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-7361539bf7b7c1a7190881c48822fb28ae71d9af7731dd79165d04664d1f0169 | A square and a circle intersect such 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 the answer in terms of $\pi$. | [
"\\(\\frac{3}{\\pi}\\)"
] | [
"A square and a circle intersect such 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 the answer in terms of $\\pi$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 26
} | 41 | 0 | 0 |
omr-ws-5670500fa51cf697dd18b7f7645d95f95b5fe266b5c27cc8cccdb63d4297dc49 | A square and a rectangle have the same perimeter and share a common corner. If the side length of the square is $s$ and the dimensions of the rectangle are $l$ and $w$, calculate the distance from the common corner to the intersection point of the diagonals of the rectangle along the diagonal of the square. Express you... | [
"\\dfrac{s\\sqrt{2}}{2}"
] | [
"A square and a rectangle have the same perimeter and share a common corner. If the side length of the square is $s$ and the dimensions of the rectangle are $l$ and $w$, calculate the distance from the common corner to the intersection point of the diagonals of the rectangle along the diagonal of the square. Expres... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6aa78ec3401d474de7f37d9fb22b51744a2f5028f640d4dc530b58dc93f3bc1e | A square and a regular hexagon are drawn inside a unit circle. What is the total area covered by the square and the hexagon? | [
"\\(\\frac{7-\\sqrt{3}}{2}\\)"
] | [
"A square and a regular hexagon are drawn inside a unit circle. What is the total area covered by the square and the hexagon?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 10
} | 10 | 0 | 0 |
omr-ws-9ca6403404b9b05e2d14df10b1f7383a9c40fc9a855525d5afb4505eec182ac7 | A square and a regular hexagon are inscribed in the same circle. If the side of the square measures $4\sqrt{2}$ cm, what is the measure of the apothem of the regular hexagon? | [
"\\( 2\\sqrt{3} \\)"
] | [
"A square and a regular hexagon are inscribed in the same circle. If the side of the square measures $4\\sqrt{2}$ cm, what is the measure of the apothem of the regular hexagon?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-596d3c045e482108325a4d650977fcaa3a3238f2d10d08c3869c4c1a0b1559a0 | A square and a regular hexagon have equal areas. The ratio of the perimeter of the square to the perimeter of the regular hexagon is $(\sqrt{k}(\sqrt[f]{w})):3$ where $k$, $f$, and $w$ are all positive integers each of which is less than $8 \frac{1}{2}$. Find the value of $k + f + w$. | [
"13"
] | [
"A square and a regular hexagon have equal areas. The ratio of the perimeter of the square to the perimeter of the regular hexagon is $(\\sqrt{k}(\\sqrt[f]{w})):3$ where $k$, $f$, and $w$ are all positive integers each of which is less than $8 \\frac{1}{2}$. Find the value of $k + f + w$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 7
} | 29 | 0 | 0 |
omr-ws-8fcc7e91574c4ab4663a2c82392ee921cc8f03cdfb49080bab63993050467803 | A square and a triangle have equal perimeters. The lengths of the three sides of the triangle are $6.1$ cm, $8.2$ cm and $9.7$ cm. What is the area of the square in square centimeters? | [
"36"
] | [
"A square and a triangle have equal perimeters. The lengths of the three sides of the triangle are $6.1$ cm, $8.2$ cm and $9.7$ cm. What is the area of the square in square centimeters?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7f40879b22fa904471e5bead7fd7959577c95b2175f03bafc9d8f65992eff73d | A square and an equilateral triangle are both inscribed in a circle of radius 1. One of the sides of the triangle is parallel to two sides of the square. What is the area of the region where they overlap? | [
"\\(\\frac{9\\sqrt{2}+2\\sqrt{6}-6\\sqrt{3}}{6}\\)"
] | [
"A square and an equilateral triangle are both inscribed in a circle of radius 1. One of the sides of the triangle is parallel to two sides of the square. What is the area of the region where they overlap?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 5,
"tir": 4
} | 14 | 0 | 0 |
omr-ws-d6628c0c064c91059740600fe85e171d4be5b5f477b9ed1ea0cfefa2619c52bd | A square and an equilateral triangle are inscribed in a circle. The seven vertices form a convex septagon $S$ inscribed in the circle (it might be a hexagon if two vertices coincide). For which positions of the triangle relative to the square does $S$ have the largest and smallest area, respectively? | [
"0^\\circ"
] | [
"A square and an equilateral triangle are inscribed in a circle. The seven vertices form a convex septagon $S$ inscribed in the circle (it might be a hexagon if two vertices coincide). For which positions of the triangle relative to the square does $S$ have the largest and smallest area, respectively?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-5e09fe6762791e10f2a16a0e9033daa4c2f47048a24d99500bc7f57711d3227e | A square and an equilateral triangle are inscribed in the same circle. The seven vertices form a convex heptagon \( S \) inscribed in the circle (which might be a hexagon if two vertices coincide). Determine the positions of the triangle relative to the square that maximize and minimize the area of \( S \). | [
"Minimum area: \\( s^2 \\left( \\frac{3}{4} + \\frac{\\sqrt{3}}{4} \\right) \\), Maximum area: \\( s^2 \\left( \\frac{\\sqrt{2}}{4} + \\frac{\\sqrt{6}}{4} + \\frac{1}{4} \\right) \\)"
] | [
"A square and an equilateral triangle are inscribed in the same circle. The seven vertices form a convex heptagon \\( S \\) inscribed in the circle (which might be a hexagon if two vertices coincide). Determine the positions of the triangle relative to the square that maximize and minimize the area of \\( S \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 3
} | 3 | 0 | 0 |
omr-ws-fd74e740e2ccf49b570eb9a70b6d3507b6592cb80b9272c0e88f6d46c815e2f7 | A square and an equilateral triangle have the same area. Let \( A \) be the area of the circle circumscribed around the square and \( B \) the area of the circle circumscribed around the triangle. Find \( \frac{A}{B} \). | [
"\\(\\frac{3\\sqrt{3}}{8}\\)"
] | [
"A square and an equilateral triangle have the same area. Let \\( A \\) be the area of the circle circumscribed around the square and \\( B \\) the area of the circle circumscribed around the triangle. Find \\( \\frac{A}{B} \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 26
} | 41 | 0 | 0 |
omr-ws-263d970890033abbd09f5356c44193ca3af0549b90619d69736eca9a365cad58 | A square and an equilateral triangle share a common side but lie in perpendicular planes. The center of the square is point A and the centroid of the triangle is point B. A new equilateral triangle ABC is drawn using segment AB. Compute the ratio of the area of triangle ABC to the area of the original equilateral trian... | [
"\\(\\frac{13}{36}\\)"
] | [
"A square and an equilateral triangle share a common side but lie in perpendicular planes. The center of the square is point A and the centroid of the triangle is point B. A new equilateral triangle ABC is drawn using segment AB. Compute the ratio of the area of triangle ABC to the area of the original equilateral ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-891491be05d33bc5521b1caff61856dc2a7b7df71c56fc3be6e1c995f991688f | A square and an equilateral triangle together have the property that the area of each is equal to the perimeter of the other. Find the area of the square. | [
"\\( 12\\sqrt[3]{4} \\)"
] | [
"A square and an equilateral triangle together have the property that the area of each is equal to the perimeter of the other. Find the area of the square."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-dcab1b738cb9ce26f5c549795b6205d64e8c747ea74796802a793a19696d52ea | A square array of dots with 10 rows and 10 columns is given. Each dot is colored either blue or red. There are 52 red dots, with 2 red dots at the corners and an additional 16 red dots on the edges. The remaining red dots are inside the array. There are 98 green line segments connecting adjacent dots of different color... | [
"37"
] | [
