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-8f5c18f290e1e0d32681d4d94041a0865d863f38438905c54f86696255fdd24c | A forestry has a $p\%$ annual growth rate of wood. Every winter an amount $x$ of wood is obtained. What must $x$ be so that in $n$ years the amount of wood in the forestry becomes $q$ times the original amount $a$? | [
"x=a\\frac{\\left(1+\\frac{p}{100}\\right)^n-q}{\\left(1+\\frac{p}{100}\\right)^n-1} \\cdot \\frac{p}{100}"
] | [
"A forestry has a $p\\%$ annual growth rate of wood. Every winter an amount $x$ of wood is obtained. What must $x$ be so that in $n$ years the amount of wood in the forestry becomes $q$ times the original amount $a$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a2142b0a6e9e668dbaa241576ba0d6eb1e926cab0590c46d977db94f50ba9d5d | A forty-foot length of chain, of uniform weight per unit of length, hangs between two buildings, attached at the same height at both ends. The shape of this chain is called a "catenary." If the distance from the points of suspension to the lowest point is twenty feet, how far apart are the buildings? | [
"40"
] | [
"A forty-foot length of chain, of uniform weight per unit of length, hangs between two buildings, attached at the same height at both ends. The shape of this chain is called a \"catenary.\" If the distance from the points of suspension to the lowest point is twenty feet, how far apart are the buildings?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 5
} | 16 | 0 | 0 |
omr-ws-74737d75b88e819b89993a37f11d527da11051f76d7e798bf8465b3080f9bd30 | A fossilized leaf contains 14% of its normal amount of carbon-14. How old is the fossil (to the nearest year)? Use 5,600 years as the half-life of carbon-14. | [
"15882"
] | [
"A fossilized leaf contains 14% of its normal amount of carbon-14. How old is the fossil (to the nearest year)? Use 5,600 years as the half-life of carbon-14."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-c6977b0365ac48dce5c1a891879093c1caf2545c4b15fa50d323d88c9e6ae4ce | A four-digit integer $m$ and the four-digit integer obtained by reversing the order of the digits of $m$ are both divisible by 45. If $m$ is divisible by 7, what is the greatest possible value of $m$? | [
"5985"
] | [
"A four-digit integer $m$ and the four-digit integer obtained by reversing the order of the digits of $m$ are both divisible by 45. If $m$ is divisible by 7, what is the greatest possible value of $m$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0073e24f1db2cdb3acc75072e85cf698552882e9c42d9595eea2030cdac02e7c | A four-digit integer \( m \) and the four-digit integer obtained by reversing the order of the digits of \( m \) are both divisible by 45. If \( m \) is also divisible by 7, what is the greatest possible value of \( m \)? | [
"5985"
] | [
"A four-digit integer \\( m \\) and the four-digit integer obtained by reversing the order of the digits of \\( m \\) are both divisible by 45. If \\( m \\) is also divisible by 7, what is the greatest possible value of \\( m \\)?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-00d2670bf930238d64a976f89020d18f699d6c82ed5ef04d661e924dd8343086 | A four-digit license plate number is equal to the square of the sum of the two two-digit numbers formed by taking the first two digits and the last two digits of the license plate number. The first digit of the license plate is not zero. What is the four-digit license plate number? | [
"9801"
] | [
"A four-digit license plate number is equal to the square of the sum of the two two-digit numbers formed by taking the first two digits and the last two digits of the license plate number. The first digit of the license plate is not zero. What is the four-digit license plate number?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-8ed6504efc40d21444fbba45ddf4674c44b2758269425531905e2f47de67aa3c | A four-digit number $\overline{abcd}$ ($a \ne 0$) is called "pora" if the following conditions are satisfied: - $a \ge b$ - $ab - cd = cd - ba$ Find all four-digit numbers that are "pora". | [
"$2011, 1111, 4022, 3122, 2222, 6033, 5133, 4233, 3333, 8044, 7144, 6244, 5344, 4444, 9155, 8255, 7355, 6455, 5555, 9366, 8466, 7566, 6666, 9577, 8677, 7777, 9788, 8888, 9999$"
] | [
"A four-digit number $\\overline{abcd}$ ($a \\ne 0$) is called \"pora\" if the following conditions are satisfied:\n- $a \\ge b$\n- $ab - cd = cd - ba$\n\nFind all four-digit numbers that are \"pora\"."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 25
} | 26 | 0 | 0 |
omr-ws-35df68a06018e2ad249d52a97d07d55693ae0cb917b7dbbcd16b0bb3f5a9f8ec | A four-digit number $\overline{abcd}$ is called "interchangeable" if there exists an integer $n > 1$ such that $n \times \overline{abcd}$ is a four-digit number that begins with $d$ and ends with $a$. For example, $1009$ is interchangeable since $1009 \times 9 = 9081$. Find the largest interchangeable number. | [
"2248"
] | [
