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-bc78a1f6d928051a98f47cf1bb36e5cce95c903b1db5756b1f07d51d094cb75b | A penny, nickel, dime, and quarter are simultaneously flipped. What is the expected value of the amount of the coins which come up heads? Express your answer in cents, rounded to the nearest tenth of a cent. | [
"20.5"
] | [
"A penny, nickel, dime, and quarter are simultaneously flipped. What is the expected value of the amount of the coins which come up heads? Express your answer in cents, rounded to the nearest tenth of a cent."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ee12642ed8d2fb40119424b92f8df8714af795ffe5b2c8ee2778314824f34ab7 | A pensioner receives a payment of £100 every month for the next 20 years. The pension provider funds this with a lump sum of money placed in a bank account receiving 6.5% p.a. interest. How much must the pension provider place in the account to ensure there are sufficient funds to cover the payments? | [
"13439.85"
] | [
"A pensioner receives a payment of £100 every month for the next 20 years. The pension provider funds this with a lump sum of money placed in a bank account receiving 6.5% p.a. interest. How much must the pension provider place in the account to ensure there are sufficient funds to cover the payments?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-777528c6efe6fb1b954d10e0f765fda2659d600fea565045b0d8400d11f4e86d | A pentagon $ABCDE$ (the vertices are designated in a successive order) is inscribed in a circle of unit radius. It is known that $|AB|=\sqrt 2$, $\angle ABE=45^\circ$, $\angle EBD=30^\circ$ and $|BC|=|CD|$. What is the area of the pentagon equal to? | [
"$\\frac{4+3\\sqrt 3}{4}$"
] | [
"A pentagon $ABCDE$ (the vertices are designated in a successive order) is inscribed in a circle of unit radius. It is known that $|AB|=\\sqrt 2$, $\\angle ABE=45^\\circ$, $\\angle EBD=30^\\circ$ and $|BC|=|CD|$. What is the area of the pentagon equal to?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-721cd46b6138bcfcf8c021fe63b207a753688da790f272cfae4b995191e89699 | A pentagon $ABCDE$ is circumscribed about a circle. The angles at the vertices $A$, $C$, and $E$ of the pentagon are each $100^\circ$. Find the measure of the angle $\angle ACE$. | [
"40^\\circ"
] | [
"A pentagon $ABCDE$ is circumscribed about a circle. The angles at the vertices $A$, $C$, and $E$ of the pentagon are each $100^\\circ$. Find the measure of the angle $\\angle ACE$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 5
} | 16 | 0 | 0 |
omr-ws-53b5ab2d3c7b586ff822c6642e7344933c99da644de3a1d065b3d534b7effe8a | A pentagon $ABCDE$ is inscribed in a circle $O$ with $\angle A = 90^\circ$ and $AB = CD$. Let $F$ be a point on segment $AE$. Let $BF$ intersect the circle $O$ again at $J \neq B$, $CE \cap DJ = K$, and $BD \cap FK = L$. If $BE = 10$ and $EF = 6$, find the length of $BL$. | [
"8"
] | [
"A pentagon $ABCDE$ is inscribed in a circle $O$ with $\\angle A = 90^\\circ$ and $AB = CD$. Let $F$ be a point on segment $AE$. Let $BF$ intersect the circle $O$ again at $J \\neq B$, $CE \\cap DJ = K$, and $BD \\cap FK = L$. If $BE = 10$ and $EF = 6$, find the length of $BL$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 5
} | 9 | 0 | 0 |
omr-ws-6e54c107d1e2cb115a7b7202d145b92b7cc0401252ae51340366d53f37d18e43 | A pentagon $ABCDE$ is inscribed in a circle $O$, with $AB = BC$ and $AE = DE$. A circle tangent to $DE$ at $E$ and passing through $A$ intersects $EC$ at $F$ and $BF$ at $G (\ne F)$. Let $DG \cap O = H (\ne D)$. If the radius of circle $O$ is $r$ and the length of $HA$ is $l$, find the value of $\frac{l^2}{r^2}$. | [
"3"
] | [
"A pentagon $ABCDE$ is inscribed in a circle $O$, with $AB = BC$ and $AE = DE$. A circle tangent to $DE$ at $E$ and passing through $A$ intersects $EC$ at $F$ and $BF$ at $G (\\ne F)$. Let $DG \\cap O = H (\\ne D)$. If the radius of circle $O$ is $r$ and the length of $HA$ is $l$, find the value of $\\frac{l^2}{r^2... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-eb6355392031285cf0b5d9b4bafab2d806947630afb31fd924ccd379a3c3ca87 | A pentagon $ABCDE$ is inscribed in a circle $\Gamma$. The quadrilateral $BCDE$ is a rectangle with $BC=DE=1$, and $AB=EA=6$. What is the diameter of $\Gamma$? | [
"4.5"
] | [
"A pentagon $ABCDE$ is inscribed in a circle $\\Gamma$. The quadrilateral $BCDE$ is a rectangle with $BC=DE=1$, and $AB=EA=6$. What is the diameter of $\\Gamma$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-72234f2856d6b86a57d0c36960fd24fc3f2a66633ae76221233bc73c33712ece | A pentagon $ABCDE$ is inscribed in a circle of radius $10$ such that $BC$ is parallel to $AD$ and $AD$ intersects $CE$ at $M$. The tangents to this circle at $B$ and $E$ meet the extension of $DA$ at a common point $P$. Given that $PB=PE=24$ and $\angle BPD$ is $30$ degrees, find the length of $BM$. | [
"13"
] | [
"A pentagon $ABCDE$ is inscribed in a circle of radius $10$ such that $BC$ is parallel to $AD$ and $AD$ intersects $CE$ at $M$. The tangents to this circle at $B$ and $E$ meet the extension of $DA$ at a common point $P$. Given that $PB=PE=24$ and $\\angle BPD$ is $30$ degrees, find the length of $BM$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 4,
"tir": 1
} | 10 | 0 | 0 |
omr-ws-58e88bf6ce34f5cd71a24ffb8ab82f0a15d4bd662a7aaacc0485733e0fdad0f4 | A pentagon $ABCDE$ is inscribed in a circle such that the tangents to the circle at $A, B, C,$ and $D$ are respectively parallel to the opposite sides $DE, EA, AB,$ and $BC$. If the side lengths of the pentagon are given by $AB = a, BC = b, CD = c, DE = d,$ and $EA = e$, find the value of $\frac{a+b+c+d+e}{5}$ given th... | [
"1"
] | [
"A pentagon $ABCDE$ is inscribed in a circle such that the tangents to the circle at $A, B, C,$ and $D$ are respectively parallel to the opposite sides $DE, EA, AB,$ and $BC$. If the side lengths of the pentagon are given by $AB = a, BC = b, CD = c, DE = d,$ and $EA = e$, find the value of $\\frac{a+b+c+d+e}{5}$ gi... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-a80e1091ca6c18bdae4778f8de3919db6534be0b9cafa545aa78335dc240d8cf | A pentagon $ABCDE$ is inscribed in a circle with $BC<CD$ and $AB<DE$. This pentagon is the base of a pyramid with vertex $S$. If $AS$ is the longest edge starting from $S$, find the minimum possible value of $\frac{BS}{CS}$. | [
"1"
] | [
"A pentagon $ABCDE$ is inscribed in a circle with $BC<CD$ and $AB<DE$. This pentagon is the base of a pyramid with vertex $S$. If $AS$ is the longest edge starting from $S$, find the minimum possible value of $\\frac{BS}{CS}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-3b2c98a01bf9d3c13b8ab3a20e2eb872b3b00b480dfc5c81af3941249b7492a7 | A pentagon $ABCDE$ is inscribed in a circle with sides $AB = BC$ and $CD = DE$. The diagonals $AD$ and $BE$ intersect at point $P$, and the diagonal $BD$ intersects segments $CA$ and $CE$ at points $Q$ and $T$ respectively. If the length of $PQ$ is $x$ and the length of $PT$ is $y$, find the value of $x^2 - y^2$ given ... | [
"0"
] | [
"A pentagon $ABCDE$ is inscribed in a circle with sides $AB = BC$ and $CD = DE$. The diagonals $AD$ and $BE$ intersect at point $P$, and the diagonal $BD$ intersects segments $CA$ and $CE$ at points $Q$ and $T$ respectively. If the length of $PQ$ is $x$ and the length of $PT$ is $y$, find the value of $x^2 - y^2$ g... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 6
} | 10 | 0 | 0 |
omr-ws-3d157495c96255ef2b5a694aa5a95d8926906fcc00c3bda8e8cd308bc5bf66ee | A pentagon $A_1A_2A_3A_4A_5$ is inscribed in a circle, with $B$ being the intersection point of $A_1A_4$ and $A_2A_5$. Given that $\angle A_4A_1A_3 = \angle A_5A_2A_4$ and $\angle A_2A_4A_1 = \angle A_3A_5A_2$, calculate the value of $\angle A_1A_3B + \angle BA_3A_5$. | [
"144^\\circ"
] | [
"A pentagon $A_1A_2A_3A_4A_5$ is inscribed in a circle, with $B$ being the intersection point of $A_1A_4$ and $A_2A_5$. Given that $\\angle A_4A_1A_3 = \\angle A_5A_2A_4$ and $\\angle A_2A_4A_1 = \\angle A_3A_5A_2$, calculate the value of $\\angle A_1A_3B + \\angle BA_3A_5$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 4
} | 5 | 0 | 0 |
omr-ws-60207c05be09ac93c6de295c8714ca91986c238ed86b7d3c15052b18b3d47251 | A pentagon (not necessarily convex) has all sides of length $1$ and the product of the cosine of any four angles is equal to zero. Find all possible values of the area of such a pentagon. | [