"A square array of dots with 10 rows and 10 columns is given. Each dot is colored either blue or red. There are 52 red dots, with 2 red dots at the corners and an additional 16 red dots on the edges. The remaining red dots are inside the array. There are 98 green line segments connecting adjacent dots of different ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-50fc58080611daad135aa34287663d65b0e6b2419e6938218341437c0bf39fa3 | A square array of dots with 10 rows and 10 columns is given. Each dot is colored either blue or red. Whenever two dots of the same color are adjacent in the same row or column, they are joined by a line segment of the same color as the dots. If they are adjacent but of different colors, they are joined by a green line ... | [
"37"
] | [
"A square array of dots with 10 rows and 10 columns is given. Each dot is colored either blue or red. Whenever two dots of the same color are adjacent in the same row or column, they are joined by a line segment of the same color as the dots. If they are adjacent but of different colors, they are joined by a green ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5cfec3d7ee439dc4a349c90dcb7a30c24922b9a6def6e4ae055564ba4ff8803a | A square array of dots with 7 rows and 7 columns is given. Each dot is colored either blue or red. Whenever two dots of the same color are adjacent in the same row or column, they are joined by a line segment of the same color as the dots. If they are adjacent but of different colors, they are joined by a purple line s... | [
"21"
] | [
"A square array of dots with 7 rows and 7 columns is given. Each dot is colored either blue or red. Whenever two dots of the same color are adjacent in the same row or column, they are joined by a line segment of the same color as the dots. If they are adjacent but of different colors, they are joined by a purple l... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-d45e974eedc69c0dbfa6c1038123daac8f2a9dc7a4c7932af111f0090b9a1af2 | A square billiard table with side length $1$ metre has a pocket at each corner. A ball is struck from one corner and hits the opposite wall at a distance of $\frac{19}{94}$ metres from the adjacent corner. If the ball keeps travelling, how many walls will it hit before it falls into a pocket? | [
"111"
] | [
"A square billiard table with side length $1$ metre has a pocket at each corner. A ball is struck from one corner and hits the opposite wall at a distance of $\\frac{19}{94}$ metres from the adjacent corner. If the ball keeps travelling, how many walls will it hit before it falls into a pocket?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 5,
"tir": 2
} | 20 | 0 | 0 |
omr-ws-ec5974e4d4b956aab9d164cb0e234a70f196d6bab812cadaebedb5fa575e2cc3 | A square board is dissected into $n^2$ rectangular cells by $n-1$ horizontal and $n-1$ vertical lines. The cells are painted in a chessboard pattern, with one diagonal consisting of $n$ black square cells. Let $A_b$ be the total area of all black cells and $A_w$ be the total area of all white cells. Find the minimum po... | [
"1"
] | [
"A square board is dissected into $n^2$ rectangular cells by $n-1$ horizontal and $n-1$ vertical lines. The cells are painted in a chessboard pattern, with one diagonal consisting of $n$ black square cells. Let $A_b$ be the total area of all black cells and $A_w$ be the total area of all white cells. Find the minim... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-0234780a205e1b63b01354cd3404021b9fa1e86571ccf1064d0c05b9e85559e0 | A square board is divided by lines parallel to the board sides ($7$ lines in each direction, not necessarily equidistant) into $64$ rectangles. Rectangles are colored into white and black in alternating order. Assume that for any pair of white and black rectangles the ratio between the area of the white rectangle and t... | [
"\\dfrac{5}{4}"
] | [
"A square board is divided by lines parallel to the board sides ($7$ lines in each direction, not necessarily equidistant) into $64$ rectangles. Rectangles are colored into white and black in alternating order. Assume that for any pair of white and black rectangles the ratio between the area of the white rectangle ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 8
} | 18 | 0 | 0 |
omr-ws-87b2468175394ef7a9499ca3c14147ac7238b3bf87a7fef451ef567e00ca4fed | A square board is divided by lines parallel to the board sides (7 lines in each direction, not necessarily equidistant) into 64 rectangles. Rectangles are colored into white and black in alternating order. Assume that for any pair of white and black rectangles, the ratio between the area of the white rectangle and the ... | [
"2"
] | [
"A square board is divided by lines parallel to the board sides (7 lines in each direction, not necessarily equidistant) into 64 rectangles. Rectangles are colored into white and black in alternating order. Assume that for any pair of white and black rectangles, the ratio between the area of the white rectangle and... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 1
} | 3 | 0 | 0 |
omr-ws-5c65adb733ed89653956732ea90579b37019bc361182807b2d39f7286723c04a | A square board of size $n \times n$ is filled with '0's and '1's. It is known that for each '0' there are at least $n$ '1's in the same row and column. Calculate the minimum possible number of '1's on the board for $n=10$. | [
"50"
] | [
"A square board of size $n \\times n$ is filled with '0's and '1's. It is known that for each '0' there are at least $n$ '1's in the same row and column. Calculate the minimum possible number of '1's on the board for $n=10$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-4b8153f8a254cdeb7ed98a45da7eb3351263b5381fe750e932f697bf04e168bb | A square board with side length 10 ft has pegs at each of its four corners. A ring with a radius of 1 ft is tossed onto the board. What is the probability that the ring encircles one of the pegs, assuming the center of the ring falls somewhere on the board? | [
"\\(\\frac{\\pi}{100}\\)"
] | [
"A square board with side length 10 ft has pegs at each of its four corners. A ring with a radius of 1 ft is tossed onto the board. What is the probability that the ring encircles one of the pegs, assuming the center of the ring falls somewhere on the board?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-423fb7eb74fadcfba5939e6cbe0b959a0b67cddf9a4543fe78ec2a1312d5f7e2 | A square box of chocolates is divided into 49 equal square cells, each containing either dark or white chocolate. At each move, Alex eats two chocolates of the same kind if they are in adjacent cells (by side or vertex). What is the maximal number of chocolates Alex can eat regardless of the distribution of chocolates ... | [
"32"
] | [
"A square box of chocolates is divided into 49 equal square cells, each containing either dark or white chocolate. At each move, Alex eats two chocolates of the same kind if they are in adjacent cells (by side or vertex). What is the maximal number of chocolates Alex can eat regardless of the distribution of chocol... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-92b7815ccdd807b59b4ac2e477f7d19fddd07239cd5d165dc8fb839aa43d59f6 | A square box of chocolates is divided into \(49\) equal square cells, each containing either dark or white chocolate. At each move, Alex eats two chocolates of the same kind if they are in adjacent cells (sharing a side or a vertex). What is the maximal number of chocolates Alex can eat regardless of the distribution o... | [
"32"
] | [
"A square box of chocolates is divided into \\(49\\) equal square cells, each containing either dark or white chocolate. At each move, Alex eats two chocolates of the same kind if they are in adjacent cells (sharing a side or a vertex). What is the maximal number of chocolates Alex can eat regardless of the distrib... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8cd64fbd0547dc2e093241b9665913dbb5aae9cacd370615cf34d91a046337f0 | A square box of size $8 \times 8$ is divided into square cells with side $1$, each containing a ball. Exactly one ball is radioactive. A detector can be used to check if the radioactive ball is in the "cross" formed by two rows intersecting at a given cell. What is the least number of such checks needed to find the rad... | [
"6"
] | [
"A square box of size $8 \\times 8$ is divided into square cells with side $1$, each containing a ball. Exactly one ball is radioactive. A detector can be used to check if the radioactive ball is in the \"cross\" formed by two rows intersecting at a given cell. What is the least number of such checks needed to find... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 5
} | 13 | 0 | 0 |