"A four-digit number $\\overline{abcd}$ is called \"interchangeable\" if there exists an integer $n > 1$ such that $n \\times \\overline{abcd}$ is a four-digit number that begins with $d$ and ends with $a$. For example, $1009$ is interchangeable since $1009 \\times 9 = 9081$. Find the largest interchangeable number... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-65656eb5d1c17194a020c81b86ea64bd156e2ee5f32cc4a65f2bf99e928cfe2b | A four-digit number $\textbf{YEAR}$ is called $\textbf{good}$ if the system of linear equations: $$\begin{cases} Yx+Ey+Az+Rw=Y \\ Rx+Yy+Ez+Aw = E \\ Ax+Ry+Yz+Ew = A \\ Ex+Ay+Rz+Yw = R \end{cases} $$ has at least two solutions. Find all such $\textbf{good}$ numbers $\textbf{YEAR}$ in the 2nd decade of the 21st century (... | [
"2013"
] | [
"A four-digit number $\\textbf{YEAR}$ is called $\\textbf{good}$ if the system of linear equations:\n\n$$\\begin{cases}\n Yx+Ey+Az+Rw=Y \\\\\n Rx+Yy+Ez+Aw = E \\\\\n Ax+Ry+Yz+Ew = A \\\\\n Ex+Ay+Rz+Yw = R\n\\end{cases}\n$$\n\nhas at least two solutions. Find all such $\\textbf{good}$ numbers $\\textbf{YEAR}$ in the... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-e3c3cc9182697a0be26e99bf9f9f34b68d386784d567ff06ea7e3123e0a7fec8 | A four-digit number $abcd$ is called cool if $a$ is divisible by $4$, the two-digit number $ab$ is divisible by $5$, the three-digit number $abc$ is divisible by $6$, and $abcd$ is divisible by $7$. How many cool numbers are there where $8$ is not one of the digits? | [
"4"
] | [
"A four-digit number $abcd$ is called cool if $a$ is divisible by $4$, the two-digit number $ab$ is divisible by $5$, the three-digit number $abc$ is divisible by $6$, and $abcd$ is divisible by $7$. How many cool numbers are there where $8$ is not one of the digits?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f5652c709fb50cf68c02b964f0d9018332aa091b97792159f326c9c2d98beb5d | A four-digit number (ranging from $0000$ to $9999$) is said to be lucky if the sum of its last two digits equals the sum of its first two digits. If a four-digit number is picked at random, what is the probability that it is a lucky number? | [
"\\(\\frac{1}{100}\\)"
] | [
"A four-digit number (ranging from $0000$ to $9999$) is said to be lucky if the sum of its last two digits equals the sum of its first two digits. If a four-digit number is picked at random, what is the probability that it is a lucky number?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4295325204c7e606399b1e1ea148e9dbd3710bf8315c3b47b60b6cc7e4687831 | A four-digit number \( N \) leaves a remainder of 10 when divided by 21, a remainder of 11 when divided by 23, and a remainder of 12 when divided by 25. What is the sum of the digits of \( N \)? | [
"16"
] | [
"A four-digit number \\( N \\) leaves a remainder of 10 when divided by 21, a remainder of 11 when divided by 23, and a remainder of 12 when divided by 25. What is the sum of the digits of \\( N \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-82bfb3b4626d3eb2e6c743795f134aa40a4dde77af44fe84e3941d16d7a3ecbd | A four-digit number \( a \) is multiplied by 3 to give a five-digit number \( b \). Together, \( a \) and \( b \) contain all the digits from 0 to 9 except for 3. Find \( a \) and \( b \) and explain how you found them. | [
"\\( a = 5694 \\), \\( b = 17082 \\)"
] | [
"A four-digit number \\( a \\) is multiplied by 3 to give a five-digit number \\( b \\). Together, \\( a \\) and \\( b \\) contain all the digits from 0 to 9 except for 3. Find \\( a \\) and \\( b \\) and explain how you found them."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 31
} | 32 | 0 | 0 |
omr-ws-f8b9316a1e9b574467eb371d5ca788b61ae8f665c16676a2acc72ab9aca10170 | A four-digit number \( a \) is multiplied by 3 to give a five-digit number \( b \). Together, \( a \) and \( b \) contain all the digits from 1 to 9 exactly once. Find \( a \) and \( b \). | [
"\\( a = 5832, b = 17496 \\) and \\( a = 5823, b = 17469 \\)."
] | [
"A four-digit number \\( a \\) is multiplied by 3 to give a five-digit number \\( b \\). Together, \\( a \\) and \\( b \\) contain all the digits from 1 to 9 exactly once. Find \\( a \\) and \\( b \\)."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 31
} | 31 | 0 | 0 |
omr-ws-e806f27b302258179f3efb1fb56d3479a32a8ece5c777ede3e081ab3a68f854b | A four-digit number \( abcd \) is called "reversible" if the number \( n \times abcd \) is a four-digit number of the form \( dnma \), where \( n > 1 \) and \( abcd \) and \( dnma \) are not multiplications but four-digit numbers. Each letter represents a digit. Find the greatest "reversible" number. | [
"2118"
] | [
"A four-digit number \\( abcd \\) is called \"reversible\" if the number \\( n \\times abcd \\) is a four-digit number of the form \\( dnma \\), where \\( n > 1 \\) and \\( abcd \\) and \\( dnma \\) are not multiplications but four-digit numbers. Each letter represents a digit. Find the greatest \"reversible\" numb... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-d50e97c554669dd428d77d0270e25723d8b7007fe0ee32e73d7eeb207c16200c | A four-digit number \( abcd \) where \( a \) is non-zero is said to be increasing if \( a < b < c < d \) and decreasing if \( a > b > c > d \). The number of four-digit increasing and decreasing numbers are \( x \) and \( y \) respectively. Find \( |x - y| \). | [
"84"
] | [
