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-b1c67b319c44b88c948b34bea837940687bc6aa1eb50076065ce97525bb66b37 | A square is divided into right triangles with legs of lengths $3$ and $4$. If the total number of triangles is $N$, find the remainder when $N$ is divided by $2$. | [
"0"
] | [
"A square is divided into right triangles with legs of lengths $3$ and $4$. If the total number of triangles is $N$, find the remainder when $N$ is divided by $2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d84f9498e43b2414b11d7685e548b5f0e7b5f9dd1652a062b00be49abb765a8a | A square is divided into several (greater than one) convex polygons with mutually different numbers of sides. Let \( n \) be the number of sides of the polygon with the fewest sides. Find the maximum possible value of \( n \). | [
"4"
] | [
"A square is divided into several (greater than one) convex polygons with mutually different numbers of sides. Let \\( n \\) be the number of sides of the polygon with the fewest sides. Find the maximum possible value of \\( n \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 2
} | 7 | 0 | 0 |
omr-ws-4ac0db16fbdc6343c046029898030593b7aee3b6aaf8566639441a4a267cbb86 | A square is divided into several convex polygons. If the total number of sides of these polygons is 32, find the maximum number of hexagons that can be among these polygons. | [
"4"
] | [
"A square is divided into several convex polygons. If the total number of sides of these polygons is 32, find the maximum number of hexagons that can be among these polygons."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-3ae63e4854b448470144e9733c2cca50c2e6d6d2326a6ecf73ce268ecd55b4f0 | A square is divided into smaller squares such that the number of smaller squares is greater than 5. Let \( f(n) \) be the minimum number of cuts required to divide the original square into exactly \( n \) smaller squares, where \( n > 5 \). Find \( f(10) \). | [
"6"
] | [
"A square is divided into smaller squares such that the number of smaller squares is greater than 5. Let \\( f(n) \\) be the minimum number of cuts required to divide the original square into exactly \\( n \\) smaller squares, where \\( n > 5 \\). Find \\( f(10) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 7
} | 11 | 0 | 0 |
omr-ws-0743087d337e83ed875baae46b2c3f5333b073823ceb8ae451d95a212ec074fb | A square is divided into some rectangles such that each rectangle has the same perimeter but different sides. Find the minimum number of such rectangles. | [
"4"
] | [
"A square is divided into some rectangles such that each rectangle has the same perimeter but different sides. Find the minimum number of such rectangles."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 2
} | 16 | 0 | 0 |
omr-ws-80169b3e839eeda0af0f4b9d9ac78e4d667f94d005d7a5d428d51e098f0989c9 | A square is divided into three congruent isosceles triangles and a shaded pentagon. What fraction of the square’s area is shaded? | [
"\\(\\frac{1}{4}\\)"
] | [
"A square is divided into three congruent isosceles triangles and a shaded pentagon. What fraction of the square’s area is shaded?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 4
} | 11 | 0 | 0 |
omr-ws-1bd2538fed0e323f39977c475cdff1b59ca4fde228905bc0f6e80703ed77aeff | A square is divided into three pieces by making two parallel cuts through opposite vertices. The three resulting pieces have equal areas. The middle piece is a parallelogram with a height of 10. What is the area of the entire square? | [
"1300"
] | [
"A square is divided into three pieces by making two parallel cuts through opposite vertices. The three resulting pieces have equal areas. The middle piece is a parallelogram with a height of 10. What is the area of the entire square?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 7,
"tir": 2
} | 19 | 0 | 0 |
omr-ws-fe0d620733c371df5f96a59a1f3e9ced342d486691b01736f7f61d5668075cdb | A square is divided into triangles using 2018 distinct points inside the square and the 4 vertices of the square, resulting in 2022 points in total. Each point is either a vertex of a triangle or lies completely outside the triangle. How many triangles are formed in this division? | [
"4038"
] | [
"A square is divided into triangles using 2018 distinct points inside the square and the 4 vertices of the square, resulting in 2022 points in total. Each point is either a vertex of a triangle or lies completely outside the triangle. How many triangles are formed in this division?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-1933fe1f6cf700aa61596d93d3eff084dd736a8e9d8ddaae870fb4880608ff21 | A square is divided into two congruent polygons. If the side length of the square is $s$, and the polygons share a boundary line segment of length $l$, find the value of $l$ when the center of the square lies on this boundary. Express your answer in terms of $s$. | [
"s\\sqrt{2}"
] | [
"A square is divided into two congruent polygons. If the side length of the square is $s$, and the polygons share a boundary line segment of length $l$, find the value of $l$ when the center of the square lies on this boundary. Express your answer in terms of $s$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-d715bdf05e323083f4929f517229d720ab829834574ee792db5d556c8cb5d832 | A square is divided into two congruent polygons. If the side length of the square is $s$, and the polygons share a boundary line segment that passes through the center of the square, find the length of this line segment in terms of $s$. | [
"s\\sqrt{2}"
] | [
"A square is divided into two congruent polygons. If the side length of the square is $s$, and the polygons share a boundary line segment that passes through the center of the square, find the length of this line segment in terms of $s$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-88cba280bf1ac54eff690d116d15523e4bc52bdbf1bb9087fd93e001a07ce0ca | A square is divided into two polygons along a straight line, and one of the two polygons is further divided into two polygons along a straight line. This process continues, with one of the resulting polygons being divided similarly each time. If after n divisions, we obtain a polygon with 47 - 46 sides, find the smalle... | [
"43"
] | [
"A square is divided into two polygons along a straight line, and one of the two polygons is further divided into two polygons along a straight line. This process continues, with one of the resulting polygons being divided similarly each time. If after n divisions, we obtain a polygon with 47 - 46 sides, find the s... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-151430f799a8b228c682b2490c3a606d6c861563888fcaf66ea3a0d6f1d1942f | A square is drawn inside a right triangle with sides 3, 4, and 5 such that one corner of the square touches the side of length 3, another corner touches the side of length 4, and the base of the square rests on the hypotenuse (side of length 5). What is the side length of the square? | [
"\\(\\frac{60}{37}\\)"
] | [
"A square is drawn inside a right triangle with sides 3, 4, and 5 such that one corner of the square touches the side of length 3, another corner touches the side of length 4, and the base of the square rests on the hypotenuse (side of length 5). What is the side length of the square?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 19
} | 34 | 0 | 0 |
omr-ws-d9d7184571e1f2edc3e27d34202ee40b2e10da23a9fb8ff737e6d60cb6bddfb4 | A square is drawn inside a right triangle with sides 3, 4, and 5 such that one corner of the square touches the side of length 3, another corner touches the side of length 4, and the base of the square rests on the hypotenuse. What is the side length of the square? | [
"\\dfrac{60}{37}"
] | [
