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-7ce29e10482391128f1aea8444b16f870613f592011835e51b5186b2b6470fde | A rectangular piece of paper measures $8$ inches by $17$ inches. It is folded so that a right angle is formed between the two segments of the original bottom edge. What is the area in square inches of the new figure? | [
"104"
] | [
"A rectangular piece of paper measures $8$ inches by $17$ inches. It is folded so that a right angle is formed between the two segments of the original bottom edge. What is the area in square inches of the new figure?"
] | [
"aops_c5_contests_amp_programs"
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"cot",
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omr-ws-409296f11196513a0bbb24cad2178fe187fed2e3921a56c66aa4f47c56c52307 | A rectangular piece of paper measures 4 units by 5 units. Several lines are drawn parallel to the edges of the paper. A rectangle determined by the intersections of some of these lines is called "basic" if (i) all four sides of the rectangle are segments of drawn line segments, and (ii) no segments of drawn lines lie i... | [
"896"
] | [
"A rectangular piece of paper measures 4 units by 5 units. Several lines are drawn parallel to the edges of the paper. A rectangle determined by the intersections of some of these lines is called \"basic\" if (i) all four sides of the rectangle are segments of drawn line segments, and (ii) no segments of drawn line... | [
"aops_c5_contests_amp_programs"
] | [
"genselect"
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"genselect": 5
} | 5 | 0 | 0 |
omr-ws-07ae8051f2c466e90058b5ab846b93b2d24b88373ebd8764653fee53f0c6563b | A rectangular piece of paper whose length is $\sqrt{3}$ times the width has area $A$. The paper is divided into equal sections along the opposite lengths, and then a dotted line is drawn from the first divider to the second divider on the opposite side. The paper is then folded flat along this dotted line to create a n... | [
"2:3"
] | [
"A rectangular piece of paper whose length is $\\sqrt{3}$ times the width has area $A$. The paper is divided into equal sections along the opposite lengths, and then a dotted line is drawn from the first divider to the second divider on the opposite side. The paper is then folded flat along this dotted line to crea... | [
"aops_c5_contests_amp_programs"
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"cot",
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omr-ws-b0d00c1217729cdb547f4616c774929847282987e940d74cd22a3d19e6258181 | A rectangular piece of paper with length $48$ and width $14$ is cut across a line, creating two shapes of equal area. One of the two shapes is discarded. What is the maximum perimeter of the remaining shape? | [
"112"
] | [
"A rectangular piece of paper with length $48$ and width $14$ is cut across a line, creating two shapes of equal area. One of the two shapes is discarded. What is the maximum perimeter of the remaining shape?"
] | [
"aops_c4_high_school_math"
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"cot"
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"cot": 15
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omr-ws-a39bdaf26a1bf748720ba3e612710fa358c0e594d6aabe9f62baf9689085dd5b | A rectangular piece of paper with length 1 is folded such that 2 opposite corners are folded together. Given that the length of the crease made by the fold has the same length as the width of the piece of paper and that the width of the paper is longer than the length, the length of the long side can be expressed as \(... | [
"22"
] | [
"A rectangular piece of paper with length 1 is folded such that 2 opposite corners are folded together. Given that the length of the crease made by the fold has the same length as the width of the piece of paper and that the width of the paper is longer than the length, the length of the long side can be expressed ... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f792559db1a2351b44faa0edd81998e8576666741938195352639e507e6097f3 | A rectangular piece of paper with vertices $A = (0,0)$, $B = (14,0)$, $C = (14,25)$, and $D = (0,25)$ is folded such that point $M$ on side $\overline{CD}$ (where $CM = MD$) is used to fold triangles $MDA$ and $MCB$ upwards so that vertices $C$ and $D$ coincide at point $E$ in space. If the radius of the inscribed sphe... | [
"121"
] | [
"A rectangular piece of paper with vertices $A = (0,0)$, $B = (14,0)$, $C = (14,25)$, and $D = (0,25)$ is folded such that point $M$ on side $\\overline{CD}$ (where $CM = MD$) is used to fold triangles $MDA$ and $MCB$ upwards so that vertices $C$ and $D$ coincide at point $E$ in space. If the radius of the inscribe... | [
"aops_c4_high_school_math"
] | [
"cot",
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"cot": 15,
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omr-ws-4bf152fb671ee64ff910bf4741a44bb5d7cd7b6d790372bf3e256ea5f37d595c | A rectangular piece of sheet metal with an area of 1200 in^2 is to be bent into a cylindrical length of stovepipe having a volume of 600 in^3. What are the dimensions of the sheet metal? Given: - The area of the sheet metal is \( h \times 2\pi r = 1200 \) in^2. - The volume of the cylinder is \( \pi r^2 h = 600 \) in^3... | [
"\\( a = 2\\pi \\) inches, \\( b = \\frac{600}{\\pi} \\) inches"
] | [
"A rectangular piece of sheet metal with an area of 1200 in^2 is to be bent into a cylindrical length of stovepipe having a volume of 600 in^3. What are the dimensions of the sheet metal?\n\nGiven:\n- The area of the sheet metal is \\( h \\times 2\\pi r = 1200 \\) in^2.\n- The volume of the cylinder is \\( \\pi r^2... | [
"aops_c4_high_school_math"
] | [
"cot",
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] | {
"cot": 6,
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"tir": 28
} | 39 | 0 | 0 |
omr-ws-5847de81bdd5a4f26aab43eff14fffafa6b82679745817190a79f674859b7855 | A rectangular plot is 6 meters longer than it is wide, and the area of the plot is 16 square meters. a) Create a trinomial expression for the area of the plot. b) Factor the polynomial and solve to find the length and width of the plot. | [
"Width = 2 meters, Length = 8 meters"
] | [
"A rectangular plot is 6 meters longer than it is wide, and the area of the plot is 16 square meters.\na) Create a trinomial expression for the area of the plot.\nb) Factor the polynomial and solve to find the length and width of the plot."
] | [
"aops_c4_high_school_math"
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"cot",
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"cot": 4,
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omr-ws-49c355144a458b339bffd38c35efadbbfb71717a465536182d1b04049fd4c0e0 | A rectangular plot measures 12 meters by 5 meters. A path of constant width runs along one side, and the area of the path is 120 square meters. Find the width of the path. | [
"\\( w \\approx 2.80329 \\) meters"
] | [
"A rectangular plot measures 12 meters by 5 meters. A path of constant width runs along one side, and the area of the path is 120 square meters. Find the width of the path."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
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"additional_problems": 1
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omr-ws-40f4f29cde60fa76503f7cf31d41fa61f02bacb2f636ca531c0a4073e325a9a5 | A rectangular plot of area 900 square meters is to be fenced in; two adjoining sides to be brick and the two others to be wooden. One meter of the wooden fence costs 10 roubles and one meter of the brick fence costs 25 roubles. A total of 2000 roubles has been alotted for the job. Will this sum be sufficient? | [
"No"
] | [
"A rectangular plot of area 900 square meters is to be fenced in; two adjoining sides to be brick and the two others to be wooden. One meter of the wooden fence costs 10 roubles and one meter of the brick fence costs 25 roubles. A total of 2000 roubles has been alotted for the job. Will this sum be sufficient?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
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"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6e13e74432b7c3ab29d6b3473b66c616cd23c6bf2f0c7852de1ea75b0b6901dc | A rectangular pool has a diagonal of 17 units and an area of 120 square units. Joey and Rachel start on opposite sides of the pool when Rachel starts chasing Joey. If Rachel runs 5 units/sec faster than Joey, how long does it take for her to catch him? | [
"\\(\\frac{23}{5}\\)"
] | [
"A rectangular pool has a diagonal of 17 units and an area of 120 square units. Joey and Rachel start on opposite sides of the pool when Rachel starts chasing Joey. If Rachel runs 5 units/sec faster than Joey, how long does it take for her to catch him?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
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"cot": 9,
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"tir": 28
} | 43 | 0 | 0 |
omr-ws-a1feda91536f04d6801ab3a368939e81601223e44b3c5269f0221d10c3c32b49 | A rectangular pool table has a hole at each of three of its corners. The lengths of the sides of the table are real numbers $a$ and $b$. A billiard ball is shot from the fourth corner along its angle bisector. The ball falls into one of the holes. What should the relation between $a$ and $b$ be for this to happen? | [
"\\(\\frac{a}{b} \\in \\mathbb{Q}\\)"
] | [
"A rectangular pool table has a hole at each of three of its corners. The lengths of the sides of the table are real numbers $a$ and $b$. A billiard ball is shot from the fourth corner along its angle bisector. The ball falls into one of the holes. What should the relation between $a$ and $b$ be for this to happen?... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
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] | {
"cot": 12,