"1"
] | [
"A pentagon (not necessarily convex) has all sides of length $1$ and the product of the cosine of any four angles is equal to zero. Find all possible values of the area of such a pentagon."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 8
} | 11 | 0 | 0 |
omr-ws-969b3aeab4b77caaf8ed6965664b70c2ce1e663d7a66d524d82088ff5033c9ca | A pentagon ABCDE has AB=BC, CD=DE, $\widehat{ABC}=120^\circ$, and $\widehat{CDE}=60^\circ$. Given that BD=2, calculate the area of the pentagon. | [
"\\sqrt{3}"
] | [
"A pentagon ABCDE has AB=BC, CD=DE, $\\widehat{ABC}=120^\\circ$, and $\\widehat{CDE}=60^\\circ$. Given that BD=2, calculate the area of the pentagon."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-e5c660ecf64ca527e7a3894df0f140c43b80167125415db31187d60df8b3aed6 | A pentagon \(ABCDE\) is inscribed in a circle such that \(AB = BC = CD = DE = EA\). Let \(F\) be an arbitrary point on the arc \(AE\) of the circumcircle. Calculate the value of \(AF + CF + EF - BF - DF\). | [
"0"
] | [
"A pentagon \\(ABCDE\\) is inscribed in a circle such that \\(AB = BC = CD = DE = EA\\). Let \\(F\\) be an arbitrary point on the arc \\(AE\\) of the circumcircle. Calculate the value of \\(AF + CF + EF - BF - DF\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-cea76c07ba639348de371aeced30d574427d38d5b5b4d31692ec58b80846cd2e | A pentagon has the following properties: - Two equal angles, - Three angles, one of which is equal to the sum of the other two, - Four angles, one of which is equal to the sum of the other three, - One angle equal to the sum of all the others. Find all the angles of the pentagon. | [
"\\(120^\\circ, 120^\\circ, 60^\\circ, 60^\\circ, 180^\\circ\\)"
] | [
"A pentagon has the following properties:\n- Two equal angles,\n- Three angles, one of which is equal to the sum of the other two,\n- Four angles, one of which is equal to the sum of the other three,\n- One angle equal to the sum of all the others.\nFind all the angles of the pentagon."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2bc5d13fa3d16be08aee98167a0ffc5609061491f58d8e8180b21f1240a5f4f1 | A pentagon is composed of a rectangle topped by an isosceles triangle. If the length of the perimeter is fixed, find the maximum possible area. Let the sides of the rectangle be \(x\) and \(y\) (with \(y\) being the side opposite the base of the triangle), and let \(\theta\) be the measure of each angle of the triangle... | [
"\\(\\frac{2-\\sqrt{3}}{4}p^2\\)"
] | [
"A pentagon is composed of a rectangle topped by an isosceles triangle. If the length of the perimeter is fixed, find the maximum possible area. Let the sides of the rectangle be \\(x\\) and \\(y\\) (with \\(y\\) being the side opposite the base of the triangle), and let \\(\\theta\\) be the measure of each angle o... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-1294d134f2d30221dd505250cdbaab9f8c0ebf787d24a6df47d8af6a605c8146 | A pentagon is formed by cutting a triangular corner from a rectangular piece of paper. The five sides of the pentagon have lengths $13,$ $19,$ $20,$ $25$ and $31,$ in some order. Find the area of the pentagon. | [
"745"
] | [
"A pentagon is formed by cutting a triangular corner from a rectangular piece of paper. The five sides of the pentagon have lengths $13,$ $19,$ $20,$ $25$ and $31,$ in some order. Find the area of the pentagon."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-02ac851c9a1c8911454508d6915a194d54fdff892b82acb51e6b4a9ffab7dfde | A pentagon is formed by cutting a triangular corner from a rectangular piece of paper. The five sides of the pentagon have lengths 13, 19, 20, 25, and 31 (not necessarily in that order). What is the area of the pentagon? | [
"745"
] | [
"A pentagon is formed by cutting a triangular corner from a rectangular piece of paper. The five sides of the pentagon have lengths 13, 19, 20, 25, and 31 (not necessarily in that order). What is the area of the pentagon?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-92e4f15d273c23958e1f863e0c59e7066eb01f04752255671dd480d8b16d1ad6 | A pentagon is formed by cutting a triangular corner from a rectangular piece of paper. The five sides of the pentagon have lengths 13, 19, 20, 25, and 31, although this is not their order around the pentagon. What is the area of the pentagon? | [
"745"
] | [
"A pentagon is formed by cutting a triangular corner from a rectangular piece of paper. The five sides of the pentagon have lengths 13, 19, 20, 25, and 31, although this is not their order around the pentagon. What is the area of the pentagon?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d1d94cd09f32d8db9aa9c3b5e301ae865a1196dd5a3fe8d57e710e14cb2b8fdd | A pentagon is inscribed in a circle and has an incircle. All 10 sets of 3 vertices from the pentagon are chosen, and the incenters of each of the 10 resulting triangles are drawn. Let $R_1$ and $R_2$ be the radii of the two concentric circles on which these 10 incenters lie, with $R_1 < R_2$. If the radius of the circu... | [
"no_valid_answer_found"
] | [
"A pentagon is inscribed in a circle and has an incircle. All 10 sets of 3 vertices from the pentagon are chosen, and the incenters of each of the 10 resulting triangles are drawn. Let $R_1$ and $R_2$ be the radii of the two concentric circles on which these 10 incenters lie, with $R_1 < R_2$. If the radius of the ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e548e4a9cf471ef488b68c68ea7f94c0452ded3a1da643371925b404bb74e10e | A pentagonal base pyramid is circumscribed around a right circular cone, whose height is equal to the radius of the base. The total surface area of the pyramid is d times greater than that of the cone. Find the volume of the pyramid if the lateral surface area of the cone is equal to $\pi\sqrt{2}$. | [
"\\dfrac{5}{3} \\sqrt{5 - 2\\sqrt{5}}"
] | [
"A pentagonal base pyramid is circumscribed around a right circular cone, whose height is equal to the radius of the base. The total surface area of the pyramid is d times greater than that of the cone. Find the volume of the pyramid if the lateral surface area of the cone is equal to $\\pi\\sqrt{2}$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 7
} | 20 | 0 | 0 |
omr-ws-805cb01983679de457b01fd9cced4d02bd5648ff2881b56dd9253c9d401a8b45 | A pentagonal pyramid is circumscribed about a right circular cone of altitude equal to the radius of the base of the cone. The total surface area of the pyramid is twice that of the cone. Find the volume of the pyramid if the lateral surface area of the cone is equal to $\pi \sqrt 2$. | [
"$2\\pi /3$"
] | [
"A pentagonal pyramid is circumscribed about a right circular cone of altitude equal to the radius of the base of the cone. The total surface area of the pyramid is twice that of the cone. Find the volume of the pyramid if the lateral surface area of the cone is equal to $\\pi \\sqrt 2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4c34347b38496f7875ba3a4a2fcdcc96269c503a67e3f97c52efdeee7b3c1859 | A perfect number is a positive integer that is equal to the sum of its proper divisors. Let $N$ be a perfect number and let $d$ be a proper divisor of $N$. Find the sum of all possible values of $\frac{N}{d}$ if $d$ is also a perfect number. | [
"0"
] | [
"A perfect number is a positive integer that is equal to the sum of its proper divisors. Let $N$ be a perfect number and let $d$ be a proper divisor of $N$. Find the sum of all possible values of $\\frac{N}{d}$ if $d$ is also a perfect number."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6a6fe0fcf5a70c75ebcc34f72dddda4d67ee6fb36bd6eac7a9701d1d2da4f58b | A perfect number is a positive integer that is equal to the sum of its proper divisors. Let $P(i,j)$ be the sum of the proper divisors of $3^i \times 5^j$ for non-negative integers $i$ and $j$. Find the value of $P(4,3) - P(3,4)$. | [
"-5614"
] | [
"A perfect number is a positive integer that is equal to the sum of its proper divisors. Let $P(i,j)$ be the sum of the proper divisors of $3^i \\times 5^j$ for non-negative integers $i$ and $j$. Find the value of $P(4,3) - P(3,4)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1f1afda1cb0b7c7c905b5e9211463e0102d8f4d561dc01be311c7995bddb4473 | A perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself. For example, $28$ is a perfect number because $1 + 2 + 4 + 7 + 14 = 28$. Let $n_i$ denote the $i$th smallest perfect number. Define $$f(x) = \sum_{i|n_x... | [
