55de3725bfd6f4d6cebb9652f6698de88afcc85299d0 | Two squares share the same center $O$ and have sides of length 1. The length of $\overline{AB}$ is $43/99$ and the area of the octagon $ABCDEFGH$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$ | [
"179585788"
] | [
"Two squares share the same center $O$ and have sides of length 1. The length of $\\overline{AB}$ is $43/99$ and the area of the octagon $ABCDEFGH$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b2c1e3085414a036297cda32a5025963acf79f15e93c44f633f077358125a46f | Two squares share the same center $O$ and have sides of length 1. The length of $\overline{AB}$ is $43/99$, and the area of the octagon $ABCDEFGH$ is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | [
"185"
] | [
"Two squares share the same center $O$ and have sides of length 1. The length of $\\overline{AB}$ is $43/99$, and the area of the octagon $ABCDEFGH$ is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 1
} | 2 | 0 | 0 |
omr-ws-46045ca5f7e479550aa4d3309acc952629708336fa5bee0484873dcfcdc47e8b | Two squares share the same center \( O \) and have sides of length 1. The length of \( \overline{AB} \) is \( \frac{43}{99} \) and the area of the octagon \( ABCDEFGH \) is \( \frac{m}{n} \), where \( m \) and \( n \) are relatively prime positive integers. Find \( m+n \). | [
"185"
] | [
"Two squares share the same center \\( O \\) and have sides of length 1. The length of \\( \\overline{AB} \\) is \\( \\frac{43}{99} \\) and the area of the octagon \\( ABCDEFGH \\) is \\( \\frac{m}{n} \\), where \\( m \\) and \\( n \\) are relatively prime positive integers. Find \\( m+n \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 6,
"tir": 2
} | 11 | 0 | 0 |
omr-ws-639182b3b9c99f5b934ac4b203aad2b0ea064d5aa0b5a15016db3f665b70e0ab | Two squares were cut out of a chessboard. In what case is it possible and in what case not to cover the remaining squares of the board with dominoes (i.e., figures of the form $2 \times 1$) without overlapping? | [
"It is possible to cover the remaining squares of the board with dominoes if the two squares cut out are of different colors, and not possible if they are of the same color."
] | [
"Two squares were cut out of a chessboard. In what case is it possible and in what case not to cover the remaining squares of the board with dominoes (i.e., figures of the form $2 \\times 1$) without overlapping?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 23
} | 45 | 0 | 0 |
omr-ws-2944dd746f729bbbb89779204a17b36bcb0ab2134e9500fe3f5352a68789763d | Two squares with side length 1 are placed such that their centers coincide. One side of the resulting octagon is \(\frac{43}{99}\). Find the area of the octagon in the form \(m + \sqrt{n}\). | [
"\\(\\frac{86}{99}\\)"
] | [
"Two squares with side length 1 are placed such that their centers coincide. One side of the resulting octagon is \\(\\frac{43}{99}\\). Find the area of the octagon in the form \\(m + \\sqrt{n}\\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3e3e32de419259897783f13e8494ed90357c281413ad0a19d24b2cb27031e549 | Two squares with side length 1 are placed such that their centers coincide. The area inside both squares forms an octagon. Given that one side of the octagon is \( \frac{43}{99} \), find the area of the octagon. | [
"\\(\\frac{86}{99}\\)",
"\\dfrac{86}{99}"
] | [
"Two squares with side length 1 are placed such that their centers coincide. The area inside both squares forms an octagon. Given that one side of the octagon is \\( \\frac{43}{99} \\), find the area of the octagon."
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 10,
"tir": 2
} | 12 | 0 | 0 |
omr-ws-82ecd93a0570fbcaba783bacd05753c0f031588f77192dc004e1324bbd773a0b | Two squares with side length 12 lie exactly on top of each other. One square is rotated around a corner point through an angle of 30 degrees relative to the other square. Determine the area of the common piece of the two squares. | [
"\\( 48\\sqrt{3} \\)"
] | [
"Two squares with side length 12 lie exactly on top of each other. One square is rotated around a corner point through an angle of 30 degrees relative to the other square. Determine the area of the common piece of the two squares."