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"tir": 7
} | 27 | 0 | 0 |
omr-ws-eb82bef21b5591a7f34a5b6c79204aead9cbfbfe95ca1c67efd55d76dbaf162d | A rectangular prism has a volume of 63 cubic units, a surface area of 64 square units, and a total edge length of 46 units. Determine the lengths of the sides of the rectangular prism. | [
"\\text{No solution exists}"
] | [
"A rectangular prism has a volume of 63 cubic units, a surface area of 64 square units, and a total edge length of 46 units. Determine the lengths of the sides of the rectangular prism."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
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omr-ws-a6fc7a081e675655189a206167d82d61bad14bda2c8612a33282045cb672ea32 | A rectangular prism has dimensions 8 inches by 2 inches by 32 inches. If a cube has the same volume as the prism, what is the surface area of the cube, in square inches? | [
"384"
] | [
"A rectangular prism has dimensions 8 inches by 2 inches by 32 inches. If a cube has the same volume as the prism, what is the surface area of the cube, in square inches?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b63052c94ab96bd0e1f58563639df64c5621d61289adf47a305800c0c1b4a1c3 | A rectangular prism has one red face, two green faces, and three blue faces. The area of the red face is $100$, the sum of the areas of the green faces is $116$, and the sum of the areas of the blue faces is $114$. Given that the volume $V$ of the rectangular prism is an integer, find $V$. | [
"280"
] | [
"A rectangular prism has one red face, two green faces, and three blue faces. The area of the red face is $100$, the sum of the areas of the green faces is $116$, and the sum of the areas of the blue faces is $114$. Given that the volume $V$ of the rectangular prism is an integer, find $V$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
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"cot": 11,
"genselect": 7
} | 18 | 0 | 0 |
omr-ws-6369d1c00e0a545a101469714602b6e422e6ec501b462193e304a015b951dd8f | A rectangular prism has one side of length $3 + \sqrt{7}$. Given that the volume and perimeter are integers and the surface area is $40$, find $x^2$ where $x$ is the length of the space diagonal. | [
"41"
] | [
"A rectangular prism has one side of length $3 + \\sqrt{7}$. Given that the volume and perimeter are integers and the surface area is $40$, find $x^2$ where $x$ is the length of the space diagonal."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-47710c5f3ae9e7c9387fa7a1646c561abe77692db377ff4aa38961f9fefeb02c | A rectangular prism has side lengths 1, 1, and 2. What is the product of the lengths of all of the diagonals? | [
"90000"
] | [
"A rectangular prism has side lengths 1, 1, and 2. What is the product of the lengths of all of the diagonals?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-683978c8812058339be004c9c1b8ee1580b4cd318dc1c329b0c1372103117b45 | A rectangular prism has three distinct faces with areas $24$, $30$, and $32$. The diagonals of each distinct face of the prism form the sides of a triangle. What is the area of this triangle? | [
"24"
] | [
"A rectangular prism has three distinct faces with areas $24$, $30$, and $32$. The diagonals of each distinct face of the prism form the sides of a triangle. What is the area of this triangle?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8055495e6fbf039000eb6f01aaeb09adc1697cd049a03a17ed6ff24c8c96ead7 | A rectangular prism is to be dissected into 16 equal pieces using exactly 4 cuts. If it is possible, determine the number of distinct ways the cuts can be made, assuming rotations and reflections are considered the same. If it is not possible, provide the minimum number of cuts needed to achieve the dissection into 16 ... | [
"5"
] | [
"A rectangular prism is to be dissected into 16 equal pieces using exactly 4 cuts. If it is possible, determine the number of distinct ways the cuts can be made, assuming rotations and reflections are considered the same. If it is not possible, provide the minimum number of cuts needed to achieve the dissection int... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2d49d28c1e670e36ea8d66f5f70c8844308601b40bcbc686db4ab6c2ba476960 | A rectangular prism with dimensions $1 \times a \times b$, where $1 < a < b < 2$, is bisected by a plane bisecting the longest edges of the prism. One of the smaller prisms is bisected in the same way. If all three resulting prisms are similar to each other and to the original box, compute $ab$. Note: Two rectangular p... | [
"2"
] | [
"A rectangular prism with dimensions $1 \\times a \\times b$, where $1 < a < b < 2$, is bisected by a plane bisecting the longest edges of the prism. One of the smaller prisms is bisected in the same way. If all three resulting prisms are similar to each other and to the original box, compute $ab$. Note: Two rectan... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-4e5b2ea3662a04c0723a0cbadb83409c51d6c919cddd3a42e8f996465b91cd19 | A rectangular prism with dimensions $20$ cm by $1$ cm by $7$ cm is made with blue $1$ cm unit cubes. The outside of the rectangular prism is coated in gold paint. If a cube is chosen at random and rolled, what is the probability that the side facing up is painted gold? | [
"\\(\\frac{167}{420}\\)"
] | [
"A rectangular prism with dimensions $20$ cm by $1$ cm by $7$ cm is made with blue $1$ cm unit cubes. The outside of the rectangular prism is coated in gold paint. If a cube is chosen at random and rolled, what is the probability that the side facing up is painted gold?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 22
} | 37 | 0 | 0 |
omr-ws-6b02fbfca580a502e25cabffd422826c819abec7431250cd47d748d2b111a9fa | A rectangular prism with dimensions $3 \times 4 \times 5$ is spun around the diagonal connecting a pair of its opposite vertices. Let $x$ be the volume of the resulting figure. Find $\lfloor{x}\rfloor$. | [
"753"
] | [
"A rectangular prism with dimensions $3 \\times 4 \\times 5$ is spun around the diagonal connecting a pair of its opposite vertices. Let $x$ be the volume of the resulting figure. Find $\\lfloor{x}\\rfloor$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 7
} | 8 | 0 | 0 |
omr-ws-a0f613d3828bb031a324d5521eb7653b47c306078a986c23f0343446afe675b7 | A rectangular prism with dimensions $77 \times 81 \times 100$ is cut into unit cubes. How many of these unit cubes does a given internal diagonal pierce? | [
"256"
] | [
"A rectangular prism with dimensions $77 \\times 81 \\times 100$ is cut into unit cubes. How many of these unit cubes does a given internal diagonal pierce?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9a3cd0041d1c06a472c41eda197b70c21bf89da0c64e2e259540265cb2983c6c | A rectangular prism with dimensions $\ell \times w \times h$ has two planes connecting the opposite sides (forming an X) from the top. There is a sphere with radius $x$ inside the prism. What is the probability that the sphere neither touches the planes nor the sides of the rectangular prism? | [
"\\left(1 - \\frac{2x}{\\sqrt{\\ell^2 + w^2}}\\right)^2 \\left(1 - \\frac{2x}{h}\\right)"
] | [
"A rectangular prism with dimensions $\\ell \\times w \\times h$ has two planes connecting the opposite sides (forming an X) from the top. There is a sphere with radius $x$ inside the prism. What is the probability that the sphere neither touches the planes nor the sides of the rectangular prism?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 8
} | 9 | 0 | 0 |
omr-ws-562ce41894918554dfed8ae1efa53e2603a3af7d5ed965938f252461a4c39e87 | A rectangular prism with dimensions $p \times q \times r$, where $p$, $q$, and $r$ are pairwise coprime, is cut into $1 \times 1 \times 1$ cubes. How many of these cubes does an internal diagonal of the prism pierce? | [
"p + q + r - 2"
] | [
"A rectangular prism with dimensions $p \\times q \\times r$, where $p$, $q$, and $r$ are pairwise coprime, is cut into $1 \\times 1 \\times 1$ cubes. How many of these cubes does an internal diagonal of the prism pierce?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-01a3c7d334e437f62779b4a38f465bdbc9d54d397114ac9d09ee4b30d4934d6e | A rectangular prism with dimensions AxBxC lies in the 3D space. You are viewing the space from an angle that makes the prism appear as a 2D shape of minimum area. What is the minimum area of the shape in terms of A, B, and C? | [
"\\min(AB, BC, AC)"
] | [
"A rectangular prism with dimensions AxBxC lies in the 3D space. You are viewing the space from an angle that makes the prism appear as a 2D shape of minimum area. What is the minimum area of the shape in terms of A, B, and C?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-5cd0afd2812c7c2bc8581106d00c37c3f6eef97994d84748685d89ebefeafa3b | A rectangular prism with positive integer side lengths formed by stacking unit cubes is called bipartisan if the same number of unit cubes can be seen on the surface as those which cannot be seen on the surface. How many non-congruent bipartisan rectangular prisms are there? | [
"2"
] | [
"A rectangular prism with positive integer side lengths formed by stacking unit cubes is called bipartisan if the same number of unit cubes can be seen on the surface as those which cannot be seen on the surface. How many non-congruent bipartisan rectangular prisms are there?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3fcb31e27fe171c69d6fca791df73fbd3d70e593fbec4f12010187abc2439ff1 | A rectangular room has a perimeter of 42 feet and an area of 108 square feet. What is the length, in feet, of the shorter side? | [
"9"
] | [
"A rectangular room has a perimeter of 42 feet and an area of 108 square feet. What is the length, in feet, of the shorter side?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-581135033dea195f9ec2804f0e424bfc318a47119e4676398c6e95a78c958a10 | A rectangular room measures 12-feet by 6-feet. How many square yards of carpet are needed to cover the floor of the room? | [