"12"
] | [
"A perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself. For example, $28$ is a perfect number because $1 + 2 + 4 + 7 + 14 = 28$. Let $n_i$ denote the $i$th smallest perfect number. Define $$f(x) = \\sum_... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2750ae045fa068fe30480191a6c30ec2fb101cc47941ce4adbcce8643c2d82a4 | A perfect partition of a positive integer $N$ is an unordered set of numbers (where numbers can be repeated) that sum to $N$ with the property that there is a unique way to express each positive integer less than $N$ as a sum of elements of the set. Repetitions of elements of the set are considered identical for the pu... | [
"6"
] | [
"A perfect partition of a positive integer $N$ is an unordered set of numbers (where numbers can be repeated) that sum to $N$ with the property that there is a unique way to express each positive integer less than $N$ as a sum of elements of the set. Repetitions of elements of the set are considered identical for t... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-a081b8771b643f117234d3ac9b2b6c7112db0c29aa50fcac29ca433a71ed16c1 | A perfect semicircular shell with a radius of 6 cm rocks back and forth between two parallel lines. The top line is 9 cm high. When the semicircle hits the top line, it reverses direction and hits the top line at another point. What is the distance between these two points? | [
"6\\sqrt{3}"
] | [
"A perfect semicircular shell with a radius of 6 cm rocks back and forth between two parallel lines. The top line is 9 cm high. When the semicircle hits the top line, it reverses direction and hits the top line at another point. What is the distance between these two points?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 7
} | 17 | 0 | 0 |
omr-ws-ae828a753d6fa8cacc2fab54f979082b1507405a8e18715044c9a3bbb23807c3 | A perfect slice of pizza is cut from a perfectly circular pizza with a radius of 1 meter. The angle of the pizza slice is 60°. A person eats the pizza at a rate of 1 cm²/sec, starting from the tip and eating straight across. After any given time, the remaining slice appears as a sector of the circle with an isosceles t... | [
"\\sqrt{t \\sqrt{3}}"
] | [
"A perfect slice of pizza is cut from a perfectly circular pizza with a radius of 1 meter. The angle of the pizza slice is 60°. A person eats the pizza at a rate of 1 cm²/sec, starting from the tip and eating straight across. After any given time, the remaining slice appears as a sector of the circle with an isosce... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 6
} | 16 | 0 | 0 |
omr-ws-5b2b5aa1273b626f239bcf70133288e84be4bf89073190d8b162255d84da716c | A perfect sphere of ice with a diameter of 2 meters is melting uniformly at a rate of 1.5 mm of depth per minute. The ice melts into a hexagonal bath with sides measuring 1.5 meters and a depth of 0.8 meters. How many minutes will it take for the bath to be completely full, assuming the volume does not change between t... | [
"158"
] | [
"A perfect sphere of ice with a diameter of 2 meters is melting uniformly at a rate of 1.5 mm of depth per minute. The ice melts into a hexagonal bath with sides measuring 1.5 meters and a depth of 0.8 meters. How many minutes will it take for the bath to be completely full, assuming the volume does not change betw... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6b3ef3d3289605da1c8397a22514b6f9f3a0a3416f97cbd57d6f4bf5e59f3ec9 | A perfect square ends with the same two digits. How many possible values of this digit are there? | [
"1"
] | [
"A perfect square ends with the same two digits. How many possible values of this digit are there?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-81aa6d72bed3f48f02f4877bebc69d382a0da2cb9bb591106a06c214886032b1 | A perfect square, a perfect cube, and a prime number all greater than one have a total sum of 25. What is their product? | [
"416"
] | [
"A perfect square, a perfect cube, and a prime number all greater than one have a total sum of 25. What is their product?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 5
} | 20 | 0 | 0 |
omr-ws-ea3bff75f963dd17caf4f3063afb307c2c2f452716b212789cc29c2405fcf6e8 | A perfectly spherical Christmas pudding of radius 1 contains a magic bean that causes a green dye to permeate all pudding within a distance \( r \) of the bean. When Pythagoras slices off the green cap with a plane cut, he exposes two green disks \( D \) of diameter 1 on each of the two pieces. After discarding the sol... | [
"\\(\\frac{\\sqrt{7}}{4}\\)"
] | [
"A perfectly spherical Christmas pudding of radius 1 contains a magic bean that causes a green dye to permeate all pudding within a distance \\( r \\) of the bean. When Pythagoras slices off the green cap with a plane cut, he exposes two green disks \\( D \\) of diameter 1 on each of the two pieces. After discardin... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 2
} | 3 | 0 | 0 |
omr-ws-09344f742279af2c1a262ffe2a273934062dd02f7dc827989e0a637f002970ff | A perfectly spherical boulder with a volume of $288\pi\,\text{m}^3$ sits on perfectly flat ground. A point-source light is placed exactly 2 meters above the top of the boulder. Find the area, in square meters, of the shadow of the boulder cast by this light onto the flat ground. | [
"252\\pi"
] | [
"A perfectly spherical boulder with a volume of $288\\pi\\,\\text{m}^3$ sits on perfectly flat ground. A point-source light is placed exactly 2 meters above the top of the boulder. Find the area, in square meters, of the shadow of the boulder cast by this light onto the flat ground."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 20
} | 35 | 0 | 0 |
omr-ws-80e265b2ce5bde9f334bba567659e9f3b6ed1aa5d440dd0feeb72e1feee5ec1b | A perfectly spherical plum is suspended 5 cm above a table, measured from the center, and is lit by a point-source of light 10 cm directly above the center of the fruit. If the shadow cast by the plum is 10 cm in diameter, find the fraction of the sphere which is illuminated. Write your answer in exact simplified form. | [
"\\dfrac{10 - \\sqrt{10}}{20}"
] | [
"A perfectly spherical plum is suspended 5 cm above a table, measured from the center, and is lit by a point-source of light 10 cm directly above the center of the fruit. If the shadow cast by the plum is 10 cm in diameter, find the fraction of the sphere which is illuminated. Write your answer in exact simplified ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fdbd6410c9ff710e8e04f2a700febaff3937f4933de9e614b8b1ddf441a7b5e7 | A perfectly spherical rigid scoop of ice cream with radius \( R \) is placed on an ice cream cone with radius \( R \) and height \( 6R \). What percentage of the ice cream is inside the cone? | [
"43.8%"
] | [
"A perfectly spherical rigid scoop of ice cream with radius \\( R \\) is placed on an ice cream cone with radius \\( R \\) and height \\( 6R \\). What percentage of the ice cream is inside the cone?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-740f723541a19c891d15dc130ac705e8c1c71e53e64f3b34239c3f51f7b965f7 | A permutation $(x_1, x_2, \ldots, x_n)$ of $(1, 2, \ldots, n)$ is said to be 'good' if there are no integers $1 \leq i < j < k \leq n$ such that $x_i, x_j, x_k$ form an increasing or decreasing sequence. For every $n$, how many good permutations are there? | [
"\\( \\begin{cases} \n1 & \\text{if } n = 1 \\\\\n2 & \\text{if } n = 2 \\\\\n4 & \\text{if } n = 3 \\\\\n6 & \\text{if } n = 4 \\\\\n0 & \\text{if } n \\geq 5 \n\\end{cases} \\)"
] | [
"A permutation $(x_1, x_2, \\ldots, x_n)$ of $(1, 2, \\ldots, n)$ is said to be 'good' if there are no integers $1 \\leq i < j < k \\leq n$ such that $x_i, x_j, x_k$ form an increasing or decreasing sequence. For every $n$, how many good permutations are there?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-d4cad16d0a0eb485fbcdb0855ae9689eda844bb4506e0cac93bf077cc3981aa6 | A permutation $\sigma: \{1,2,\ldots,n\}\to\{1,2,\ldots,n\}$ is called straight if and only if for each integer $k$, $1\leq k\leq n-1$, the following inequality is fulfilled: \[ |\sigma(k)-\sigma(k+1)|\leq 2. \] Find the smallest positive integer $n$ for which there exist at least 2003 straight permutations. | [
"16"
] | [