"A square is drawn inside a right triangle with sides 3, 4, and 5 such that one corner of the square touches the side of length 3, another corner touches the side of length 4, and the base of the square rests on the hypotenuse. What is the side length of the square?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cddc420f5b7b655b5ca4cfed18a29b81b823cdf436e19e9a1b8ed8ea437146f5 | A square is drawn inside a triangle with sides 3, 4, and 5 such that one corner of the square touches the side of length 3, another corner touches the side of length 4, and the base of the square rests on the longest side of the triangle. What is the side length of the square? | [
"\\(\\frac{60}{37}\\)"
] | [
"A square is drawn inside a triangle with sides 3, 4, and 5 such that one corner of the square touches the side of length 3, another corner touches the side of length 4, and the base of the square rests on the longest side of the triangle. What is the side length of the square?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 39,
"tir": 20
} | 59 | 0 | 0 |
omr-ws-00e9838421d508ad17f483c8f91f6aacd2f58c9edc86317f0e02ff121d478aa0 | A square is drawn such that one of its sides coincides with the line $y = 5$, and so that the endpoints of this side lie on the parabola $y = x^2 + 3x + 2$. What is the area of the square? | [
"21"
] | [
"A square is drawn such that one of its sides coincides with the line $y = 5$, and so that the endpoints of this side lie on the parabola $y = x^2 + 3x + 2$. What is the area of the square?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4b87faa0bd84451cdc7b8aaf2f4c37076e51821c5e9f613fae29ebe02b1f3b19 | A square is formed by drawing 4 diagonals of a regular octagon. The edges of the square have length 1. What is the area of the octagon? | [
"\\sqrt{2}"
] | [
"A square is formed by drawing 4 diagonals of a regular octagon. The edges of the square have length 1. What is the area of the octagon?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a396bba809ad7d03f441b4e633909a84fc3e1c4ae1694f26ade90711aaf62d8f | A square is formed by four congruent rectangles and a smaller square inside. The gray rectangles each have a perimeter of 24 cm. Consider the cases where the measurement of the inner square side is an integer of centimeters. How many such integer measures are possible? | [
"5"
] | [
"A square is formed by four congruent rectangles and a smaller square inside. The gray rectangles each have a perimeter of 24 cm. Consider the cases where the measurement of the inner square side is an integer of centimeters. How many such integer measures are possible?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-27cd93ab6f3f6c7c4793836aedc278fe83159cfd1d82f8cb4fa5b44938adef93 | A square is formed by four congruent rectangles and a smaller square inside. The gray rectangles each have a perimeter of 24 cm. How much does the side of the largest square measure? | [
"12 cm"
] | [
"A square is formed by four congruent rectangles and a smaller square inside. The gray rectangles each have a perimeter of 24 cm. How much does the side of the largest square measure?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-8cdfa7df6289f92171d94de54a741cbf44c22260bdd12050fd36c8db36e71894 | A square is hanging in mid-air at a certain angle. By 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 is hanging in mid-air at a certain angle. By 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_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 2
} | 3 | 0 | 0 |
omr-ws-844985a7d61579fb2fc8b73d343e60ff96b4b02e70ed3b3f5397d77f6ff6db5c | A square is inscribed in a circle of radius \( R \). A circle is inscribed in this square, then a square in this circle, and so on \( n \) times. Find the limit of the sum of the areas of all the squares as \( n \) tends to infinity. | [
"\\(4R^2\\)"
] | [
"A square is inscribed in a circle of radius \\( R \\). A circle is inscribed in this square, then a square in this circle, and so on \\( n \\) times. Find the limit of the sum of the areas of all the squares as \\( n \\) tends to infinity."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 31
} | 45 | 0 | 0 |
omr-ws-7340c25ee2c79d108efa2a1af67efcfd7b0decea7aa81d4a061b4690464d4407 | A square is inscribed in a circle with radius 2 cm. Then a smaller circle is inscribed in the square, after which a smaller square is inscribed in this smaller circle. If this process is continued indefinitely, find the total area of all the squares produced in this manner. | [
"16"
] | [
"A square is inscribed in a circle with radius 2 cm. Then a smaller circle is inscribed in the square, after which a smaller square is inscribed in this smaller circle. If this process is continued indefinitely, find the total area of all the squares produced in this manner."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-baa812d84b97111854d8dd145c4234a69efa29b51d15294de8b603b1948109ea | A square is inscribed in a circle. Semicircles are constructed with diameters equal to the sides of the square inside the square. Four paired intersections of these semicircles form a "flower" shape. If the radius of the circumscribed circle is $r$, calculate the difference between the area of the circumscribed circle ... | [
"0"
] | [
"A square is inscribed in a circle. Semicircles are constructed with diameters equal to the sides of the square inside the square. Four paired intersections of these semicircles form a \"flower\" shape. If the radius of the circumscribed circle is $r$, calculate the difference between the area of the circumscribed ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-3f113c74fc2671000e4f075af57607ecd82483b0a76fe630c86956bba223c13d | A square is inscribed in a circular sector bounded by radii $OA$ and $OB$ with central angle $\alpha $ $(\alpha <\pi /2)$ so that two adjacent vertices lie on radius $OA$, the third vertex on the radius $OB$, and the fourth vertex on the arc $AB$. Find the ratio of the areas of the square and the sector. | [
"$2\\sin ^2 \\alpha /[\\alpha (1+\\sin 2\\alpha +\\sin ^2 \\alpha )]$, where $\\alpha $ is the measure of the angle $AOB$ in radians"
] | [
"A square is inscribed in a circular sector bounded by radii $OA$ and $OB$ with central angle $\\alpha $ $(\\alpha <\\pi /2)$ so that two adjacent vertices lie on radius $OA$, the third vertex on the radius $OB$, and the fourth vertex on the arc $AB$. Find the ratio of the areas of the square and the sector."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ea2a07483264f7dbf362c9784d280cfb26101d870f49a5d68bdb3adf0c4a0594 | A square is inscribed in a regular octagon such that the vertices of the square are the midpoints of the sides of the octagon. If a point is selected inside the octagon, what is the probability that it lies within the square? | [
"\\(\\frac{1+\\sqrt{2}}{4}\\)"
] | [
"A square is inscribed in a regular octagon such that the vertices of the square are the midpoints of the sides of the octagon. If a point is selected inside the octagon, what is the probability that it lies within the square?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 7,
"tir": 23
} | 45 | 0 | 0 |
omr-ws-185bbe4bc2bd439bd966560c2e2b069b17354bdb1db1917d91e79fa96ed8492a | A square is inscribed in a sector with a central angle of 60°. Find the ratio of the area of the square to the area of the sector. | [
"\\(\\dfrac{18(7-2\\sqrt{3})}{37\\pi}\\)"
] | [
"A square is inscribed in a sector with a central angle of 60°. Find the ratio of the area of the square to the area of the sector."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2f4a1e19b8c0f2c02902028b9ffe827940d6b5ec052f1a992b2b4240c887b9b4 | A square is inscribed in a triangle such that two vertices of the square lie on one side of the triangle and one vertex each on the other two sides. It is given that the center of the square coincides with the centroid (intersection of the medians) of the triangle. Find the angles of the triangle. | [
"45^\\circ"
] | [