"8"
] | [
"A rectangular room measures 12-feet by 6-feet. How many square yards of carpet are needed to cover the floor of the room?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-74e5d20aca4a97dfd1624616416f60b664eb92f19c79a90c5dd43b9a9f43d49a | A rectangular room measures 30 feet in length and 12 feet in height, with ends 12 feet in width. A spider is located one foot up from the floor at the middle of one end, and the center of the spider's web is one foot down from the ceiling at the middle of the other end. What is the shortest distance the spider can walk... | [
"\\( 7\\sqrt{26} \\)"
] | [
"A rectangular room measures 30 feet in length and 12 feet in height, with ends 12 feet in width. A spider is located one foot up from the floor at the middle of one end, and the center of the spider's web is one foot down from the ceiling at the middle of the other end. What is the shortest distance the spider can... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 3
} | 3 | 0 | 0 |
omr-ws-e44ae47fa44ad6ffd85c44ef0a237b1092bd93b219059c8e199e1d734bab16aa | A rectangular room measures 30 feet in length and 12 feet in height, with ends that are 12 feet in width. A fly rests at a point one foot down from the ceiling at the middle of one end. A smudge of food is located one foot up from the floor at the middle of the other end. The fly can walk a maximum distance of 40 feet.... | [
"10\\sqrt{10}"
] | [
"A rectangular room measures 30 feet in length and 12 feet in height, with ends that are 12 feet in width. A fly rests at a point one foot down from the ceiling at the middle of one end. A smudge of food is located one foot up from the floor at the middle of the other end. The fly can walk a maximum distance of 40 ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 7
} | 10 | 0 | 0 |
omr-ws-6f6d38564008474c79ff217381e6786ca425c67955812f52764a0d26f1761c1f | A rectangular shed measuring 4 ft by 5 ft is positioned along a wall. A goat is tethered to a corner marked P with an 8-ft-long rope. What is the area of the region over which the goat can move? | [
"\\dfrac{217}{4} \\pi",
"\\dfrac{217}{4}\\pi"
] | [
"A rectangular shed measuring 4 ft by 5 ft is positioned along a wall. A goat is tethered to a corner marked P with an 8-ft-long rope. What is the area of the region over which the goat can move?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 26,
"genselect": 7
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omr-ws-7bdfff37052e9337886443d31702294c6edc2ba70b8d695a6e471a218cb2a8bd | A rectangular sheet of brass is twice as long as it is wide. Squares, $3 \times 3$, are cut from each corner, and the ends are turned up to form an open box having a volume of $648$. What were the dimensions of the original sheet of brass? | [
"30.56 x 15.28"
] | [
"A rectangular sheet of brass is twice as long as it is wide. Squares, $3 \\times 3$, are cut from each corner, and the ends are turned up to form an open box having a volume of $648$. What were the dimensions of the original sheet of brass?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-822d22e48d78145c454c349b0b9162f561b86b53281980d7f8d871c43cfb6f3f | A rectangular sheet of cardstock was cut along its diagonal. In one of the pieces remaining, in the form of a right triangle, two cuts were made, parallel to the sides minors, by means of these sides. In the end, a rectangle with a perimeter of $129$ cm was left. What was the perimeter of the sheet before cutting? | [
"258"
] | [
"A rectangular sheet of cardstock was cut along its diagonal. In one of the pieces remaining, in the form of a right triangle, two cuts were made, parallel to the sides minors, by means of these sides. In the end, a rectangle with a perimeter of $129$ cm was left. What was the perimeter of the sheet before cutting?... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-1fa1a4bb9024147a2f723487300fc077e3e80a78e53f10a32055ba92c202f21b | A rectangular sheet of paper \(ABCD\) was bent so that vertex \(B\) was taken to point \(B'\) on side \(AD\). The fold is \(EF\), with \(E\) on \(AB\) and \(F\) on \(CD\). Given \(AE = 8\), \(BE = 17\), and \(CF = 3\), calculate \(AB'\) and \(AD\). | [
"\\dfrac{70}{3}"
] | [
"A rectangular sheet of paper \\(ABCD\\) was bent so that vertex \\(B\\) was taken to point \\(B'\\) on side \\(AD\\). The fold is \\(EF\\), with \\(E\\) on \\(AB\\) and \\(F\\) on \\(CD\\). Given \\(AE = 8\\), \\(BE = 17\\), and \\(CF = 3\\), calculate \\(AB'\\) and \\(AD\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-6cb1bd7cb101c64627cbb332d099797431f48192c105d8a0ab147b784436e1c5 | A rectangular sheet of paper has dimensions 11 inches by 12 inches. A triangular piece is cut from the corner to leave a pentagon with side lengths 11, 12, 7, 5, and 9 inches. What is the area of the pentagon in square inches? | [
"126"
] | [
"A rectangular sheet of paper has dimensions 11 inches by 12 inches. A triangular piece is cut from the corner to leave a pentagon with side lengths 11, 12, 7, 5, and 9 inches. What is the area of the pentagon in square inches?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bb34d5da790b866438d2666ffd68cc118f11f89251e7045bf4b9f9527eba20d1 | A rectangular sheet of paper has dimensions 12 by 18. The paper is folded so that one pair of opposite corners now coincide. The area of the folded paper can be expressed as \(a+b\sqrt{c}\) where \(a, b,\) and \(c\) are integers and \(c\) is not divisible by the square of any prime. Find \(a+b+c\). | [
"no_valid_answer_found"
] | [
"A rectangular sheet of paper has dimensions 12 by 18. The paper is folded so that one pair of opposite corners now coincide. The area of the folded paper can be expressed as \\(a+b\\sqrt{c}\\) where \\(a, b,\\) and \\(c\\) are integers and \\(c\\) is not divisible by the square of any prime. Find \\(a+b+c\\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d2ee3db35511f68315ebccfebed83153668428e3f138ea9ab47305857af00877 | A rectangular sheet of paper is 5 cm long. When it is folded so that two opposite vertices are applied one on top of the other, the length of the fold is $\sqrt{6}$ cm. What is the width of this sheet? | [
"\\(\\sqrt{5}\\)"
] | [
"A rectangular sheet of paper is 5 cm long. When it is folded so that two opposite vertices are applied one on top of the other, the length of the fold is $\\sqrt{6}$ cm. What is the width of this sheet?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-20a3945f16b3fe8b6792f251cd552d717d112f8f42e1f99def0fc6db8eaa9084 | A rectangular sheet of paper is divided by a single cut into a triangle and a pentagon. The lengths of the sides of the pentagon are $17$, $25$, $28$, $33$, and $43$, in some order. Calculate the area of the pentagon. | [
"1359"
] | [
"A rectangular sheet of paper is divided by a single cut into a triangle and a pentagon. The lengths of the sides of the pentagon are $17$, $25$, $28$, $33$, and $43$, in some order. Calculate the area of the pentagon."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-641751f9562fcc058de73303a3519a843207a68c6864cc30c6ed7a98857c6eea | A rectangular sheet of paper is folded along a single line so that one corner lies on top of another. In the resulting figure, $60\%$ of the area is two sheets thick and $40\%$ is one sheet thick. What is the ratio of the length of the longer side of the rectangle to the length of the shorter side? | [
"\\dfrac{3}{2}"
] | [
"A rectangular sheet of paper is folded along a single line so that one corner lies on top of another. In the resulting figure, $60\\%$ of the area is two sheets thick and $40\\%$ is one sheet thick. What is the ratio of the length of the longer side of the rectangle to the length of the shorter side?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-1a77b5791f32dbfb81b7b39c1646278b22d55affe3250083252c1f4a42b18dd9 | A rectangular sheet of paper is folded so that its diagonally opposite corners meet. The fold divides the paper into two halves of equal area, but when folded in this way there are two triangular regions which do not overlap the other half. The ratio of the sum of the areas in these two triangles to the area of the pap... | [
"\\sqrt{5} - 2"
] | [
"A rectangular sheet of paper is folded so that its diagonally opposite corners meet. The fold divides the paper into two halves of equal area, but when folded in this way there are two triangular regions which do not overlap the other half. The ratio of the sum of the areas in these two triangles to the area of th... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-998c1d0d1a76522c8f20070500e4c051dabe9094957bfa5940cbd2ac0f94ef97 | A rectangular sheet of paper is folded so that the two diagonally opposite corners come together. If the crease formed is of the same length as the longer side of the sheet, what is the ratio of the longer side of the sheet to the shorter side? | [
"\\(\\sqrt{2}\\)"
] | [
"A rectangular sheet of paper is folded so that the two diagonally opposite corners come together. If the crease formed is of the same length as the longer side of the sheet, what is the ratio of the longer side of the sheet to the shorter side?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0146c7a36c179136c62c383e1ec736d68b258af021a67f0fd7f831cbcfbe29c7 | A rectangular sheet of paper is folded so that two diagonally opposite corners come together. If the crease formed is the same length as the longer side of the sheet, what is the ratio of the longer side of the sheet to the shorter side? | [
"\\(\\sqrt{\\frac{1 + \\sqrt{5}}{2}}\\)"
] | [
"A rectangular sheet of paper is folded so that two diagonally opposite corners come together. If the crease formed is the same length as the longer side of the sheet, what is the ratio of the longer side of the sheet to the shorter side?"