omr-ws-acf31930caa54349aa907cde4619420b09e78bbfaf19540caf5f6c9d1873d546 | A square box with side length 1997 is divided into square cells of side length 1, each containing a ball. Exactly one ball is radioactive. A detector can check any $2 \times 2$ square of cells to determine if it contains the radioactive ball. What is the least number of such checks needed to find the radioactive ball f... | [
"1000000"
] | [
"A square box with side length 1997 is divided into square cells of side length 1, each containing a ball. Exactly one ball is radioactive. A detector can check any $2 \\times 2$ square of cells to determine if it contains the radioactive ball. What is the least number of such checks needed to find the radioactive ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-22d11bf6473f5bbdbe8ca5b3bf0c8610d52e45b2f73401e665879849875bc2f5 | A square carpet of side length 9 feet is designed with one large shaded square and eight smaller, congruent shaded squares, as shown. [asy] draw((0,0)--(9,0)--(9,9)--(0,9)--(0,0)); fill((1,1)--(2,1)--(2,2)--(1,2)--cycle,gray(.8)); fill((4,1)--(5,1)--(5,2)--(4,2)--cycle,gray(.8)); fill((7,1)--(8,1)--(8,2)--(7,2)--cycle,... | [
"17"
] | [
"A square carpet of side length 9 feet is designed with one large shaded square and eight smaller, congruent shaded squares, as shown. [asy]\n\ndraw((0,0)--(9,0)--(9,9)--(0,9)--(0,0));\n\nfill((1,1)--(2,1)--(2,2)--(1,2)--cycle,gray(.8));\n\nfill((4,1)--(5,1)--(5,2)--(4,2)--cycle,gray(.8));\n\nfill((7,1)--(8,1)--(8,... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e1c240aecd0e30ba28453aa9deef1237b5bcb0cb4ccce89434b377ec70d9a210 | A square carrom board has pockets at the vertices and a striker initially placed at a corner. It is struck at an angle in the interval $(0,90)$ with one of the walls of the board. Assume elastic collisions, no friction, and a point-sized striker. a) Find the set of numbers $n$ such that there are exactly $n+1$ angles t... | [
"\\{n \\mid n + 2 \\text{ is a prime number}\\}"
] | [
"A square carrom board has pockets at the vertices and a striker initially placed at a corner. It is struck at an angle in the interval $(0,90)$ with one of the walls of the board. Assume elastic collisions, no friction, and a point-sized striker.\n\na) Find the set of numbers $n$ such that there are exactly $n+1$ ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-37c869411fe4f9ae9792905d081fdf1f5d7175f06dbc89432f080841a1c04ddd | A square carrom board has pockets at the vertices and a striker initially placed at a corner. It is struck at an angle in the interval $(0,90)$ with one of the walls of the board. Assume elastic collisions, no friction, and a point-sized striker. c) Find the number of angles at which the striker can be struck so that i... | [
"\\phi(n + 2)"
] | [
"A square carrom board has pockets at the vertices and a striker initially placed at a corner. It is struck at an angle in the interval $(0,90)$ with one of the walls of the board. Assume elastic collisions, no friction, and a point-sized striker.\n\nc) Find the number of angles at which the striker can be struck s... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-b040f982bd002acc12fac4b2b3f6d9e30d01319bdde249aa938c86a805d4319f | A square consists of \( 7 \times 7 \) squares. A beetle lives in each box. After the signal, all the beetles simultaneously crawl to other squares -- each to a new square that has a common corner (but not a common edge) with the previous home. It may happen that several beetles crawl to the same box. What is the smalle... | [
"7"
] | [
"A square consists of \\( 7 \\times 7 \\) squares. A beetle lives in each box. After the signal, all the beetles simultaneously crawl to other squares -- each to a new square that has a common corner (but not a common edge) with the previous home. It may happen that several beetles crawl to the same box. What is th... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 7
} | 10 | 0 | 0 |
omr-ws-10753c79d944cd457d0c07636ab670e476b1c991cec594d7574b0d176f82fb41 | A square dartboard $\{(x, y)| - 5 \le x \le 5, -5 \le y \le 5\}$ has the following scoring rules: (i) If a dart lands within one unit of the origin, the player gets $10$ points. (ii) If a dart lands between one and three units from the origin, the player gets points equal to the quadrant number. (iii) If a dart lands m... | [
"\\dfrac{9\\pi}{10}"
] | [
"A square dartboard $\\{(x, y)| - 5 \\le x \\le 5, -5 \\le y \\le 5\\}$ has the following scoring rules:\n(i) If a dart lands within one unit of the origin, the player gets $10$ points.\n(ii) If a dart lands between one and three units from the origin, the player gets points equal to the quadrant number.\n(iii) If ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-43c64deb3fc0bab2a2b47aa8c44d859e4ffb261530d7f5e5f9a24a1e5a517e71 | A square estate of $100 \times 100$ square miles is divided into ten thousand $1 \times 1$ square mile plots. Each of three sons is assigned a point inside the estate. A $1 \times 1$ square plot is bequeathed to the son whose assigned point is closest to the center of that plot. Let $N$ be the maximum number of disconn... | [
"100"
] | [
"A square estate of $100 \\times 100$ square miles is divided into ten thousand $1 \\times 1$ square mile plots. Each of three sons is assigned a point inside the estate. A $1 \\times 1$ square plot is bequeathed to the son whose assigned point is closest to the center of that plot. Let $N$ be the maximum number of... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 5
} | 7 | 0 | 0 |
omr-ws-64a3f4c7462414abe2c32b5d15eda236d3fe60874c97a55039feebfdb2e0ba5f | A square field with a side of $180$ m was sown with rye. On any plot of $40$ m $\times$ $100$ m, the sides of which are parallel to the sides of the field, at least $91\%$ of its area is sown. What is the smallest percentage of the field area that could have been sown with rye? | [
"100%"
] | [
"A square field with a side of $180$ m was sown with rye. On any plot of $40$ m $\\times$ $100$ m, the sides of which are parallel to the sides of the field, at least $91\\%$ of its area is sown. What is the smallest percentage of the field area that could have been sown with rye?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7392f6b08b2e32f917213b1288c67227a65cacedc01524793d044fcc9d1c20fe | A square field with side length 60 meters is to be divided into three regions of equal area, with each region sharing a well at vertex $B$. Points $M$ and $N$ are on sides $AD$ and $CD$, respectively. How far must points $M$ and $N$ be from vertex $D$ to ensure that each region has the same area? | [
"20 meters"
] | [
"A square field with side length 60 meters is to be divided into three regions of equal area, with each region sharing a well at vertex $B$. Points $M$ and $N$ are on sides $AD$ and $CD$, respectively. How far must points $M$ and $N$ be from vertex $D$ to ensure that each region has the same area?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 6
} | 18 | 0 | 0 |
omr-ws-5b4e9fd8e85d4ec326be60ea007bad1eb7c7d27c49539c7a9acbaa5ddbe531d2 | A square flag consists of a white background, two red strips of uniform width stretching from opposite corners, and a blue square in the area where the two red strips intersect. If the entire cross (both the red arms and the blue center) takes up 36% of the area of the flag, what percent of the area of the flag is blue... | [
"2"
] | [
"A square flag consists of a white background, two red strips of uniform width stretching from opposite corners, and a blue square in the area where the two red strips intersect. If the entire cross (both the red arms and the blue center) takes up 36% of the area of the flag, what percent of the area of the flag is... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 7,
"tir": 8
} | 25 | 0 | 0 |
omr-ws-0ce0d67f7621bdc2d36e02f216212248cafa88e9c8133795808e29bcdb19a9af | A square flag has a red cross of uniform width with a blue square in the center on a white background. The cross (both the red arms and the blue center) takes up 36% of the area of the flag. What percent of the area of the flag is blue? | [
"2"
] | [
"A square flag has a red cross of uniform width with a blue square in the center on a white background. The cross (both the red arms and the blue center) takes up 36% of the area of the flag. What percent of the area of the flag is blue?"
] | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-29820e7d555dae6ec5fb206be8c1b4a9c0d8d5f1e63d2d9f051deaa4639a7977 | A square formation of Army cadets, 50 feet on a side, marches forward at a constant pace. The mascot, a dog, starts at the center of the rear. The dog marches forward to the center of the front rank, then instantly turns around and returns to his starting position at the center of the rear. The cadets have marched 50 f... | [
"\\( 50(1+\\sqrt{2}) \\)"
] | [
"A square formation of Army cadets, 50 feet on a side, marches forward at a constant pace. The mascot, a dog, starts at the center of the rear. The dog marches forward to the center of the front rank, then instantly turns around and returns to his starting position at the center of the rear. The cadets have marched... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-998c8eb790b936c2f47d74f8b4b750c2064dc40f9124ba60df15341bdd526046 | A square formation of soldiers, each side 50 meters long, is walking with a constant speed. A dog starts at the bottom of the formation and runs to the top soldier, then runs back to the bottom soldier. By the time the dog returns to the bottom, the formation has advanced 50 meters. How far did the dog travel? | [
"\\(50 + 50\\sqrt{2}\\)"
] | [
"A square formation of soldiers, each side 50 meters long, is walking with a constant speed. A dog starts at the bottom of the formation and runs to the top soldier, then runs back to the bottom soldier. By the time the dog returns to the bottom, the formation has advanced 50 meters. How far did the dog travel?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 14
} | 36 | 0 | 0 |
omr-ws-ba708f7fc2d47c0d73ed2867837d99b0fe8a858c807b259398da4eb3d1e42259 | A square garden is divided into a grid of $25 \times 25$ equal small square plots to plant 25 different types of flowers. In each small square plot, exactly one type of flower is planted, and the types planted are symmetrical across the main diagonal of the garden (the line connecting the top-left corner to the bottom-... | [
"25"
] | [
"A square garden is divided into a grid of $25 \\times 25$ equal small square plots to plant 25 different types of flowers. In each small square plot, exactly one type of flower is planted, and the types planted are symmetrical across the main diagonal of the garden (the line connecting the top-left corner to the b... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-12bdec47ee6796420ea842bd3512986439b0c6ab0b99730f6e18a09b58965808 | A square grid $2n \times 2n$ is constructed using matches, where each match is a segment of length 1. Peter can make a move by selecting a vertex that is the endpoint of 3 or 4 matches and deleting two matches that form a segment of length 2. Determine the least possible number of matches that could remain after a seri... | [
"\\(2(2n-1)\\)"
] | [
"A square grid $2n \\times 2n$ is constructed using matches, where each match is a segment of length 1. Peter can make a move by selecting a vertex that is the endpoint of 3 or 4 matches and deleting two matches that form a segment of length 2. Determine the least possible number of matches that could remain after ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9c1e466debc9b9b9c31ce82f36697a268d8a72ae8612f28345e3b3ddd315a1eb | A square grid is composed of \( n^2 \equiv 1 \pmod{4} \) unit cells, each containing a locust. Each locust jumps the same number of cells either horizontally or vertically without leaving the grid. Let \( k \) be the number of cells each locust jumps. Determine the smallest possible value of \( k \) such that at least ... | [
"1"
] | [
"A square grid is composed of \\( n^2 \\equiv 1 \\pmod{4} \\) unit cells, each containing a locust. Each locust jumps the same number of cells either horizontally or vertically without leaving the grid. Let \\( k \\) be the number of cells each locust jumps. Determine the smallest possible value of \\( k \\) such t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-b5e349ea1609f52a6d43cd54e4265c01225fb891e721ddbcb7460cc955c43c9d | A square grid of side length 2023 is filled with non-overlapping dominos, leaving a single gap. You repeatedly move dominos to fill the gap (and create a new gap). No domino can be moved more than once. How many times can you move a domino at most? | [
"2046264"
] | [
"A square grid of side length 2023 is filled with non-overlapping dominos, leaving a single gap. You repeatedly move dominos to fill the gap (and create a new gap). No domino can be moved more than once. How many times can you move a domino at most?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-f06bea2ff3d7692b9f4803ae830ab1a0bac05a3cf500bfb8420850e3740500d1 | A square hangs in mid-air at a certain angle. Extending the sides of the square until they meet the ground at points P, A, G, and E in that order, it is found that PA=2 and GE=3. What is the area of the square? | [
"\\(\\frac{36}{13}\\)"
] | [
"A square hangs in mid-air at a certain angle. Extending the sides of the square until they meet the ground at points P, A, G, and E in that order, it is found that PA=2 and GE=3. What is the area of the square?"
] | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-39ac2f5748c43a62266c34c1aeb48a78b9f445d3b43ce4fa45fdcdf10c0a9225 | A square has a diagonal of length $10\sqrt{2}$ centimeters. What is the number of square centimeters in the area of the square? | [
"100"
] | [
"A square has a diagonal of length $10\\sqrt{2}$ centimeters. What is the number of square centimeters in the area of the square?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2298092f1e04c8517e478de3c3bca4a16fd05cf3afadc5d12015af0b2de4edc3 | A square has a side length of 10. Four smaller squares with side length 2 are drawn in each corner of the larger square. What is the maximum area of a square that can be drawn within the larger square, excluding the areas occupied by the four smaller squares? | [
"60"
] | [
"A square has a side length of 10. Four smaller squares with side length 2 are drawn in each corner of the larger square. What is the maximum area of a square that can be drawn within the larger square, excluding the areas occupied by the four smaller squares?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7e0467d52bbc4ca2165d009dc3e83bb58e2a5bd7cf52ba1afff3b67fa7bb9d25 | A square has all of its vertices on lattice points and has a side length of $\sqrt{p}$, where $p$ is a prime number. Find the number of lattice points that are contained within the square. | [
"\\( p - 1 \\)"
] | [
"A square has all of its vertices on lattice points and has a side length of $\\sqrt{p}$, where $p$ is a prime number. Find the number of lattice points that are contained within the square."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 11,
"tir": 5
} | 16 | 0 | 0 |
omr-ws-c89b886068caa885692cb3adafdd9ea5426618812f4af3f78716db339a6d9809 | A square has an area of $25.$ A rectangle has the same width as the square. The length of the rectangle is double its width. What is the area of the rectangle? | [
"50"
] | [
"A square has an area of $25.$ A rectangle has the same width as the square. The length of the rectangle is double its width. What is the area of the rectangle?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-54c947929b37350cff7b43390953240a1f522c045eac791f7e96165e183ef0a9 | A square has an area of $25x^2 + 20x + 4$. Find the restrictions on $x$ such that the side length of the square is positive. | [
"\\( x > -\\frac{2}{5} \\)"
] | [
"A square has an area of $25x^2 + 20x + 4$. Find the restrictions on $x$ such that the side length of the square is positive."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-cd675fc71fac69e238a08c324d786563d8c994de98b2e4f473bc3cc0f1b7eb0b | A square has area $\frac14$. What is the side length of the square? | [