] | [
"aops_c6_high_school_olympiads"
] | [
"genselect"
] | {
"genselect": 7
} | 7 | 0 | 0 |
omr-ws-b98045d802e17f6a00d349e5199cebcd3a4f06dd287ff81d2ef3f9a97dcdf003 | Two squares with side lengths $a$ and $b$ are arranged such that two adjacent vertices of each square are on a circle $k$ and the other two vertices are on a chord $s$ of $k$. Determine the distance of the center $M$ of the circle $k$ from the chord $s$ depending on $a$ and $b$. | [
"\\( \\frac{5(b - a)}{8} \\)"
] | [
"Two squares with side lengths $a$ and $b$ are arranged such that two adjacent vertices of each square are on a circle $k$ and the other two vertices are on a chord $s$ of $k$. Determine the distance of the center $M$ of the circle $k$ from the chord $s$ depending on $a$ and $b$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-df89ba44284d41656b2d5f1be6bc136501f414b560506b296edb4ec275ef3541 | Two squares with side lengths $a$ and $b$ are arranged such that two adjacent vertices of each square are on a circle $k$ and the other two vertices are on a chord $s$ of $k$. Investigate what values the ratio of the side lengths $a$ and $b$ of the squares can take. | [
"\\(\\frac{1}{5} \\leq \\frac{a}{b} \\leq 1\\)"
] | [
"Two squares with side lengths $a$ and $b$ are arranged such that two adjacent vertices of each square are on a circle $k$ and the other two vertices are on a chord $s$ of $k$. Investigate what values the ratio of the side lengths $a$ and $b$ of the squares can take."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-24d9e0f6c5ef183cb51fdf4e95fed73b7bbb0e2b40f224e0a4bb8a18b3a29c02 | Two squares with side lengths of 17 cm overlap to form a rectangle with dimensions 17 cm by 30 cm. Find the area of the overlapping region. | [
"68"
] | [
"Two squares with side lengths of 17 cm overlap to form a rectangle with dimensions 17 cm by 30 cm. Find the area of the overlapping region."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-34a97bc79f370b0d17b9e78065a32f3172aefd1e043c70c4215b713e1fbe779b | Two squares with sides \( a \) and \( b \) are placed inside a unit square without overlapping. Find the maximum possible value of \( a + b \). | [
"1"
] | [
"Two squares with sides \\( a \\) and \\( b \\) are placed inside a unit square without overlapping. Find the maximum possible value of \\( a + b \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 6
} | 13 | 0 | 0 |
omr-ws-04af2d250d70af4f2e6b98e9bb80eec2cf3219259d7cc75d0bf651e08e664730 | Two squares with the same side length have their centers coinciding. One of the squares is randomly rotated by an angle. Evaluate the expected area of the new geometric figure formed by the overlap and non-overlap regions of the two squares. | [
"\\( 2s^2 \\)"
] | [
"Two squares with the same side length have their centers coinciding. One of the squares is randomly rotated by an angle. Evaluate the expected area of the new geometric figure formed by the overlap and non-overlap regions of the two squares."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-29e543e549fdffba898bbb433d7c3b7d51047b4bbd6eb009e35a231e1cf9c901 | Two squares, each 1 meter by 1 meter, are placed one on top of the other with their centers coinciding, but the top square is rotated through 45 degrees relative to the bottom square. What is the area of the portion of the lower square that remains visible? | [
"\\(\\frac{3 - 2\\sqrt{2}}{4}\\)"
] | [
"Two squares, each 1 meter by 1 meter, are placed one on top of the other with their centers coinciding, but the top square is rotated through 45 degrees relative to the bottom square. What is the area of the portion of the lower square that remains visible?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fc1794da962cd2220b12dc9b74e76d8132929c307d5ec236aa47a5f61f16d715 | Two squids are given distinct positive integers no larger than $2023$ as their IDs. They take turns alternatively, where on each turn a squid can either announce a positive integer or declare which squid has the larger ID. If the declaration is correct, they win; otherwise, they lose. Find the smallest positive integer... | [
"10"
] | [
"Two squids are given distinct positive integers no larger than $2023$ as their IDs. They take turns alternatively, where on each turn a squid can either announce a positive integer or declare which squid has the larger ID. If the declaration is correct, they win; otherwise, they lose. Find the smallest positive in... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d089a2273d22f9a8bce99882f8e61dd5dd828054b3a744cb17ffd8390cf87bb7 | Two squirrels, Bushy and Jumpy, have collected $2021$ walnuts. Jumpy numbers the walnuts from 1 through $2021$ and digs $2021$ holes in a circular pattern. Bushy places one walnut into each hole without regard to the numbering. They take turns reordering the walnuts: Bushy chooses $1232$ walnuts to reorder, and Jumpy c... | [