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 63,
"tir": 55
} | 118 | 0 | 0 |
omr-ws-8500efd8eefc7b53974a118bd60a2fce8b08e43ffec899ecf48c341061d3be17 | A rectangular sheet of paper is to contain 72 square inches of printed matter with 2-inch margins at the top and bottom and 1-inch margins on each side. What dimensions for the sheet will use the least amount of paper? | [
"8 \\text{ inches by } 16 \\text{ inches}"
] | [
"A rectangular sheet of paper is to contain 72 square inches of printed matter with 2-inch margins at the top and bottom and 1-inch margins on each side. What dimensions for the sheet will use the least amount of paper?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-9530d4ae687fd71796c5cfea805d523bf2df8bd15279eca4d02cd6e25eae9c48 | A rectangular sheet of paper measuring $m \times n$ was cut into squares and right isosceles triangles under the following conditions: 1) all squares are equal to each other, all triangles are also equal to each other, 2) the diagonal of the square is equal to the leg of a right triangle and equal to twice the side of ... | [
"\\dfrac{mn}{2}"
] | [
"A rectangular sheet of paper measuring $m \\times n$ was cut into squares and right isosceles triangles under the following conditions:\n1) all squares are equal to each other, all triangles are also equal to each other,\n2) the diagonal of the square is equal to the leg of a right triangle and equal to twice the ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e2b4d642978e4656601b7ed94bd1ac4dc94f5b2f47d9e53a1712299c93204407 | A rectangular sheet of paper was folded three times. After the first fold, the resulting rectangle (Rectangle 1) has a perimeter that is 20 cm more than the perimeter of the rectangle after the second fold (Rectangle 2). Rectangle 2 has a perimeter that is 16 cm more than the perimeter of the rectangle after the third ... | [
"504"
] | [
"A rectangular sheet of paper was folded three times. After the first fold, the resulting rectangle (Rectangle 1) has a perimeter that is 20 cm more than the perimeter of the rectangle after the second fold (Rectangle 2). Rectangle 2 has a perimeter that is 16 cm more than the perimeter of the rectangle after the t... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5f16dd09b54d7e2565f6922f484150050b217f92fff8071198222798585ba1ab | A rectangular sheet of paper, ABCD, has AD = 1 and AB = r, where 1 < r < 2. The paper is folded along a line through A so that the edge AD falls onto the edge AB. Without unfolding, the paper is folded again along a line through B so that the edge CB also lies on AB. The result is a triangular piece of paper. A region ... | [
"\\dfrac{r^2}{4}"
] | [
"A rectangular sheet of paper, ABCD, has AD = 1 and AB = r, where 1 < r < 2. The paper is folded along a line through A so that the edge AD falls onto the edge AB. Without unfolding, the paper is folded again along a line through B so that the edge CB also lies on AB. The result is a triangular piece of paper. A re... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 8
} | 12 | 0 | 0 |
omr-ws-aa2457677a4c1027d235097102c27b4f8571005ba48ced3584341f2ceb96630e | A rectangular sheet with sides \(a\) and \(b\) is folded along a diagonal. Compute the area of the overlapping triangle. | [
"\\(\\frac{a^2b+b^3}{4a}\\)"
] | [
"A rectangular sheet with sides \\(a\\) and \\(b\\) is folded along a diagonal. Compute the area of the overlapping triangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 2
} | 4 | 0 | 0 |
omr-ws-c0ef2c27149e60f31af7eb7d640f44c6c498cc18cbcc9efee502e7bd9a92231f | A rectangular solid \( S \) with dimensions \( p \times q \times r \), where \( p, q, \) and \( r \) are relatively prime in pairs, is cut into \( pqr \) unit cubes by planes parallel to its faces. How many of these unit cubes would be cut if a corner \( X \) of \( S \) were cut off by a plane going through three neigh... | [
"\\(\\frac{pq + pr + qr - 1}{2}\\)"
] | [
"A rectangular solid \\( S \\) with dimensions \\( p \\times q \\times r \\), where \\( p, q, \\) and \\( r \\) are relatively prime in pairs, is cut into \\( pqr \\) unit cubes by planes parallel to its faces. How many of these unit cubes would be cut if a corner \\( X \\) of \\( S \\) were cut off by a plane goin... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-dc64e4a53f06a72bd18823e88379f1232d932c227fb9cc7e19d1d43a067f40a2 | A rectangular square of size \( m \times n \) is drawn on grid paper. Find the number of squares that the diagonal of this rectangle divides into two parts. | [
"m + n - \\gcd(m, n)"
] | [
"A rectangular square of size \\( m \\times n \\) is drawn on grid paper. Find the number of squares that the diagonal of this rectangle divides into two parts."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3d7d81cb84747e51af591d50050a226109395403d051ba699f2fe77ecad6d3f9 | A rectangular storage box has a base that is 3 times as long as it is wide. The top is made from a material that costs 35 times the price per square meter of the material used for the rest of the box. Find the dimensions that minimize the cost if the volume must be 3 cubic meters. | [
"\\( w = \\frac{1}{3}, l = 1, h = 9 \\)"
] | [
"A rectangular storage box has a base that is 3 times as long as it is wide. The top is made from a material that costs 35 times the price per square meter of the material used for the rest of the box. Find the dimensions that minimize the cost if the volume must be 3 cubic meters."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 6,
"tir": 25
} | 31 | 0 | 0 |
omr-ws-095f44b99eb8ade8e34186f39b9e8a648a1b0d07ec98e3a8030bc67c70592391 | A rectangular storage box has a base that is three times as long as it is wide. The top is made of a material that costs 35 times the price per square meter of the material used for the rest of the box. Find the dimensions that minimize the cost if the volume of the box must be 3 cubic meters. | [
"\\( w = \\frac{1}{3} \\), \\( l = 1 \\), \\( h = 9 \\)",
"\\( w = \\frac{1}{3} \\, \\text{m}, \\, l = 1 \\, \\text{m}, \\, h = 9 \\, \\text{m} \\)"
] | [
"A rectangular storage box has a base that is three times as long as it is wide. The top is made of a material that costs 35 times the price per square meter of the material used for the rest of the box. Find the dimensions that minimize the cost if the volume of the box must be 3 cubic meters."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 8,
"tir": 54
} | 62 | 0 | 0 |
omr-ws-9124540ec296866410cc5be0f16a61c8e17bbeded797310c1b87eb8222648e02 | A rectangular strip ABCD of paper is 5 cm wide and 50 cm long, yellow on one side and green on the other. First, the strip is folded so that vertex B coincides with the midpoint M of side CD. Then, it is folded again so that vertex D coincides with the midpoint N of side AB. What is the area in cm² of the visible yello... | [
"60 cm²"
] | [
"A rectangular strip ABCD of paper is 5 cm wide and 50 cm long, yellow on one side and green on the other. First, the strip is folded so that vertex B coincides with the midpoint M of side CD. Then, it is folded again so that vertex D coincides with the midpoint N of side AB. What is the area in cm² of the visible ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d72981360d35234d42fe937fd4388a488b9f514e7a14fe876d22b9136bd504a9 | A rectangular strip of paper $ABCD$ is folded along lines $EF$ and $HG$ such that vertices $A$ and $B$ are taken to the same point $A'$ on the perpendicular bisector of segment $AB$, and the angle $\angle HA'E$ is a right angle. The pentagon $A'EFGH$ is obtained. The lower edges of the strip (segments $FC'$ and $GD'$) ... | [
"3"
] | [