"A square is inscribed in a triangle such that two vertices of the square lie on one side of the triangle and one vertex each on the other two sides. It is given that the center of the square coincides with the centroid (intersection of the medians) of the triangle. Find the angles of the triangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-a47209b38912fbbd3ef69f11ee52884ff7f105df1c3650eb75c17850eb8e5e1c | A square is inscribed in an ellipse with axes of length $x$ and $y$. Find the largest possible area of the square. | [
"\\(\\frac{4x^2 y^2}{x^2 + y^2}\\)"
] | [
"A square is inscribed in an ellipse with axes of length $x$ and $y$. Find the largest possible area of the square."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-4b747d4f3effef9e8b54304449ebdd709906ea4c4879736292fc3d5727491634 | A square is inscribed in an ellipse with the equation \(\frac{x^2}{16} + \frac{y^2}{9} = 1\). Let the area of the union of the square inscribed in the ellipse and the rhombus formed by connecting the endpoints of the minor axis and the major axis be \(a\). If \(a\) can be expressed as \(\frac{m}{n}\) where \(m\) and \(... | [
"721"
] | [
"A square is inscribed in an ellipse with the equation \\(\\frac{x^2}{16} + \\frac{y^2}{9} = 1\\). Let the area of the union of the square inscribed in the ellipse and the rhombus formed by connecting the endpoints of the minor axis and the major axis be \\(a\\). If \\(a\\) can be expressed as \\(\\frac{m}{n}\\) wh... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 4
} | 16 | 0 | 0 |
omr-ws-a70e0f22c7f9a453452d5614cac2e29d93707f904242d808b34476597493268d | A square is inscribed in another square such that each vertex of the smaller square lies on a side of the larger square. The area of the smaller square is \(\frac{25}{49}\) times the area of the larger square. Find the ratio in which each vertex of the smaller square divides each side of the larger square. | [
"\\(\\frac{3}{4}\\)"
] | [
"A square is inscribed in another square such that each vertex of the smaller square lies on a side of the larger square. The area of the smaller square is \\(\\frac{25}{49}\\) times the area of the larger square. Find the ratio in which each vertex of the smaller square divides each side of the larger square."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 46,
"genselect": 8,
"tir": 4
} | 58 | 0 | 0 |
omr-ws-94889fa106e77cca6e388b73a4284be629997aec1a5e06ac116c5bf4b0305e63 | A square is inscribed inside a circle with area $64\pi$. What is the area of the square? | [
"128"
] | [
"A square is inscribed inside a circle with area $64\\pi$. What is the area of the square?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-e4c8c82d46a9ba2e8a54bb61e6e67e646c62e6dd3f0a81d434065920f0389a75 | A square is inscribed into a circular sector, bounded by the radii $OA$ and $OB$, with central angle $\alpha $ $(\alpha <\pi /2)$, so that its two adjacent vertices lie on the radius $OA$, the third vertex lies on the radius $OB$ and the fourth vertex lies on the arc $AB$. Find the ratio of the areas of the square and ... | [
"$\\frac{2\\sin ^2 \\alpha }{\\alpha (1+\\sin 2\\alpha +\\sin ^2 \\alpha )}$"
] | [
"A square is inscribed into a circular sector, bounded by the radii $OA$ and $OB$, with central angle $\\alpha $ $(\\alpha <\\pi /2)$, so that its two adjacent vertices lie on the radius $OA$, the third vertex lies on the radius $OB$ and the fourth vertex lies on the arc $AB$. Find the ratio of the areas of the squ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9cfd1463526e091f9e58c1b492b06c2bfd529d97e340a9988f9ac6933862419e | A square is located on the plane, and a point is drawn with invisible ink. A person in special glasses sees the point. If you draw a line, then he will say whether the point lies on that line, and if it does not, then on which side of it lies the point. What is the least number of lines you need to draw to find out if ... | [
"2"
] | [
"A square is located on the plane, and a point is drawn with invisible ink. A person in special glasses sees the point. If you draw a line, then he will say whether the point lies on that line, and if it does not, then on which side of it lies the point. What is the least number of lines you need to draw to find ou... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-194f654bdcc1d5d18e464de192729d057a9c020ca40863119aad2d418d2c247c | A square is made up of four congruent rectangles and a smaller square inside. The rectangles have a perimeter of $24$ cm each. Consider the cases in which the length of the side of the interior square is an integer number of centimeters. How many such lengths are possible? | [
"5"
] | [
"A square is made up of four congruent rectangles and a smaller square inside. The rectangles have a perimeter of $24$ cm each. Consider the cases in which the length of the side of the interior square is an integer number of centimeters. How many such lengths are possible?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 2
} | 5 | 0 | 0 |
omr-ws-f71f43cc631855b6c32949c39c61d167873bdb87b0ee9b1d800ab862e02ad181 | A square is made up of four congruent rectangles and a smaller square inside. The rectangles have a perimeter of $24$ cm each. How long is the side of the larger square? | [
"12"
] | [
"A square is made up of four congruent rectangles and a smaller square inside. The rectangles have a perimeter of $24$ cm each. How long is the side of the larger square?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-42100c86fa5b2e0dede7d168ee4a8929e5a28a491a9d37775492b012f8d72b73 | A square is partitioned by 2 perpendicular lines into 4 regions, 3 of which have an area of 1. Calculate the area of the square. | [
"4"
] | [
"A square is partitioned by 2 perpendicular lines into 4 regions, 3 of which have an area of 1. Calculate the area of the square."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 2
} | 8 | 0 | 0 |
omr-ws-f44f1f7015be92be5c4dc674c83add279f766dcefca34f07b61a6de3329c5db5 | A square is partitioned into $n^2 \geq 4$ rectangles using $2(n-1)$ lines, with $n-1$ lines parallel to one side of the square and $n-1$ lines parallel to the other side. Let $S(n)$ be the maximum number of rectangles that can be chosen from this partition such that for each pair of chosen rectangles, one can be placed... | [
"9"
] | [
"A square is partitioned into $n^2 \\geq 4$ rectangles using $2(n-1)$ lines, with $n-1$ lines parallel to one side of the square and $n-1$ lines parallel to the other side. Let $S(n)$ be the maximum number of rectangles that can be chosen from this partition such that for each pair of chosen rectangles, one can be ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7f787c2fa4d7df016afae9a760bac6c324979068f2c30f836f5a0affc71f26af | A square is partitioned into \( n \) convex polygons. Find the maximum number of edges in the resulting figure. | [
"\\( 2n + 2 \\)"
] | [
"A square is partitioned into \\( n \\) convex polygons. Find the maximum number of edges in the resulting figure."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-410c4c5a45a6ee2506ff8ee5fe55bc862661d263f6fd256e18f2ea2c9de9e09d | A square is reshaped into a rectangle whose length is twice its width. What is the ratio of the area of the square to the area of the rectangle? | [
"9:8"
] | [
"A square is reshaped into a rectangle whose length is twice its width. What is the ratio of the area of the square to the area of the rectangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-68a4669317aec0957650701d36e8c898b6729bfbc5c44ec115690573f1af6eab | A square is subdivided into $K^2$ equal smaller squares. A broken line passes through the centers of all the smaller squares. Find the minimum number of links in this broken line. | [
"\\( K^2 - 1 \\)"
] | [