"A four-digit number \\( abcd \\) where \\( a \\) is non-zero is said to be increasing if \\( a < b < c < d \\) and decreasing if \\( a > b > c > d \\). The number of four-digit increasing and decreasing numbers are \\( x \\) and \\( y \\) respectively. Find \\( |x - y| \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-3284f494adae86108b7e3dc20f11c4ecfa05ff5c65f3afa41bbce6ca7531aa78 | A four-digit number \( n = \overline{abcd} \), where \( a, b, c, \) and \( d \) are digits and \( a \neq 0 \), is said to be guanaco if the product \( \overline{ab} \times \overline{cd} \) is a positive divisor of \( n \). Find all guanaco numbers. | [
"1352, 1734"
] | [
"A four-digit number \\( n = \\overline{abcd} \\), where \\( a, b, c, \\) and \\( d \\) are digits and \\( a \\neq 0 \\), is said to be guanaco if the product \\( \\overline{ab} \\times \\overline{cd} \\) is a positive divisor of \\( n \\). Find all guanaco numbers."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 32
} | 32 | 0 | 0 |
omr-ws-32b6225bd9c9af5a148cafb8ad996d205a95acd3b4d6b2cdef123dbd6c0b0bd6 | A four-digit number is "well-meaning" if its second digit is the mean of its first and third digits, and its third digit is the mean of its second and fourth digits. How many well-meaning four-digit numbers are there? | [
"30"
] | [
"A four-digit number is \"well-meaning\" if its second digit is the mean of its first and third digits, and its third digit is the mean of its second and fourth digits. How many well-meaning four-digit numbers are there?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4a72b98c2ac71f196d864d22edb85163d5fac0b84b59ba5d31a7d8062f5c9657 | A four-digit number leaves a remainder of 10 when divided by 21, a remainder of 11 when divided by 23, and a remainder of 12 when divided by 25. What is the sum of the digits of this number? | [
"16"
] | [
"A four-digit number leaves a remainder of 10 when divided by 21, a remainder of 11 when divided by 23, and a remainder of 12 when divided by 25. What is the sum of the digits of this number?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 24
} | 39 | 0 | 0 |
omr-ws-31a0c17c2b556d2b700e1acb99c8ba9d8b93d63cac6e37cce32def46c58b4af2 | A four-digit number uses each of the digits 1, 2, 3, and 4 exactly once. What is the probability that the number is not a multiple of 4? | [
"\\(\\frac{3}{4}\\)"
] | [
"A four-digit number uses each of the digits 1, 2, 3, and 4 exactly once. What is the probability that the number is not a multiple of 4?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-789fe6ef83e7fb29363443a0d8ceaee792bf4f297e6197918fb80dc3cd178215 | A four-digit number which is a perfect square is created by writing Anne's age in years followed by Tom's age in years. Similarly, in 31 years, their ages in the same order will again form a four-digit perfect square. Determine the present ages of Anne and Tom. | [
"Anne's age is 12 and Tom's age is 25."
] | [
"A four-digit number which is a perfect square is created by writing Anne's age in years followed by Tom's age in years. Similarly, in 31 years, their ages in the same order will again form a four-digit perfect square. Determine the present ages of Anne and Tom."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 10,
"tir": 29
} | 39 | 0 | 0 |
omr-ws-8ec49b6e97a1a1b3962c4e24fe27296764e1ebbd47890a005754cfd05f2d1b7d | A four-digit number will be formed from the digits $0, 1, 2, 3, 4, 5$ provided that the digits in the number are not repeated, and the number formed is a multiple of $3$. What is the probability that the number formed has a value less than $3000$? | [
"0.4 (or 40%) adjusted for the multiple of 3 condition. \n\nSince the exact count is not provided, the answer is:"
] | [
"A four-digit number will be formed from the digits $0, 1, 2, 3, 4, 5$ provided that the digits in the number are not repeated, and the number formed is a multiple of $3$. What is the probability that the number formed has a value less than $3000$?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-2b7abf746d39c1ec419f062b8977541667a537dc0d7df3c99e44b79ae620dd26 | A four-digit positive integer is called "quasisential" if its digits are all positive integer powers of $2,$ but the sum of its digits is not an integer power of $2.$ How many four-digit positive integers are "quasisential"? | [
"228"
] | [
"A four-digit positive integer is called \"quasisential\" if its digits are all positive integer powers of $2,$ but the sum of its digits is not an integer power of $2.$ How many four-digit positive integers are \"quasisential\"?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 2
} | 15 | 0 | 0 |
omr-ws-d13e7c09ae9751a0cee42399282f1adaa038424359b92071f39c6e7291466a7c | A four-element set $\{a, b, c, d\}$ of positive integers is called "good" if there are two of them such that their product is a multiple of the greatest common divisor of the remaining two. Find the greatest possible value of $n$ such that any four-element set with elements less than or equal to $n$ is good. | [
"230"
] | [
"A four-element set $\\{a, b, c, d\\}$ of positive integers is called \"good\" if there are two of them such that their product is a multiple of the greatest common divisor of the remaining two. Find the greatest possible value of $n$ such that any four-element set with elements less than or equal to $n$ is good."
... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dc014030f77d96460304fa967a3c697e42ec8b8b54af18aa0e9be62507d34dad | A four-sided polygon is inscribed in a circle with diameter \( AC = 2 \). Given that \( \mathbf{AC} = \frac{3}{2} \mathbf{AB} + \frac{5}{2} \mathbf{AD} \) and \( AC \) and \( BD \) intersect at point \( E \), find the length of \( BD \). | [
"\\dfrac{4\\sqrt{5}}{5}"
] | [
"A four-sided polygon is inscribed in a circle with diameter \\( AC = 2 \\). Given that \\( \\mathbf{AC} = \\frac{3}{2} \\mathbf{AB} + \\frac{5}{2} \\mathbf{AD} \\) and \\( AC \\) and \\( BD \\) intersect at point \\( E \\), find the length of \\( BD \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e7117c8541a2f1fc0a15c5fcdf4eb127b1c7341b246690241a61ba5ebe94ab1d | A four-term sequence is formed by adding each term of a four-term arithmetic sequence of positive integers to the corresponding term of a four-term geometric sequence of positive integers. The first three terms of the resulting four-term sequence are 57, 60, and 91. What is the fourth term of this sequence? | [
"206"
] | [
"A four-term sequence is formed by adding each term of a four-term arithmetic sequence of positive integers to the corresponding term of a four-term geometric sequence of positive integers. The first three terms of the resulting four-term sequence are 57, 60, and 91. What is the fourth term of this sequence?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 48,
"genselect": 5
} | 53 | 0 | 0 |
omr-ws-6abbf0557f2e6734dff7f2bc362156064f628b102f42fd632f07ab813441cc7b | A fraction a/b is said to approximate a number if it is closer to the number than a/b+1 or a/b-1. x/y approximates π and x/z approximates e. The ratio (z-y):x = k:1. Compute k to 4 decimal places. | [
"0.0496"
] | [
"A fraction a/b is said to approximate a number if it is closer to the number than a/b+1 or a/b-1. x/y approximates π and x/z approximates e. The ratio (z-y):x = k:1. Compute k to 4 decimal places."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 6
} | 18 | 0 | 0 |
omr-ws-18c47641310c26ed479af15481057d11ce9350dfc3a307c6198c02fe0aacf884 | A fractional linear transformation maps the annulus $r<|z|<1$ (where $r>0$) onto the domain bounded by the two circles $\bigg|z-\frac{1}{4}\bigg|=\frac{1}{4}$ and $|z|=1$. Find $r$. | [
"\\( \\sqrt{5} - 2 \\)"
] | [
"A fractional linear transformation maps the annulus $r<|z|<1$ (where $r>0$) onto the domain bounded by the two circles $\\bigg|z-\\frac{1}{4}\\bigg|=\\frac{1}{4}$ and $|z|=1$. Find $r$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-24009217e723045a70fae1c237fc30599a2bd35a13d4f93a10bf201741e183fe | A frame of a cube with an edge $1$ is given. $8$ spiders sit on this frame, while the distance between any two (measured by the shortest path along the edges of the cube) not less than $R$. For what maximum $R$ is this possible? | [
"3"
] | [
"A frame of a cube with an edge $1$ is given. $8$ spiders sit on this frame, while the distance between any two (measured by the shortest path along the edges of the cube) not less than $R$. For what maximum $R$ is this possible?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-99b05d798f4572b131a486d76bb527d01270448e2be8b4baf299ac1ac6a7bf2d | A frame of a cube with an edge of 1 is given. 8 spiders sit on this frame, while the distance between any two (measured by the shortest path along the edges of the cube) is not less than \( R \). For what maximum \( R \) is this possible? | [
"2"
] | [
"A frame of a cube with an edge of 1 is given. 8 spiders sit on this frame, while the distance between any two (measured by the shortest path along the edges of the cube) is not less than \\( R \\). For what maximum \\( R \\) is this possible?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-be03aa0c55d786e9aef59661f48165bbdc097efe975b99724e8849fb66a4b7b8 | A frame of a tetrahedron is assembled from sticks of length $1$. Baba Yaga plants $7$ spiders per frame. From the distances between pairs of spiders (measured the shortest path along the edges of the tetrahedron) Kashchei chooses the smallest distance $R$ and pays Yaga $R$ kg of gold. What is the largest number can Yag... | [
"1"
] | [
"A frame of a tetrahedron is assembled from sticks of length $1$. Baba Yaga plants $7$ spiders per frame. From the distances between pairs of spiders (measured the shortest path along the edges of the tetrahedron) Kashchei chooses the smallest distance $R$ and pays Yaga $R$ kg of gold. What is the largest number ca... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6b67963f8206e62aa4135cb3f65db8f0a1374735e3127d2059e1395b60718939 | A free-falling body is known to cover 4.9 m in the first second and 9.8 m more than during the preceding second for every subsequent second. Suppose two bodies begin falling from the same height, one after the other, with an interval of 5 s. In what time will they be at a distance of 220.5 m from each other? | [
"In 7 s after the first body begins to fall"
] | [
"A free-falling body is known to cover 4.9 m in the first second and 9.8 m more than during the preceding second for every subsequent second. Suppose two bodies begin falling from the same height, one after the other, with an interval of 5 s. In what time will they be at a distance of 220.5 m from each other?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1193f1ff142d1d1f8a0b4db335cc3454e0a2716e1a44184afbfe924ae650cd99 | A freight train and a passenger train start toward each other from two points that are 500 miles apart. After 3 hours, the trains are still 80 miles apart. If the speed of the passenger train is 20 miles per hour greater than the speed of the freight train, find the speed of each train. | [
