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-9a04a7c00bd9a4eebb24f2291046443276493bd3935106ea324d3482e2e289fc | A rectangle with side lengths 137 and 1806 is divided into unit squares, and 137 sentries are placed, each in their own unit square. Each sentry guards every unit square in the same row and column it is in, but no sentry can guard a square that another sentry is in. There are $k$ ways to place the sentries. What is the... | [
"416"
] | [
"A rectangle with side lengths 137 and 1806 is divided into unit squares, and 137 sentries are placed, each in their own unit square. Each sentry guards every unit square in the same row and column it is in, but no sentry can guard a square that another sentry is in. There are $k$ ways to place the sentries. What i... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-d1018ee254eb0e5491db7dbaf9b40ed0492cd7f0ac85a293a4304370f9a110a0 | A rectangle with side lengths 3 and 4 is rotated around its diagonal. The volume of the resulting figure can be expressed in the form $\frac{a\pi}{b}$, where $a$ and $b$ are relatively prime positive integers. Find the remainder when $a+b$ is divided by 1000. | [
"589"
] | [
"A rectangle with side lengths 3 and 4 is rotated around its diagonal. The volume of the resulting figure can be expressed in the form $\\frac{a\\pi}{b}$, where $a$ and $b$ are relatively prime positive integers. Find the remainder when $a+b$ is divided by 1000."
] | [
"aops_c5_contests_amp_programs"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-a50701facbf822bc1fa07c7b471ea64140a566248ac7430205abb9156d2f30f1 | A rectangle with side lengths \(a_0, b_0\) is dissected into smaller rectangles with side lengths \(a_k, b_k\) for \(1 \le k \le n\). The sides of the smaller rectangles are parallel to the corresponding sides of the larger rectangle. If \(a_0 = \frac{\pi}{3}\) and \(b_0 = \frac{\pi}{4}\), and the dissection results in... | [
"\\dfrac{\\sqrt{6}}{4}"
] | [
"A rectangle with side lengths \\(a_0, b_0\\) is dissected into smaller rectangles with side lengths \\(a_k, b_k\\) for \\(1 \\le k \\le n\\). The sides of the smaller rectangles are parallel to the corresponding sides of the larger rectangle. If \\(a_0 = \\frac{\\pi}{3}\\) and \\(b_0 = \\frac{\\pi}{4}\\), and the ... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-17c5c9f94e4f78b0833062ae02ea6bbb7db511a9c5ab314e3029f7ffccabfaea | A rectangle with sides $n$ and $p$ is divided into $np$ unit squares. Initially, $m$ unit squares are painted black, and the rest are painted white. The process is as follows: if a white unit square has at least two sides in common with black unit squares, it turns black. Find the smallest integer $m$ such that there e... | [
"\\max(n, p)"
] | [
"A rectangle with sides $n$ and $p$ is divided into $np$ unit squares. Initially, $m$ unit squares are painted black, and the rest are painted white. The process is as follows: if a white unit square has at least two sides in common with black unit squares, it turns black. Find the smallest integer $m$ such that th... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 2
} | 6 | 0 | 0 |
omr-ws-5408b90083384be7b00610ba12742972124503ae9587f76c2b3f72502c5585d2 | A rectangle with sides \( A \) and \( B \) is divided into a finite number of squares with side lengths \( x_1, x_2, \ldots, x_n \). If \( A = 12 \) and \( B = 18 \), find the sum of all possible values of \( x_i \) such that \( \frac{x_i}{A} \) and \( \frac{x_i}{B} \) are rational. | [
"12"
] | [
"A rectangle with sides \\( A \\) and \\( B \\) is divided into a finite number of squares with side lengths \\( x_1, x_2, \\ldots, x_n \\). If \\( A = 12 \\) and \\( B = 18 \\), find the sum of all possible values of \\( x_i \\) such that \\( \\frac{x_i}{A} \\) and \\( \\frac{x_i}{B} \\) are rational."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-d1807c0b033b10efe1cd50efb44587f872701d63295bf12ae5d7f9b7f97342c5 | A rectangle with sides \( m \ge 3 \) and \( n \ge 3 \) consists of \( mn \) unit squares. If in each of those unit squares exactly one number is written, such that any \( 3 \times 3 \) square within the rectangle is a magic square, find the maximum number of different numbers that can be written in the rectangle. | [
"9"
] | [
"A rectangle with sides \\( m \\ge 3 \\) and \\( n \\ge 3 \\) consists of \\( mn \\) unit squares. If in each of those unit squares exactly one number is written, such that any \\( 3 \\times 3 \\) square within the rectangle is a magic square, find the maximum number of different numbers that can be written in the ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-00aa53f7a322f2c8e6cb8a3485a7df8264ab969122828719e930213e3ca07765 | A rectangle with sides \((2m-1)\) cm and \((2n-1)\) cm is divided into squares by drawing parallel lines such that the distance between two consecutive parallel lines is 1 cm. How many rectangles can be drawn on the figure such that both the length and breadth are in odd centimeters? | [
"\\(m^2 n^2\\)"
] | [
"A rectangle with sides \\((2m-1)\\) cm and \\((2n-1)\\) cm is divided into squares by drawing parallel lines such that the distance between two consecutive parallel lines is 1 cm. How many rectangles can be drawn on the figure such that both the length and breadth are in odd centimeters?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 10
} | 25 | 0 | 0 |
omr-ws-8d5d5feffc5fca69b286f61ecf511088dc418f0a82433e361a80237dd812cc0e | A rectangle with sides \(a\) and \(b\) (\(a > b\)) has mirrors on three sides and an opening in the middle of the fourth side. A light ray enters the opening. At what angle to the normal should it enter in order to come back through the same point that it entered if it only bounces once off each mirror? | [
"\\arctan\\left(\\frac{a}{b}\\right)"
] | [
"A rectangle with sides \\(a\\) and \\(b\\) (\\(a > b\\)) has mirrors on three sides and an opening in the middle of the fourth side. A light ray enters the opening. At what angle to the normal should it enter in order to come back through the same point that it entered if it only bounces once off each mirror?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-509585a898857ebdaf4115f07b239aecb709dc5788f69a8d98954fcf34d9f6d6 | A rectangle with sides \(a\) and \(b\) (\(a > b\)) has mirrors on three sides and an opening in the middle of the fourth side. A light ray enters the opening. For what angles would the light ray eventually come back through the same point that it entered if it is allowed to bounce off as many times as necessary? | [
"\\theta = \\arctan\\left(\\frac{m b}{n a}\\right)"
] | [
"A rectangle with sides \\(a\\) and \\(b\\) (\\(a > b\\)) has mirrors on three sides and an opening in the middle of the fourth side. A light ray enters the opening. For what angles would the light ray eventually come back through the same point that it entered if it is allowed to bounce off as many times as necess... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 4
} | 13 | 0 | 0 |
omr-ws-4ef75d100616f99f6d9a99bce8de4aa2bf0a6a2df9ef3ce9fe098a79a206ec1d | A rectangle with sides of $5$ and $12$ has one of the diagonals drawn. Two circles are inscribed in each of the triangles formed. Compute the distance between the centers of these two inscribed circles. | [
"$\\sqrt{65}$"
] | [
"A rectangle with sides of $5$ and $12$ has one of the diagonals drawn. Two circles are inscribed in each of the triangles formed. Compute the distance between the centers of these two inscribed circles."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-82ceb0d0db6e2c0aac962ffe6030fe962e09900940e1c6cc9b46b04b554c04fe | A rectangle with sides parallel to the $x$-axis and $y$-axis has opposite vertices at $(15, 3)$ and $(16, 5)$. A line is drawn through points $A(0, 0)$ and $B(3, 1)$. Another line is drawn through points $C(0, 10)$ and $D(2, 9)$. How many points on the rectangle lie on at least one of the two lines? | [
"1"
] | [
"A rectangle with sides parallel to the $x$-axis and $y$-axis has opposite vertices at $(15, 3)$ and $(16, 5)$. A line is drawn through points $A(0, 0)$ and $B(3, 1)$. Another line is drawn through points $C(0, 10)$ and $D(2, 9)$. How many points on the rectangle lie on at least one of the two lines?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 17
} | 17 | 0 | 0 |
omr-ws-efdfb0425a5e3d70c47411dcf4b2494c1707888ba0df5dbaaf6683832596bea8 | A rectangle with the greatest area is inscribed in a sector of a circle of radius $R$ (the central angle of the sector being $\alpha $). Calculate the area. | [