"\\frac12"
] | [
"A square has area $\\frac14$. What is the side length of the square?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4567ae0a38bfece9dcfeb9dd7beab682d258299fe0ab4ef23ec2a9ac7eb48e90 | A square has been built on each side of a right-angled triangle with a hypotenuse of $4 \textrm{cm}$. The resulting figure can be inscribed in a circle. Find the area of the triangle in $\textrm{cm}^2$. | [
"4"
] | [
"A square has been built on each side of a right-angled triangle with a hypotenuse of $4 \\textrm{cm}$. The resulting figure can be inscribed in a circle. Find the area of the triangle in $\\textrm{cm}^2$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 2
} | 11 | 0 | 0 |
omr-ws-63ea8f3f8b0d90e108760b079999a670529f009d5d49b64eb8127fdc038f934a | A square has been divided into 2022 rectangles with no two of them having a common interior point. What is the maximal number of distinct lines that can be determined by the sides of these rectangles? | [
"2025"
] | [
"A square has been divided into 2022 rectangles with no two of them having a common interior point. What is the maximal number of distinct lines that can be determined by the sides of these rectangles?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1710b49d9600279680a0db63e8136d54764eee6c8bfa84cee36a985183b3072d | A square has consecutive vertices at \( A(3,3) \) and \( B(2,4) \). Determine the equation of the circumcircle of the square. | [
"(x - 2)^2 + (y - 3)^2 = 1"
] | [
"A square has consecutive vertices at \\( A(3,3) \\) and \\( B(2,4) \\). Determine the equation of the circumcircle of the square."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 6
} | 15 | 0 | 0 |
omr-ws-33be1e485198e5263f83f452ee5bec5637a089a62b3fef735118689757b45091 | A square has its base on the x-axis, and one vertex on each branch of the curve \( y = \frac{1}{x^2} \). What is the area of the square? | [
"\\( 2\\sqrt[3]{2} \\)"
] | [
"A square has its base on the x-axis, and one vertex on each branch of the curve \\( y = \\frac{1}{x^2} \\). What is the area of the square?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-524fed1863326402d4b3fc59266deec0bfeeee06850f8274d71ede8be7c9fd33 | A square has one side on the x-axis, and lines through its vertices make $45$ degree angles with the x-axis. If the y-coordinate of point P is 1998, determine the length of one side of the square. | [
"1332"
] | [
"A square has one side on the x-axis, and lines through its vertices make $45$ degree angles with the x-axis. If the y-coordinate of point P is 1998, determine the length of one side of the square."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 2
} | 5 | 0 | 0 |
omr-ws-a4fb03295285ac3d53bcb79ce111378dae9ab10a9f8de2474ce281cb793a0a49 | A square has sides of length $2$. Set $S$ is the set of all line segments that have length $2$ and whose endpoints are on adjacent sides of the square. The midpoints of the line segments in set $S$ enclose a region whose area to the nearest hundredth is $k$. Find $100k$. | [
"086"
] | [
"A square has sides of length $2$. Set $S$ is the set of all line segments that have length $2$ and whose endpoints are on adjacent sides of the square. The midpoints of the line segments in set $S$ enclose a region whose area to the nearest hundredth is $k$. Find $100k$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 5,
"tir": 1
} | 9 | 0 | 0 |
omr-ws-f1e3c9c12a3fd12af89cc30c8700ce7875cd588abc0dacee6048f9cfa5b0113f | A square has sides of length 10, and a circle centered at one of its vertices has radius 10. What is the area of the union of the regions enclosed by the square and the circle? | [
"\\( 100 + 75\\pi \\)"
] | [
"A square has sides of length 10, and a circle centered at one of its vertices has radius 10. What is the area of the union of the regions enclosed by the square and the circle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 31
} | 46 | 0 | 0 |
omr-ws-334940fc7e3f34a8d1f40435e2e2e8516e56b22b39caa155107c7a53a14ca987 | A square has sides of length 10, and a circle centered at one of its vertices has radius 10. What is the area of the union of the regions enclosed by the square and the circle? Express your answer in terms of $\pi$. | [
"100+75\\pi"
] | [
"A square has sides of length 10, and a circle centered at one of its vertices has radius 10. What is the area of the union of the regions enclosed by the square and the circle? Express your answer in terms of $\\pi$."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-aa6ede34a547d63c64bf974144dd90e72dfe187c85dd8f4a127a765025662003 | A square has sides of length 2. Let \( S \) be the set of all line segments that have length 2 and whose endpoints are on adjacent sides of the square. Let \( L \) be the set of the midpoints of all segments in \( S \). Find the area enclosed by \( L \). | [
"\\pi"
] | [
"A square has sides of length 2. Let \\( S \\) be the set of all line segments that have length 2 and whose endpoints are on adjacent sides of the square. Let \\( L \\) be the set of the midpoints of all segments in \\( S \\). Find the area enclosed by \\( L \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 7
} | 20 | 0 | 0 |
omr-ws-e68b66ef6a939f906e716e30ffca0a85f44edf4e158895a967a103db2765f699 | A square has sides of length 2. Set $\cal S$ is the set of all line segments that have length 2 and whose endpoints are on adjacent sides of the square. The midpoints of the line segments in set $\cal S$ enclose a region whose area to the nearest hundredth is $k$. Find $100k$. | [
"200"
] | [
"A square has sides of length 2. Set $\\cal S$ is the set of all line segments that have length 2 and whose endpoints are on adjacent sides of the square. The midpoints of the line segments in set $\\cal S$ enclose a region whose area to the nearest hundredth is $k$. Find $100k$."
] | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5ece637ec139f34e3b73c0c38d6db2f93bf062aa2138c9abc31f8fd4193f811d | A square has sides of length 2. Set \( S \) is the set of all line segments that have length 2 and whose endpoints are on adjacent sides of the square. The midpoints of the line segments in set \( S \) enclose a region whose area to the nearest hundredth is \( k \). Find \( 100k \). | [
"314",
"314.16"
] | [
"A square has sides of length 2. Set \\( S \\) is the set of all line segments that have length 2 and whose endpoints are on adjacent sides of the square. The midpoints of the line segments in set \\( S \\) enclose a region whose area to the nearest hundredth is \\( k \\). Find \\( 100k \\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 18,
"genselect": 1
} | 19 | 0 | 0 |
omr-ws-4dd15a27a5433fc8e9f35e41b64a6e5a706d176f20d9631cbf231466d8bc4727 | A square has sides of length 4 units each. A stripe of width 1 unit is drawn inside the square, centered on the diagonal. The area of this stripe can be expressed in the form \(a + \sqrt{b}\) square units, where \(a\) and \(b\) are rational numbers. What is the value of \(a + b\)? Express your answer as a decimal to th... | [
"31.5"
] | [
"A square has sides of length 4 units each. A stripe of width 1 unit is drawn inside the square, centered on the diagonal. The area of this stripe can be expressed in the form \\(a + \\sqrt{b}\\) square units, where \\(a\\) and \\(b\\) are rational numbers. What is the value of \\(a + b\\)? Express your answer as a... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 1,
"tir": 2
} | 9 | 0 | 0 |
omr-ws-cd96d5b6e5a447c36a1d51d28dcdefe30eaad96ab75749a38b01b812f0639e2a | A square has two diagonals, and a convex pentagon has five diagonals. How many diagonals does a convex decagon have? | [
"35"
] | [