"Speed of freight train = 60 mph, Speed of passenger train = 80 mph"
] | [
"A freight train and a passenger train start toward each other from two points that are 500 miles apart. After 3 hours, the trains are still 80 miles apart. If the speed of the passenger train is 20 miles per hour greater than the speed of the freight train, find the speed of each train."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 25
} | 25 | 0 | 0 |
omr-ws-4509d1554dd232e2934da982c97bdd20946a68c07532c006eaac8e331e082c99 | A freight train consists of four- and eight- wheel tank-cars with oil. The train weighs 940 tons. It is required to determine the number of the eight- and four- wheel tank-cars and also their weight, if it is given that the number of the four-wheel cars is 5 more than that of the eight-wheel cars; the eight-wheel car w... | [
"There are 10 eight-wheel tank-cars, each weighing 24 tons, and 15 four-wheel cars, each weighing 8 tons"
] | [
"A freight train consists of four- and eight- wheel tank-cars with oil. The train weighs 940 tons. It is required to determine the number of the eight- and four- wheel tank-cars and also their weight, if it is given that the number of the four-wheel cars is 5 more than that of the eight-wheel cars; the eight-wheel ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3bed54917ac98c67d1405d86d35122a51c27aa1d1582e970ed0741d0efbfac22 | A freight train departed from Moscow at $x$ hours and $y$ minutes and arrived at Saratov at $y$ hours and $z$ minutes. The length of its trip was $z$ hours and $x$ minutes. Find all possible values of $x$. | [
"\\( x = 0 \\) or \\( x = 12 \\)"
] | [
"A freight train departed from Moscow at $x$ hours and $y$ minutes and arrived at Saratov at $y$ hours and $z$ minutes. The length of its trip was $z$ hours and $x$ minutes. Find all possible values of $x$."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 11
} | 11 | 0 | 0 |
omr-ws-1cf6da72f2b1e38777296d315abf7466413fa436f30969562b6659588183c5b7 | A freight train departed from point A at $x$ hours $y$ minutes and arrived at point B at $y$ hours $x$ minutes. It is known that it took $x$ hours and $x$ minutes to get from point A to point B. Find all possible values of $x$. | [
"0 or 12"
] | [
"A freight train departed from point A at $x$ hours $y$ minutes and arrived at point B at $y$ hours $x$ minutes. It is known that it took $x$ hours and $x$ minutes to get from point A to point B. Find all possible values of $x$."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 15
} | 15 | 0 | 0 |
omr-ws-d7ce1e535b3dfec9d26361b7504496f8bc782df3bd8f1445eafc5b0bd00d5f5a | A freight train going from $A$ to $B$ arrives at station $C$ together with a passenger train going from $B$ to $A$ with a speed $m$ times that of the freight train. The two trains stop at $C$ for $t$ hours and then continue on their ways, each train increasing its speed 25\% over the original speed it had prior to arri... | [
"$5m(t-t_2)-5m^{-1}(t-t_1)$"
] | [
"A freight train going from $A$ to $B$ arrives at station $C$ together with a passenger train going from $B$ to $A$ with a speed $m$ times that of the freight train. The two trains stop at $C$ for $t$ hours and then continue on their ways, each train increasing its speed 25\\% over the original speed it had prior t... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-45742eb6ac4e7800bc94a00774d052544fca3d190d3cb275470dfe9b6d1a8685 | A freight train travels 1 mile in 1 minute 30 seconds. At this rate, how many miles will the train travel in 1 hour? | [
"40"
] | [
"A freight train travels 1 mile in 1 minute 30 seconds. At this rate, how many miles will the train travel in 1 hour?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-39af00394f10f132fcf899a82e5ac8df042367ca89c23885d3704769da450926 | A frequency distribution of the scores for Mr. Sampson's algebra class is shown. What percent of the class received a score in the $60\%$-$69\%$ range? \begin{tabular}{|c|c|} Test Scores & Frequencies\\ \hline $90\% - 100\%$& IIII\\ $80\% - 89\%$& IIII IIII\\ $70\% - 79\%$& IIII II\\ $60\% - 69\%$ & IIII I\\ Below $60\... | [
"20\\%"
] | [
"A frequency distribution of the scores for Mr. Sampson's algebra class is shown. What percent of the class received a score in the $60\\%$-$69\\%$ range? \\begin{tabular}{|c|c|}\nTest Scores & Frequencies\\\\\n\\hline\n$90\\% - 100\\%$& IIII\\\\\n$80\\% - 89\\%$& IIII IIII\\\\\n$70\\% - 79\\%$& IIII II\\\\\n$60\\%... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f05b78dd925a84ab759946a58ac082c728c78643829625cd85065895649fb7c4 | A friction-less board has the shape of an equilateral triangle with side length 1 meter. A ball is fired from vertex $A$ towards side $BC$ and bounces off the walls nine times before hitting a vertex for the first time. The bounces follow the law of reflection (angle of incidence equals angle of reflection). The distan... | [
"31"
] | [
"A friction-less board has the shape of an equilateral triangle with side length 1 meter. A ball is fired from vertex $A$ towards side $BC$ and bounces off the walls nine times before hitting a vertex for the first time. The bounces follow the law of reflection (angle of incidence equals angle of reflection). The d... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-57f4a786eaf7066680e5b3565fcc908c076a740c35e3e8d8271c53b756ab9a92 | A frictionless board has the shape of an equilateral triangle with side length $1$ meter and bouncing walls along the sides. A tiny super bouncy ball is fired from vertex $A$ towards side $BC$. The ball bounces off the walls nine times before it hits a vertex for the first time. The bounces are such that the angle of i... | [