"$S_{max}=R^2\\tan\\frac{\\alpha }{2}$"
] | [
"A rectangle with the greatest area is inscribed in a sector of a circle of radius $R$ (the central angle of the sector being $\\alpha $). Calculate the area."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f16f07638d8c951215e4cc21d2c658279d21483fef6e66b1019720b0b65ebfe9 | A rectangle with vertices $A(-6, 1), B(-2,1), C(-2,7),$ and $D(-6,7)$ is rotated $45^{\circ}$ clockwise about the point $(-6,1)$. What is the area of the portion of this rectangle that lies above the $x$-axis? | [
"\\( 15 + 4\\sqrt{2} \\)"
] | [
"A rectangle with vertices $A(-6, 1), B(-2,1), C(-2,7),$ and $D(-6,7)$ is rotated $45^{\\circ}$ clockwise about the point $(-6,1)$. What is the area of the portion of this rectangle that lies above the $x$-axis?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-f1df1b728a8e6290ed2e5009dc1f311afd4dde47fa87ecb0e79e9d1f940add39 | A rectangle's length is twice its width, and its area is 18 square units. What is the length of the rectangle? | [
"12"
] | [
"A rectangle's length is twice its width, and its area is 18 square units. What is the length of the rectangle?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5f5be924c681e89441b4a702fe72bc518dd7b8e6deb070063914867c5e376a7a | A rectangle, which is not a square, has sides with integer lengths, measured in centimeters. If its perimeter is \( n \) centimeters and its area is \( n \) square centimeters, determine \( n \). | [
"18"
] | [
"A rectangle, which is not a square, has sides with integer lengths, measured in centimeters. If its perimeter is \\( n \\) centimeters and its area is \\( n \\) square centimeters, determine \\( n \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bdaefa2ae6aa59dd67b247462c9042c165323155ad5a2707a5074ed16593c61c | A rectangle, whose one side length is twice the other side, is inscribed inside a triangle with sides $3$ cm, $4$ cm, and $5$ cm, such that the longer sides of the rectangle lie entirely on the longest side of the triangle. The other two vertices of the rectangle lie respectively on the other two sides of the triangle.... | [
"\\(\\frac{60}{49}, \\frac{120}{49}\\)"
] | [
"A rectangle, whose one side length is twice the other side, is inscribed inside a triangle with sides $3$ cm, $4$ cm, and $5$ cm, such that the longer sides of the rectangle lie entirely on the longest side of the triangle. The other two vertices of the rectangle lie respectively on the other two sides of the tria... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 14
} | 16 | 0 | 0 |
omr-ws-20545feeeb19f9f75cb0d2ac342703a974d252a2c781e30b28f5d33d78229aeb | A rectangle-shaped puzzle is assembled with 2000 pieces that are all equal rectangles, similar to the large rectangle, with the sides of the small rectangles parallel to those of the large one. The shortest side of each piece measures 1. Determine the minimum possible value of the area of the large rectangle. | [
"2500"
] | [
"A rectangle-shaped puzzle is assembled with 2000 pieces that are all equal rectangles, similar to the large rectangle, with the sides of the small rectangles parallel to those of the large one. The shortest side of each piece measures 1. Determine the minimum possible value of the area of the large rectangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 3
} | 14 | 0 | 0 |
omr-ws-44d1339e0beee44292d3785b5698fea6c0e8ee2d718b79acb19301b432b3b6b6 | A rectangular $2n$-sided polygon with center $O$ is painted using $n$ colors such that: 1. Each vertex is painted with exactly one color. 2. Each color is used to paint exactly two vertices. Two ways of painting are considered the same if one can be obtained from the other by a rotation about $O$. How many distinct way... | [
"\\(\\left\\lfloor \\frac{2n-1}{2} \\right\\rfloor \\cdot \\frac{(2n-2)!}{2^{n-1}}\\)"
] | [
"A rectangular $2n$-sided polygon with center $O$ is painted using $n$ colors such that:\n1. Each vertex is painted with exactly one color.\n2. Each color is used to paint exactly two vertices.\nTwo ways of painting are considered the same if one can be obtained from the other by a rotation about $O$. How many dist... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3354c5e31fac78d5ba57697d5562212bf79297302d0e81e20df2cdba08a9f90f | A rectangular aquarium is \(\frac{2}{3}\) filled with water and then is tilted on its side until the water level coincides with one edge of the bottom. During the tilting process, 6 quarts of water are poured out. What is the volume of the tank in quarts? | [
"36"
] | [
"A rectangular aquarium is \\(\\frac{2}{3}\\) filled with water and then is tilted on its side until the water level coincides with one edge of the bottom. During the tilting process, 6 quarts of water are poured out. What is the volume of the tank in quarts?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-a5f3766e049d51b4ff3c9bce76bc71df2f2dd64c2fbfd6bf17e19594548c3e99 | A rectangular area must be designed so that it is surrounded by a guard net on three sides and a long brick wall on the fourth side. What are the most advantageous dimensions of the area that will maximize the surface area, if $l$ running meters of net are available? | [
"The side of the area adjacent to the wall must be twice as large as the other side"
] | [
"A rectangular area must be designed so that it is surrounded by a guard net on three sides and a long brick wall on the fourth side. What are the most advantageous dimensions of the area that will maximize the surface area, if $l$ running meters of net are available?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9f5589b681e9b444db6b4a617f3de84d6fc6aeeced1263a6e31637e0564e465e | A rectangular area of 9000 m$^2$ must be surrounded by a fence, with two opposite sides being made of brick and the other two of wood. One meter of wooden fencing costs 10 roubles while one meter of brick walling costs 25 roubles. What is the least amount of money that must be allotted to the construction of such a fen... | [
"6000 rubles"
] | [
"A rectangular area of 9000 m$^2$ must be surrounded by a fence, with two opposite sides being made of brick and the other two of wood. One meter of wooden fencing costs 10 roubles while one meter of brick walling costs 25 roubles. What is the least amount of money that must be allotted to the construction of such ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-025054a561d2a1ae98e5e1146662a510c9991b02819bc757ec4c7e556627e1fa | A rectangular array of chairs is an arrangement of the chairs in rows and columns such that each row contains the same number of chairs as every other row and each column contains the same number of chairs as every other column. If there must be at least two chairs in every row and column and all of the chairs in the r... | [
"7"
] | [
"A rectangular array of chairs is an arrangement of the chairs in rows and columns such that each row contains the same number of chairs as every other row and each column contains the same number of chairs as every other column. If there must be at least two chairs in every row and column and all of the chairs in... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-39e5ccf6fe3e79abbfb6d85bf056f02bc56c38d7625bb6e8f2ebe2cbd2d1dabe | A rectangular array of positive integers has 4 rows. The sum of the entries in each column is 20. Within each row, all entries are distinct. What is the maximum possible number of columns? | [
"9"
] | [
"A rectangular array of positive integers has 4 rows. The sum of the entries in each column is 20. Within each row, all entries are distinct. What is the maximum possible number of columns?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 27,
"genselect": 7
} | 34 | 0 | 0 |
omr-ws-a05520189e8941207cb095829685a44d2b0e6fb318ef51ddef718dad6b5709d7 | A rectangular band formation is a formation with $m$ band members in each of $r$ rows, where $m$ and $r$ are integers. A particular band has less than 100 band members. The director arranges them in a rectangular formation and finds that he has two members left over. If he increases the number of members in each row by... | [
"98"
] | [