"A rectangular strip of paper $ABCD$ is folded along lines $EF$ and $HG$ such that vertices $A$ and $B$ are taken to the same point $A'$ on the perpendicular bisector of segment $AB$, and the angle $\\angle HA'E$ is a right angle. The pentagon $A'EFGH$ is obtained. The lower edges of the strip (segments $FC'$ and $... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-50bc3c29f9614de6c7cd8e2bfe5626c3a230b53919963cf555d232d95940a3d6 | A rectangular strip of width \( w \) and infinite length is folded exactly once. Find the smallest possible area of the region where the strip overlaps with itself. | [
"0"
] | [
"A rectangular strip of width \\( w \\) and infinite length is folded exactly once. Find the smallest possible area of the region where the strip overlaps with itself."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bfbf5600b664c5be2e4c4b2507d5dde4fd4efbd564b3c64614122c29655aea5b | A rectangular swimming pool has the same depth everywhere. The combined surface area of the floor of the pool and the four sides is $4800$ square feet. What is the maximum volume of the swimming pool in cubic feet? | [
"32000"
] | [
"A rectangular swimming pool has the same depth everywhere. The combined surface area of the floor of the pool and the four sides is $4800$ square feet. What is the maximum volume of the swimming pool in cubic feet?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-47c3902cff323a090ed06b2e4435f8532a326c32b841f2535ad4b4cc829b7c60 | A rectangular swimming pool with dimensions 'a' and 'b' is situated in a grass field. A goat is tied to a corner of the pool using a rope. Determine the maximum possible area in which the goat can graze without the grazing areas in the clockwise and counterclockwise directions overlapping. | [
"\\(\\pi(a^2 + b^2 + \\frac{3ab}{2})\\)",
"\\dfrac{\\pi}{2}(a^2 + b^2)"
] | [
"A rectangular swimming pool with dimensions 'a' and 'b' is situated in a grass field. A goat is tied to a corner of the pool using a rope. Determine the maximum possible area in which the goat can graze without the grazing areas in the clockwise and counterclockwise directions overlapping."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 1
} | 3 | 0 | 0 |
omr-ws-fad0aa33752a40e25cfb973c301c1d45f4b63c25a07a2497da92cc512c49dfcd | A rectangular swimming pool with dimensions \(a\) and \(b\) is situated in a grass field. A goat is tied to one corner of the pool using a rope. Determine the maximum possible area in which the goat can graze without the grazing areas in the clockwise and counterclockwise directions overlapping. | [
"\\dfrac{\\pi}{8} \\left(3a^2 + 3b^2 - ab + (a + b)\\sqrt{a^2 + b^2}\\right)"
] | [
"A rectangular swimming pool with dimensions \\(a\\) and \\(b\\) is situated in a grass field. A goat is tied to one corner of the pool using a rope. Determine the maximum possible area in which the goat can graze without the grazing areas in the clockwise and counterclockwise directions overlapping."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 5
} | 8 | 0 | 0 |
omr-ws-995d0071f93cbcae74c64463543cf8f8cd8b6d9a9364f7f925ebf7b29a401686 | A rectangular table measuring 3.55m in length by 1.8m in width with a height of 0.85m is in a room with a single light source directly above the middle of the table. The shadow on the width of the floor (the side parallel to the width of the table) is measured at 2.6m. How high is the light source above the top of the ... | [
"0.57375"
] | [
"A rectangular table measuring 3.55m in length by 1.8m in width with a height of 0.85m is in a room with a single light source directly above the middle of the table. The shadow on the width of the floor (the side parallel to the width of the table) is measured at 2.6m. How high is the light source above the top of... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f0c819c18689c4f805141f12f577f8f1c93bf233d770a93ede7994d9f68cb3da | A rectangular table with 9 rows and 2008 columns is filled with numbers 1 through 2008 such that each number appears exactly 9 times. The difference between any two numbers in the same column is not greater than 3. What is the maximum value of the minimum sum of a column? | [
"24"
] | [
"A rectangular table with 9 rows and 2008 columns is filled with numbers 1 through 2008 such that each number appears exactly 9 times. The difference between any two numbers in the same column is not greater than 3. What is the maximum value of the minimum sum of a column?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-a43792b2ed38860952122a88997a62cde08072a27329e4fe979a2230868a90a2 | A rectangular table with 9 rows and 2008 columns is filled with the numbers 1 through 2008 such that each number appears exactly 9 times. Additionally, the difference between any two numbers in the same column is at most 3. What is the maximum value of the minimum sum of the numbers in any column? | [
"24"
] | [
"A rectangular table with 9 rows and 2008 columns is filled with the numbers 1 through 2008 such that each number appears exactly 9 times. Additionally, the difference between any two numbers in the same column is at most 3. What is the maximum value of the minimum sum of the numbers in any column?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1a0e71bc68a642a4e531fb180fda63cf01d63f5023788f4d8cc89cda6a8b71ec | A rectangular tile measures 3 inches by 4 inches. What is the fewest number of these tiles that are needed to completely cover a rectangular region that is 2 feet by 5 feet? | [
"120"
] | [
"A rectangular tile measures 3 inches by 4 inches. What is the fewest number of these tiles that are needed to completely cover a rectangular region that is 2 feet by 5 feet?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-29d981735ce1490c1410acf63698c98bfe5295aeaa29bd7703dff169ce5dc6e5 | A rectangular top measuring $12$ cm by $16$ cm becomes diagonally folded so that point $B$ coincides with point $A$. The line segment $[MN]$ shows the fold. Determine all possible positions of a point $P$ on the edge of the sheet such that the area of triangle $MNP$ is equal to $54$ cm$^2$. | [
"\\text{No such points exist}"
] | [
"A rectangular top measuring $12$ cm by $16$ cm becomes diagonally folded so that point $B$ coincides with point $A$. The line segment $[MN]$ shows the fold. Determine all possible positions of a point $P$ on the edge of the sheet such that the area of triangle $MNP$ is equal to $54$ cm$^2$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 6
} | 11 | 0 | 0 |
omr-ws-72425bdd5bc2a2ff65938fc237d61603054738f29273c7cd706eb9983799b7a6 | A rectangular warehouse is to be built with a height of 3 meters. The materials available are sufficient to build walls covering 55 square meters. The design includes 4 doors (each 150cm x 210cm) and 5 windows (each 110cm x 110cm). Determine the dimensions of the warehouse that maximize the internal area, given that th... | [
"\\( l = 12800\\sqrt{\\frac{5}{907}} - 300 \\), \\( w = 10\\sqrt{4535} - 300 \\), \\( h = 300 \\) cm"
] | [
"A rectangular warehouse is to be built with a height of 3 meters. The materials available are sufficient to build walls covering 55 square meters. The design includes 4 doors (each 150cm x 210cm) and 5 windows (each 110cm x 110cm). Determine the dimensions of the warehouse that maximize the internal area, given th... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fbb039204df69d9837d91c1b5731ad9ac423dbe67dd00c9411945fc8b076034f | A rectangular warren ABCD (AB < AD) with a perimeter of 240 meters is divided into two sections: a smaller rectangle ABNM for horse stables and the rest for raising cattle. The area for horse stables is one third of the rest. Find the perimeter of the warren used to raise cattle. | [
"200"
] | [