"A permutation $\\sigma: \\{1,2,\\ldots,n\\}\\to\\{1,2,\\ldots,n\\}$ is called straight if and only if for each integer $k$, $1\\leq k\\leq n-1$, the following inequality is fulfilled:\n\\[ |\\sigma(k)-\\sigma(k+1)|\\leq 2. \\]\nFind the smallest positive integer $n$ for which there exist at least 2003 straight per... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 5
} | 5 | 0 | 0 |
omr-ws-eeccf173eace5c8f790a59ba81713ec72981c7bf9030cefc32c99d3d2282177d | A permutation $a_1, a_2, \cdots, a_6$ of the numbers $1, 2, \cdots, 6$ can be transformed to $1, 2, \cdots, 6$ by transposing two numbers exactly four times. Find the number of such permutations. | [
"274"
] | [
"A permutation $a_1, a_2, \\cdots, a_6$ of the numbers $1, 2, \\cdots, 6$ can be transformed to $1, 2, \\cdots, 6$ by transposing two numbers exactly four times. Find the number of such permutations."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 45,
"genselect": 8,
"tir": 9
} | 62 | 0 | 0 |
omr-ws-b7d8776205c00e15a79182834dbbaf11e9db21f6d1bb6893274981c0b772199d | A permutation \( P \) of \(\{0, 1, \dots, 2022\}\) is called a square permutation if for all \( i \), \( P(i) + i \) is a perfect square. Let \( S \) be the sum of all \( i \) such that \( 9 \mid P(i) + i \). Find the remainder when \( S \) is divided by \( 1000 \). | [
"234"
] | [
"A permutation \\( P \\) of \\(\\{0, 1, \\dots, 2022\\}\\) is called a square permutation if for all \\( i \\), \\( P(i) + i \\) is a perfect square. Let \\( S \\) be the sum of all \\( i \\) such that \\( 9 \\mid P(i) + i \\). Find the remainder when \\( S \\) is divided by \\( 1000 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c5d2628e6e67c5b75da6d44e7e15d9a8bf7ea40b0f93199c9e92e02b8954f562 | A permutation \( \pi \) of \( \mathbf{Z}/n\mathbf{Z} \) is called good if for every \( x \in \mathbf{Z}/n\mathbf{Z} \), the difference \( \pi(x) - x \) is a constant modulo \( n \). Let \( n \) be an odd integer greater than 1. Find the number of good permutations of \( \mathbf{Z}/n\mathbf{Z} \). | [
"n"
] | [
"A permutation \\( \\pi \\) of \\( \\mathbf{Z}/n\\mathbf{Z} \\) is called good if for every \\( x \\in \\mathbf{Z}/n\\mathbf{Z} \\), the difference \\( \\pi(x) - x \\) is a constant modulo \\( n \\). Let \\( n \\) be an odd integer greater than 1. Find the number of good permutations of \\( \\mathbf{Z}/n\\mathbf{Z}... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-afb8e8e0e93a66d0fd47b15a69b029dfd106c96a1de4dbdf216573e4647a10be | A permutation \( \pi \) of \( \mathbf{Z}/n\mathbf{Z} \) is called good if for every \( x \in \mathbf{Z}/n\mathbf{Z} \), the sum \( x + \pi(x) \) is also a permutation of \( \mathbf{Z}/n\mathbf{Z} \). For \( n = 15 \), find the number of permutations \( \pi \) of \( \mathbf{Z}/n\mathbf{Z} \) that are not good. | [
""
] | [
"A permutation \\( \\pi \\) of \\( \\mathbf{Z}/n\\mathbf{Z} \\) is called good if for every \\( x \\in \\mathbf{Z}/n\\mathbf{Z} \\), the sum \\( x + \\pi(x) \\) is also a permutation of \\( \\mathbf{Z}/n\\mathbf{Z} \\). For \\( n = 15 \\), find the number of permutations \\( \\pi \\) of \\( \\mathbf{Z}/n\\mathbf{Z}... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 1 |
omr-ws-f16262717e11ade435e4a5cc6bf8e8efe2f2724a57a02aaa2916de0eacad227c | A permutation \( a_1, a_2, \ldots, a_n \) of \( 1, 2, \ldots, n \) is said to be indecomposable if \( n \) is the least positive integer \( j \) for which \( \{a_1, a_2, \ldots, a_j\} = \{1, 2, \ldots, j\} \). Let \( f(n) \) be the number of indecomposable permutations of \( 1, 2, \ldots, n \). Find a recurrence relati... | [
"\\( f(n) = n! - \\sum_{i=1}^{n-1} f(i)(n-i)! \\)"
] | [
"A permutation \\( a_1, a_2, \\ldots, a_n \\) of \\( 1, 2, \\ldots, n \\) is said to be indecomposable if \\( n \\) is the least positive integer \\( j \\) for which \\( \\{a_1, a_2, \\ldots, a_j\\} = \\{1, 2, \\ldots, j\\} \\). Let \\( f(n) \\) be the number of indecomposable permutations of \\( 1, 2, \\ldots, n \... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 26
} | 74 | 0 | 0 |
omr-ws-2071ccd33f8840cd49bd9d6f5ad9386a9d765e22fa18aeb1c146b090d74f4d6d | A permutation \((a_1, \dots, a_n)\) from \((1, \dots, n)\) is called disorderly if for every \(1 \leq i < j < k \leq n\), \(n \nmid a_i + a_k - 2a_j\). Find all natural numbers \(n \geq 3\) such that there exists a disorderly permutation from \((1, \dots, n)\). | [
"2^k"
] | [
"A permutation \\((a_1, \\dots, a_n)\\) from \\((1, \\dots, n)\\) is called disorderly if for every \\(1 \\leq i < j < k \\leq n\\), \\(n \\nmid a_i + a_k - 2a_j\\). Find all natural numbers \\(n \\geq 3\\) such that there exists a disorderly permutation from \\((1, \\dots, n)\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 8
} | 18 | 0 | 0 |
omr-ws-5906e7d758f0a1feebb38f750804a465541575216b9c36ae06355cd9266b075b | A permutation \((a_1, a_2, \ldots, a_n)\) of the integers \((1, 2, \ldots, n)\) is said to be fantastic if \(a_1 + a_2 + \cdots + a_k\) is divisible by \(k\) for each \(k\) from 1 to \(n\). Find all fantastic permutations. | [
"(3, 1, 2)"
] | [
"A permutation \\((a_1, a_2, \\ldots, a_n)\\) of the integers \\((1, 2, \\ldots, n)\\) is said to be fantastic if \\(a_1 + a_2 + \\cdots + a_k\\) is divisible by \\(k\\) for each \\(k\\) from 1 to \\(n\\). Find all fantastic permutations."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-eaf1cc7850bd13deeb6225fbcb9fccc16d1db6887f96757841e63183df6696a3 | A permutation \((a_1, a_2, a_3, a_4, a_5)\) of \((1, 2, 3, 4, 5)\) is heavy-tailed if \(a_1 + a_2 < a_4 + a_5\). What is the number of heavy-tailed permutations? | [
"48"
] | [
"A permutation \\((a_1, a_2, a_3, a_4, a_5)\\) of \\((1, 2, 3, 4, 5)\\) is heavy-tailed if \\(a_1 + a_2 < a_4 + a_5\\). What is the number of heavy-tailed permutations?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 28,
"genselect": 6
} | 34 | 0 | 0 |
omr-ws-4b4e0a7f3b991f188d0db5eb45325c3d35b5c7804bd7f8ea7c0afd3b8f56b814 | A permutation \(a_{1}, a_{2}, \ldots, a_{7}\) of \(1, 2, \ldots, 7\) is said to be bumpy if no three consecutive elements in the sequence form an increasing or a decreasing sequence. How many bumpy sequences are there? | [
"544"
] | [
"A permutation \\(a_{1}, a_{2}, \\ldots, a_{7}\\) of \\(1, 2, \\ldots, 7\\) is said to be bumpy if no three consecutive elements in the sequence form an increasing or a decreasing sequence. How many bumpy sequences are there?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 7,
"tir": 16
} | 35 | 0 | 0 |
omr-ws-d946c8a4b54660e95f8a3b30c104167b25a4d7379e79105da76eb953d77e7f48 | A permutation of $[n]$ is a list of numbers from $1$ to $n$ written in a row. A permutation is said to be jumpy if no two adjacent elements differ by exactly $1$. A jumpy permutation $\pi$ is said to be boring if the following holds: There exists an element $i$ in $\pi$ such that if we remove $i$ from $\pi$ and re-numb... | [
"2"
] | [
"A permutation of $[n]$ is a list of numbers from $1$ to $n$ written in a row. A permutation is said to be jumpy if no two adjacent elements differ by exactly $1$. A jumpy permutation $\\pi$ is said to be boring if the following holds: There exists an element $i$ in $\\pi$ such that if we remove $i$ from $\\pi$ and... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 3
} | 12 | 0 | 0 |
omr-ws-f39c6fe43aafd5a6acdfceb95a41337f8ac55e226b388a9cbaa95388737324ce | A permutation of $\{1, 2, \cdots, n\}$ is called *magic* if each element $k$ has at least $\left\lfloor \frac{k}{2} \right\rfloor$ numbers less than it to its left. For each $n$, find the number of *magical* permutations. | [
"\\[\n\\begin{cases} \n(k!)^2, & \\text{if } n = 2k \\\\\n(k!)^2 \\cdot (k + 1), & \\text{if } n = 2k + 1 \n\\end{cases}\n\\]"
] | [
"A permutation of $\\{1, 2, \\cdots, n\\}$ is called *magic* if each element $k$ has at least $\\left\\lfloor \\frac{k}{2} \\right\\rfloor$ numbers less than it to its left. For each $n$, find the number of *magical* permutations."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 29,
"genselect": 5,
"tir": 11
} | 45 | 0 | 0 |
omr-ws-9afa28eeebe092905d9997d51d2b072091d6ccf611e2f59105758ab9a7fe1bb0 | A permutation of ${ 1,2,3,.........,16}$ is called $Blocksum- simple$ if there exists an integer $n$ such that the sum of any 4 consecutive integers in the permutation is either $n$ or $n+1$. How many $Blocksum- simple$ permutations are there? | [
"48"
] | [