"A rectangular band formation is a formation with $m$ band members in each of $r$ rows, where $m$ and $r$ are integers. A particular band has less than 100 band members. The director arranges them in a rectangular formation and finds that he has two members left over. If he increases the number of members in each r... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f98f26f1330cd68a11f05c1f51ed44847f3d70994aa93419460724af96dcd8c5 | A rectangular bath $20\ {\rm cm}\times 90\ {\rm cm}\times 25\ {\rm cm}$ (a rectangular parallelepiped) is used to wash photographic negatives. Water flows in through one pipe and, at the same time, out through another pipe to ensure its constant agitation in the bath. It requires 5 minutes less time to empty the bath t... | [
"$2\\frac{1}{4}$ l/min; 3 l/min"
] | [
"A rectangular bath $20\\ {\\rm cm}\\times 90\\ {\\rm cm}\\times 25\\ {\\rm cm}$ (a rectangular parallelepiped) is used to wash photographic negatives. Water flows in through one pipe and, at the same time, out through another pipe to ensure its constant agitation in the bath. It requires 5 minutes less time to emp... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-da453f37c4fb97cdc5e6bc41212abd627df57fed77343a66c3e4862737d257ed | A rectangular beam is cut from a cylindrical log of radius 30 cm. The strength of a beam of width \( w \) and height \( h \) is proportional to \( wh^2 \). Find the width and height of the beam of maximum strength. | [
"\\( w = 20\\sqrt{3}, h = 20\\sqrt{6} \\)"
] | [
"A rectangular beam is cut from a cylindrical log of radius 30 cm. The strength of a beam of width \\( w \\) and height \\( h \\) is proportional to \\( wh^2 \\). Find the width and height of the beam of maximum strength."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 30
} | 43 | 0 | 0 |
omr-ws-2098e3307acaa4824dbb7ba99b269c65dfd1814c34b04e2b4873614a236f4ef3 | A rectangular billiards table is 2 meters by 3 meters. A ball is on the short edge at a distance of $\frac{\sqrt{3}}{2}$ meters from a corner. It is hit in such a way that it strikes each of the other walls once and returns to its starting point. How many meters did it travel? | [
"\\(2\\sqrt{13}\\)"
] | [
"A rectangular billiards table is 2 meters by 3 meters. A ball is on the short edge at a distance of $\\frac{\\sqrt{3}}{2}$ meters from a corner. It is hit in such a way that it strikes each of the other walls once and returns to its starting point. How many meters did it travel?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 4,
"tir": 12
} | 24 | 0 | 0 |
omr-ws-35ae09dec3d7f35efe162ed4f0e490b312e18014b6b7c315552d9c05e47e6ce4 | A rectangular block with dimensions \(a \times 1995 \times c\), where \(a \leq 1995 \leq c\), is cut into two non-empty parts by a plane parallel to one of the faces. One of the parts is similar to the original block. How many possibilities are there for the pair \((a, c)\)? | [
"8"
] | [
"A rectangular block with dimensions \\(a \\times 1995 \\times c\\), where \\(a \\leq 1995 \\leq c\\), is cut into two non-empty parts by a plane parallel to one of the faces. One of the parts is similar to the original block. How many possibilities are there for the pair \\((a, c)\\)?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-046f38c324e72bd6fc8f84c3eb18add81b0e6360909fa525e61fe7512404173a | A rectangular board is said to be magnificent if, for each line $L$ and for each pair of columns $C$ and $D$, there is another line $M$ exactly equal to $L$, except in columns $C$ and $D$, where $M$ has symbols different from those of $L$. What is the smallest possible number of rows on a magnificent board with $2023$ ... | [
"2024"
] | [
"A rectangular board is said to be magnificent if, for each line $L$ and for each pair of columns $C$ and $D$, there is another line $M$ exactly equal to $L$, except in columns $C$ and $D$, where $M$ has symbols different from those of $L$. What is the smallest possible number of rows on a magnificent board with $2... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 5
} | 11 | 0 | 0 |
omr-ws-2eb547968e84d9850f75c38b52fb7e6ca7c5c8a50eabcfa574605d3996f56383 | A rectangular board with length \(10\) cm and width \(8\) cm has each corner cut off by a square of side \(x\) cm. The remaining shape is folded to form a cuboid without a lid. If the volume of the cuboid is at least \(48\) cubic cm, find the range of \(x\). | [
"\\(1 \\leq x \\leq 2\\)"
] | [
"A rectangular board with length \\(10\\) cm and width \\(8\\) cm has each corner cut off by a square of side \\(x\\) cm. The remaining shape is folded to form a cuboid without a lid. If the volume of the cuboid is at least \\(48\\) cubic cm, find the range of \\(x\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 22
} | 37 | 0 | 0 |
omr-ws-bb8858c6979bafde8d938b34b872f61382a7af2645531322a592e2d74b5d5730 | A rectangular box $P$ is inscribed in a sphere of radius $r$. The surface area of $P$ is 384, and the sum of the lengths of its 12 edges is 112. What is $r$? | [
"10"
] | [
"A rectangular box $P$ is inscribed in a sphere of radius $r$. The surface area of $P$ is 384, and the sum of the lengths of its 12 edges is 112. What is $r$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a5bdbe19523044c55b1ff2a5799c4ad542c5372cfc745f196de323801890606c | A rectangular box $\mathcal{P}$ has distinct edge lengths $a, b,$ and $c$. The sum of the lengths of all $12$ edges of $\mathcal{P}$ is $13$, the sum of the areas of all $6$ faces of $\mathcal{P}$ is $\frac{11}{2}$, and the volume of $\mathcal{P}$ is $\frac{1}{2}$. What is the length of the longest interior diagonal co... | [
"\\(\\frac{9}{4}\\)"
] | [
"A rectangular box $\\mathcal{P}$ has distinct edge lengths $a, b,$ and $c$. The sum of the lengths of all $12$ edges of $\\mathcal{P}$ is $13$, the sum of the areas of all $6$ faces of $\\mathcal{P}$ is $\\frac{11}{2}$, and the volume of $\\mathcal{P}$ is $\\frac{1}{2}$. What is the length of the longest interior ... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-752eefcb8d33290109e6a09b271c6074f1b78522f3634dd0207629a158edd24f | A rectangular box \( P \) is inscribed in a sphere of radius \( r \). The surface area of \( P \) is 384, and the sum of the lengths of its 12 edges is 112. What is \( r \)? | [
"10"
] | [
"A rectangular box \\( P \\) is inscribed in a sphere of radius \\( r \\). The surface area of \\( P \\) is 384, and the sum of the lengths of its 12 edges is 112. What is \\( r \\)?"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 31,
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} | 38 | 0 | 0 |
omr-ws-8d601c5ebe232e8ed7c1e3ee014c29690855b2fe5c75dc6ef802cfaa5081f203 | A rectangular box \( P \) with integer side lengths is inscribed in a sphere of radius \( r \). The volume of \( P \) is 30 and the sum of the lengths of its 12 edges is 120. What is the volume of the sphere with radius \( r \)? | [
"\\(\\frac{152 \\pi \\sqrt{38}}{3}\\)"
] | [
"A rectangular box \\( P \\) with integer side lengths is inscribed in a sphere of radius \\( r \\). The volume of \\( P \\) is 30 and the sum of the lengths of its 12 edges is 120. What is the volume of the sphere with radius \\( r \\)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
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"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e58631a7154d4b159b32b3a64f7c79085b6e36a0ad79c83190c90def989e181f | A rectangular box has a total surface area of $94$ square inches and the sum of the lengths of all its edges is $48$ inches. What is the sum of the lengths in inches of all of its interior diagonals? | [
"\\(20 \\sqrt{2}\\)"
] | [
"A rectangular box has a total surface area of $94$ square inches and the sum of the lengths of all its edges is $48$ inches. What is the sum of the lengths in inches of all of its interior diagonals?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
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"cot": 16,
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} | 40 | 0 | 0 |
omr-ws-cc4580e5bea367644e132d2d13ea112e01324292945edbb6f2a60d4495a670d1 | A rectangular box has a volume of $4320$ cubic inches and a surface area of $1704$ square inches. The sum of the lengths of its $12$ edges is $208$ inches. What would be the volume of the box, in cubic inches, if its length, width and height were each increased by one inch? | [
"5225 \\text{ cubic inches}"
] | [
"A rectangular box has a volume of $4320$ cubic inches and a surface area of $1704$ square inches. The sum of the lengths of its $12$ edges is $208$ inches. What would be the volume of the box, in cubic inches, if its length, width and height were each increased by one inch?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0fa0f44ecbb04dbcc4b996971fb03a8df07a534cb1da83cbdf81f3bef70bcffe | A rectangular box has a volume of $64$. Find the sum of the dimensions of the box with the minimum surface area. | [