"A rectangular warren ABCD (AB < AD) with a perimeter of 240 meters is divided into two sections: a smaller rectangle ABNM for horse stables and the rest for raising cattle. The area for horse stables is one third of the rest. Find the perimeter of the warren used to raise cattle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-7b2b02214884017b514ea8d4b26218bcbc8bb22af53a1d02e4d38c3ac69ed581 | A rectangular window is divided by a vertical bar into two halves. The left half has a window pane with an area of one-sixth of the total window area and a perimeter of 92 cm. The right half has a window pane with an area of one-eighth of the total window area and a perimeter of 84 cm. Find the perimeters of the entire... | [
"180 cm"
] | [
"A rectangular window is divided by a vertical bar into two halves. The left half has a window pane with an area of one-sixth of the total window area and a perimeter of 92 cm. The right half has a window pane with an area of one-eighth of the total window area and a perimeter of 84 cm. Find the perimeters of the e... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1053a1a251625e2a7ed81e7ac58213fe75392d7b2418a4d7a85bcc62f81b4d62 | A rectangular window with a semicircle on top has a perimeter of 20 feet. What dimensions will maximize the area to let in the most light? | [
"\\(\\frac{20}{3} \\times \\frac{20 - 5\\pi}{3}\\)"
] | [
"A rectangular window with a semicircle on top has a perimeter of 20 feet. What dimensions will maximize the area to let in the most light?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-09f81d27502a7115552bab6d50c649ddae5baa3595b89cbde6e2e306e1fd69e2 | A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. The remainder of the yard has a trapezoidal shape with parallel sides of lengths 15 meters and 25 meters. What fraction of the yard is occupied by the flower beds? | [
"\\(\\frac{1}{5}\\)"
] | [
"A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. The remainder of the yard has a trapezoidal shape with parallel sides of lengths 15 meters and 25 meters. What fraction of the yard is occupied by the flower beds?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 22
} | 37 | 0 | 0 |
omr-ws-b745655d15b72fe9f58214c96c51f494bf50bd2705bf92c5ca9bc05049a117fa | A recursion is defined as $A_n = A_{n+1} + A_{n-1}$. Given that $A_1 = 1$ and $A_2 = 2$, find the sum of the first 300 terms. | [
"0"
] | [
"A recursion is defined as $A_n = A_{n+1} + A_{n-1}$. Given that $A_1 = 1$ and $A_2 = 2$, find the sum of the first 300 terms."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-655da29b63b1e80e03aec4d1f6cd38ef34b6fe6546fa4571eedfd0f2e5e2c51c | A recursive function \(a_n\) is defined as \(a_{xy} = a_{x-1}a_{y-1}\) for all \(x, y > 1\). If \(a_1\) and \(a_2\) are given, compute the number of terms in the first 1000 terms of the sequence that cannot be determined by the information given. | [
"499"
] | [
"A recursive function \\(a_n\\) is defined as \\(a_{xy} = a_{x-1}a_{y-1}\\) for all \\(x, y > 1\\). If \\(a_1\\) and \\(a_2\\) are given, compute the number of terms in the first 1000 terms of the sequence that cannot be determined by the information given."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-28b63b3c8f3ea983b4cc85d18ddad06caa407519866330ceca0f5a3592d50162 | A recursive sequence $a_0, a_1, \ldots$ satisfies $a_0 = 1$ and \[ a_n = \frac{\cos (n \arccos (\tfrac 23)) - (a_1 a_{n - 1} + \ldots + a_{n - 1}a_1)}{2a_0} \] for all positive integers $n.$ If the value of $\sum_{k=0}^\infty \frac{a_k}{2^k}$ can be expressed as $\frac{a \sqrt b}{c}$ for $a, c$ coprime and $b$ squarefr... | [
"23"
] | [
"A recursive sequence $a_0, a_1, \\ldots$ satisfies $a_0 = 1$ and \\[ a_n = \\frac{\\cos (n \\arccos (\\tfrac 23)) - (a_1 a_{n - 1} + \\ldots + a_{n - 1}a_1)}{2a_0} \\] for all positive integers $n.$ If the value of $\\sum_{k=0}^\\infty \\frac{a_k}{2^k}$ can be expressed as $\\frac{a \\sqrt b}{c}$ for $a, c$ coprim... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-58589114bedfc134232120b64a75eddf1891871eb7a692c25719d7a90fb3d4e1 | A recursive sequence \( r_n \) is defined as \( r_1 = 1 \) and \( r_n = nr_{n-1} + n \). Let \( S \) be the sum of all \( n \) such that \( 1 \leq n \leq 1000 \) and \( 40 \mid n \). Find the last two digits of \( S \). | [
"00"
] | [
"A recursive sequence \\( r_n \\) is defined as \\( r_1 = 1 \\) and \\( r_n = nr_{n-1} + n \\). Let \\( S \\) be the sum of all \\( n \\) such that \\( 1 \\leq n \\leq 1000 \\) and \\( 40 \\mid n \\). Find the last two digits of \\( S \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-87479ac93f60fbf6d48b39edf3d3e98e29a64c8fb3a3475a1d4236db45869ada | A recursive sequence has terms in the form \( \frac{a_n}{b_n} \) and is defined such that all even-numbered terms are the reciprocal of the immediately preceding term, and in all odd-numbered terms, \( a_n \) and \( b_n \) are equal to \( a_{n-2} + 1 \) and \( b_{n-2} + 1 \) respectively. If \( t_1 = \frac{2}{3} \), fi... | [
"1"
] | [
"A recursive sequence has terms in the form \\( \\frac{a_n}{b_n} \\) and is defined such that all even-numbered terms are the reciprocal of the immediately preceding term, and in all odd-numbered terms, \\( a_n \\) and \\( b_n \\) are equal to \\( a_{n-2} + 1 \\) and \\( b_{n-2} + 1 \\) respectively. If \\( t_1 = \... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-d164675fd850028d7117521bbd0253383d4176cd1960203753aa4705d1fb7a87 | A recursive sequence is defined as follows: - \( a_1 \) is a positive integer. - \( a_{n+1} = \frac{a_n}{n} \) if \( a_n \) is divisible by \( n \). - \( a_{n+1} = a_n + 1 \) if \( a_n \) is not divisible by \( n \). Given that \( a_{11} = 99 \), how many possible values are there for \( a_1 \)? | [
"144"
] | [
"A recursive sequence is defined as follows:\n- \\( a_1 \\) is a positive integer.\n- \\( a_{n+1} = \\frac{a_n}{n} \\) if \\( a_n \\) is divisible by \\( n \\).\n- \\( a_{n+1} = a_n + 1 \\) if \\( a_n \\) is not divisible by \\( n \\).\n\nGiven that \\( a_{11} = 99 \\), how many possible values are there for \\( a_... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 4
} | 14 | 0 | 0 |
omr-ws-c5bc544173a813dac2adcfb370585c562b56e7ceb0745b650cea0dc692bda034 | A recursive sequence is defined by $f(n) = f(n-2) + f(n-3)$. The sequence begins with $2, 3, 4, 5, 7, 9, \ldots$. Find a closed form for the $N$th term of the sequence. | [
"A \\alpha^n + B \\beta^n + C \\gamma^n"
] | [
"A recursive sequence is defined by $f(n) = f(n-2) + f(n-3)$. The sequence begins with $2, 3, 4, 5, 7, 9, \\ldots$. Find a closed form for the $N$th term of the sequence."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-02cb0217e6d96f241c05430feef5795c198448c29fe69d878619519c3c2c119d | A red ball and a green ball are randomly and independently tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin $k$ is $2^{-k}$ for $k = 1, 2, 3, \ldots$. What is the probability that the red ball is tossed into a higher numbered bin than the green ball? | [
"\\(\\frac{1}{3}\\)"
] | [
"A red ball and a green ball are randomly and independently tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin $k$ is $2^{-k}$ for $k = 1, 2, 3, \\ldots$. What is the probability that the red ball is tossed into a higher numbered bin than the green... | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-5a8e650ee5ebc48e324e7cdc6cf3dfd2d1fb3b396e99fd00e65dcc32371899b7 | A red ball and a green ball are randomly and independently tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin \( k \) is \( 2^{-k} \) for \( k = 1, 2, 3, \ldots \). What is the probability that the red ball is tossed into a higher-numbered bin than the... | [
"\\(\\frac{1}{3}\\)"
] | [
"A red ball and a green ball are randomly and independently tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin \\( k \\) is \\( 2^{-k} \\) for \\( k = 1, 2, 3, \\ldots \\). What is the probability that the red ball is tossed into a higher-numbered ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-03c2b83c22a75a9e024f620e8f192b9153bb0ddd820daeece643fec51446c22b | A red bus departs from Town A at 6:20 am and arrives in Town B at 11:50 am. A blue bus departs from Town B at 3:35 am and arrives in Town A at 9:20 am. If each bus moves at a constant speed without stopping, at what time will the two buses meet? | [