"1234"
] | [
"Two squirrels, Bushy and Jumpy, have collected $2021$ walnuts. Jumpy numbers the walnuts from 1 through $2021$ and digs $2021$ holes in a circular pattern. Bushy places one walnut into each hole without regard to the numbering. They take turns reordering the walnuts: Bushy chooses $1232$ walnuts to reorder, and Ju... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-004bd29e8388fcae4068e5f59f838a8ffd9aa7b891b691cfbb65cf32bf867e87 | Two squirrels, Bushy and Jumpy, have collected 2021 walnuts for the winter. Jumpy numbers the walnuts from 1 through 2021 and digs 2021 little holes in a circular pattern in the ground around their favorite tree. The next morning, Jumpy notices that Bushy had placed one walnut into each hole, but had paid no attention ... | [
"10"
] | [
"Two squirrels, Bushy and Jumpy, have collected 2021 walnuts for the winter. Jumpy numbers the walnuts from 1 through 2021 and digs 2021 little holes in a circular pattern in the ground around their favorite tree. The next morning, Jumpy notices that Bushy had placed one walnut into each hole, but had paid no atten... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c6664b86087edc36a8b00a2b7f5cd36be4e1a75b889ad69e1176a2d38322e0c7 | Two standard 6-sided dice are rolled. What is the probability that the sum rolled is a perfect square? | [
"\\dfrac{7}{36}"
] | [
"Two standard 6-sided dice are rolled. What is the probability that the sum rolled is a perfect square?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-21eb5e7daffd3e88140fe0a3d93c481c904518b6c2b34ec714fe8317860166ed | Two standard 6-sided dice are tossed. What is the probability that the sum of the numbers shown on the dice is a prime number? Express your answer as a common fraction. | [
"\\frac{5}{12}"
] | [
"Two standard 6-sided dice are tossed. What is the probability that the sum of the numbers shown on the dice is a prime number? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d4d0ff8da14b3a63b27d7c7daf33111d5df152aa9bdf4b3191f8e7c8fb2f35ef | Two standard dice are rolled. What is the expected number of 1's obtained? Express your answer as a common fraction. | [
"\\frac{1}{3}"
] | [
"Two standard dice are rolled. What is the expected number of 1's obtained? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f10c619533f2263262343028556436597f8a2bbb0f164bb9a453d20a6d5756d7 | Two standard six-sided dice are rolled. On average, what is the product of the numbers showing on the two dice? | [
"12"
] | [
"Two standard six-sided dice are rolled. On average, what is the product of the numbers showing on the two dice?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-69fac4a95077490c0ff8f58512bcf52e81410779aa179d56ee6713ca1e0a3ebe | Two stations $A$ and $B$ are on a rectilinear highway passing from west to east, $B$ lying 9 km to the east of $A$. A car starts from $A$ and moves uniformly eastwards at a speed of 40 km/h. A motorcycle simultaneously starts from $B$ in the same direction and moves with a constant acceleration of 32 km/h$^2$. Determin... | [
"16 km"
] | [
"Two stations $A$ and $B$ are on a rectilinear highway passing from west to east, $B$ lying 9 km to the east of $A$. A car starts from $A$ and moves uniformly eastwards at a speed of 40 km/h. A motorcycle simultaneously starts from $B$ in the same direction and moves with a constant acceleration of 32 km/h$^2$. Det... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-67c7f1bee807306208ad282ac7c6a1bbb14a794d6eeb6a1992644b74e6898c1f | Two stations, P and R, are 20 km apart on a straight road. Station P sees a fire at an angle of 60 degrees with the road, while station R sees the fire at an angle of 45 degrees with the road. Determine which station should send its men and how far they will have to travel to reach the fire. | [
"14.6 km"
] | [
"Two stations, P and R, are 20 km apart on a straight road. Station P sees a fire at an angle of 60 degrees with the road, while station R sees the fire at an angle of 45 degrees with the road. Determine which station should send its men and how far they will have to travel to reach the fire."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 32
} | 32 | 0 | 0 |