"21"
] | [
"A frictionless board has the shape of an equilateral triangle with side length $1$ meter and bouncing walls along the sides. A tiny super bouncy ball is fired from vertex $A$ towards side $BC$. The ball bounces off the walls nine times before it hits a vertex for the first time. The bounces are such that the angle... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-1795460d1de620bbb213156cc0d9bceb5729d63586b0a347a2bbae522f9026b5 | A frictionless particle is sliding down a helix given by \( r(\theta) = (a \cos(\theta))\mathbf{i} + (a \sin(\theta))\mathbf{j} + (b\theta)\mathbf{k} \) with \( a, b > 0 \) under the influence of gravity. The speed of a falling particle after a distance \( z \) is \( \sqrt{2gz} \). a) Find \( \frac{d\theta}{dt} \) at \... | [
"\\( 2\\sqrt{\\frac{\\pi g b}{a^2 + b^2}} \\)"
] | [
"A frictionless particle is sliding down a helix given by \\( r(\\theta) = (a \\cos(\\theta))\\mathbf{i} + (a \\sin(\\theta))\\mathbf{j} + (b\\theta)\\mathbf{k} \\) with \\( a, b > 0 \\) under the influence of gravity. The speed of a falling particle after a distance \\( z \\) is \\( \\sqrt{2gz} \\).\n\na) Find \\(... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 5
} | 7 | 0 | 0 |
omr-ws-c26f1c17943870b098686764bb370fed58c7f53a0e6bdc173f332df0719e7f5e | A frictionless shell is fired from the ground with speed $v$ at an unknown angle to the vertical. It hits a plane at a height $h$. Calculate the maximum radius $R$ from the point directly below the point of impact where the gun can be situated, given by $R = \frac{v}{g} (v^2 - 2gh)^{\frac{1}{2}}$. Find $R$ when $v = 10... | [
"970"
] | [
"A frictionless shell is fired from the ground with speed $v$ at an unknown angle to the vertical. It hits a plane at a height $h$. Calculate the maximum radius $R$ from the point directly below the point of impact where the gun can be situated, given by $R = \\frac{v}{g} (v^2 - 2gh)^{\\frac{1}{2}}$. Find $R$ when ... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 5
} | 11 | 0 | 0 |
omr-ws-605573cefd164a1f9af323b132474ff8567641cbcbc7e2c864fe367b9ac80265 | A friendly football match lasts 90 minutes. In this problem, we consider one of the teams, coached by Sir Alex, which plays with 11 players at all times. a) Sir Alex wants each of his players to play the same integer number of minutes, but each player has to play less than 60 minutes in total. What is the minimum numbe... | [
"a) 18, b) 17"
] | [
"A friendly football match lasts 90 minutes. In this problem, we consider one of the teams, coached by Sir Alex, which plays with 11 players at all times.\n\na) Sir Alex wants each of his players to play the same integer number of minutes, but each player has to play less than 60 minutes in total. What is the minim... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-124d0464d610d9d47e39899fd78d5323df854a6b02478df094ac84cec8be840e | A frisbee is thrown in an 8-by-16 foot room according to the equation \( x = -(y-4)^2 + 16 \). The frisbee travels at a speed of 2 feet per second and spontaneously combusts after 10 seconds. Will the frisbee hit the window located at the top left corner (0, 8)? | [
"No"
] | [
"A frisbee is thrown in an 8-by-16 foot room according to the equation \\( x = -(y-4)^2 + 16 \\). The frisbee travels at a speed of 2 feet per second and spontaneously combusts after 10 seconds. Will the frisbee hit the window located at the top left corner (0, 8)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 4
} | 5 | 0 | 0 |
omr-ws-76a4fcfc7a9eb42d3ec718b663178dc9e70750f7dd2f57572eeeaa4d7de64972 | A frog (of negligible size) moves by jumping on an infinite grid plane 1cm × 1cm. It can make the following jumps: - 6cm to the right, - 8cm upwards, - 9cm to the left, - 10cm downwards. A 10×10 square on the grid is colored red. How many of the 100 red points can the frog reach if it starts at a corner of this square? | [
"20"
] | [
"A frog (of negligible size) moves by jumping on an infinite grid plane 1cm × 1cm. It can make the following jumps:\n- 6cm to the right,\n- 8cm upwards,\n- 9cm to the left,\n- 10cm downwards.\nA 10×10 square on the grid is colored red. How many of the 100 red points can the frog reach if it starts at a corner of th... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 1
} | 24 | 0 | 0 |
omr-ws-72c48d75d5a161ef05723cba0d117617158b3ff9ee6d573d2fa3c07f3887caa7 | A frog and a grasshopper are 2 meters apart. Each second, the frog moves either 25 cm or 50 cm towards the grasshopper, and the grasshopper jumps either 25 cm or 50 cm towards the frog. The frog will eat the grasshopper if the two reach each other on the ground. In how many ways can the grasshopper be eaten by the frog... | [
"14"
] | [
"A frog and a grasshopper are 2 meters apart. Each second, the frog moves either 25 cm or 50 cm towards the grasshopper, and the grasshopper jumps either 25 cm or 50 cm towards the frog. The frog will eat the grasshopper if the two reach each other on the ground. In how many ways can the grasshopper be eaten by the... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-df00d0664463c16b8cccc9934140f6aa7ea8d46af40bee1cb0e7ae9b2a291faa | A frog at $(1, 2)$ makes jumps of length $1$ parallel to the coordinate axes, chosen randomly. The sequence ends when the frog reaches a side of the square with vertices $(0,0), (0,4), (4,4),$ and $(4,0)$. What is the probability that the sequence of jumps ends on a vertical side of the square? | [