"A permutation of ${ 1,2,3,.........,16}$ is called $Blocksum- simple$ if there exists an integer $n$ such that the sum of any 4 consecutive integers in the permutation is either $n$ or $n+1$. How many $Blocksum- simple$ permutations are there?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 1
} | 23 | 0 | 0 |
omr-ws-d49a10349e792a744de752aadc9cd6322e312b6d0739904d21a9e7f2f75585a8 | A permutation of the first $n$ positive integers is valid if, for all $i > 1$, $i$ comes after $\left\lfloor \frac{i}{2} \right\rfloor$ in the permutation. What is the probability that a random permutation of the first $14$ integers is valid? | [
"\\dfrac{1}{31752}"
] | [
"A permutation of the first $n$ positive integers is valid if, for all $i > 1$, $i$ comes after $\\left\\lfloor \\frac{i}{2} \\right\\rfloor$ in the permutation. What is the probability that a random permutation of the first $14$ integers is valid?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3716abfecb282097cf67bcc73e3e4da7a64c03c1cc55cdcb6283e37659d23b67 | A permutation of the first 1000 positive integers \((a_1, a_2, \ldots, a_{1000})\) is called a "good permutation" if for any positive integers \( m, n \) (where \( m, n \geq 1000 \)), if \( n \) is a multiple of \( m \), then \( a_n \) is a multiple of \( a_m \). Find the smallest positive integer \( k \) such that the... | [
"501"
] | [
"A permutation of the first 1000 positive integers \\((a_1, a_2, \\ldots, a_{1000})\\) is called a \"good permutation\" if for any positive integers \\( m, n \\) (where \\( m, n \\geq 1000 \\)), if \\( n \\) is a multiple of \\( m \\), then \\( a_n \\) is a multiple of \\( a_m \\). Find the smallest positive intege... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 3
} | 7 | 0 | 0 |
omr-ws-12e225eafbee7212e519c1ea8665a5d41f4ebcc8f4b4bcb1702af459e8e75373 | A permutation of the integers $1$ through $2022$ is chosen at random. Let $N$ be the number of consecutive pairs of integers in the permutation that sum to $2022$. Let $M$ be the number of consecutive pairs of integers in the permutation with an absolute difference of $1012$. Given $N \ne M$, find the probability $N > ... | [
"\\(\\frac{1}{2}\\)"
] | [
"A permutation of the integers $1$ through $2022$ is chosen at random. Let $N$ be the number of consecutive pairs of integers in the permutation that sum to $2022$. Let $M$ be the number of consecutive pairs of integers in the permutation with an absolute difference of $1012$. Given $N \\ne M$, find the probability... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 27
} | 41 | 0 | 0 |
omr-ws-cd4e1e8092a29a2c6ee87d78c588337ebf48fc5eb3164a0f33d313c310017330 | A permutation of the integers 1901, 1902, ..., 2000 is a sequence \(a_1, a_2, \ldots, a_{100}\) in which each of those integers appears exactly once. Given such a permutation, we form the sequence of partial sums \(s_1 = a_1\), \(s_2 = a_1 + a_2\), \(s_3 = a_1 + a_2 + a_3\), ..., \(s_{100} = a_1 + a_2 + \cdots + a_{100... | [
"\\dfrac{99! \\times 34! \\times (33!)^2}{66!}"
] | [
"A permutation of the integers 1901, 1902, ..., 2000 is a sequence \\(a_1, a_2, \\ldots, a_{100}\\) in which each of those integers appears exactly once. Given such a permutation, we form the sequence of partial sums \\(s_1 = a_1\\), \\(s_2 = a_1 + a_2\\), \\(s_3 = a_1 + a_2 + a_3\\), ..., \\(s_{100} = a_1 + a_2 + ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-03d0a02845a8c23490f90175e04c4cbb424fdba88e226e3b3dc06978e1414e84 | A perpendicular bisector of a side of an acute angle triangle divides one of its altitudes in the ratio $2 : 1$, counting from the top. Find one of the angles of the triangle. | [
"\\(60^\\circ\\)"
] | [
"A perpendicular bisector of a side of an acute angle triangle divides one of its altitudes in the ratio $2 : 1$, counting from the top. Find one of the angles of the triangle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 6
} | 20 | 0 | 0 |
omr-ws-4e3087ab0745f432f3794b7ceed9b6aae851fd5e24a41544ab0dd605c85b35fd | A perpendicular is drawn from the midpoint of a side of a rhombus to a point above the plane of the rhombus. The upper end of this perpendicular is at a distance equal to half the side of the rhombus from the larger diagonal, which is 16 cm. Find the length of this perpendicular. | [
"4"
] | [
"A perpendicular is drawn from the midpoint of a side of a rhombus to a point above the plane of the rhombus. The upper end of this perpendicular is at a distance equal to half the side of the rhombus from the larger diagonal, which is 16 cm. Find the length of this perpendicular."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-265399fa2a1c0cd9aec6a1f72d3b2b9eee7cf9098a7d29550d7aece9b5baaccd | A perpendicular, equal to $a$, is dropped from the foot of the altitude of a regular triangular pyramid to a lateral face. Find the volume of the pyramid, given that the plane angle at the vertex of the pyramid is equal to $\alpha $. | [
"$9a^3\\cos ^2 (\\alpha /2)\\cot(\\alpha /2)[3-4\\sin ^2 (\\alpha /2)]$"
] | [
"A perpendicular, equal to $a$, is dropped from the foot of the altitude of a regular triangular pyramid to a lateral face. Find the volume of the pyramid, given that the plane angle at the vertex of the pyramid is equal to $\\alpha $."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3c6255e533eec29045947954cdd747e1630696520480549d602b79a5853fd3c5 | A perpendicular, equal to $p$, is drawn from the foot of the altitude of a regular triangular pyramid to a lateral edge. Find the volume of the pyramid if the dihedral angle between a lateral face and the base is equal to $\alpha $. | [
"$1/8p^3 \\sqrt 3\\cot^2 \\alpha (4+\\tan^2 \\alpha )^{3/2}$"
] | [
"A perpendicular, equal to $p$, is drawn from the foot of the altitude of a regular triangular pyramid to a lateral edge. Find the volume of the pyramid if the dihedral angle between a lateral face and the base is equal to $\\alpha $."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5d361d059218ca768186ec2d6442e5e14d36f0f22aca23af8bee794744678d27 | A person $X$ is standing at a point $P$ on a flat plane and starts walking. At each step, $X$ walks exactly $1$ unit in one of the directions: North, South, East, or West. After $6$ steps, $X$ returns to the starting point $P$. Find the number of distinct paths $X$ can take. | [
"400"
] | [
"A person $X$ is standing at a point $P$ on a flat plane and starts walking. At each step, $X$ walks exactly $1$ unit in one of the directions: North, South, East, or West. After $6$ steps, $X$ returns to the starting point $P$. Find the number of distinct paths $X$ can take."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-f44d7d8065fed95052ce27984c0ac5ac8d0995c939e5e8a626db7c1cb34542d6 | A person 6 feet tall standing near a streetlight casts a 10-foot shadow on the ground. If the base of the streetlight is 45 feet from the tip of the shadow, what is the height, in feet, of the streetlight? | [
"27"
] | [
"A person 6 feet tall standing near a streetlight casts a 10-foot shadow on the ground. If the base of the streetlight is 45 feet from the tip of the shadow, what is the height, in feet, of the streetlight?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 9
} | 24 | 0 | 0 |
omr-ws-9aca717e75ec9c6d40ec243433442f1d9bcc2661be6331bfcf7ea796c2541a6c | A person borrowed a wad of money and spent it as follows: 1. $18 on a new CD. 2. Half of the remaining money on a new dress. 3. $11 on lunch. 4. One-third of the remaining money on a book. 5. $12 on gas. 6. One-fourth of the remaining money on a discounted CD. 7. $2 to someone. 8. $10 to her mom. List the items she bou... | [
"124"
] | [
"A person borrowed a wad of money and spent it as follows:\n1. $18 on a new CD.\n2. Half of the remaining money on a new dress.\n3. $11 on lunch.\n4. One-third of the remaining money on a book.\n5. $12 on gas.\n6. One-fourth of the remaining money on a discounted CD.\n7. $2 to someone.\n8. $10 to her mom.\n\nList t... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-7451256bf27b6de428480825844527d39b1d8aa7a4398c665794baeba905dfc4 | A person borrowed a wad of money and spent it as follows: 1. \$18 on a new CD. 2. Half of the remaining money on a new dress. 3. \$11 on lunch. 4. One-third of the remaining money on a book. 5. \$12 on gas. 6. One-fourth of the remaining money on a discounted CD. 7. \$2 to someone. 8. \$10 to her mother. List the items... | [
"124"
] | [