"A square is subdivided into $K^2$ equal smaller squares. A broken line passes through the centers of all the smaller squares. Find the minimum number of links in this broken line."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a899e31af3f0d176280edb4ffa038c7eea998703a0463901e17c666539cd55fa | A square is tiled by 24 congruent rectangles. Points A and B are the midpoints of a pair of opposite sides of the square. On each side of line AB, four rectangles are chosen at random and colored black. The square is then folded over line AB. Compute the probability that exactly one pair of black rectangles is coincide... | [
"\\(\\frac{224}{495}\\)"
] | [
"A square is tiled by 24 congruent rectangles. Points A and B are the midpoints of a pair of opposite sides of the square. On each side of line AB, four rectangles are chosen at random and colored black. The square is then folded over line AB. Compute the probability that exactly one pair of black rectangles is coi... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 28
} | 42 | 0 | 0 |
omr-ws-b83c938142ed48401fb8af9d7e2cb57cf7355844acac68889734617502f04657 | A square is tiled by 24 congruent rectangles. Points A and B are the midpoints of a pair of opposite sides of the square. On each side of line \(AB\), four rectangles are chosen at random and colored black. The square is then folded over line \(AB\). Compute the probability that exactly one pair of black rectangles is ... | [
"\\dfrac{224}{495}"
] | [
"A square is tiled by 24 congruent rectangles. Points A and B are the midpoints of a pair of opposite sides of the square. On each side of line \\(AB\\), four rectangles are chosen at random and colored black. The square is then folded over line \\(AB\\). Compute the probability that exactly one pair of black recta... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-beba66e4abb7b68aaaad0d6c1a70308d2a8daf6d94da9d563fdc8c4579b4b5a1 | A square is tiled by 24 congruent rectangles. Points \( A \) and \( B \) are the midpoints of a pair of opposite sides of the square. On each side of line \( AB \), four rectangles are chosen at random and colored black. The square is then folded over line \( AB \). Compute the probability that exactly one pair of blac... | [
"\\dfrac{224}{495}"
] | [
"A square is tiled by 24 congruent rectangles. Points \\( A \\) and \\( B \\) are the midpoints of a pair of opposite sides of the square. On each side of line \\( AB \\), four rectangles are chosen at random and colored black. The square is then folded over line \\( AB \\). Compute the probability that exactly one... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 45,
"genselect": 8
} | 53 | 0 | 0 |
omr-ws-c69efb1a6b06aadd5ce21a6432de24d92eb29b9e0af333a7c4d60e8a4fca559f | A square is triangulated in such a way that no three vertices are collinear. For every vertex (including vertices of the square), the number of sides issuing from it is counted. If it is possible for all these numbers to be even, find the minimum number of triangles in such a triangulation. If it is not possible, enter... | [
"8"
] | [
"A square is triangulated in such a way that no three vertices are collinear. For every vertex (including vertices of the square), the number of sides issuing from it is counted. If it is possible for all these numbers to be even, find the minimum number of triangles in such a triangulation. If it is not possible, ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-ec5a5c59a943edb83bceddd0015e91d082f83acc674357dc367ab8ce5659dde2 | A square matrix \( 2005 \times 2005 \) is filled with 1s and 0s. It is known that for each 1, the quantity of 1s in the same column or in the same row is at most 40%. What is the maximal number of 1s in the matrix? | [
"1608010"
] | [
"A square matrix \\( 2005 \\times 2005 \\) is filled with 1s and 0s. It is known that for each 1, the quantity of 1s in the same column or in the same row is at most 40%. What is the maximal number of 1s in the matrix?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 2
} | 10 | 0 | 0 |
omr-ws-945d746e983fa30a3224b771da5bf1a7161931052264477862164cf08358da40 | A square of a whole number has four equal digits at its end. What are these digits? | [
"These digits are four noughts"
] | [
"A square of a whole number has four equal digits at its end. What are these digits?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0e65114e84952786ee35ce4dfd5e52568ffa37196bbdacc2ed847575542695ce | A square of area $108$ is inscribed inside a square of area $144$. There are two segments, labeled $m$ and $n$. The value of $m$ can be expressed as $a + b \sqrt{c}$, where $a, b, c$ are positive integers and $c$ is square-free. Find $a+b+c$. | [
"11"
] | [
"A square of area $108$ is inscribed inside a square of area $144$. There are two segments, labeled $m$ and $n$. The value of $m$ can be expressed as $a + b \\sqrt{c}$, where $a, b, c$ are positive integers and $c$ is square-free. Find $a+b+c$."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 45,
"tir": 13
} | 58 | 0 | 0 |
omr-ws-9f984fefa8e38fbab365f7015bfedbbaf21833861754772d280fa5a9782d8876 | A square of area $2$ is inscribed in a square of area $3$, creating four congruent triangles. What is the ratio of the shorter leg to the longer leg in one of the shaded right triangles? | [
"\\(2 - \\sqrt{3}\\)"
] | [
"A square of area $2$ is inscribed in a square of area $3$, creating four congruent triangles. What is the ratio of the shorter leg to the longer leg in one of the shaded right triangles?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-326ecfa5cb7c53d5bec808107f8e538247507df262753db647caaa61e6f7e483 | A square of area $20$ cm$^2$ fits inside a semicircle. Find the area of the largest square that will fit inside the full circle. | [
"50"
] | [
"A square of area $20$ cm$^2$ fits inside a semicircle. Find the area of the largest square that will fit inside the full circle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 11
} | 26 | 0 | 0 |
omr-ws-aae34ecacf74e2aae9e7751f032737150ca74cbbca0ce872f53b4626fec6b83e | A square of area $40$ is inscribed in a semicircle. What is the area of the semicircle? | [
"25\\pi"
] | [
"A square of area $40$ is inscribed in a semicircle. What is the area of the semicircle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-55c1dab3de5ea5027e4ce0101d4cf5a66d3a6e711755792830650e574baa8671 | A square of area 40 is inscribed in a semicircle. What is the area of the semicircle? | [
"25π"
] | [
"A square of area 40 is inscribed in a semicircle. What is the area of the semicircle?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 31
} | 79 | 0 | 0 |
omr-ws-7d3d6fab766d046edf631debf5fa1714c41c51aa20416766e058d5c19bbe9277 | A square of area \(A\) is divided into 102 smaller squares, 101 of which have sides of length 1. Determine all possible values for \(A\). | [
"121"
] | [
"A square of area \\(A\\) is divided into 102 smaller squares, 101 of which have sides of length 1. Determine all possible values for \\(A\\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-97e8bf717c90f3d43d4be80c114e82e636c6e3ba5c129e1d6ec203c96a8483f8 | A square of dimensions \((n-1) \times (n-1)\) is divided into \((n-1)^2\) unit squares. Each of the \(n^2\) vertices of these squares is to be colored either red or blue. Find the number of different colorings such that each unit square has exactly 2 red vertices. (Two colorings are considered different if at least one... | [
"2^{n+1} - 2"
] | [
"A square of dimensions \\((n-1) \\times (n-1)\\) is divided into \\((n-1)^2\\) unit squares. Each of the \\(n^2\\) vertices of these squares is to be colored either red or blue. Find the number of different colorings such that each unit square has exactly 2 red vertices. (Two colorings are considered different if ... | [
"aops_c6_high_school_olympiads"
] | [
"genselect"
] | {
"genselect": 8
} | 8 | 0 | 0 |