"7:48 am"
] | [
"A red bus departs from Town A at 6:20 am and arrives in Town B at 11:50 am. A blue bus departs from Town B at 3:35 am and arrives in Town A at 9:20 am. If each bus moves at a constant speed without stopping, at what time will the two buses meet?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-264c5109ff9f66424faf36960f03c61b2c7d25b6820b4f78632d694734613bd0 | A red equilateral triangle \( T \) with side length 1 is drawn on a plane. For a positive real \( c \), three blue equilateral triangles with side length \( c \) are placed on the plane to completely cover \( T \). Find the minimum value of \( c \). | [
"\\(\\frac{1}{\\sqrt{3}}\\)"
] | [
"A red equilateral triangle \\( T \\) with side length 1 is drawn on a plane. For a positive real \\( c \\), three blue equilateral triangles with side length \\( c \\) are placed on the plane to completely cover \\( T \\). Find the minimum value of \\( c \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-93c51b82f4349e450626217779c1f9860bef6edb2cd0c7385036d7086bd28fca | A red paper is shaped like a regular hexagon with side length 2. A blue paper is shaped like a square with a diagonal of 4. The square is placed on top of the hexagon so that two opposite vertices of the square coincide with two opposite vertices of the hexagon. Find the area of the region of the hexagon that is not co... | [
"\\(6 - 2\\sqrt{3}\\)"
] | [
"A red paper is shaped like a regular hexagon with side length 2. A blue paper is shaped like a square with a diagonal of 4. The square is placed on top of the hexagon so that two opposite vertices of the square coincide with two opposite vertices of the hexagon. Find the area of the region of the hexagon that is n... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 7,
"tir": 20
} | 38 | 0 | 0 |
omr-ws-6863024f7ec630b7ee9be3af4e22b7c865431f989e1ac8654507c8f6377aea09 | A reflection takes $\begin{pmatrix} -1 \\ 7 \end{pmatrix}$ to $\begin{pmatrix} 5 \\ -5 \end{pmatrix}.$ Which vector does the reflection take $\begin{pmatrix} -4 \\ 3 \end{pmatrix}$ to? | [
"\\begin{pmatrix} 0 \\\\ -5 \\end{pmatrix}"
] | [
"A reflection takes $\\begin{pmatrix} -1 \\\\ 7 \\end{pmatrix}$ to $\\begin{pmatrix} 5 \\\\ -5 \\end{pmatrix}.$ Which vector does the reflection take $\\begin{pmatrix} -4 \\\\ 3 \\end{pmatrix}$ to?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b1d1f8705e3b6ff3dd87f7f670b4392166e724a3a292cb48f2da5dc3d6a0229b | A region $S$ in the complex plane is defined by $S = \{ x+iy : -1 \le x \le 1, -1 \le y \le 1 \}$. A complex number $z = x + iy$ is chosen uniformly at random from $S$. What is the probability that $\left(\frac{3}{4} + \frac{3}{4}i\right)z$ is also in $S$? | [
"\\(\\frac{4}{9}\\)"
] | [
"A region $S$ in the complex plane is defined by $S = \\{ x+iy : -1 \\le x \\le 1, -1 \\le y \\le 1 \\}$. A complex number $z = x + iy$ is chosen uniformly at random from $S$. What is the probability that $\\left(\\frac{3}{4} + \\frac{3}{4}i\\right)z$ is also in $S$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-eef41bcfe90df4d1b3182dbbb6b2cd000abb62368b37bb8abcf7dcdf5aa378c5 | A region $S$ in the complex plane is defined by \begin{align*} S = \{x + iy: - 1\le x\le1, - 1\le y\le1\}. \end{align*}A complex number $z = x + iy$ is chosen uniformly at random from $S$. What is the probability that $\left(\frac34 + \frac34i\right)z$ is also in $S$? | [
"\\frac 79"
] | [
"A region $S$ in the complex plane is defined by \\begin{align*}\nS = \\{x + iy: - 1\\le x\\le1, - 1\\le y\\le1\\}.\n\\end{align*}A complex number $z = x + iy$ is chosen uniformly at random from $S$. What is the probability that $\\left(\\frac34 + \\frac34i\\right)z$ is also in $S$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5cf6ee60398908d07e86828e8aad47c340551b0b741157ce34bfb7f52a213051 | A region \( S \) in the complex plane is defined by \[ S = \{x + iy: -1 \le x \le 1, -1 \le y \le 1\}. \] A complex number \( z = x + iy \) is chosen uniformly at random from \( S \). What is the probability that \( \left(\frac{3}{4} + \frac{3}{4}i\right)z \) is also in \( S \)? | [
"\\(\\frac{17}{18}\\)",
"\\(\\frac{7}{9}\\)"
] | [
"A region \\( S \\) in the complex plane is defined by\n\\[\nS = \\{x + iy: -1 \\le x \\le 1, -1 \\le y \\le 1\\}.\n\\]\nA complex number \\( z = x + iy \\) is chosen uniformly at random from \\( S \\). What is the probability that \\( \\left(\\frac{3}{4} + \\frac{3}{4}i\\right)z \\) is also in \\( S \\)?",
"A re... | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"additional_problems",
"cot",
"tir"
] | {
"additional_problems": 1,
"cot": 47,
"tir": 30
} | 78 | 0 | 0 |
omr-ws-aed582b795b65defa7f567547506f755c73df48fd9bdffeb48fc2d3207d052c0 | A region in the coordinate plane is bounded by the equations $x = 0$, $x = 6$, $y = 0$, and $y = 8$. A line through $(3, 4)$ with slope $4$ cuts the region in half. Another line going through the same point cuts the region into fourths, each with the same area. What is the slope of this line? | [
"-\\dfrac{4}{9}"
] | [
"A region in the coordinate plane is bounded by the equations $x = 0$, $x = 6$, $y = 0$, and $y = 8$. A line through $(3, 4)$ with slope $4$ cuts the region in half. Another line going through the same point cuts the region into fourths, each with the same area. What is the slope of this line?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-d331161fae1d6305ffdbf3d30650f1afbb00571325c9554d1172c62687ed9a74 | A region is enclosed by the curves \( x^2 = 4y \), \( x^2 = -4y \), \( x = 4 \), and \( x = -4 \). \( V_1 \) is the volume of the solid obtained by rotating this region around the y-axis. | [
"\\( 64\\pi \\)"
] | [
"A region is enclosed by the curves \\( x^2 = 4y \\), \\( x^2 = -4y \\), \\( x = 4 \\), and \\( x = -4 \\). \\( V_1 \\) is the volume of the solid obtained by rotating this region around the y-axis."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 1,
"tir": 22
} | 29 | 0 | 0 |
omr-ws-06849f362196ce777caf948ad72a9e57dd2cb91c5fcc9489b946ed0b1ec9bc25 | A regular $(5 \times 5)$-array of lights is defective such that toggling the switch for one light causes each adjacent light in the same row and column, as well as the light itself, to change state. Initially, all the lights are switched off. After a certain number of toggles, exactly one light is switched on. Find all... | [
"(2,2), (2,4), (3,3), (4,2), (4,4)"
] | [
"A regular $(5 \\times 5)$-array of lights is defective such that toggling the switch for one light causes each adjacent light in the same row and column, as well as the light itself, to change state. Initially, all the lights are switched off. After a certain number of toggles, exactly one light is switched on. Fi... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 1,
"tir": 1
} | 4 | 0 | 0 |
omr-ws-bd95de14230577697a9532173cd67d058d93979487107d586600caf9217ead92 | A regular $100$-gon is cut into several parallelograms and two triangles. If the area of one of the triangles is $A$, find the area of the other triangle in terms of $A$. | [
"A"
] | [
"A regular $100$-gon is cut into several parallelograms and two triangles. If the area of one of the triangles is $A$, find the area of the other triangle in terms of $A$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-918a31d5656dd6f576583285424a7056a5627849e72c9fd5818fbfdc4d391bf2 | A regular $12$-gon has vertices $A_1, A_2, \ldots, A_{12}$. If $A_3$, $A_6$, $A_9$, and $A_{12}$ are the midpoints of the four sides of a square with area $1$, compute the area of the $12$-gon. | [