"12"
] | [
"A rectangular box has a volume of $64$. Find the sum of the dimensions of the box with the minimum surface area."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-90053ddd177e139332c21ffb8e3c11f7398a1b8bd4337a96cf81333cc7a0b7f4 | A rectangular box has a volume of 108 cubic feet. How many cubic yards are in the volume of this box? | [
"4"
] | [
"A rectangular box has a volume of 108 cubic feet. How many cubic yards are in the volume of this box?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8eeb4304fdf21624e4cd7e21d7eb88e2cb47f3e562ee5d12bb25d496b5284f58 | A rectangular box has a volume of 48, and the sum of the lengths of its twelve edges is 48. Find the largest integer that could be the length of an edge of the box. | [
"6"
] | [
"A rectangular box has a volume of 48, and the sum of the lengths of its twelve edges is 48. Find the largest integer that could be the length of an edge of the box."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-ae6c2d656d086bf72107bc84d2046502199da27ea391aa1e3b06574c6f7d5187 | A rectangular box has interior dimensions 6-inches by 5-inches by 10-inches. The box is filled with as many solid 3-inch cubes as possible, with all of the cubes entirely inside the rectangular box. What percent of the volume of the box is taken up by the cubes? | [
"54"
] | [
"A rectangular box has interior dimensions 6-inches by 5-inches by 10-inches. The box is filled with as many solid 3-inch cubes as possible, with all of the cubes entirely inside the rectangular box. What percent of the volume of the box is taken up by the cubes?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4f4f4f80155f3ab6d7a87b32bf79d687c17a84cfa9d3c26888ffba6c0c93eaaf | A rectangular box has sides of integer length and a surface area of 2010 square units. What is the maximum volume it could have? | [
"5850"
] | [
"A rectangular box has sides of integer length and a surface area of 2010 square units. What is the maximum volume it could have?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c7daa5ea5efe6b8b437a4b2db7f2fd5ff6ccc4a73b9e6c1079b0ca879a5597af | A rectangular box has sides of integer length and a volume of 2013 cubic units. What is its smallest possible surface area? | [
"1774"
] | [
"A rectangular box has sides of integer length and a volume of 2013 cubic units. What is its smallest possible surface area?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bcfa1c4ca87a5a888c9f6eb824e2ff77277045d818305538053abce2085883b9 | A rectangular box has width $12$ inches, length $16$ inches, and height $\frac{m}{n}$ inches, where $m$ and $n$ are relatively prime positive integers. Three faces of the box meet at a corner of the box. The center points of those three faces are the vertices of a triangle with an area of $30$ square inches. Find $m+n$... | [
"041",
"41"
] | [
"A rectangular box has width $12$ inches, length $16$ inches, and height $\\frac{m}{n}$ inches, where $m$ and $n$ are relatively prime positive integers. Three faces of the box meet at a corner of the box. The center points of those three faces are the vertices of a triangle with an area of $30$ square inches. Find... | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 3
} | 35 | 0 | 0 |
omr-ws-b9c8f7ab6b744c1718c4719eac1f4a58df45ad2e1ab30cd1776847a07050ccbc | A rectangular box measures $a \times b \times c$, where $a$, $b$, and $c$ are integers and $1 \le a \le b \le c$. The volume and the surface area of the box are numerically equal. How many ordered triples $(a, b, c)$ are possible? | [
"10"
] | [
"A rectangular box measures $a \\times b \\times c$, where $a$, $b$, and $c$ are integers and $1 \\le a \\le b \\le c$. The volume and the surface area of the box are numerically equal. How many ordered triples $(a, b, c)$ are possible?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0416f0365bd9186c10dd72cbeead90a86e9e21c37919c431826cb2fdaef2b087 | A rectangular box measures $a \times b \times c$, where $a$, $b$, and $c$ are integers and $1 \leq a \leq b \leq c$. The volume and the surface area of the box are numerically equal. How many ordered triples $(a, b, c)$ are possible? | [
"10"
] | [
"A rectangular box measures $a \\times b \\times c$, where $a$, $b$, and $c$ are integers and $1 \\leq a \\leq b \\leq c$. The volume and the surface area of the box are numerically equal. How many ordered triples $(a, b, c)$ are possible?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cf864b2ebb020ef773b3dc92b9c3a195dd9ff02baa945f2b8af0fe22189fe314 | A rectangular box measures $a \times b \times c$, where $a,$ $b,$ and $c$ are integers and $1 \leq a \leq b \leq c$. The volume and surface area of the box are numerically equal. How many ordered triples $(a,b,c)$ are possible? | [
"10"
] | [
"A rectangular box measures $a \\times b \\times c$, where $a,$ $b,$ and $c$ are integers and $1 \\leq a \\leq b \\leq c$. The volume and surface area of the box are numerically equal. How many ordered triples $(a,b,c)$ are possible?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-54cf8b3c92e2cfe82ca50d5f0c2ae59b68059ff38fd04df3e292ee6a9879a412 | A rectangular box measures \(a \times b \times c\), where \(a\), \(b\), and \(c\) are integers and \(1 \le a \le b \le c\). The volume and the surface area of the box are numerically equal. How many ordered triples \((a, b, c)\) are possible? | [
"10"
] | [
"A rectangular box measures \\(a \\times b \\times c\\), where \\(a\\), \\(b\\), and \\(c\\) are integers and \\(1 \\le a \\le b \\le c\\). The volume and the surface area of the box are numerically equal. How many ordered triples \\((a, b, c)\\) are possible?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-c2625039da5ecc5ca18b0fac6eada672a32fbbaf6172966c11dd2b6c5f0e9c92 | A rectangular box with a volume of 48 cubic feet is to be constructed such that its length is twice its width. The material for the top and the four sides costs three times as much per square foot as the material for the bottom. Find the height of the box in feet that minimizes the cost. | [
"\\(\\frac{8}{3}\\)"
] | [
"A rectangular box with a volume of 48 cubic feet is to be constructed such that its length is twice its width. The material for the top and the four sides costs three times as much per square foot as the material for the bottom. Find the height of the box in feet that minimizes the cost."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-38083655919b3c0fcf54a9f8ec45df7b28e326e7a0282a9ad904705b4f979ffc | A rectangular box with no top has a volume of 1000 cubic centimeters. Find its minimum possible surface area. | [
"300\\sqrt[3]{4}"
] | [
"A rectangular box with no top has a volume of 1000 cubic centimeters. Find its minimum possible surface area."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f9e0583236974fc9e9a827b797f31aab605aebd89840dfde86d86c30a475cde6 | A rectangular box with no top is to be twice as long as it is wide and must hold 12 cubic feet. Find the height of the box that minimizes the surface area. | [
"\\(\\frac{6}{\\sqrt[3]{81}}\\)"
] | [
"A rectangular box with no top is to be twice as long as it is wide and must hold 12 cubic feet. Find the height of the box that minimizes the surface area."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-54e9e9d769e8ee8d936535d4e1a0882b1f9703ef0a871577839256de5a1b336d | A rectangular box with side lengths $1$, $2$, and $16$ is cut into two congruent smaller boxes with integer side lengths. Compute the square of the largest possible length of the space diagonal of one of the smaller boxes. | [
"258"
] | [
"A rectangular box with side lengths $1$, $2$, and $16$ is cut into two congruent smaller boxes with integer side lengths. Compute the square of the largest possible length of the space diagonal of one of the smaller boxes."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-917a9b01d114befa31fbfa83306d84fffc7f5a18d8d1edfd953136d29460bf53 | A rectangular box without a lid is to be made to have a volume of 1 $m^3$. Find the minimum surface area of this box. | [
"\\( 3\\sqrt[3]{4} \\)",
"\\(3\\sqrt[3]{4}\\)"
] | [
"A rectangular box without a lid is to be made to have a volume of 1 $m^3$. Find the minimum surface area of this box."