omr-ws-231cc9fe5236f21e5329f8d1b7b440833b78a72c5e0336c909d09e6634905079 | A square of dimensions \((n-1) \times (n-1)\) is divided into \((n-1)^2\) unit squares. Each of the \(n^2\) vertices of these squares is to be colored red or blue. Find the number of different colorings such that each unit square has exactly two red vertices. (Two coloring schemes are regarded as different if at least ... | [
"2^{n+1} - 2"
] | [
"A square of dimensions \\((n-1) \\times (n-1)\\) is divided into \\((n-1)^2\\) unit squares. Each of the \\(n^2\\) vertices of these squares is to be colored red or blue. Find the number of different colorings such that each unit square has exactly two red vertices. (Two coloring schemes are regarded as different ... | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 7
} | 7 | 0 | 0 |
omr-ws-d491f08352414aa06daeaa511a2d327377e7055624868e5af35fa67992d31a65 | A square of perimeter 20 is inscribed in a square of perimeter 28. What is the greatest distance between a vertex of the inner square and a vertex of the outer square? | [
"\\(\\sqrt{65}\\)"
] | [
"A square of perimeter 20 is inscribed in a square of perimeter 28. What is the greatest distance between a vertex of the inner square and a vertex of the outer square?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 46,
"tir": 29
} | 75 | 0 | 0 |
omr-ws-1eccf818f976d5a13b76457c5e192e3ff7c875954106aefce321b9bf6908e2ab | A square of side $ n$ is divided into $ n^2$ unit squares, each of which is colored red, yellow, or green. Find the smallest value of $ n$ such that, for any coloring, there exist a row and a column having at least three equally colored unit squares. | [
"7"
] | [
"A square of side $ n$ is divided into $ n^2$ unit squares, each of which is colored red, yellow, or green. Find the smallest value of $ n$ such that, for any coloring, there exist a row and a column having at least three equally colored unit squares."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 18
} | 18 | 0 | 0 |
omr-ws-07372351ccec78bbe07a9d49c5a6411b976cde001c3970c428f966f4fe764af0 | A square of side $1$ is covered with $m^2$ rectangles. Let $P$ be the maximum perimeter of these rectangles. Calculate the minimum possible value of $mP$. | [
"4"
] | [
"A square of side $1$ is covered with $m^2$ rectangles. Let $P$ be the maximum perimeter of these rectangles. Calculate the minimum possible value of $mP$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cee619dcf47b8aedf0ebbab12aaa19e9e6185ab6ff7e061ba3fd9e8d1db68c98 | A square of side $1$ is divided into two parts by a straight line. Then one of those two parts is divided into two parts by another straight line, leaving three pieces altogether. This process continues, with each step dividing one piece into two using a straight line. After $1999$ such divisions, the square is divided... | [
"\\dfrac{1}{1024}"
] | [
"A square of side $1$ is divided into two parts by a straight line. Then one of those two parts is divided into two parts by another straight line, leaving three pieces altogether. This process continues, with each step dividing one piece into two using a straight line. After $1999$ such divisions, the square is di... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-2272b38a5f92cf233dd68f69c29d14b38a62b1dad0994b71cbda2485076d920f | A square of side $120$ cm has been divided into six rectangles of equal area (one of the which is shaded). Calculate the perimeter of the shaded rectangle, in cm. | [
"200"
] | [
"A square of side $120$ cm has been divided into six rectangles of equal area (one of the which is shaded). Calculate the perimeter of the shaded rectangle, in cm."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b387998ae19772f3ad3dc9a8a142343f1d8ddafe28380cf82b9ec489e8569301 | A square of side $2a$ lies in the first quadrant of the $xy$-plane such that two consecutive vertices are always on the $x$- and $y$-axes respectively. Let $P$ be a point within or on the boundary of the square. If $P$ describes a portion of a conic, find the equation of this conic in terms of $a$ when $P$ is the midpo... | [
"2 - a^2"
] | [
"A square of side $2a$ lies in the first quadrant of the $xy$-plane such that two consecutive vertices are always on the $x$- and $y$-axes respectively. Let $P$ be a point within or on the boundary of the square. If $P$ describes a portion of a conic, find the equation of this conic in terms of $a$ when $P$ is the ... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-67d9d7136e478819cad72b04a35d8f2010c968af8ab8f4e5fc6022ddc439be4a | A square of side $2a$ lies in the first quadrant of the $xy$-plane such that two consecutive vertices are always on the $x$-axis and $y$-axis, respectively. Find the locus of the midpoint of the square. | [
"\\( y = x \\)"
] | [
"A square of side $2a$ lies in the first quadrant of the $xy$-plane such that two consecutive vertices are always on the $x$-axis and $y$-axis, respectively. Find the locus of the midpoint of the square."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 5,
"tir": 8
} | 23 | 0 | 0 |
omr-ws-887047a69040739d48625b4a1ac3a12a9974ec93bdb58ac5447d063a9f986654 | A square of side $5$ is divided into $25$ unit squares. Let $A$ be the set of the $16$ interior points of the initial square which are vertices of the unit squares. What is the largest number of points of $A$ such that no three of them form an isosceles right triangle? | [
"6"
] | [
"A square of side $5$ is divided into $25$ unit squares. Let $A$ be the set of the $16$ interior points of the initial square which are vertices of the unit squares. What is the largest number of points of $A$ such that no three of them form an isosceles right triangle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 4
} | 15 | 0 | 0 |
omr-ws-9f390989f935146fdfba7f9189048c70a3c6fd191c7e4e091b20f3005448ae92 | A square of side 1 is decomposed into 9 equal squares of side 1/3, and the one in the center is painted black. Each of the remaining eight squares is further divided into nine equal squares, and the center square of each is painted black. This process is repeated 1000 times. Calculate the total area of the black region... | [
"1 - \\left( \\dfrac{8}{9} \\right)^{1000}"
] | [
"A square of side 1 is decomposed into 9 equal squares of side 1/3, and the one in the center is painted black. Each of the remaining eight squares is further divided into nine equal squares, and the center square of each is painted black. This process is repeated 1000 times. Calculate the total area of the black r... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7c58416a2c22f5fb9ef9181bdcacde2b542b4fedc8d6e5701359c0ab91794e93 | A square of side 2 inches has a semicircle with radius 1 inch centered at the midpoint of one edge, AB. A line XY parallel to AB is drawn with X on segment AD, intersecting the semicircle at points P and Q, where P is closer to X. At what height should XY be placed to minimize the shaded area? | [
"\\(\\frac{\\sqrt{3}}{2}\\)"
] | [
"A square of side 2 inches has a semicircle with radius 1 inch centered at the midpoint of one edge, AB. A line XY parallel to AB is drawn with X on segment AD, intersecting the semicircle at points P and Q, where P is closer to X. At what height should XY be placed to minimize the shaded area?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 1,
"tir": 3
} | 5 | 0 | 0 |
omr-ws-a00600cd9d8a9edaed442bcd4778d695d75402c3c77b1371c84e1c341b3fdeff | A square of side \( N = n^2 + 1 \), where \( n \in \mathbb{N}^* \), is partitioned into unit squares (of side 1), along \( N \) rows and \( N \) columns. The \( N^2 \) unit squares are colored using \( N \) colors, with \( N \) squares of each color. For a given coloring, let \( f(n) \) be the maximum number of colors ... | [
"n + 1"
] | [