"\\dfrac{3}{4}"
] | [
"A regular $12$-gon has vertices $A_1, A_2, \\ldots, A_{12}$. If $A_3$, $A_6$, $A_9$, and $A_{12}$ are the midpoints of the four sides of a square with area $1$, compute the area of the $12$-gon."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-99c267ecd39c860bc35e49318c1627b786d2eb8162b48d207fa025e67214eabc | A regular $17$-gon with vertices $V_1, V_2, \ldots, V_{17}$ and sides of length $3$ has a point $P$ on $V_1V_2$ such that $V_1P = 1$. A chord from $V_1$ to $V_2$ containing $P$ is rotated within the interior of the heptadecagon around $V_2$, then $V_3$, and so on, until $P$ returns to its original position. Find the to... | [
"\\( 45\\pi \\)"
] | [
"A regular $17$-gon with vertices $V_1, V_2, \\ldots, V_{17}$ and sides of length $3$ has a point $P$ on $V_1V_2$ such that $V_1P = 1$. A chord from $V_1$ to $V_2$ containing $P$ is rotated within the interior of the heptadecagon around $V_2$, then $V_3$, and so on, until $P$ returns to its original position. Find ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-611c8c1725fb745fa888ee9c7e1ee8c4683749ffab7089f47b852b7d2c6724fc | A regular $2008$-gon is located in the Argand plane such that $(x_1,y_1)=(p,0)$ and $(x_{1005},y_{1005})=(p+2,0)$ where $p$ is a prime and vertices $(x_1,y_1),(x_2,y_2),\cdots,(x_{2008},y_{2008})$ are arranged in counterclockwise order. Let $$S=\prod_{k=1}^{1004}(x_{2k-1}+iy_{2k-1})=(x_1+iy_1)(x_3+iy_3)\cdots(x_{2007}+... | [
"2"
] | [
"A regular $2008$-gon is located in the Argand plane such that $(x_1,y_1)=(p,0)$ and $(x_{1005},y_{1005})=(p+2,0)$ where $p$ is a prime and vertices $(x_1,y_1),(x_2,y_2),\\cdots,(x_{2008},y_{2008})$ are arranged in counterclockwise order. Let\n$$S=\\prod_{k=1}^{1004}(x_{2k-1}+iy_{2k-1})=(x_1+iy_1)(x_3+iy_3)\\cdots(... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-da1125448cf1e9f402c6f97ba714aae01e2d540240b1c024f735bc79e96bdab1 | A regular $2008$-gon is located in the Cartesian plane such that $(x_1,y_1)=(p,0)$ and $(x_{1005},y_{1005})=(p+2,0)$, where $p$ is prime and the vertices \[(x_1,y_1),(x_2,y_2),(x_3,y_3),\cdots,(x_{2008},y_{2008})\] are arranged in counterclockwise order. Let \begin{align*}S&=(x_1+y_1i)(x_3+y_3i)(x_5+y_5i)\cdots(x_{2007... | [
"2"
] | [
"A regular $2008$-gon is located in the Cartesian plane such that $(x_1,y_1)=(p,0)$ and $(x_{1005},y_{1005})=(p+2,0)$, where $p$ is prime and the vertices \\[(x_1,y_1),(x_2,y_2),(x_3,y_3),\\cdots,(x_{2008},y_{2008})\\] are arranged in counterclockwise order. Let \\begin{align*}S&=(x_1+y_1i)(x_3+y_3i)(x_5+y_5i)\\cdo... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-c41158d6b097bf1fe05372674bd3252f4057f20e11e47f283f7d29f42f2300cf | A regular $2012$-gon is inscribed in a circle. Find the maximal $k$ such that we can choose $k$ vertices from the $2012$ vertices and construct a convex $k$-gon without parallel sides. | [
"1509"
] | [
"A regular $2012$-gon is inscribed in a circle. Find the maximal $k$ such that we can choose $k$ vertices from the $2012$ vertices and construct a convex $k$-gon without parallel sides."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4abb0f70b36cc079e17194ad3aefe0945c257e4421ed588d6974ac510636459a | A regular $2018$-gon is inscribed in a circle. The numbers $1, 2, \ldots, 2018$ are arranged on the vertices of the $2018$-gon such that the sum of any two neighboring numbers equals the sum of the two numbers that are on the antipodes of those two vertices (with respect to the given circle). Determine the number of di... | [
"\\(2 \\cdot 1008!\\)"
] | [
"A regular $2018$-gon is inscribed in a circle. The numbers $1, 2, \\ldots, 2018$ are arranged on the vertices of the $2018$-gon such that the sum of any two neighboring numbers equals the sum of the two numbers that are on the antipodes of those two vertices (with respect to the given circle). Determine the number... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7042bd58abb664b758b98f4de5803e1852684b9379d19ece7afecf0a810ea990 | A regular $2n$-gon is inscribed in a circle, and a regular $n$-gon is circumscribed about the same circle. The difference between the areas of the polygons is $P$. Determine the radius of the circle. | [
"$R=\\sqrt {\\frac{P}{n\\left(\\tan\\frac{180^\\circ}{n}-\\sin\\frac{180^\\circ}{n}\\right)}} =\\frac{1}{\\sin\\frac{90^\\circ}{n}} \\sqrt {\\frac{P\\cot\\frac{180^\\circ}{n}}{2n}}$"
] | [
"A regular $2n$-gon is inscribed in a circle, and a regular $n$-gon is circumscribed about the same circle. The difference between the areas of the polygons is $P$. Determine the radius of the circle."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-780d8729e7cabb5d195d5604b3aaf00d3acfaec9254a8e3a67b7f31d6d262e40 | A regular $61$-gon with vertices $A_1, A_2, A_3, \ldots, A_{61}$ is inscribed in a circle with radius $r$. Given that $$(\overline{A_1A_2})(\overline{A_1A_3})(\overline{A_1A_4}) \cdots (\overline{A_1A_{61}}) = r,$$ and $r^{2006} = \frac{p}{q}$ where $p$ and $q$ are coprime positive integers, find the remainder when $p+... | [
"142"
] | [
"A regular $61$-gon with vertices $A_1, A_2, A_3, \\ldots, A_{61}$ is inscribed in a circle with radius $r$. Given that $$(\\overline{A_1A_2})(\\overline{A_1A_3})(\\overline{A_1A_4}) \\cdots (\\overline{A_1A_{61}}) = r,$$ and $r^{2006} = \\frac{p}{q}$ where $p$ and $q$ are coprime positive integers, find the remain... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1d9177c69e370a7824d82207be57ea06e0e2179d964d4eb374a0aafa3e5528e8 | A regular $n$-gon has area $1$ square unit, and every regular $n$-gon inscribed in it has area at least $0.99$ square units. What is the minimum value of $n$? | [
"32"
] | [
"A regular $n$-gon has area $1$ square unit, and every regular $n$-gon inscribed in it has area at least $0.99$ square units. What is the minimum value of $n$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8f0ce6b8ccf607b18578037cd68a00e3bda35771dc2d62d76b27709fdf7e5b4e | A regular $n$-gon is inscribed in a circle of radius $1$. Let $X$ be the set of all arcs $PQ$, where $P$ and $Q$ are distinct vertices of the $n$-gon. Five elements $L_1, L_2, \ldots, L_5$ of $X$ are chosen at random (with replacement). Calculate the expected length of the intersection $L_1 \cap L_2 \cap L_3 \cap L_4 \... | [
"2\\pi \\left( \\dfrac{7}{22} \\right)^5"
] | [
"A regular $n$-gon is inscribed in a circle of radius $1$. Let $X$ be the set of all arcs $PQ$, where $P$ and $Q$ are distinct vertices of the $n$-gon. Five elements $L_1, L_2, \\ldots, L_5$ of $X$ are chosen at random (with replacement). Calculate the expected length of the intersection $L_1 \\cap L_2 \\cap L_3 \\... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-8676f74df5d74897d081c94a3e6cf708a84df981d1c6bd4c7273eba247c2c02e | A regular $n$-gon is inscribed in a circle of radius $1$. Let $a_1, \cdots, a_{n-1}$ be the distances from one of the vertices of the polygon to all the other vertices. Calculate the value of \[ (5-a_1^2)\cdots(5-a_{n-1}^2) \] for $n=10$. | [
"3025"
] | [
"A regular $n$-gon is inscribed in a circle of radius $1$. Let $a_1, \\cdots, a_{n-1}$ be the distances from one of the vertices of the polygon to all the other vertices. Calculate the value of\n\\[\n(5-a_1^2)\\cdots(5-a_{n-1}^2)\n\\]\nfor $n=10$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-4610421d5e81d9bbdea524e214b45bf7e47d733d0212540736c5c3ba6ad0d446 | A regular $n$-gon is inscribed in a circle of radius $n$. Find the sum of all the sides and diagonals of the polygon in terms of $n$. | [
"n^2 \\cot \\left( \\frac{\\pi}{2n} \\right)"
] | [
"A regular $n$-gon is inscribed in a circle of radius $n$. Find the sum of all the sides and diagonals of the polygon in terms of $n$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
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