"A square has two diagonals, and a convex pentagon has five diagonals. How many diagonals does a convex decagon have?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a3fa24783733da4387a832d596a4120139f0ab686cdb1c5727a0e330dee48748 | A square has two of its vertices on the $OX$ and $OY$ axes, respectively. The coordinates of another vertex are (13, 7). Find the coordinates of the other vertices. | [
"(7, -6)",
"(7, 0), (0, 6), (6, 13) or (-7, 0), (0, -20), (20, -13) or (-6, 0), (0, 13), (7, -6) or (-20, 0), (0, -13), (-7, 20)"
] | [
"A square has two of its vertices on the $OX$ and $OY$ axes, respectively. The coordinates of another vertex are (13, 7). Find the coordinates of the other vertices."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 9,
"tir": 15
} | 24 | 0 | 0 |
omr-ws-c029668022f9120e4557d4dfc4d9df9f59d1ef59e60ef7fc197822278c75275a | A square has vertices at $(0,0)$, $(12,0)$, $(0,15)$, and $(12,15)$. Lines of the form $y=k$ where $k$ is an integer between $1$ and $14$ inclusive, and lines of the form $x=w$ where $w$ is between $1$ and $11$ inclusive, are drawn. How many of the resulting small squares intersect a diagonal of the rectangle? | [
"46"
] | [
"A square has vertices at $(0,0)$, $(12,0)$, $(0,15)$, and $(12,15)$. Lines of the form $y=k$ where $k$ is an integer between $1$ and $14$ inclusive, and lines of the form $x=w$ where $w$ is between $1$ and $11$ inclusive, are drawn. How many of the resulting small squares intersect a diagonal of the rectangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 3,
"tir": 1
} | 8 | 0 | 0 |
omr-ws-22740fdd3668bf14440827678e45f2650ce9a64db08316b10b92c4ca055724a8 | A square hole of depth \( h \) with a base of length \( a \) is given. A dog is tied to the center of the square at the bottom of the hole by a rope of length \( L > \sqrt{2a^2 + h^2} \). The dog walks on the ground around the hole, and the edges of the hole are smooth so that the rope can freely slide along it. Find t... | [
"\\pi (L^2 - h^2) - a^2"
] | [
"A square hole of depth \\( h \\) with a base of length \\( a \\) is given. A dog is tied to the center of the square at the bottom of the hole by a rope of length \\( L > \\sqrt{2a^2 + h^2} \\). The dog walks on the ground around the hole, and the edges of the hole are smooth so that the rope can freely slide alon... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 8
} | 13 | 0 | 0 |
omr-ws-031bd88040c5132ca036d8f10acc958842aac5484f0ec0f90266cd23d5f75938 | A square house is partitioned into an $n \times n$ grid, where each cell is a room. All neighboring rooms have a door connecting them, and each door can either be normal or inversive. If USJL walks over an inversive door, he would become inverted-USJL, and vice versa. USJL must choose a room to begin and walk through e... | [
"0"
] | [
"A square house is partitioned into an $n \\times n$ grid, where each cell is a room. All neighboring rooms have a door connecting them, and each door can either be normal or inversive. If USJL walks over an inversive door, he would become inverted-USJL, and vice versa. USJL must choose a room to begin and walk thr... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 4
} | 8 | 0 | 0 |
omr-ws-347e1059dc0dd6c3981ba1e2d609a67abe6a7b585ab1a17e8a4b5345c6c0bbd9 | A square in the coordinate plane has vertices whose $y$-coordinates are $0$, $1$, $4$, and $5$. What is the area of the square? | [
"17"
] | [
"A square in the coordinate plane has vertices whose $y$-coordinates are $0$, $1$, $4$, and $5$. What is the area of the square?"
] | [
"aops_c5_contests_amp_programs"
] | [
"tir"
] | {
"tir": 4
} | 4 | 0 | 0 |
omr-ws-682217f6eeb8fab5b03a7e9214ac57cd47d2ce30b1b2f888410ad12e1b3d122a | A square is broken up into square cells so that it is $2010 \times 2010$. This square is divided into 3-celled L-shaped "corners." Mark one cell in each corner so that each row and each column has the same number of marked cells. Calculate the number of marked cells in each row. | [
"670"
] | [
"A square is broken up into square cells so that it is $2010 \\times 2010$. This square is divided into 3-celled L-shaped \"corners.\" Mark one cell in each corner so that each row and each column has the same number of marked cells. Calculate the number of marked cells in each row."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7cbc1a8780e62efbdaef9276bb24b82e04d802326ad51310d625241b72ad23a0 | A square is centered at the origin with its sides parallel to the axes. A projectile is aimed at the center of the square, and its impact point coordinates are independent standard normal random variables. If the probability that the projectile lands inside the square is 0.85, find the length of the side of the square. | [
"3.524"
] | [
"A square is centered at the origin with its sides parallel to the axes. A projectile is aimed at the center of the square, and its impact point coordinates are independent standard normal random variables. If the probability that the projectile lands inside the square is 0.85, find the length of the side of the sq... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 4
} | 18 | 0 | 0 |
omr-ws-f85d9f6d9f4ae29c90f83aead1fb0d13320f6813e502d601321a07959d22069f | A square is centered at the origin with sides parallel to the axes. A pulse of light is fired from the origin at an angle $\theta$ above the $x$-axis, where $0 \le \theta \le \frac{\pi}{2}$. The light beam travels through the plane and bounces off the sides of the square, which act like mirrors. For what angles $\theta... | [
"\\(\\theta = \\arctan\\left(\\frac{p}{q}\\right)\\) for integers \\(p\\) and \\(q\\)."
] | [
"A square is centered at the origin with sides parallel to the axes. A pulse of light is fired from the origin at an angle $\\theta$ above the $x$-axis, where $0 \\le \\theta \\le \\frac{\\pi}{2}$. The light beam travels through the plane and bounces off the sides of the square, which act like mirrors. For what ang... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 6,
"tir": 4
} | 25 | 0 | 0 |
omr-ws-79abaae67bc1a60962a060beff36a954075098bb002c06e01fcbd29ac024757f | A square is centered at the origin with sides parallel to the axes. A pulse of light is fired from the origin at an angle $\theta$ above the $x$-axis, where $0 \le \theta \le \frac{\pi}{2}$. The light beam travels through the plane and bounces off the sides of the square, which act like mirrors. The light beam obeys th... | [
"6"
] | [
"A square is centered at the origin with sides parallel to the axes. A pulse of light is fired from the origin at an angle $\\theta$ above the $x$-axis, where $0 \\le \\theta \\le \\frac{\\pi}{2}$. The light beam travels through the plane and bounces off the sides of the square, which act like mirrors. The light be... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-892c6f4d2d404ce52a56223a1a4e032c88b9d134deb7a84ff73933ddb1d61add | A square is constructed on the hypotenuse of a right triangle as on a side (outside the triangle). The center of the square is connected to the vertex of the right angle of the triangle. What segments is the hypotenuse divided into if the legs are 21 cm and 28 cm? | [
"20"
] | [
"A square is constructed on the hypotenuse of a right triangle as on a side (outside the triangle). The center of the square is connected to the vertex of the right angle of the triangle. What segments is the hypotenuse divided into if the legs are 21 cm and 28 cm?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-d4cce74ddb43ce46f93d69c3a63dcbee8d11f48aba5ae5a471b0a6177a7ce2c1 | A square is cut into disjoint rectangles. Any horizontal line, not passing through the sides of the rectangles, intersects exactly $n$ of them, and each vertical line, not passing through the sides of the rectangles, intersects exactly $m$ of them. Find the smallest possible number of the rectangles. | [
"$2m + 2n - 4$ if $m,n \\geq 2$; otherwise, $mn$."