"A square of side \\( N = n^2 + 1 \\), where \\( n \\in \\mathbb{N}^* \\), is partitioned into unit squares (of side 1), along \\( N \\) rows and \\( N \\) columns. The \\( N^2 \\) unit squares are colored using \\( N \\) colors, with \\( N \\) squares of each color. For a given coloring, let \\( f(n) \\) be the ma... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-27918ef3ec201d9228f95abcf7445f61533914369dbad0e11547a29138c6ab02 | A square of side \( a \) is inscribed in \( \triangle GFH \) with one side of the square along \( GH \). If \( FG = t \), find the value of \( a \) in terms of \( t \) and simplify the expression. What is the simplified form of \( a \) when \( t = \frac{2 \cos 18^\circ \cos^2 36^\circ}{\cos 6^\circ} \)? | [
"\\sin 6^\\circ"
] | [
"A square of side \\( a \\) is inscribed in \\( \\triangle GFH \\) with one side of the square along \\( GH \\). If \\( FG = t \\), find the value of \\( a \\) in terms of \\( t \\) and simplify the expression. What is the simplified form of \\( a \\) when \\( t = \\frac{2 \\cos 18^\\circ \\cos^2 36^\\circ}{\\cos 6... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fadcfdd6c5021c190a3574ae2bd7fb5c04265a96f9cdbc8affa0e19f6af9e73f | A square of side length $5$ is inscribed in a square of side length $7$. If we construct a grid of $1 \times 1$ squares for both squares, we find that the two grids have $8$ lattice points in common. If we do the same construction by inscribing a square of side length $1489$ in a square of side length $2009$, and const... | [
"8"
] | [
"A square of side length $5$ is inscribed in a square of side length $7$. If we construct a grid of $1 \\times 1$ squares for both squares, we find that the two grids have $8$ lattice points in common. If we do the same construction by inscribing a square of side length $1489$ in a square of side length $2009$, and... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 5
} | 16 | 0 | 0 |
omr-ws-3022b905531fdbefaee0d3780c33536dbb9b03adfa085f14d0acc79bb5869444 | A square of side length $\frac{1}{\pi}$ is being rolled along a flat surface. The distance traveled by a vertex in one revolution of the square can be expressed in the form $\frac{(a+\sqrt{b})\pi}{c}$, such that $a$ and $c$ are relatively prime and $b$ is not the square of any prime. Find $a+b+c$. | [
"9"
] | [
"A square of side length $\\frac{1}{\\pi}$ is being rolled along a flat surface. The distance traveled by a vertex in one revolution of the square can be expressed in the form $\\frac{(a+\\sqrt{b})\\pi}{c}$, such that $a$ and $c$ are relatively prime and $b$ is not the square of any prime. Find $a+b+c$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-ba8d4fd6c575086466f052af741efcf87724ce1364b2df98c54a595957925bfe | A square of side length 1 and a circle of radius $\dfrac{\sqrt{3}}{3}$ share the same center. What is the area inside the circle but outside the square? | [
"\\(\\dfrac{2\\pi}{9} - \\dfrac{\\sqrt{3}}{3}\\)"
] | [
"A square of side length 1 and a circle of radius $\\dfrac{\\sqrt{3}}{3}$ share the same center. What is the area inside the circle but outside the square?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 46,
"tir": 20
} | 66 | 0 | 0 |
omr-ws-f5898caebd9f0a8b0d40b084480104a61567f965c884f580f89266e27596fcce | A square of side length 1 and a circle of radius $\frac{\sqrt{3}}{3}$ share the same center. What is the area inside the circle but outside the square? | [
"\\dfrac{2\\pi - 3\\sqrt{3}}{9}"
] | [
"A square of side length 1 and a circle of radius $\\frac{\\sqrt{3}}{3}$ share the same center. What is the area inside the circle but outside the square?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-76d07b125b2403fefa71cfd6a35b95321e1e2c0a0475f6f6dd275957b0379f8a | A square of side length 1 and a circle of radius $\sqrt{3}/3$ share the same center. What is the area inside the circle but outside the square? | [
"\\(\\frac{2\\pi}{9} - \\frac{\\sqrt{3}}{3}\\)"
] | [
"A square of side length 1 and a circle of radius $\\sqrt{3}/3$ share the same center. What is the area inside the circle but outside the square?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 6,
"tir": 26
} | 46 | 0 | 0 |
omr-ws-a7771b31bc041e93bdfdeb4e827b46f502fd4a68044220abe719b4f1b7948217 | A square of side length 1 and a circle of radius \(\frac{\sqrt{3}}{3}\) share the same center. What is the area inside the circle but outside the square? | [
"\\dfrac{2\\pi}{9} - \\dfrac{\\sqrt{3}}{3}"
] | [
"A square of side length 1 and a circle of radius \\(\\frac{\\sqrt{3}}{3}\\) share the same center. What is the area inside the circle but outside the square?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-aada0fe7e428f94344ded497439efebb7a5521a4d5b73294ea46e142dc7f942c | A square of side length 10 cm is rolled along a line in a clockwise direction until a corner point P returns to the line for the first time. Determine the area under the curve traced by point P to the nearest square centimeter. | [
"414"
] | [
"A square of side length 10 cm is rolled along a line in a clockwise direction until a corner point P returns to the line for the first time. Determine the area under the curve traced by point P to the nearest square centimeter."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-4de0b89b1d4d16c909cb4589a94b12ffc8d10ff55dcc95e5af7f2b93df1333a9 | A square of side length 2 has semicircles drawn on each of its sides. An elastic band is stretched tightly around the square and the 4 semicircles. What is the length of the elastic band in this stretched position? | [
"\\(4\\sqrt{2} + 2\\pi\\)"
] | [
"A square of side length 2 has semicircles drawn on each of its sides. An elastic band is stretched tightly around the square and the 4 semicircles. What is the length of the elastic band in this stretched position?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 1
} | 2 | 0 | 0 |
omr-ws-95d17fd634acc27deae96d4ed2f4689705caeb9ab7bc70734a1c33d51cf0984f | A square of side length 2 has semicircles drawn on each side. An “elastic band” is stretched tightly around the figure. What is the length of the elastic band in this position? | [
"4\\pi"
] | [
"A square of side length 2 has semicircles drawn on each side. An “elastic band” is stretched tightly around the figure. What is the length of the elastic band in this position?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a194eeca846808b557303b50e82fc57ab92b5db42205d77f81a5b378845902ac | A square of side length 2 rotates 45 degrees about its center. Find the area of the region that it travels over. | [
"\\(\\pi + 8 - 4\\sqrt{2}\\)"
] | [
"A square of side length 2 rotates 45 degrees about its center. Find the area of the region that it travels over."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 6
} | 14 | 0 | 0 |
omr-ws-ca37ba9dd947940a270ba99dba2ebdf6a8ead26053d30aed440e636f06b52b6f | A square of side length 5 begins nestled in a corner of a square of side length 8. The small and large square share corner \( P \), and \( Q \) is the opposite corner of the small square. The small square begins to roll around inside the large square, pivoting on its corners as it hits the sides of the large square. Ev... | [
"d\\sqrt{2}"
] | [
"A square of side length 5 begins nestled in a corner of a square of side length 8. The small and large square share corner \\( P \\), and \\( Q \\) is the opposite corner of the small square. The small square begins to roll around inside the large square, pivoting on its corners as it hits the sides of the large s... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 2
} | 12 | 0 | 0 |
omr-ws-78eed64530d2b845b8898cfe7ed7e3010efe10565e2f1578e5cc4aa2e46fe909 | A square of side length 8 has its corners cut off to make a regular octagon. Find the area of the octagon to the nearest whole number. | [
"53"
] | [
"A square of side length 8 has its corners cut off to make a regular octagon. Find the area of the octagon to the nearest whole number."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 22