"A person borrowed a wad of money and spent it as follows:\n1. \\$18 on a new CD.\n2. Half of the remaining money on a new dress.\n3. \\$11 on lunch.\n4. One-third of the remaining money on a book.\n5. \\$12 on gas.\n6. One-fourth of the remaining money on a discounted CD.\n7. \\$2 to someone.\n8. \\$10 to her moth... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-2fc2565451578a2fa98b38b1432729d0af04ce7818dedf3c800ae83b9614ffbb | A person can answer a question correctly on the AMC with probability \( N \). What is the smallest possible \( N \) such that the expected value of the final score is 120 or more? (No guessing allowed) | [
"\\(\\frac{4}{5}\\)"
] | [
"A person can answer a question correctly on the AMC with probability \\( N \\). What is the smallest possible \\( N \\) such that the expected value of the final score is 120 or more? (No guessing allowed)"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 4,
"tir": 2
} | 10 | 0 | 0 |
omr-ws-f42dfedf5f58a932dc4393bbb42abd7e0495a3e19bf99e18bfcca3e78735fcd1 | A person can climb a staircase of 8 steps by taking 1, 2, or 3 steps at a time. How many ways can the person climb to the 8th step without stepping on the 7th step? | [
"37"
] | [
"A person can climb a staircase of 8 steps by taking 1, 2, or 3 steps at a time. How many ways can the person climb to the 8th step without stepping on the 7th step?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-f769beec05cfcad4237f5f5a13ddf2dca89f40d58db54180e72ce569126a593a | A person cashed a cheque for $200$ pounds, asking for some one-pound coins, $10$ times as many two-pound coins, and the balance in five-pound notes. How many of each did the person get? | [
"\\( a = 5 \\), \\( b = 50 \\), \\( c = 19 \\)"
] | [
"A person cashed a cheque for $200$ pounds, asking for some one-pound coins, $10$ times as many two-pound coins, and the balance in five-pound notes. How many of each did the person get?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 30
} | 31 | 0 | 0 |
omr-ws-1d8e53b45efc2899d3ff5b0a8de79ec46e4c82b0f1dc9e2218ca3f3d26323cb7 | A person chooses two numbers \(a\) and \(b\) such that \(|a|, |b| \le 1\). What is the probability that \(|a|^2 + |b|^2 \le 1\)? | [
"\\(\\frac{1}{2}\\)"
] | [
"A person chooses two numbers \\(a\\) and \\(b\\) such that \\(|a|, |b| \\le 1\\). What is the probability that \\(|a|^2 + |b|^2 \\le 1\\)?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-48283d88f5ebb45d05a77b1b159ead26468a06ea0db1b2f6773a8941aaa4c892 | A person conducts a random walk on a line, starting at $0$. Each minute, he randomly moves either $1$ unit in the positive direction or $1$ unit in the negative direction. What is his expected distance from the origin after $6$ moves? | [
"\\(\\frac{15}{8}\\)"
] | [
"A person conducts a random walk on a line, starting at $0$. Each minute, he randomly moves either $1$ unit in the positive direction or $1$ unit in the negative direction. What is his expected distance from the origin after $6$ moves?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-16e1673cfe01528fca378e57ce943f51fb022fbe96e4e26e10aeeb1b589ef3b5 | A person counted 74 heads and 196 legs at a horse racing area. How many humans and horses are there? | [
"50 humans and 24 horses"
] | [
"A person counted 74 heads and 196 legs at a horse racing area. How many humans and horses are there?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 23
} | 38 | 0 | 0 |
omr-ws-374841fdfa2082b26583c54916b5918caa42043631e7b81d37f75788b57716fd | A person decides to save money in a savings account for retirement. At the beginning of each year, \$1,000 is invested at 14% compounded annually. How much is the retirement fund after 40 years? | [
"1529905"
] | [
"A person decides to save money in a savings account for retirement. At the beginning of each year, \\$1,000 is invested at 14% compounded annually. How much is the retirement fund after 40 years?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 3
} | 4 | 0 | 0 |
omr-ws-d035bf73689ce74a08666728e9be52ba047ef8b0e0bf95fd008bfeadda17cc82 | A person dives from a bridge into a river and swims upstream for 1 hour at a constant speed. She then turns around and swims downstream at the same speed. As she passes under the bridge, a bystander tells her that her hat fell into the river when she dived. She continues downstream at the same speed and catches up with... | [
"\\(\\frac{1}{2}\\)"
] | [
"A person dives from a bridge into a river and swims upstream for 1 hour at a constant speed. She then turns around and swims downstream at the same speed. As she passes under the bridge, a bystander tells her that her hat fell into the river when she dived. She continues downstream at the same speed and catches up... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-a6a0dc3fcc8029bb63745674bbec54da4816fe36bcbc0e6057ac5792edb2d968 | A person drives a motorized vehicle, and the fuel consumption is given by a graph. Initially, the vehicle contains $3$ liters of fuel. After two hours, $1$ liter of fuel remains. a. If $1$ liter of fuel allows the vehicle to cover $32$ km, what is the total distance covered? Explain your answer. b. After two hours of t... | [
"32"
] | [
"A person drives a motorized vehicle, and the fuel consumption is given by a graph. Initially, the vehicle contains $3$ liters of fuel. After two hours, $1$ liter of fuel remains.\na. If $1$ liter of fuel allows the vehicle to cover $32$ km, what is the total distance covered? Explain your answer.\nb. After two hou... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-02fffe635443bb6318318c794a12c97536b13f1f4eea80ec6ad7a8d49b1abe60 | A person drives his car at a rate of 25 km/h and spends rupees 2 per km on petrol. If he drives at a faster speed of 40 km/h, the petrol cost increases to rupees 5 per km. Given that he has rupees 100 to spend on petrol, what is the maximum distance he can travel within one hour? Express this as a linear programming pr... | [
"30"
] | [
"A person drives his car at a rate of 25 km/h and spends rupees 2 per km on petrol. If he drives at a faster speed of 40 km/h, the petrol cost increases to rupees 5 per km. Given that he has rupees 100 to spend on petrol, what is the maximum distance he can travel within one hour? Express this as a linear programmi... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 1,
"tir": 23
} | 26 | 0 | 0 |
omr-ws-c58d6b3c653c4f9b3b8b3187f85484498e4da81cfcf08404c52dac3dc25be023 | A person flips a coin with equal probability of $\frac{1}{2}$ for heads and tails. Let event A be getting 5 heads in a row, and event B be getting 4 tails in a row. The person flips the coin until either event A or event B occurs, and then stops. Find the probability of event A occurring. | [
"\\dfrac{15}{46}"
] | [
"A person flips a coin with equal probability of $\\frac{1}{2}$ for heads and tails. Let event A be getting 5 heads in a row, and event B be getting 4 tails in a row. The person flips the coin until either event A or event B occurs, and then stops. Find the probability of event A occurring."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-31b493a3b5360e78adb0216f2604aaa6cf896210747ca7543290c94381893b6a | A person flying a kite holds the string 5 ft above the ground. When 200 ft of string has been let out, and the angle of the string above the earth is 61 degrees, a pin falls out of the kite and straight to the ground. What is the distance in feet that the pin lands from the person flying the kite? | [
"97"
] | [
"A person flying a kite holds the string 5 ft above the ground. When 200 ft of string has been let out, and the angle of the string above the earth is 61 degrees, a pin falls out of the kite and straight to the ground. What is the distance in feet that the pin lands from the person flying the kite?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f8195ca985640ebfb1a553b0316d8288727b82950fabbc3255bc8f3bc67e56ad | A person found $n > 1$ sets consisting of real numbers and computed the sum of elements for each set. The sums were $1!, 2!, \cdots, n!$ in some order. If $S$ is the number of distinct sums written on the board, find the minimum possible value of $S - \frac{\pi(n)}{2}$ for $n = 15$. | [
"12"
] | [
"A person found $n > 1$ sets consisting of real numbers and computed the sum of elements for each set. The sums were $1!, 2!, \\cdots, n!$ in some order. If $S$ is the number of distinct sums written on the board, find the minimum possible value of $S - \\frac{\\pi(n)}{2}$ for $n = 15$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c6fc4c7e45874f7605b89e75d30e8bd5497062067e873d27ed812adea705dd93 | A person has 10 pairs of shoes in a bag. She picks 3 shoes. What is the probability that there will be at least 1 pair of shoes? | [