"\\dfrac{5}{8}"
] | [
"A frog at $(1, 2)$ makes jumps of length $1$ parallel to the coordinate axes, chosen randomly. The sequence ends when the frog reaches a side of the square with vertices $(0,0), (0,4), (4,4),$ and $(4,0)$. What is the probability that the sequence of jumps ends on a vertical side of the square?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6925dced83e25eff3320879876327f5a8360eb313c9f8588c4cb9580198a3b1f | A frog begins at $P_0 = (0,0)$ and makes a sequence of jumps according to the following rule: from $P_n = (x_n, y_n),$ the frog jumps to $P_{n+1},$ which may be any of the points $(x_n + 7, y_n + 2),$ $(x_n + 2, y_n + 7),$ $(x_n - 5, y_n - 10),$ or $(x_n - 10, y_n - 5).$ There are $M$ points $(x, y)$ with $|x| + |y| \l... | [
"373"
] | [
"A frog begins at $P_0 = (0,0)$ and makes a sequence of jumps according to the following rule: from $P_n = (x_n, y_n),$ the frog jumps to $P_{n+1},$ which may be any of the points $(x_n + 7, y_n + 2),$ $(x_n + 2, y_n + 7),$ $(x_n - 5, y_n - 10),$ or $(x_n - 10, y_n - 5).$ There are $M$ points $(x, y)$ with $|x| +... | [
"MATH_training_set"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 8,
"tir": 14
} | 27 | 0 | 0 |
omr-ws-eb8f9449863cc954c50759d982329ff62ebfd583b2f10ca511533e038b49e1ba | A frog begins at $P_0 = (0,0)$ and makes a sequence of jumps according to the following rule: from $P_n=(x_n,y_n)$, the frog jumps to $P_{n+1}$, which may be any of the points $(x_n+7, y_n+2)$, $(x_n+2,y_n+7)$, $(x_n-5, y_n-10)$, or $(x_n-10,y_n-5)$. There are $M$ points $(x,y)$ with $|x|+|y| \le 100$ that can be reach... | [
"373"
] | [
"A frog begins at $P_0 = (0,0)$ and makes a sequence of jumps according to the following rule: from $P_n=(x_n,y_n)$, the frog jumps to $P_{n+1}$, which may be any of the points $(x_n+7, y_n+2)$, $(x_n+2,y_n+7)$, $(x_n-5, y_n-10)$, or $(x_n-10,y_n-5)$. There are $M$ points $(x,y)$ with $|x|+|y| \\le 100$ that can be... | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 8,
"tir": 16
} | 31 | 0 | 0 |
omr-ws-1098392c08b87fc98bff0beef61455be705b4f14f641ba7df545c017d6734a77 | A frog can "jump around" a point by rotating $120^\circ$ anti-clockwise about that point. Starting at point $X$, the frog jumps around points $A$, $B$, and $C$ alternately. Given $XA = 10$ and $AB = 15$, after 420 jumps, the frog returns to its original position at point $X$. Determine the maximum possible value $p$ an... | [
"600"
] | [
"A frog can \"jump around\" a point by rotating $120^\\circ$ anti-clockwise about that point. Starting at point $X$, the frog jumps around points $A$, $B$, and $C$ alternately. Given $XA = 10$ and $AB = 15$, after 420 jumps, the frog returns to its original position at point $X$. Determine the maximum possible valu... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 6,
"tir": 2
} | 21 | 0 | 0 |
omr-ws-2526a5b4bbf6b58ffd686c4123f84451efb1cf576b2f059f5db82028951a54ad | A frog can jump forward any number of lily pads but never backward. Starting on the first of a sequence of twelve lily pads, in how many ways can the frog jump to the last lily pad? | [
"\\(2^{10}\\)"
] | [
"A frog can jump forward any number of lily pads but never backward. Starting on the first of a sequence of twelve lily pads, in how many ways can the frog jump to the last lily pad?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-5954a8ca814402118438a301f3842bd06e10eff848b7d4e66c2582049abda5b4 | A frog can jump from chair \( k \) to chair \( n \) if \( |k - n| = 5 \) or \( |k - n| = 8 \), where \( k, n \in \{1, 2, \ldots, 12\} \). Determine all possible starting chair numbers from which the frog can jump to each of the 12 chairs exactly once. | [
"8"
] | [
"A frog can jump from chair \\( k \\) to chair \\( n \\) if \\( |k - n| = 5 \\) or \\( |k - n| = 8 \\), where \\( k, n \\in \\{1, 2, \\ldots, 12\\} \\). Determine all possible starting chair numbers from which the frog can jump to each of the 12 chairs exactly once."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-bd315593fd6f729f1527c97ed6c925db6d43f795e14162ec274271a765fa06e1 | A frog can jump in a grid plane with the following jumps: 6 cm to the right, 8 cm upwards, 9 cm to the left, and 10 cm down. Starting from a corner of a 10×10 square grid of red dots, how many of these 100 red dots can the frog reach? | [
"20"
] | [
"A frog can jump in a grid plane with the following jumps: 6 cm to the right, 8 cm upwards, 9 cm to the left, and 10 cm down. Starting from a corner of a 10×10 square grid of red dots, how many of these 100 red dots can the frog reach?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 1
} | 14 | 0 | 0 |
omr-ws-316e38784ee725a80b1157fa7a3bc302557507e19c430b0cde9e0b9d6078f5e8 | A frog can jump over a point in the following way: if a frog at point $M$ jumps over point $N$, it lands on the point of reflection of point $M$ over $N$. A frog is initially at point $X$. Points $A$, $B$, and $C$ are such that $XA = 36$ cm, $XB = 84$ cm, and $XC = 48$ cm. The frog first jumps over $A$, then over $B$, ... | [
"\\( 113064 \\)"
] | [
"A frog can jump over a point in the following way: if a frog at point $M$ jumps over point $N$, it lands on the point of reflection of point $M$ over $N$. A frog is initially at point $X$. Points $A$, $B$, and $C$ are such that $XA = 36$ cm, $XB = 84$ cm, and $XC = 48$ cm. The frog first jumps over $A$, then over ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c23f3db9b5e12eb4de00d6412b6febb1e98095c151a0f455873cad794147f7ab | A frog is at position $1$ on the number line. Every move, it can jump one or two units right. However, for every positive integer $K$, it can only visit position $2K$ if it has not visited position $K$. Find the number of distinct paths the frog can take to get to position $N$. | [