} | 54 | 0 | 0 |
omr-ws-970991d2e1b33c15bba6eee4c0336d025fca16fba5b00525115bbadac1ceaba1 | A square of side length \( a \) is drawn. Four identical circles, each with radius \( a \) and centered at the vertices of the square, are also drawn. Find the area of each region formed by the intersection of the square and the circles. | [
"\\dfrac{\\pi a^2}{4}"
] | [
"A square of side length \\( a \\) is drawn. Four identical circles, each with radius \\( a \\) and centered at the vertices of the square, are also drawn. Find the area of each region formed by the intersection of the square and the circles."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 4
} | 12 | 0 | 0 |
omr-ws-15ba20565fde104360b666160092399544e3188532dc0535781f4f2c89154716 | A square of side length \( a \) is drawn. Four identical circles, each with radius \( a \) and centered at the vertices of the square, are also drawn. How many regions does the square get divided into? | [
"9"
] | [
"A square of side length \\( a \\) is drawn. Four identical circles, each with radius \\( a \\) and centered at the vertices of the square, are also drawn. How many regions does the square get divided into?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-29731a6fb12c729ba3b5e3c277bfc25042e5738e78cd607cc83db15bf7cd793b | A square of side length \( a \) is drawn. Four identical circles, each with radius \( a \) and centered at the vertices of the square, are drawn. Find the area of each region formed by the intersection of the square and the circles. | [
"\\dfrac{\\pi a^2}{4}"
] | [
"A square of side length \\( a \\) is drawn. Four identical circles, each with radius \\( a \\) and centered at the vertices of the square, are drawn. Find the area of each region formed by the intersection of the square and the circles."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 5
} | 8 | 0 | 0 |
omr-ws-3fc39bdb0e332f1be75466fda096e845134a51299c896d8bbd2491ff4e5208c6 | A square of side length \( a \) is drawn. Four identical circles, each with radius \( a \) and centered at the vertices of the square, are drawn. How many regions does the square get divided into? | [
"9"
] | [
"A square of side length \\( a \\) is drawn. Four identical circles, each with radius \\( a \\) and centered at the vertices of the square, are drawn. How many regions does the square get divided into?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 8,
"tir": 1
} | 18 | 0 | 0 |
omr-ws-8520ae4ed4579e29e52a70fd119ef93cd563abd2976f08dcf9ebf5b16b3184bd | A square of side length \( s \) has three circles of equal radius \( r \) inscribed inside it. What is the area of the square that is not occupied by the circles? | [
"\\( s^2 - 3\\pi r^2 \\)"
] | [
"A square of side length \\( s \\) has three circles of equal radius \\( r \\) inscribed inside it. What is the area of the square that is not occupied by the circles?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 1,
"tir": 4
} | 10 | 0 | 0 |
omr-ws-a7ea681eec1361b798916628df6f4516abe5506483956ed95f702c36ac9e8784 | A square of side length \( s \) is inscribed in a semicircle of radius \( r \). Compute \( \frac{s}{r} \). | [
"\\(\\frac{2\\sqrt{5}}{5}\\)"
] | [
"A square of side length \\( s \\) is inscribed in a semicircle of radius \\( r \\). Compute \\( \\frac{s}{r} \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-52ad4f1a16c8d5deb765ef1413c038e33fcc1d074fc21132bc24d6e46192784a | A square of side length \( s \) is inscribed symmetrically inside a sector of a circle with radius \( r \) and central angle \( x \), such that two vertices lie on the straight sides of the sector and two vertices lie on the circular arc of the sector. Determine the exact value of \( \frac{s}{r} \). | [
"\\(\\frac{2 \\sin \\frac{x}{2}}{\\sqrt{1 + 4 \\sin \\frac{x}{2} \\cos \\frac{x}{2} + 4 \\sin^2 \\frac{x}{2}}}\\)"
] | [
"A square of side length \\( s \\) is inscribed symmetrically inside a sector of a circle with radius \\( r \\) and central angle \\( x \\), such that two vertices lie on the straight sides of the sector and two vertices lie on the circular arc of the sector. Determine the exact value of \\( \\frac{s}{r} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 1
} | 2 | 0 | 0 |
omr-ws-7230eb2b618a72593ab0074d5187927f55bb79efb94cda767fc67066b14eab3a | A square of side length \( s \) lies in a plane perpendicular to a line \( L \). One vertex of the square lies on \( L \). As this square moves a distance \( h \) along \( L \), the square turns one revolution about \( L \) to generate a corkscrew-like column with square cross sections. Find the volume of the column. | [
"\\( s^2 h \\)"
] | [
"A square of side length \\( s \\) lies in a plane perpendicular to a line \\( L \\). One vertex of the square lies on \\( L \\). As this square moves a distance \\( h \\) along \\( L \\), the square turns one revolution about \\( L \\) to generate a corkscrew-like column with square cross sections. Find the volume... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 11
} | 25 | 0 | 0 |
omr-ws-5533c8e84ca2b55f7ad358771c6b028bc005f2dd795a654f97dd8c3768b66744 | A square of side length \( s \) lies in a plane perpendicular to a line \( L \). One vertex of the square lies on \( L \). As this square moves a distance \( h \) along \( L \), the square turns two revolutions about \( L \) to generate a corkscrew-like column with square cross sections. Find the volume of the column. | [
"\\( s^2 h \\)"
] | [
"A square of side length \\( s \\) lies in a plane perpendicular to a line \\( L \\). One vertex of the square lies on \\( L \\). As this square moves a distance \\( h \\) along \\( L \\), the square turns two revolutions about \\( L \\) to generate a corkscrew-like column with square cross sections. Find the volum... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 8
} | 22 | 0 | 0 |
omr-ws-756961402d72531274de0c839c984776a6bdea8eb8f279459b42736c318d32d4 | A square of side length \(2^n\) is divided into unit squares. One of the unit squares is deleted. If the remaining area can be perfectly tiled with \(L\)-trominos, calculate the total number of \(L\)-trominos used when \(n = 4\). | [
"85"
] | [
"A square of side length \\(2^n\\) is divided into unit squares. One of the unit squares is deleted. If the remaining area can be perfectly tiled with \\(L\\)-trominos, calculate the total number of \\(L\\)-trominos used when \\(n = 4\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e6d8671f7c22d8ea2fae54695492912518c774ac127346a593e6d9a9576c4b5a | A square of size $2 \times 2$ with one of its cells occupied by a tower is called a castle. What is the maximal number of castles that can be placed on a board of size $7 \times 7$ so that the castles have no common cells and all the towers stand on the diagonals of the board? | [
"6"
] | [
"A square of size $2 \\times 2$ with one of its cells occupied by a tower is called a castle. What is the maximal number of castles that can be placed on a board of size $7 \\times 7$ so that the castles have no common cells and all the towers stand on the diagonals of the board?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8808c3f5fa984c26afcb749dce01d1f065c25a31bef21c07611d1da20eefe51a | A square of size $N \times N$ is divided into $N^2$ unit squares, and the center of each unit square is colored either white or black. Determine the maximal value of $N$ such that there does not exist a rectangle with vertices at the centers of these unit squares, where the sides of the rectangle are parallel to the si... | [
"4"
] | [