] | [
"aops_c6_high_school_olympiads",
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 55
} | 87 | 0 | 0 |
omr-ws-302b3c0f1568047aa3d90276ac1884676de16b087af60c1aaa386871a47f3e24 | A rectangular building consists of 30 square rooms arranged in a $2 \times 15$ grid. Each room has three doors, each leading to another room (not necessarily different). How many ways are there to distribute the doors between the rooms so that it is possible to get from any room to any other one without leaving the bui... | [
"1"
] | [
"A rectangular building consists of 30 square rooms arranged in a $2 \\times 15$ grid. Each room has three doors, each leading to another room (not necessarily different). How many ways are there to distribute the doors between the rooms so that it is possible to get from any room to any other one without leaving t... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-c998e363ebed5a084a21d465ebf1854efefd4512d291f6c2965d8e5e4a69ea40 | A rectangular card is folded through one corner such that the adjacent corner falls on a side, thus forming 3 right triangles with areas in an arithmetic progression. If the area of the smallest triangle is \( 3 \, \text{cm}^2 \), what is the area of the largest triangle? | [
"12"
] | [
"A rectangular card is folded through one corner such that the adjacent corner falls on a side, thus forming 3 right triangles with areas in an arithmetic progression. If the area of the smallest triangle is \\( 3 \\, \\text{cm}^2 \\), what is the area of the largest triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 1
} | 9 | 0 | 0 |
omr-ws-6ed7dee6da161c0f3f30ef1bc98142c6ccaa556fdaca5547a682e9ae004b6cce | A rectangular cow pasture is enclosed on three sides by a fence and the fourth side is part of the side of a barn that is $400$ feet long. The fence costs $\$5$ per foot, and $\$1,\!200$ altogether. To the nearest foot, find the length of the side parallel to the barn that will maximize the area of the pasture. | [
"120"
] | [
"A rectangular cow pasture is enclosed on three sides by a fence and the fourth side is part of the side of a barn that is $400$ feet long. The fence costs $\\$5$ per foot, and $\\$1,\\!200$ altogether. To the nearest foot, find the length of the side parallel to the barn that will maximize the area of the pasture.... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6d3d49ce8d643927b8a7e31f0b65c68297dc9f9599cd2b02826ab859f7d98d88 | A rectangular field has dimensions 120 meters by 192 meters. You want to divide it into equal square plots with integer side lengths. Determine the smallest number of such square plots and the number of posts needed if a post is placed in each corner of each plot. | [
"40 plots, 54 posts"
] | [
"A rectangular field has dimensions 120 meters by 192 meters. You want to divide it into equal square plots with integer side lengths. Determine the smallest number of such square plots and the number of posts needed if a post is placed in each corner of each plot."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 28
} | 28 | 0 | 0 |
omr-ws-7f8f79403126491d80aaddfea09f0c085e3ac17e009a4c0fba1d6f0e8b12c8c9 | A rectangular field has dimensions 7 km by 12 km, divided into two parts: a lower part with dimensions 3 km by 12 km (sand) and a higher part with dimensions 4 km by 12 km (grass). A man starts at the lower left corner and wants to reach the upper right corner. The man runs at half the speed in the sand compared to his... | [
"\\( x = 1.5834 \\) kilometers from the left side of the field through the sand part."
] | [
"A rectangular field has dimensions 7 km by 12 km, divided into two parts: a lower part with dimensions 3 km by 12 km (sand) and a higher part with dimensions 4 km by 12 km (grass). A man starts at the lower left corner and wants to reach the upper right corner. The man runs at half the speed in the sand compared t... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 27
} | 28 | 0 | 0 |
omr-ws-21ffba5d9b32a8217a488030291f2b0425ed86de6f476968bfb83c9196e33466 | A rectangular field is half as wide as it is long, and it is completely enclosed by 54 meters of fencing. What is the number of square meters in the area of the field? | [
"162"
] | [
"A rectangular field is half as wide as it is long, and it is completely enclosed by 54 meters of fencing. What is the number of square meters in the area of the field?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d5e18190282c1cf4d3e349a46674ca4bfbc1d6f8b989b9d1056f090ea12c0a3e | A rectangular field is to be enclosed by a fence and divided into two parts by another fence. Find the maximum area that can be enclosed and separated in this way with 800 meters of fencing. | [
"\\(\\frac{80000}{3}\\)"
] | [
"A rectangular field is to be enclosed by a fence and divided into two parts by another fence. Find the maximum area that can be enclosed and separated in this way with 800 meters of fencing."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-3aaaa42bb85b47caf78ec5f6b676540382b58f0df2c738ac96cde0e7f82be994 | A rectangular field measures 24 meters by 52 meters. An agricultural researcher has 1994 meters of fence to partition the field into congruent, square test plots. The entire field must be partitioned, and the sides of the squares must be parallel to the edges of the field. What is the largest number of square test plot... | [
"702"
] | [
"A rectangular field measures 24 meters by 52 meters. An agricultural researcher has 1994 meters of fence to partition the field into congruent, square test plots. The entire field must be partitioned, and the sides of the squares must be parallel to the edges of the field. What is the largest number of square test... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 36,
"genselect": 7,
"tir": 6
} | 49 | 0 | 0 |
omr-ws-c64eefa435339baead2a1635fb9dbfa99ee82dedaae3c65bc398a564e65045a3 | A rectangular floor is covered by a certain number of equally large square tiles. The tiles along the edge are red, and the rest are white. There are equally many red and white tiles. How many tiles can there be? | [
"48, 60"
] | [
"A rectangular floor is covered by a certain number of equally large square tiles. The tiles along the edge are red, and the rest are white. There are equally many red and white tiles. How many tiles can there be?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 23
} | 24 | 0 | 0 |
omr-ws-7374ff2208a8ec99fbdd4efbed14aa1f95f13a68e4a33866f9b01986d1306d67 | A rectangular floor is made up of identical square tiles, with the floor being 40 tiles long and 30 tiles wide. If a straight line is drawn diagonally across the floor from one corner to the other, how many tiles will it cross? | [
"60"
] | [
"A rectangular floor is made up of identical square tiles, with the floor being 40 tiles long and 30 tiles wide. If a straight line is drawn diagonally across the floor from one corner to the other, how many tiles will it cross?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 17
} | 32 | 0 | 0 |
omr-ws-c64a107e38290296d89fbee66a64fcc0431fc78eb859d1ea30088ece6f2c87bd | A rectangular floor measuring 12 x 20 is covered by 1x1 square tiles. A chalk line segment is drawn from one corner to the diagonally opposite corner. How many square tiles have a chalk line segment on them? | [
"28"
] | [
"A rectangular floor measuring 12 x 20 is covered by 1x1 square tiles. A chalk line segment is drawn from one corner to the diagonally opposite corner. How many square tiles have a chalk line segment on them?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-069479d14baa02927fd331729a90d7b4d9c66a97d9f7131938a90801a1645fca | A rectangular floor that is $10$ feet wide and $17$ feet long is tiled with $170$ one-foot square tiles. A bug walks from one corner to the opposite corner in a straight line. Including the first and the last tile, how many tiles does the bug visit? | [
"26"
] | [
"A rectangular floor that is $10$ feet wide and $17$ feet long is tiled with $170$ one-foot square tiles. A bug walks from one corner to the opposite corner in a straight line. Including the first and the last tile, how many tiles does the bug visit?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 23
} | 55 | 0 | 0 |
omr-ws-1ee39dd93324663f3ff400cb48cdbf66016e8e5bc1eaf4a32e56e77a8b5098c7 | A rectangular floor that is 10 feet wide and 17 feet long is tiled with 170 one-foot square tiles. A bug walks from one corner to the opposite corner in a straight line. Including the first and the last tile, how many tiles does the bug visit? | [
"26"
] | [