"\\(\\frac{3}{19}\\)"
] | [
"A person has 10 pairs of shoes in a bag. She picks 3 shoes. What is the probability that there will be at least 1 pair of shoes?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-2012ea50601e453a446ecf204adcb27483f18b80f9de61982f00e79f4fe0983a | A person has 100 identically looking coins, where 30 are real and 70 are fake. The real coins all have the same weight, while the fake coins each have a different weight and are all heavier than the real coins. Using only one balance scale to compare two arbitrary groups of coins, determine the least number of weighing... | [
"4",
"70"
] | [
"A person has 100 identically looking coins, where 30 are real and 70 are fake. The real coins all have the same weight, while the fake coins each have a different weight and are all heavier than the real coins. Using only one balance scale to compare two arbitrary groups of coins, determine the least number of wei... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 4
} | 11 | 0 | 0 |
omr-ws-cd0d9d33c288b451d22d4a3233f9f009aaa6608c706fcd84d7b98106633aabed | A person has 12 CDs of the following types: A (3), B (4), C (2), D (1), E (2). If 5 CDs are randomly selected, what is the probability that either 3 A's or 3 B's will be selected? | [
"\\(\\frac{37}{198}\\)"
] | [
"A person has 12 CDs of the following types: A (3), B (4), C (2), D (1), E (2). If 5 CDs are randomly selected, what is the probability that either 3 A's or 3 B's will be selected?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-afa88d6fa0fbfc48a5e0be4d3a8877bbc8f860fa2f359d0aff3a1374e1a5e035 | A person has 250 bones. Each fall breaks 200 bones, and a bone can be broken multiple times. After 5 falls, what is the probability that all 250 bones are broken? | [
"e^{-\\frac{2}{25}}"
] | [
"A person has 250 bones. Each fall breaks 200 bones, and a bone can be broken multiple times. After 5 falls, what is the probability that all 250 bones are broken?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 3
} | 17 | 0 | 0 |
omr-ws-737819a1dc6d75562fa8c64ab13d05fbaaa000b508791f85e8712d005357d45a | A person has 3 children. The product of their ages is 72. The sum of their ages is equal to a specific number (the bar tab). After being told that the oldest child can play the piano, the ages can be uniquely determined. What are the ages of the children? | [
"3, 3, 8"
] | [
"A person has 3 children. The product of their ages is 72. The sum of their ages is equal to a specific number (the bar tab). After being told that the oldest child can play the piano, the ages can be uniquely determined. What are the ages of the children?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 30
} | 43 | 0 | 0 |
omr-ws-a605e1516b302ea80b660b57deee13224165617cd83ff09d7a1322239248ed57 | A person has 5 pairs of socks, all of identical color but each pair made from a different fabric. They randomly pair off the socks in the dark. What is the probability that exactly two matching pairs and three unmatched pairs are formed? | [
"\\(\\frac{4}{189}\\)"
] | [
"A person has 5 pairs of socks, all of identical color but each pair made from a different fabric. They randomly pair off the socks in the dark. What is the probability that exactly two matching pairs and three unmatched pairs are formed?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a651d27531e3e5896e912c851ce3a15d14453489313cd76dcff68fa5c716ac5b | A person has 6 friends. Each evening, for 5 days, he or she invites 3 of them so that the same group is never invited twice. How many ways are there to do this? | [
"\\( \\frac{20!}{15!} \\)"
] | [
"A person has 6 friends. Each evening, for 5 days, he or she invites 3 of them so that the same group is never invited twice. How many ways are there to do this?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-f56e157eef8a7cf179ecb524004b8a78be6b5cfd87a9235922714b0f404e5d52 | A person has a beekeeping, initially with \( A \) bees. The annual birth rate of the bee population is constant and equal to \( P\% \). Each year, \( X \) bees die. The population of bees is equal to the product \( Q \cdot A \) at the end of \( N \) years. Determine the value of \( X \) in terms of \( Q \), \( N \), \(... | [
"X = \\frac{A P \\left( \\left(1 + \\frac{P}{100}\\right)^N - Q \\right)}{100 \\left( \\left(1 + \\frac{P}{100}\\right)^N - 1 \\right)}"
] | [
"A person has a beekeeping, initially with \\( A \\) bees. The annual birth rate of the bee population is constant and equal to \\( P\\% \\). Each year, \\( X \\) bees die. The population of bees is equal to the product \\( Q \\cdot A \\) at the end of \\( N \\) years. Determine the value of \\( X \\) in terms of \... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b43a5a2a6358802e6b63aa4d1bb6be72c26c9092690d005eeb8573bb3aaf7dfa | A person has a sum of \( n \) dollars. Each day, they can buy one of the following: an ice cream for 1 dollar, a sandwich for 2 dollars, or a cake for 2 dollars. In how many ways can they spend the entire sum? | [
"\\dfrac{2^{n+1} + (-1)^n}{3}"
] | [
"A person has a sum of \\( n \\) dollars. Each day, they can buy one of the following: an ice cream for 1 dollar, a sandwich for 2 dollars, or a cake for 2 dollars. In how many ways can they spend the entire sum?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-1c6b3b2253d9f3db25cd793b1b52a4f24b42ef29a2be6bdad7941cd1046a7e1c | A person has seven friends and invites a different subset of three friends to dinner every night for one week (seven days). In how many ways can this be done so that all friends are invited at least once? | [
"31169376000"
] | [
"A person has seven friends and invites a different subset of three friends to dinner every night for one week (seven days). In how many ways can this be done so that all friends are invited at least once?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 31
} | 31 | 0 | 0 |
omr-ws-dc98369afd361ba1fdf6aa7124a38e4ee109adf869f7722182b3be6face6099e | A person has six friends and during a certain vacation, he met them during several dinners. He found that he dined: - with all six friends exactly on one day, - with every five of them on 2 days, - with every four of them on 3 days, - with every three of them on 4 days, - with every two of them on 5 days. Further, ever... | [
"1"
] | [
"A person has six friends and during a certain vacation, he met them during several dinners. He found that he dined:\n- with all six friends exactly on one day,\n- with every five of them on 2 days,\n- with every four of them on 3 days,\n- with every three of them on 4 days,\n- with every two of them on 5 days.\nFu... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 32
} | 32 | 0 | 0 |
omr-ws-03fb09ec1190af1568e9dd5d5ac62eb9073ce342dafecba91fd3034b277243bd | A person has steps to climb. From step \( i \), they can go to step \( i+1 \), \( i+2 \), ..., \( i+k-1 \), or \( i+k \). How many ways can they climb to the top starting from step 1? | [
"2^{k-1}"
] | [
"A person has steps to climb. From step \\( i \\), they can go to step \\( i+1 \\), \\( i+2 \\), ..., \\( i+k-1 \\), or \\( i+k \\). How many ways can they climb to the top starting from step 1?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 2
} | 11 | 0 | 0 |
omr-ws-9ff826c05d00630340129f579d66eea3ab47d6cc3e74804aba604db5c2c94917 | A person has to answer a question with 4 alternatives, where one or more may be correct. The probabilities are as follows: - The probability that he guesses is $\frac{1}{3}$. - The probability that he copies is $\frac{1}{6}$. - The probability that he copies the correct answer is $\frac{1}{8}$. If the answer he marked ... | [
"\\(\\frac{360}{391}\\)"
] | [
"A person has to answer a question with 4 alternatives, where one or more may be correct. The probabilities are as follows:\n- The probability that he guesses is $\\frac{1}{3}$.\n- The probability that he copies is $\\frac{1}{6}$.\n- The probability that he copies the correct answer is $\\frac{1}{8}$.\n\nIf the ans... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-111b38f535a1aeb94fe8f2416fb45e32f038cdbedda971b5f55c6746947c812d | A person has to change all the bulbs in an avenue of 20 posts, with each post placed 10 meters apart. The truck is 10 meters from the first post. How many meters will the person walk in total if they walk as little as possible to change all the bulbs? Note: The answer choices provided are in yards, but the problem stat... | [