"A square of size $N \\times N$ is divided into $N^2$ unit squares, and the center of each unit square is colored either white or black. Determine the maximal value of $N$ such that there does not exist a rectangle with vertices at the centers of these unit squares, where the sides of the rectangle are parallel to ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-771c3227340936f505a774173960ebc86b136e06f225b7fd090d9bc4c288df42 | A square of size \( n \times n \) was cut into squares of sizes \( 40 \times 40 \) and \( 49 \times 49 \) such that squares of both types appeared. What is the minimal possible value of \( n \)? | [
"1960"
] | [
"A square of size \\( n \\times n \\) was cut into squares of sizes \\( 40 \\times 40 \\) and \\( 49 \\times 49 \\) such that squares of both types appeared. What is the minimal possible value of \\( n \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 1
} | 12 | 0 | 0 |
omr-ws-b17995abe22b9bebb745dff186d03f269294615c1e32c7d3986ed1c0cbcbe55c | A square of unit side length is divided into regions by line segments parallel to its sides. The total length of the interior segments is 18. Let $A$ be the area of the largest region formed. Calculate the minimum possible value of $100A$. | [
"1"
] | [
"A square of unit side length is divided into regions by line segments parallel to its sides. The total length of the interior segments is 18. Let $A$ be the area of the largest region formed. Calculate the minimum possible value of $100A$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-90e9b3529516f4420cd93b92942e1dd13d914eb6caf2ccea3644bc336bd5fbef | A square on the coordinate plane has two of its vertices located at points $(9,15)$ and $(16,1)$. This square is positioned so that a nonzero portion of the square lies in the fourth quadrant. What is the area of the region inside both the square and the first quadrant? | [
"\\(\\frac{715}{4}\\)",
"\\dfrac{385}{4}"
] | [
"A square on the coordinate plane has two of its vertices located at points $(9,15)$ and $(16,1)$. This square is positioned so that a nonzero portion of the square lies in the fourth quadrant. What is the area of the region inside both the square and the first quadrant?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 24,
"genselect": 7,
"tir": 2
} | 33 | 0 | 0 |
omr-ws-c5ab7e99689ad87805f5deb830d9cb5f997b5812e66af50d4fd84b70b193494f | A square paper of side $n$ is divided into $n^2$ unit square cells. A maze is drawn on the paper with unit walls between some cells in such a way that one can reach every cell from every other cell without crossing any wall. Find, in terms of $n$, the largest possible total length of the walls. | [
"\\((n-1)^2\\)"
] | [
"A square paper of side $n$ is divided into $n^2$ unit square cells. A maze is drawn on the paper with unit walls between some cells in such a way that one can reach every cell from every other cell without crossing any wall. Find, in terms of $n$, the largest possible total length of the walls."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-2ff1f3ed1ca038e76e5a65ba52950083314a90bf789906c96709a6fc49c0b7ec | A square paper of side length \( n \) is divided into \( n \times n \) unit square cells. A maze is drawn on the paper with unit walls between some cells such that one can reach every cell from every other cell without crossing any wall. Find, in terms of \( n \), the largest possible total length of the walls. | [
"\\( (n+1)^2 \\)"
] | [
"A square paper of side length \\( n \\) is divided into \\( n \\times n \\) unit square cells. A maze is drawn on the paper with unit walls between some cells such that one can reach every cell from every other cell without crossing any wall. Find, in terms of \\( n \\), the largest possible total length of the wa... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 5
} | 9 | 0 | 0 |
omr-ws-1547b2754c723cbc260eb6dc140ff7c9153fb4cac0703109e1e54c8189ab18e9 | A square passes through the points (0,12), (-4,7), (8,0), and (10,9). Find the area of the square. | [
"\\dfrac{968}{5}"
] | [
"A square passes through the points (0,12), (-4,7), (8,0), and (10,9). Find the area of the square."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-91400ab12476305e053b4cf3e55f00ae75047cb04845c522becff200c3aae617 | A square piece of aluminum foil with a side length of 10 cm is used to close the top of a cup. The foil is peeled off from side $BC$ perpendicular to it within one second at a constant speed. The visible area of the foil first decreases and then increases again. Graphically represent the contents of the visible foil ar... | [
"F(t) = 100 \\left| \\cos(\\pi t) \\right|"
] | [
"A square piece of aluminum foil with a side length of 10 cm is used to close the top of a cup. The foil is peeled off from side $BC$ perpendicular to it within one second at a constant speed. The visible area of the foil first decreases and then increases again. Graphically represent the contents of the visible fo... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-1377904354f8710a39d21663c48989fb0e73bd922811e58bcabe535d7895a341 | A square piece of paper $FBHD$ with side length $a$ is given. Points $G$ and $A$ are marked on $FB$ such that $FG = GA = AB$, and points $E$ and $C$ are marked on $BH$ such that $BE = EC = CH$. The paper is folded along $DG$, $DA$, $DC$, and $AC$ so that $G$ overlaps with $B$, and $F$ and $H$ overlap with $E$. Compute ... | [
"\\dfrac{a^3}{18\\sqrt{2}}"
] | [
"A square piece of paper $FBHD$ with side length $a$ is given. Points $G$ and $A$ are marked on $FB$ such that $FG = GA = AB$, and points $E$ and $C$ are marked on $BH$ such that $BE = EC = CH$. The paper is folded along $DG$, $DA$, $DC$, and $AC$ so that $G$ overlaps with $B$, and $F$ and $H$ overlap with $E$. Com... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-0591160f2e9951621ad0ef4f277815cdd13f945eb18d034b4cd60ec768830682 | A square piece of paper ABCD is folded such that point D is placed at some point D' on BC, and AD is carried into A'D', crossing AB at E. For a specific choice of D' on BC where BD' = 3 and BC = 10, calculate the radius of the circle inscribed in triangle EBD'. Express your answer as a simplified fraction. | [
"\\dfrac{21}{17}"
] | [
"A square piece of paper ABCD is folded such that point D is placed at some point D' on BC, and AD is carried into A'D', crossing AB at E. For a specific choice of D' on BC where BD' = 3 and BC = 10, calculate the radius of the circle inscribed in triangle EBD'. Express your answer as a simplified fraction."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-bd392a36340f2d3d4a5fcd0738fbc6be5bc27abe47e3f929589bdf17fe704e87 | A square piece of paper has side length $1$ and vertices $A, B, C,$ and $D$ in that order. The paper is folded so that vertex $C$ meets edge $\overline{AD}$ at point $C'$, and edge $\overline{AB}$ at point $E$. Given that $C'D = \frac{1}{3}$, what is the perimeter of triangle $\triangle AEC'$? | [
"2"
] | [
"A square piece of paper has side length $1$ and vertices $A, B, C,$ and $D$ in that order. The paper is folded so that vertex $C$ meets edge $\\overline{AD}$ at point $C'$, and edge $\\overline{AB}$ at point $E$. Given that $C'D = \\frac{1}{3}$, what is the perimeter of triangle $\\triangle AEC'$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-16976a65a41d3a8d0df845fba2f82764e7e20b68c939344e75ff6d1eebb502d1 | A square piece of paper has side length $1$ and vertices $A, B, C,$ and $D$ in that order. The paper is folded so that vertex $C$ meets edge $\overline{AD}$ at point $C'$, and edge $\overline{BC}$ intersects edge $\overline{AB}$ at point $E$. Given that $C'D = \frac{1}{3}$, what is the perimeter of $\triangle AEC'$? | [
"2"
] | [
"A square piece of paper has side length $1$ and vertices $A, B, C,$ and $D$ in that order. The paper is folded so that vertex $C$ meets edge $\\overline{AD}$ at point $C'$, and edge $\\overline{BC}$ intersects edge $\\overline{AB}$ at point $E$. Given that $C'D = \\frac{1}{3}$, what is the perimeter of $\\triangle... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 48
} | 48 | 0 | 0 |
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