] | [
"A square is cut into disjoint rectangles. Any horizontal line, not passing through the sides of the rectangles, intersects exactly $n$ of them, and each vertical line, not passing through the sides of the rectangles, intersects exactly $m$ of them. Find the smallest possible number of the rectangles."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-950f0316b3f51f7e0286fbc7df79f65a522b88b5bec794531bef2d730e8f9fea | A square is dissected into \( n \) triangles of equal area. If \( n \) is the smallest even number for which this is possible, find the sum of the areas of two such triangles when the side length of the square is 2 units. | [
"4"
] | [
"A square is dissected into \\( n \\) triangles of equal area. If \\( n \\) is the smallest even number for which this is possible, find the sum of the areas of two such triangles when the side length of the square is 2 units."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b314701716d3acb17e86055ffbf8cf06f80627c87e5fc510a36328f2bbd5093a | A square is dissected into smaller squares. Let $N$ be the total number of smaller squares. Find the smallest possible value of $N$ such that $N > 1988$. | [
"1989"
] | [
"A square is dissected into smaller squares. Let $N$ be the total number of smaller squares. Find the smallest possible value of $N$ such that $N > 1988$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-aeb39db72a31360a4d817c020b41307e613fc2d3404ed488a0a58871ad5b688d | A square is divided by horizontal and vertical lines that form $n^2$ squares each with side $1$. What is the greatest possible value of $n$ such that it is possible to select $n$ squares such that any rectangle with area $n$ formed by the horizontal and vertical lines would contain at least one of the selected $n$ squa... | [
"4"
] | [
"A square is divided by horizontal and vertical lines that form $n^2$ squares each with side $1$. What is the greatest possible value of $n$ such that it is possible to select $n$ squares such that any rectangle with area $n$ formed by the horizontal and vertical lines would contain at least one of the selected $n$... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-726dd76311846e977eb61f0758212b600e357912214d3939d1e4627b5f6977cd | A square is divided into $100$ rectangles by $9$ straight lines parallel to one of its sides and $9$ lines parallel to the other. If exactly $9$ of these rectangles are squares, what is the minimum number of these $9$ squares that must have the same size? | [
"2"
] | [
"A square is divided into $100$ rectangles by $9$ straight lines parallel to one of its sides and $9$ lines parallel to the other. If exactly $9$ of these rectangles are squares, what is the minimum number of these $9$ squares that must have the same size?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4a03ae7fed507690adf39ac966ba7ccf313a0fdac8590e3cc80009eeb1958e66 | A square is divided into $16$ equal squares, obtaining a set of $25$ different vertices. What is the least number of vertices one must remove from this set so that no $4$ points of the remaining set are the vertices of any square with sides parallel to the sides of the initial square? | [
"13"
] | [
"A square is divided into $16$ equal squares, obtaining a set of $25$ different vertices. What is the least number of vertices one must remove from this set so that no $4$ points of the remaining set are the vertices of any square with sides parallel to the sides of the initial square?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-562d913e757471886136dadebdf88a28f0136c15832e893af494050f505fdcf9 | A square is divided into $2010 \times 2010$ cells. This square is further divided into L-shaped "corners" consisting of 3 cells each. Mark one cell in each L-shaped corner such that each row and each column has the same number of marked cells. Calculate the number of marked cells in each row. | [
"670"
] | [
"A square is divided into $2010 \\times 2010$ cells. This square is further divided into L-shaped \"corners\" consisting of 3 cells each. Mark one cell in each L-shaped corner such that each row and each column has the same number of marked cells. Calculate the number of marked cells in each row."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-48f23ea2d895e8c76682e217c79429edcc3c28a4cd2c552576dec21cc7dc9873 | A square is divided into $n^2$ equal smaller squares, and the diagonals of each smaller square are drawn. Determine, in terms of $n$, the total number of isosceles right-angled triangles of all sizes. | [
"\\dfrac{2}{3}n(n+1)(2n+1)"
] | [
"A square is divided into $n^2$ equal smaller squares, and the diagonals of each smaller square are drawn. Determine, in terms of $n$, the total number of isosceles right-angled triangles of all sizes."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 7
} | 13 | 0 | 0 |
omr-ws-35c63108bf141d82b17920a745ddc7a382a926dbfba7b109ec197492e6a3be76 | A square is divided into $n^2$ smaller squares each with side length 1 by horizontal and vertical lines. What is the greatest possible value of $n$ such that it is possible to select $n$ squares such that any rectangle with area $n$ formed by the horizontal and vertical lines would contain at least one of the selected ... | [
"5"
] | [
"A square is divided into $n^2$ smaller squares each with side length 1 by horizontal and vertical lines. What is the greatest possible value of $n$ such that it is possible to select $n$ squares such that any rectangle with area $n$ formed by the horizontal and vertical lines would contain at least one of the sele... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 1,
"tir": 2
} | 7 | 0 | 0 |
omr-ws-e2f31450b771b97911c24bcb69c892bbef778fe3a999094430e65587890ca8d1 | A square is divided into $n^4$ fields like a chessboard. $n^3$ game pieces are placed on these squares, with at most one piece per square, and the same number of pieces in each row. The arrangement is symmetrical with respect to one of the diagonals of the square, called $d$. If $n$ is an odd integer, calculate the min... | [
"n^2"
] | [
"A square is divided into $n^4$ fields like a chessboard. $n^3$ game pieces are placed on these squares, with at most one piece per square, and the same number of pieces in each row. The arrangement is symmetrical with respect to one of the diagonals of the square, called $d$. If $n$ is an odd integer, calculate th... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-2e551761116491d3735091b49362d4601089c3731ddb5cd13f0462c9b2a3ed0d | A square is divided into $n^4$ fields like a chessboard. $n^3$ game pieces are placed on these squares, with at most one piece per square, and the same number of pieces in each row. The arrangement is symmetrical with respect to one of the diagonals of the square, called $d$. If $n$ is even, find the number of ways to ... | [
"0"
] | [
"A square is divided into $n^4$ fields like a chessboard. $n^3$ game pieces are placed on these squares, with at most one piece per square, and the same number of pieces in each row. The arrangement is symmetrical with respect to one of the diagonals of the square, called $d$. If $n$ is even, find the number of way... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ac367da03c552e33578b2c6fc8baf9b7b7e622aacc2c847aadfc991e0331c169 | A square is divided into 169 identical small squares, and each small square contains either a 0 or a 1. It is not allowed to have the arrangements 101, 111, or 1001 in any row or column. What is the maximum possible number of 1's in the table? | [
"69"
] | [
"A square is divided into 169 identical small squares, and each small square contains either a 0 or a 1. It is not allowed to have the arrangements 101, 111, or 1001 in any row or column. What is the maximum possible number of 1's in the table?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-45bf1db1dbe5cac7de7e02572240df20b1a433263da0f44e8503c13f59f6cb5a | A square is divided into 25 smaller equal squares by drawing lines parallel to its sides. What is the maximum number of diagonals that can be drawn in these smaller squares such that no two diagonals share a common point? | [
"13"
] | [
"A square is divided into 25 smaller equal squares by drawing lines parallel to its sides. What is the maximum number of diagonals that can be drawn in these smaller squares such that no two diagonals share a common point?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 8,
"tir": 1
} | 14 | 0 | 0 |
omr-ws-cb036ea1122e87fdb141911a6ba267756273c25367bc72f1d3ccfadb1efdcff2 | A square is divided into 25 unit squares by drawing lines parallel to the sides of the square. Some diagonals of unit squares are drawn such that no two diagonals share any points. What is the maximum number of diagonals that can be drawn with this property? | [
"16"
] | [
"A square is divided into 25 unit squares by drawing lines parallel to the sides of the square. Some diagonals of unit squares are drawn such that no two diagonals share any points. What is the maximum number of diagonals that can be drawn with this property?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-3ffd74337a28d88ac1e8a2cb64458277b37abd8ea15a6eaf27ae3afa894c7d80 | A square is divided into 4 equal rectangles and a smaller square. The area of the shaded square is 36, and the area of each of the equal rectangles is 720. Calculate the lengths of the sides of the rectangles. | [
"24 and 30"
] | [