omr-ws-838046cbcab0035d22b3a273752de4c0cc3ea9df4f0da7c9497c6b583f36e1fa | Two stock pens are built next to each other, with one side of each being against an existing wall. The width (perpendicular to the wall) is \( x \) meters. The total amount of fence material to be used is 100 meters. How wide will the pens be if the width is one-third the overall length? | [
"\\(\\frac{50}{3}\\)"
] | [
"Two stock pens are built next to each other, with one side of each being against an existing wall. The width (perpendicular to the wall) is \\( x \\) meters. The total amount of fence material to be used is 100 meters. How wide will the pens be if the width is one-third the overall length?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 18
} | 31 | 0 | 0 |
omr-ws-d9a049d86b69dd7881b6658a288c04ca08601f9a93336d1ade11926ccf98f2c2 | Two stock pens are built next to each other, with one side of each being against an existing wall. The width (perpendicular to the wall) is \( x \) meters. The total amount of fence material to be used is 100 meters. How wide will the pens be if they are 30 meters long overall? | [
"\\(\\frac{70}{3}\\)"
] | [
"Two stock pens are built next to each other, with one side of each being against an existing wall. The width (perpendicular to the wall) is \\( x \\) meters. The total amount of fence material to be used is 100 meters. How wide will the pens be if they are 30 meters long overall?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 23
} | 38 | 0 | 0 |
omr-ws-3d9583887d05992045acc70e14aa79a9deae8750c112554ca852ca4e98671ec0 | Two stores have warehouses with wheat. The first warehouse has 16 more tons of wheat than the second. Every night at midnight, the owner of each store steals a quarter of the wheat from the rival's warehouse and adds it to his own. After 10 days, the thieves are caught. Determine which warehouse has more wheat at this ... | [
"\\dfrac{1}{64}"
] | [
"Two stores have warehouses with wheat. The first warehouse has 16 more tons of wheat than the second. Every night at midnight, the owner of each store steals a quarter of the wheat from the rival's warehouse and adds it to his own. After 10 days, the thieves are caught. Determine which warehouse has more wheat at ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f6b6d0fc788f928918770269865566462203840a3fa71c46f5d20c54f9aed5dd | Two straight lines $a$ and $b$ are given, along with points $A$ and $B$. Point $X$ slides along line $a$, and point $Y$ slides along line $b$, such that $AX \parallel BY$. Find the locus of the intersection point of $AY$ with $XB$. | [
"\\text{A straight line}"
] | [
"Two straight lines $a$ and $b$ are given, along with points $A$ and $B$. Point $X$ slides along line $a$, and point $Y$ slides along line $b$, such that $AX \\parallel BY$. Find the locus of the intersection point of $AY$ with $XB$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 1
} | 12 | 0 | 0 |
omr-ws-f10bbc44dd51d10f80bff9eb47773144b59822312b64fe7bd7c62c9098ef1ad4 | Two straight lines \(L_1\) and \(L_2\) intersect at \(60^\circ\) at point \(O\). An equilateral triangle \(PQR\) is nestled between the lines with vertex \(P\) at \(O\). The triangle can be moved such that \(Q\) remains on \(L_1\) and \(P\) remains on \(L_2\). Suppose \(P\) moves along \(L_2\) until the triangle is aga... | [
"\\( y = \\sqrt{12 - 9x^2} \\)"
] | [
"Two straight lines \\(L_1\\) and \\(L_2\\) intersect at \\(60^\\circ\\) at point \\(O\\). An equilateral triangle \\(PQR\\) is nestled between the lines with vertex \\(P\\) at \\(O\\). The triangle can be moved such that \\(Q\\) remains on \\(L_1\\) and \\(P\\) remains on \\(L_2\\). Suppose \\(P\\) moves along \\(... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-754467ab1f6f2a71472451a91846902e43e4acda63ce56d002e9eddecb0f0fec | Two straight lines are drawn from a point \( O \) such that the angle between them is acute. Points \( A, B, \) and \( C \) are on one line such that \( OA > OB > OC \) and points \( R, S, \) and \( T \) are on the other line such that \( OR > OS > OT \). Calculate the value of \( \frac{AR + BS + CT}{BR + CS + AT} \) i... | [
"no_valid_answer_found"
] | [
"Two straight lines are drawn from a point \\( O \\) such that the angle between them is acute. Points \\( A, B, \\) and \\( C \\) are on one line such that \\( OA > OB > OC \\) and points \\( R, S, \\) and \\( T \\) are on the other line such that \\( OR > OS > OT \\). Calculate the value of \\( \\frac{AR + BS + C... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-65c211ef63837939aab04b85923462cc5801da3ab7f6f5a0faa247a1c7b79f64 | Two straight lines divide a square of side length 1 into four regions. Let $P$ be the maximum perimeter of any one of these regions. Find the minimum possible value of $P$. | [