"A rectangular floor that is 10 feet wide and 17 feet long is tiled with 170 one-foot square tiles. A bug walks from one corner to the opposite corner in a straight line. Including the first and the last tile, how many tiles does the bug visit?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-427e94518c035770b0c269571f5f6d2fcd1bf5b5b9831a3e16de8bb83b9a1170 | A rectangular flower garden measures 13 feet in length by 8 feet in width. A landscaper orders 1 cubic yard of premixed cement to create a border of uniform width around the garden. If the border is to have a depth of 3 inches, how wide will the border be? | [
"\\( x = -126 + 18\\sqrt{73} \\)"
] | [
"A rectangular flower garden measures 13 feet in length by 8 feet in width. A landscaper orders 1 cubic yard of premixed cement to create a border of uniform width around the garden. If the border is to have a depth of 3 inches, how wide will the border be?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
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"additional_problems": 1
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omr-ws-597d4a6cb9bf00db867ed8f223840d7e82e1edf1dd1598cb821e7d1a669279f4 | A rectangular garden 50 feet long and 10 feet wide is enclosed by a fence. To make the garden larger, while using the same fence, its shape is changed to a square. By how many square feet does this enlarge the garden? | [
"400"
] | [
"A rectangular garden 50 feet long and 10 feet wide is enclosed by a fence. To make the garden larger, while using the same fence, its shape is changed to a square. By how many square feet does this enlarge the garden?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e20ac3da3fd8dfe4679d4fcfd236223ee0456abcf462ed92355583c29998bbbc | A rectangular garden is built, and a rectangular walkway with fixed width is built around it. If the rectangles formed by the garden and walkway both have integral width, sides, and diagonals, what is the smallest possible area of the walkway? | [
"60"
] | [
"A rectangular garden is built, and a rectangular walkway with fixed width is built around it. If the rectangles formed by the garden and walkway both have integral width, sides, and diagonals, what is the smallest possible area of the walkway?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
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"cot": 5,
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omr-ws-4804fcf78cd93e29efd8d23857e402550f1550553d19c367bf988837544d5308 | A rectangular garden with a perimeter of 36 meters has one side along the bank of a river that measures 8 meters. The cost of fencing materials along the river is 45 per meter, while the rest of the fencing costs 37.50 per meter. Calculate the total cost to enclose the garden. | [
"1410"
] | [
"A rectangular garden with a perimeter of 36 meters has one side along the bank of a river that measures 8 meters. The cost of fencing materials along the river is 45 per meter, while the rest of the fencing costs 37.50 per meter. Calculate the total cost to enclose the garden."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
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} | 24 | 0 | 0 |
omr-ws-eb125eaa1c9aa0a13eeccd1e2d6ec63f59f744bf8a28843c606be0bf64a57e18 | A rectangular garden with width \( x \) and length \( y \) is surrounded by a 2-inch wide walkway. What is the area of the walkway in square feet? | [
"\\(\\frac{6x + 6y + 4}{9}\\)"
] | [
"A rectangular garden with width \\( x \\) and length \\( y \\) is surrounded by a 2-inch wide walkway. What is the area of the walkway in square feet?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
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"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9cb4f57ba44269776294af4708ce3680d1a313a6756637f22b65596e667a824d | A rectangular glass $ABCD$ is supported on the corner of a balcony. The glass touches the horizontal ramps at points $P$ and $Q$ respectively and the vertical column at point $S$. The base $AB$ of the glass rests completely on the balcony floor. Given that $OP = OQ = 30$ cm, $OS = 15$ cm, and $RO = 80$ cm, determine th... | [
"30\\sqrt{2} \\text{ cm} \\times 15\\sqrt{3} \\text{ cm}"
] | [
"A rectangular glass $ABCD$ is supported on the corner of a balcony. The glass touches the horizontal ramps at points $P$ and $Q$ respectively and the vertical column at point $S$. The base $AB$ of the glass rests completely on the balcony floor. Given that $OP = OQ = 30$ cm, $OS = 15$ cm, and $RO = 80$ cm, determi... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 8
} | 12 | 0 | 0 |
omr-ws-15175b15d4ee53f9c79318862f45774e87ce8ca5a2719b18670639c3de5ff3b0 | A rectangular grid with integer side lengths greater than 1 is given. Smaller rectangles with odd integer areas and integer side lengths greater than 1 are cut one by one until only a single unit square remains. Find the least possible area of the initial grid. | [
"15"
] | [
"A rectangular grid with integer side lengths greater than 1 is given. Smaller rectangles with odd integer areas and integer side lengths greater than 1 are cut one by one until only a single unit square remains. Find the least possible area of the initial grid."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-199cdfe25f5b14d535a6cedddb858239917eacaea722755e68c93f9baa7ab28d | A rectangular grid with integer side lengths greater than 1 is given. Smaller rectangles with odd integer areas and integer side lengths greater than 1 are cut out one by one. Finally, one single unit square is left. Find the least possible area of the initial grid before the cuttings. | [
"121"
] | [
"A rectangular grid with integer side lengths greater than 1 is given. Smaller rectangles with odd integer areas and integer side lengths greater than 1 are cut out one by one. Finally, one single unit square is left. Find the least possible area of the initial grid before the cuttings."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-940797d21788efd873d22a92546ca49eb5ca81d6a2b38f9a27d454b78079ed48 | A rectangular lawn is uniformly covered by grass of constant height. Andy’s mower cuts a strip of grass 1 metre wide. He mows the lawn using the following pattern: first he mows the grass in the rectangular “ring” \(A_1\) of width 1 metre running around the edge of the lawn, then he mows the 1-metre-wide ring \(A_2\) i... | [
"124"
] | [
"A rectangular lawn is uniformly covered by grass of constant height. Andy’s mower cuts a strip of grass 1 metre wide. He mows the lawn using the following pattern: first he mows the grass in the rectangular “ring” \\(A_1\\) of width 1 metre running around the edge of the lawn, then he mows the 1-metre-wide ring \\... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-a7c3b2af6f025f92bacaacbe1d1b673d90b18af341407c8eb84ed4a6ac151383 | A rectangular paper with a length of 5 cm is folded such that two diagonally opposite vertices coincide. The length of the crease formed is $\sqrt{6}$ cm. Find the breadth of the rectangle. | [
"$\\sqrt{5}$"
] | [
"A rectangular paper with a length of 5 cm is folded such that two diagonally opposite vertices coincide. The length of the crease formed is $\\sqrt{6}$ cm. Find the breadth of the rectangle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-d6ecba3158610a7955179f6f89ddddad0d9cfeadae3642bffe36955b148289f6 | A rectangular parallelepiped has dimensions \(a\), \(b\), and \(c\) that satisfy the relation \(3a + 4b + 10c = 500\), and the length of the main diagonal is \(20\sqrt{5}\). Find the volume and the total surface area of the parallelepiped. | [
"2624"
] | [
"A rectangular parallelepiped has dimensions \\(a\\), \\(b\\), and \\(c\\) that satisfy the relation \\(3a + 4b + 10c = 500\\), and the length of the main diagonal is \\(20\\sqrt{5}\\). Find the volume and the total surface area of the parallelepiped."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e8d3ee5d71b30db07d1a2593827b99f56174175fb4c52a687e020811b813e1f4 | A rectangular parallelepiped has sides of integer lengths. The sum of the number of its vertices, the length of its edges, the area of its faces, and its volume is 2015. What is the volume of the parallelepiped? | [
"957"
] | [
"A rectangular parallelepiped has sides of integer lengths. The sum of the number of its vertices, the length of its edges, the area of its faces, and its volume is 2015. What is the volume of the parallelepiped?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 5
} | 20 | 0 | 0 |
omr-ws-e4b4de31409002ff5d9ea7bbc821d0bf0798a9eba01c3a5cba4e403507509844 | A rectangular parallelepiped is to be enclosed in a cylindrical container with a hemispherical lid. The total height of the container from the base to the top of the lid is 60 centimetres, and its base has a radius of 30 centimetres. Find the volume of the largest rectangular parallelepiped that can be completely enclo... | [
"108000 cm^3"