"200"
] | [
"A person has to change all the bulbs in an avenue of 20 posts, with each post placed 10 meters apart. The truck is 10 meters from the first post. How many meters will the person walk in total if they walk as little as possible to change all the bulbs?\n\nNote: The answer choices provided are in yards, but the prob... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-af971a9ef4534d00ee06220ded0aed9fd855b37486ab859de727c612eac57ffb | A person is 40 miles away from a hotel, positioned such that the shortest path to the hotel forms a 30-60-90 triangle with a road 20 miles away. The person can walk at 3 mph in the grass and 5 mph on the road. Determine the path that minimizes the time taken to reach the hotel. | [
"12.26 hours"
] | [
"A person is 40 miles away from a hotel, positioned such that the shortest path to the hotel forms a 30-60-90 triangle with a road 20 miles away. The person can walk at 3 mph in the grass and 5 mph on the road. Determine the path that minimizes the time taken to reach the hotel."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 17
} | 19 | 0 | 0 |
omr-ws-4d07a3b9801340a78b1788c7be235d4e7d4e22b83f250a2006c33a0b9379c15d | A person is at a bin and needs to transfer balls from a line of 10 balls, each 1 unit apart, to the bin. The person can carry 2 balls at a time and walks a total of 20 units. What is the maximum number of balls that can be transferred? | [
"5"
] | [
"A person is at a bin and needs to transfer balls from a line of 10 balls, each 1 unit apart, to the bin. The person can carry 2 balls at a time and walks a total of 20 units. What is the maximum number of balls that can be transferred?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 8
} | 18 | 0 | 0 |
omr-ws-2c93ee50aedea6f447cc8eb812d4315f5c68dd4fca8fe49410a0996f4048735b | A person is born in the second half of the 19th century. If he will be $x$ years old in $x^2$ years, find his age in that year. | [
"44"
] | [
"A person is born in the second half of the 19th century. If he will be $x$ years old in $x^2$ years, find his age in that year."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-4d475b50bf4a6dfb43ea03a64a26dcc26fbbd537e6dbe36dc383874ec98015ab | A person is in the desert at point A, 10 km from a long, straight road and 45 km from a base camp at point B. The base camp is 10 km from the road, on the same side of the road as the person. On the road, the person can travel at a rate of 50 kph, but in the desert sands, the person can travel at 30 kph. The person wan... | [
"1.5"
] | [
"A person is in the desert at point A, 10 km from a long, straight road and 45 km from a base camp at point B. The base camp is 10 km from the road, on the same side of the road as the person. On the road, the person can travel at a rate of 50 kph, but in the desert sands, the person can travel at 30 kph. The perso... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-980f1ee62535629200a5cbc412155d865a01ed920670255e7a57e08838cfb196 | A person is instructed to take 48 pills over 30 days, with the condition that they take at least 1 pill and at most 6 pills each day. Let $S$ be the sum of all possible values of $k$ such that there exist $k$ consecutive days during which the total number of pills taken is exactly 11. Find $S$. | [
"65"
] | [
"A person is instructed to take 48 pills over 30 days, with the condition that they take at least 1 pill and at most 6 pills each day. Let $S$ be the sum of all possible values of $k$ such that there exist $k$ consecutive days during which the total number of pills taken is exactly 11. Find $S$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f389595cc7c0c613871042eae6a1173b3c0f761b5d828240fbd1e29bfecc334e | A person is lost in a 3 by 3 unit square maze, starting at the south-west corner and aiming to reach the north-east corner. The total length of all internal walls is four units, and all squares are reachable from any other square. Each move, whether it hits a wall or not, is counted as one move. Determine the expected ... | [
"20"
] | [
"A person is lost in a 3 by 3 unit square maze, starting at the south-west corner and aiming to reach the north-east corner. The total length of all internal walls is four units, and all squares are reachable from any other square. Each move, whether it hits a wall or not, is counted as one move. Determine the expe... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 6
} | 11 | 0 | 0 |
omr-ws-51b108db1e30b0bbe6a29e96f36d6d2de9a290b4324fb48800e8acf4bdb6b35f | A person is riding in a hot air balloon. For the first hour and a half, the wind current is a constant 22.0 miles per hour in the direction N 37.5° E. Then the wind current changes to 18.5 miles per hour and heads the balloon in the direction S 52.5° E for another 2 hours. What is the bearing of the balloon from its st... | [
"S 10.34° E"
] | [
"A person is riding in a hot air balloon. For the first hour and a half, the wind current is a constant 22.0 miles per hour in the direction N 37.5° E. Then the wind current changes to 18.5 miles per hour and heads the balloon in the direction S 52.5° E for another 2 hours. What is the bearing of the balloon from i... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-10d0875c52a36a246597994c89aa50556f747575a15228d138fa72f31ceb7f6f | A person is stacking 10 hemispheres such that some (or all, or none) make up spheres, and the rest face in the same direction. If the stacks are made randomly with equal probability under this constraint, what is the probability that the stack contains an even number of spheres? | [
"\\(\\frac{44}{89}\\)"
] | [
"A person is stacking 10 hemispheres such that some (or all, or none) make up spheres, and the rest face in the same direction. If the stacks are made randomly with equal probability under this constraint, what is the probability that the stack contains an even number of spheres?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 2,
"tir": 12
} | 16 | 0 | 0 |
omr-ws-a9888b635d535049ad4065d045adf84b82182570861397277f0c866b539523b2 | A person is walking down an escalator moving downwards and takes a total of 26 steps to reach the bottom in 30 seconds. When running down the escalator, the same person takes a total of 34 steps and reaches the bottom in 18 seconds. How many steps are on the escalator at a time? | [
"46"
] | [
"A person is walking down an escalator moving downwards and takes a total of 26 steps to reach the bottom in 30 seconds. When running down the escalator, the same person takes a total of 34 steps and reaches the bottom in 18 seconds. How many steps are on the escalator at a time?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-666f91c00c97eb326c4982feb041ae43747430173b48f2d5dfb56d813f3f17c7 | A person is walking due north from point P at a rate of 8 m/s. After 10 minutes, a boy walks from point M (1000 ft away from P) towards point P at a rate of 10 m/s. After 15 minutes from the boy's start, how far apart are the person and the boy? | [
"12000"
] | [
"A person is walking due north from point P at a rate of 8 m/s. After 10 minutes, a boy walks from point M (1000 ft away from P) towards point P at a rate of 10 m/s. After 15 minutes from the boy's start, how far apart are the person and the boy?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 6
} | 16 | 0 | 0 |
omr-ws-266022e249079c9680b8477eae8244177996f3d1c57c979c69770e94760fb91f | A person keeps flipping an unfair coin until it flips \(3\) tails in a row. The probability of it landing on heads is \(\frac{2}{3}\) and the probability it lands on tails is \(\frac{1}{3}\). What is the expected value of the number of times the coin flips? | [
"39"
] | [
"A person keeps flipping an unfair coin until it flips \\(3\\) tails in a row. The probability of it landing on heads is \\(\\frac{2}{3}\\) and the probability it lands on tails is \\(\\frac{1}{3}\\). What is the expected value of the number of times the coin flips?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 4
} | 19 | 0 | 0 |
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