"2"
] | [
"Two straight lines divide a square of side length 1 into four regions. Let $P$ be the maximum perimeter of any one of these regions. Find the minimum possible value of $P$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-16296bcd0c1aff77cd006ee3aff24529566d21360f72c0ee725c30461e191ec0 | Two straight lines intersect at an angle of $40^\circ$. A circle with center at point $O$ touches these lines. A line, different from the given ones, is tangent to the same circle and intersects the given lines at points $B$ and $C$. What can the angle $\angle BOC$ be equal to? | [
"110^\\circ"
] | [
"Two straight lines intersect at an angle of $40^\\circ$. A circle with center at point $O$ touches these lines. A line, different from the given ones, is tangent to the same circle and intersects the given lines at points $B$ and $C$. What can the angle $\\angle BOC$ be equal to?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-4142f823d25552bd1333aa7ec3e05f810cd67ba8ceb1db78e49bb643da5b4547 | Two straight lines intersect at point $O$ at an angle $a$. Points $A$, $B$, and $C$ are on one line in the order $A$, $B$, $C$, and $M$ is an arbitrary point on the other line, different from $O$. Let $\angle AMB = \gamma$ and $\angle BMC = \phi$. Consider the function $F(M) = \cot \gamma + \cot \phi$. Denote the small... | [
"\\tan^2 \\frac{a}{2}"
] | [
"Two straight lines intersect at point $O$ at an angle $a$. Points $A$, $B$, and $C$ are on one line in the order $A$, $B$, $C$, and $M$ is an arbitrary point on the other line, different from $O$. Let $\\angle AMB = \\gamma$ and $\\angle BMC = \\phi$. Consider the function $F(M) = \\cot \\gamma + \\cot \\phi$. Den... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 4
} | 7 | 0 | 0 |
omr-ws-2c784e98fd743e1a2733f2fcd3f4284bb55ca9528ae7680f87d4ac944f5cb828 | Two straight lines intersecting at an angle of $46^\circ$ are the axes of symmetry of the figure $F$ on the plane. What is the smallest number of axes of symmetry this figure can have? | [
"18"
] | [
"Two straight lines intersecting at an angle of $46^\\circ$ are the axes of symmetry of the figure $F$ on the plane. What is the smallest number of axes of symmetry this figure can have?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2ec16d4b2a4119e8365a890bdfa0c3c640d89e005c9215d90d84a6f4f4f62199 | Two straight lines, each of length 8 units, intersect at a vertex. The other two endpoints of these lines are connected to form a triangle. What is the highest possible area of this triangle? | [
"32"
] | [
"Two straight lines, each of length 8 units, intersect at a vertex. The other two endpoints of these lines are connected to form a triangle. What is the highest possible area of this triangle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1120d06a7c8b0c90d417b5746217c8352612c6059da8d1b98f45de8c21719378 | Two straight pipes (circular cylinders), with radii $1$ and $\frac{1}{4}$, lie parallel and in contact on a flat floor. What is the sum of the possible radii of a third parallel pipe lying on the same floor and in contact with both? | [
"\\(\\frac{10}{9}\\)"
] | [
"Two straight pipes (circular cylinders), with radii $1$ and $\\frac{1}{4}$, lie parallel and in contact on a flat floor. What is the sum of the possible radii of a third parallel pipe lying on the same floor and in contact with both?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 23
} | 38 | 0 | 0 |
omr-ws-657e6e95c76c48cb31299dd8e0d45fe07d2878131dfcb1923e859f95605b42df | Two straight roads intersect at right angles near the Krupps factory at Baden-Baden. Fabio drives his scooter towards the intersection at a rate of 50 kph. Ilsa drives her scooter on the other road away from the intersection at a speed of 30 kph. When Fabio is 2 kilometers from the intersection and Ilsa is 4 kilometers... | [
"\\( 2\\sqrt{5} \\) kph"
] | [
"Two straight roads intersect at right angles near the Krupps factory at Baden-Baden. Fabio drives his scooter towards the intersection at a rate of 50 kph. Ilsa drives her scooter on the other road away from the intersection at a speed of 30 kph. When Fabio is 2 kilometers from the intersection and Ilsa is 4 kilom... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 11
} | 12 | 0 | 0 |
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