] | [
"A rectangular parallelepiped is to be enclosed in a cylindrical container with a hemispherical lid. The total height of the container from the base to the top of the lid is 60 centimetres, and its base has a radius of 30 centimetres. Find the volume of the largest rectangular parallelepiped that can be completely ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5736d509ea8303217ef3b6ff56143b0ce4c3913bb3ea4ce60bcf838bbc0a7e79 | A rectangular parallelepiped with sides $10$ cm, $10$ cm, and $20$ cm is hanging on a string in front of you. A ladybug is planted on one of the vertices of the parallelepiped. What is the fastest way for the ladybug to get to the opposite vertex of the parallelepiped, and how many centimeters will it crawl? | [
"\\(20\\sqrt{2}\\) cm"
] | [
"A rectangular parallelepiped with sides $10$ cm, $10$ cm, and $20$ cm is hanging on a string in front of you. A ladybug is planted on one of the vertices of the parallelepiped. What is the fastest way for the ladybug to get to the opposite vertex of the parallelepiped, and how many centimeters will it crawl?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-cca16b8ae7af4a3f4489e0ed4e64ebe1b1ecfd036b9ead2b66ea57413a14e4e3 | A rectangular parallelepiped with square bases has side length $a$ and height $h$. A cylinder is inscribed in this parallelepiped such that its bases are in opposite square faces and its surface touches the remaining four parallelogrammic faces. A plane is drawn through the center of the circle which is the base of the... | [
"\\dfrac{1}{4}"
] | [
"A rectangular parallelepiped with square bases has side length $a$ and height $h$. A cylinder is inscribed in this parallelepiped such that its bases are in opposite square faces and its surface touches the remaining four parallelogrammic faces. A plane is drawn through the center of the circle which is the base o... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8776131bf570f65782e5f7d23e807fbcb37ad9ea512910b7e96a269cf7daa91b | A rectangular parallelepiped-shaped cardboard box is to be enclosed in a cylindrical container with a hemispherical lid. The total height of the container from the base to the top of the lid is 60 centimeters, and the base has a radius of 30 centimeters. Find the volume of the largest box that can be completely enclose... | [
"\\(108000 \\text{ cm}^3\\)"
] | [
"A rectangular parallelepiped-shaped cardboard box is to be enclosed in a cylindrical container with a hemispherical lid. The total height of the container from the base to the top of the lid is 60 centimeters, and the base has a radius of 30 centimeters. Find the volume of the largest box that can be completely en... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-54417109af7c4d120e19a6de5d7e2f78f08995cd00b8e2d90d6e9653044e3bb8 | A rectangular parking lot has a diagonal of $25$ meters and an area of $168$ square meters. What is the perimeter of the parking lot? | [
"62"
] | [
"A rectangular parking lot has a diagonal of $25$ meters and an area of $168$ square meters. What is the perimeter of the parking lot?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bc9434451189d109b9d3b4087399dc5708da49f540f61515094d4f2a03c5c70e | A rectangular pen uses the wall of a house as one side and is enclosed by 1000 m of fencing. Describe how the area of the pen changes as the length \( L \) increases. | [
"The area of the pen increases as \\( L \\) increases from 50 to 400, and then starts to decrease."
] | [
"A rectangular pen uses the wall of a house as one side and is enclosed by 1000 m of fencing. Describe how the area of the pen changes as the length \\( L \\) increases."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-66b3e36381c86581e37b44d0115583c68ccff16f0fa1a3c57aada4262e44762b | A rectangular pen uses the wall of a house as one side and is enclosed by 1000 m of fencing. Estimate the maximum area of the pen and the dimensions of the rectangle that would give this maximum possible area. | [
"Maximum area = 62500 m², dimensions = 250 m by 375 m"
] | [
"A rectangular pen uses the wall of a house as one side and is enclosed by 1000 m of fencing. Estimate the maximum area of the pen and the dimensions of the rectangle that would give this maximum possible area."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b20b0036514a1b4783bde7b8439515eaf340c9feaf5966d5ba1f0422c54e169f | A rectangular pen uses the wall of a house as one side and is enclosed by 1000 m of fencing. Express the area of the pen in terms of the length \( L \) only. | [
"\\( 500L - L^2 \\)"
] | [
"A rectangular pen uses the wall of a house as one side and is enclosed by 1000 m of fencing. Express the area of the pen in terms of the length \\( L \\) only."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 2,
"tir": 4
} | 9 | 0 | 0 |
omr-ws-1eee82c744f45951ba08334d3ae2f4b782d05f73fb167d512730ed609bb9a690 | A rectangular picture with dimensions 5in. by 8in. is surrounded by a frame of uniform width. If the area of the frame (not including the space for the picture) is 114 square inches greater than the area of the picture, find the width of the frame. | [
"3 inches"
] | [
"A rectangular picture with dimensions 5in. by 8in. is surrounded by a frame of uniform width. If the area of the frame (not including the space for the picture) is 114 square inches greater than the area of the picture, find the width of the frame."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 14
} | 14 | 0 | 0 |
omr-ws-6d1016e3ec9f09b5340586b4657a084e708b252e5d65712c3f774aba89cc8f08 | A rectangular piece of carpet fits perfectly in two different storerooms such that each corner of the carpet touches a different wall. The widths of the storerooms are 38 feet and 50 feet, respectively, and both storerooms have the same length. The dimensions of the carpet are integral numbers of feet. Determine the di... | [
"(50, 25)"
] | [
"A rectangular piece of carpet fits perfectly in two different storerooms such that each corner of the carpet touches a different wall. The widths of the storerooms are 38 feet and 50 feet, respectively, and both storerooms have the same length. The dimensions of the carpet are integral numbers of feet. Determine t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 7,
"tir": 6
} | 13 | 0 | 0 |
omr-ws-0ce280347bdb62a43c54bf0e4c20928025eb4388cb74bee956e4d53c8b73b9e4 | A rectangular piece of carpet fits perfectly in two different storerooms such that each corner touches a different wall. The dimensions of the storerooms are 38 feet by 55 feet. What are the dimensions of the carpet? | [
"38 feet by 55 feet"
] | [
"A rectangular piece of carpet fits perfectly in two different storerooms such that each corner touches a different wall. The dimensions of the storerooms are 38 feet by 55 feet. What are the dimensions of the carpet?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-82c8355c0a9cfe4e3ab0a353048729b72dd9c0f4adae62bb5dab9c2ba9654a1b | A rectangular piece of carpet is laid flat on the floor of two different storerooms such that each corner of the carpet touches a different wall of the room. The dimensions of the storerooms are 38 feet by 55 feet and 50 feet by 55 feet. Determine the dimensions of the carpet. | [
"\\(25 \\times 50\\)"
] | [
"A rectangular piece of carpet is laid flat on the floor of two different storerooms such that each corner of the carpet touches a different wall of the room. The dimensions of the storerooms are 38 feet by 55 feet and 50 feet by 55 feet. Determine the dimensions of the carpet."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 17,
"tir": 11
} | 28 | 0 | 0 |
omr-ws-ac4abc0203ec65a123982ddea958ed89afc246cd5ad1912c9cb97c17c0f3346e | A rectangular piece of land with an area of 5000 m² is to be enclosed by a wall and then divided into three equal regions by partition walls parallel to one of its sides. If the total length of the wall is 445 m, calculate the possible dimensions of the land. | [
"\\( 197.15 \\, \\text{m} \\) and \\( 25.35 \\, \\text{m} \\)"
] | [
"A rectangular piece of land with an area of 5000 m² is to be enclosed by a wall and then divided into three equal regions by partition walls parallel to one of its sides. If the total length of the wall is 445 m, calculate the possible dimensions of the land."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-51267778f08056a39c73ecc9a65d3365985dc4dbd4a522e7a12575e6ded1a4dd | A rectangular piece of length 4 cm and width 3 cm is folded so that a pair of diagonally opposite vertices coincide. How long is the crease formed? | [
"2.5"
] | [
"A rectangular piece of length 4 cm and width 3 cm is folded so that a pair of diagonally opposite vertices coincide. How long is the crease formed?"
] | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 5
} | 5 | 0 | 0 |
omr-ws-40496201b1b7aac07e474c5fe8974a6eda40760b21e6ecf5aefa0e88370dd19b | A rectangular piece of paper $ABCD$ has sides of lengths $AB = 1$ and $BC = 2$. The rectangle is folded in half such that $AD$ coincides with $BC$ and $EF$ is the folding line. Then, the paper is folded along a line $BM$ such that the corner $A$ falls on line $EF$. What is the measure of $\angle ABM$ in degrees? | [
"30 degrees"
] | [
"A rectangular piece of paper $ABCD$ has sides of lengths $AB = 1$ and $BC = 2$. The rectangle is folded in half such that $AD$ coincides with $BC$ and $EF$ is the folding line. Then, the paper is folded along a line $BM$ such that the corner $A$ falls on line $EF$. What is the measure of $\\angle ABM$ in degrees?"... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b7bd8f97535e8931d73199d4bb614be37bfcb9077eac8f9ab8856c3847d12921 | A rectangular piece of paper 30 cm by 21 cm is folded so that opposite corners coincide. How long is the crease? | [
"\\(\\frac{21\\sqrt{149}}{10}\\)"
] | [
"A rectangular piece of paper 30 cm by 21 cm is folded so that opposite corners coincide. How long is the crease?"
] | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 8
} | 8 | 0 | 0 |
omr-ws-33690b3ff8bcdad48090344587ccef0011bc578450f73d692a6857f216b742d0 | A rectangular piece of paper 6 inches wide is folded so that one corner touches the opposite side. The length of the crease \( L \) in terms of angle \( \theta \) is to be determined. | [
"\\( 3 \\sec^2 \\theta \\csc \\theta \\)"
] | [
"A rectangular piece of paper 6 inches wide is folded so that one corner touches the opposite side. The length of the crease \\( L \\) in terms of angle \\( \\theta \\) is to be determined."
] | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7b1017886e7399c0a0ecc77a507e1ba6b0f82826621404dbc6f923aa18913050 | A rectangular piece of paper 8 inches wide is folded so that one corner touches the opposite side. Find the length of the crease \( L \) in terms of the angle \( x \). | [
"\\( \\frac{8}{\\sin 2x \\cos x} \\)"
] | [
"A rectangular piece of paper 8 inches wide is folded so that one corner touches the opposite side. Find the length of the crease \\( L \\) in terms of the angle \\( x \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-a7bfda40ea6dfa6b1bf3cb0e88994066c245fdd6bae33ad97b438da09b9cfe63 | A rectangular piece of paper \( ADEF \) is folded so that corner \( D \) meets the opposite edge \( EF \) at \( D' \), forming a crease \( BC \) where \( B \) lies on edge \( AD \) and \( C \) lies on edge \( DE \). Given \( AD = 25 \) cm, \( DE = 16 \) cm, and \( AB = 5 \) cm, find \( BC^2 \). | [
"500"
] | [
"A rectangular piece of paper \\( ADEF \\) is folded so that corner \\( D \\) meets the opposite edge \\( EF \\) at \\( D' \\), forming a crease \\( BC \\) where \\( B \\) lies on edge \\( AD \\) and \\( C \\) lies on edge \\( DE \\). Given \\( AD = 25 \\) cm, \\( DE = 16 \\) cm, and \\( AB = 5 \\) cm, find \\( BC^... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-4fd81cfb940caa6679bec1b2fe6abdba191df71e64dda9873cc0101f05a17d13 | A rectangular piece of paper \(ABCD\) has \(AB = CD = 12\) and \(BC = DA = 16\). Folding point \(B\) onto point \(D\) results in pentagon \(DAEFC\). Find the area of the heptagon formed by folding point \(C\) onto point \(E\). | [
"67"
] | [
"A rectangular piece of paper \\(ABCD\\) has \\(AB = CD = 12\\) and \\(BC = DA = 16\\). Folding point \\(B\\) onto point \\(D\\) results in pentagon \\(DAEFC\\). Find the area of the heptagon formed by folding point \\(C\\) onto point \\(E\\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-40f5de129f1d7dd72653060cece90389f5ea5670e6948c2aa2c1212b4eb50cea | A rectangular piece of paper has integer side lengths. When folded so that a pair of diagonally opposite vertices coincide, the crease is of length 65. Find a possible value of the perimeter of the paper. | [
"170"
] | [
"A rectangular piece of paper has integer side lengths. When folded so that a pair of diagonally opposite vertices coincide, the crease is of length 65. Find a possible value of the perimeter of the paper."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f3f71639b4f1350aea4074f901d990100ceb570a6b6c9fd5b3c620aeaab8be09 | A rectangular piece of paper is divided into polygonal areas by making straight cuts. Each cut divides one existing area into two new areas. What is the minimum number of cuts needed to ensure that there are at least 251 polygons with 11 sides among the resulting polygons? | [
"2007"
] | [
"A rectangular piece of paper is divided into polygonal areas by making straight cuts. Each cut divides one existing area into two new areas. What is the minimum number of cuts needed to ensure that there are at least 251 polygons with 11 sides among the resulting polygons?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 2
} | 11 | 0 | 0 |
omr-ws-9d26f75f148b3f9d23d84f69ed63fb63b5048e75c06925b02d6fef01f1fdfb13 | A rectangular piece of paper is folded so that two of its diagonally opposite corners meet. The length of the fold is equal to the length of the paper. Find the ratio of the length to the width. | [
"\\sqrt{\\dfrac{1 + \\sqrt{5}}{2}}"
] | [
"A rectangular piece of paper is folded so that two of its diagonally opposite corners meet. The length of the fold is equal to the length of the paper. Find the ratio of the length to the width."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
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