"A square is divided into 4 equal rectangles and a smaller square. The area of the shaded square is 36, and the area of each of the equal rectangles is 720. Calculate the lengths of the sides of the rectangles."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 16
} | 19 | 0 | 0 |
omr-ws-849fc3ef86610395af2a5b30ab443866c700bc64528356f052bd3701720cfae8 | A square is divided into 9 congruent squares. The distinct integers \( a, b, c \) occupy the first row, \( d, e, f \) the second row, and \( g, h, i \) the third row (all 9 integers are distinct). The elements in each row, diagonal, and column have the same product \( N \). What is the least possible value of \( N \)? | [
"216"
] | [
"A square is divided into 9 congruent squares. The distinct integers \\( a, b, c \\) occupy the first row, \\( d, e, f \\) the second row, and \\( g, h, i \\) the third row (all 9 integers are distinct). The elements in each row, diagonal, and column have the same product \\( N \\). What is the least possible value... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 6
} | 20 | 0 | 0 |
omr-ws-2fbc6b0390011a10ea90b43091a0b9040d3db17c5672123617a7544f03d7f0ac | A square is divided into \( n \) polygons by lines. What is the largest possible sum of the internal angles of all these polygons? | [
"\\((n+1) \\times 180^\\circ \\times (n-1)\\)"
] | [
"A square is divided into \\( n \\) polygons by lines. What is the largest possible sum of the internal angles of all these polygons?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3422c5e97ddb15a4a36af25a6be1f9999e47a6d1e708c327db1ccf250195ad75 | A square is divided into \( n \) triangles of equal area. Find the smallest possible value of \( n \) such that this division is possible. | [
"2"
] | [
"A square is divided into \\( n \\) triangles of equal area. Find the smallest possible value of \\( n \\) such that this division is possible."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2f3397e108792453b37e066a0df93004679d076cff54f06c213350643c84c2b9 | A square is divided into a 100x100 grid. Let \(a_1, a_2, \ldots, a_{100}\) and \(b_1, b_2, \ldots, b_{100}\) be 200 distinct real numbers. In the grid cell at the \(i\)-th row and \(j\)-th column, the number \(a_i + b_j\) is written. Given that the product of the 100 numbers in every row is equal to 1, calculate the pr... | [
"-1"
] | [
"A square is divided into a 100x100 grid. Let \\(a_1, a_2, \\ldots, a_{100}\\) and \\(b_1, b_2, \\ldots, b_{100}\\) be 200 distinct real numbers. In the grid cell at the \\(i\\)-th row and \\(j\\)-th column, the number \\(a_i + b_j\\) is written. Given that the product of the 100 numbers in every row is equal to 1,... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8aa373b0ec7de9fa90840f3eba3e1aec68250aadfb0eb097d681eaac66ffa995 | A square is divided into a finite number of isosceles trapezoids, none of which are rectangles. Let $N$ be the minimum number of such trapezoids needed for such a division. Calculate $N$. | [
"4"
] | [
"A square is divided into a finite number of isosceles trapezoids, none of which are rectangles. Let $N$ be the minimum number of such trapezoids needed for such a division. Calculate $N$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 1
} | 8 | 0 | 0 |
omr-ws-abf470424a207d66651d7259b3ce34edb5e419fe584073ec23c9a453acd36b6e | A square is divided into congruent rectangles with sides of integer lengths. A rectangle is "good" if it has at least one point in common with a given diagonal of the square. If the side length of the square is 100 units and the side lengths of the congruent rectangles are 10 units by 20 units, calculate the total area... | [
"2000"
] | [
"A square is divided into congruent rectangles with sides of integer lengths. A rectangle is \"good\" if it has at least one point in common with a given diagonal of the square. If the side length of the square is 100 units and the side lengths of the congruent rectangles are 10 units by 20 units, calculate the tot... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-3bb5ee8eb3937b6d70d7a167203f27b0728a3c65684718f629dbfe8d2b365a4b | A square is divided into congruent rectangles with sides of integer lengths. A rectangle is considered important if it has at least one point in common with a given diagonal of the square. If the side length of the square is $12$ units and the side lengths of the congruent rectangles are $3$ units by $4$ units, calcula... | [
"72"
] | [
"A square is divided into congruent rectangles with sides of integer lengths. A rectangle is considered important if it has at least one point in common with a given diagonal of the square. If the side length of the square is $12$ units and the side lengths of the congruent rectangles are $3$ units by $4$ units, ca... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 1
} | 7 | 0 | 0 |
omr-ws-09acffdad430b166114367a4a28e7c1b6a5cc07cb749ed3f43af3c73fdd48dbc | A square is divided into congruent rectangles with sides of integer lengths. A rectangle is considered important if it has at least one point in common with a given diagonal of the square. If the side length of the square is 100 units and the side lengths of the congruent rectangles are 10 units by 20 units, calculate ... | [
"2000"
] | [
"A square is divided into congruent rectangles with sides of integer lengths. A rectangle is considered important if it has at least one point in common with a given diagonal of the square. If the side length of the square is 100 units and the side lengths of the congruent rectangles are 10 units by 20 units, calcu... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 1
} | 6 | 0 | 0 |
omr-ws-ac5fcb54d88d27addb08c201e9853313c006a8c6e4e0a5252d5c33b76893a700 | A square is divided into congruent rectangles with sides of integer lengths. Let \( S_1 \) be the set of rectangles that have at least one point in common with one diagonal of the square, and let \( S_2 \) be the set of rectangles that have at least one point in common with the other diagonal of the square. If the side... | [
"20"
] | [
"A square is divided into congruent rectangles with sides of integer lengths. Let \\( S_1 \\) be the set of rectangles that have at least one point in common with one diagonal of the square, and let \\( S_2 \\) be the set of rectangles that have at least one point in common with the other diagonal of the square. If... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-38fbcf3c0a31ae0970a70b52064ef35038b4550da3cf4c0de4b4f5866d5212ce | A square is divided into eight congruent triangles by the diagonals and the perpendicular bisectors of its sides. How many ways are there to color the triangles red and blue if two ways that are reflections or rotations of each other are considered the same? | [
"43"
] | [
"A square is divided into eight congruent triangles by the diagonals and the perpendicular bisectors of its sides. How many ways are there to color the triangles red and blue if two ways that are reflections or rotations of each other are considered the same?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-b278ad1b3281e26282bc350ad5a3c5eea430cec25d66a6e0bb577c7a046a18cc | A square is divided into four parts by two perpendicular lines such that three of these parts have areas equal to 1. If the side length of the square is $s$, find the value of $s^2$. | [
"4"
] | [
"A square is divided into four parts by two perpendicular lines such that three of these parts have areas equal to 1. If the side length of the square is $s$, find the value of $s^2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c6504adaf8c20b9751589e4ca308273f749b3839fbbd840a20b0149cebbce17e | A square is divided into four parts by two perpendicular lines. Three of these parts have areas of 1 each. Calculate the area of the fourth part. | [
"1"
] | [
"A square is divided into four parts by two perpendicular lines. Three of these parts have areas of 1 each. Calculate the area of the fourth part."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-7bd8cd425d05f6c60f2d480155a4ff2438c7af7716639ffb538eb3bd4ed4e3ec | A square is divided into four triangles by its diagonals. A circle is inscribed in each triangle, and then a square is inscribed in each circle. Compute the ratio of the total area of the smaller squares to that of the large square. | [
"\\( 4 \\left(3 - 2\\sqrt{2}\\right) \\)"
] | [
"A square is divided into four triangles by its diagonals. A circle is inscribed in each triangle, and then a square is inscribed in each circle. Compute the ratio of the total area of the smaller squares to that of the large square."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 4,
"tir": 21
} | 27 | 0 | 0 |
omr-ws-c5c0b4cbd7a23e0b0a06edeae60253141b4709745b7dcf6abf80a02d1d1989e8 | A square is divided into red and blue rectangles such that the sum of the areas of the red rectangles equals the sum of the areas of the blue rectangles. For each blue rectangle, write the ratio of the length of its vertical side to the length of its horizontal side. For each red rectangle, write the ratio of the lengt... | [
"\\(\\frac{5}{2}\\)"
] | [
"A square is divided into red and blue rectangles such that the sum of the areas of the red rectangles equals the sum of the areas of the blue rectangles. For each blue rectangle, write the ratio of the length of its vertical side to the length of its horizontal side. For each red rectangle, write the ratio of the ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 28,
"tir": 2
} | 30 | 0 | 0 |
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