"2"
] | [
"A frog is at position $1$ on the number line. Every move, it can jump one or two units right. However, for every positive integer $K$, it can only visit position $2K$ if it has not visited position $K$. Find the number of distinct paths the frog can take to get to position $N$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c5e15e926a329e9527e76b17824bf51168582bf64fc764fb251a5b2d0130a7e8 | A frog is at the center of a circular shaped island with radius \( r \). The frog jumps \( 1/2 \) meters at first. After the first jump, it turns right or left at exactly \( 90^\circ \), and it always jumps one half of its previous jump. After a finite number of jumps, what is the least \( r \) that yields the frog can... | [
"\\(\\frac{\\sqrt{5}}{3}\\)"
] | [
"A frog is at the center of a circular shaped island with radius \\( r \\). The frog jumps \\( 1/2 \\) meters at first. After the first jump, it turns right or left at exactly \\( 90^\\circ \\), and it always jumps one half of its previous jump. After a finite number of jumps, what is the least \\( r \\) that yield... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 43,
"tir": 25
} | 68 | 0 | 0 |
omr-ws-c3cd0560d2d8fc1d4955275341a6e70617ddaec85c611072a5c907216c712314 | A frog is at the origin point of the Cartesian plane. Each time, it jumps one or two units to the right. Let $\frac{a}{b}$ be the probability of the frog getting to the point $(10,0)$ after a few jumps, where $a$ and $b$ are positive integers and $\gcd(a, b) = 1$. Determine $a + b$. | [
"1707"
] | [
"A frog is at the origin point of the Cartesian plane. Each time, it jumps one or two units to the right. Let $\\frac{a}{b}$ be the probability of the frog getting to the point $(10,0)$ after a few jumps, where $a$ and $b$ are positive integers and $\\gcd(a, b) = 1$. Determine $a + b$."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-4cac576126b4fc647e36f7ee3ce6b6cb3ac30f39436c8f3390a56fee1b5b9df7 | A frog is initially at the point $(0,0)$ in a Cartesian plane. Each step, the frog can jump to a nearest lattice point. If the frog never arrives at the point $(0,0)$ in its first 7 steps, find the total number of ways the frog can reach $(0,0)$ in 8 steps. | [
"1876"
] | [
"A frog is initially at the point $(0,0)$ in a Cartesian plane. Each step, the frog can jump to a nearest lattice point. If the frog never arrives at the point $(0,0)$ in its first 7 steps, find the total number of ways the frog can reach $(0,0)$ in 8 steps."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-7f44fee6cb4877f3f689d625b9ce10d5a05261926a26b7785b3a3d03baee724e | A frog is jumping on $N$ stones numbered from $1$ to $N$ from left to right. The frog jumps to the previous stone (to the left) with probability $p$ and to the next stone (to the right) with probability $1-p$. If the frog jumps to the left from the leftmost stone or to the right from the rightmost stone, it will fall i... | [
"\\dfrac{243}{364}"
] | [
"A frog is jumping on $N$ stones numbered from $1$ to $N$ from left to right. The frog jumps to the previous stone (to the left) with probability $p$ and to the next stone (to the right) with probability $1-p$. If the frog jumps to the left from the leftmost stone or to the right from the rightmost stone, it will f... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ecf956cf1e2e1b9ead3ccd7996341ac3f0073f3d71d50bac92e3458f8f2590d3 | A frog is jumping on \( N \) stones numbered from 1 to \( N \) from left to right. The frog jumps to the previous stone (to the left) with probability \( p \) and to the next stone (to the right) with probability \( 1-p \). If the frog jumps to the left from the leftmost stone or to the right from the rightmost stone, ... | [
"\\dfrac{2 \\cdot 3^N}{3^{N+1} - 1}"
] | [
"A frog is jumping on \\( N \\) stones numbered from 1 to \\( N \\) from left to right. The frog jumps to the previous stone (to the left) with probability \\( p \\) and to the next stone (to the right) with probability \\( 1-p \\). If the frog jumps to the left from the leftmost stone or to the right from the righ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7f3cb2186e01dbc95fe53564233904a1ad6832612b815360cc097cd4d3211b31 | A frog is jumping on the vertices of a square ABCD. It starts at vertex A and in each jump moves to an adjacent vertex with a probability of 1/2 for each. The frog stops when it has reached all vertices at least once. Determine the probability of each vertex being the last vertex the frog reaches. | [
"\\dfrac{1}{3}"
] | [
"A frog is jumping on the vertices of a square ABCD. It starts at vertex A and in each jump moves to an adjacent vertex with a probability of 1/2 for each. The frog stops when it has reached all vertices at least once. Determine the probability of each vertex being the last vertex the frog reaches."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-5d2e519a0bea25e9dcf1c35a392a7ec44abcf7b640ce67171a272d94a3371b14 | A frog is located at $(0,0,0)$. The frog can move to $(x+1,y,z)$, $(x,y+1,z)$, or $(x,y,z+1)$ from its current position $(x,y,z)$. How many ways can the frog move to $(5,5,5)$ if it is bounded by the planes $y = x$ and $y = x + 1$? | [
"3003"
] | [
"A frog is located at $(0,0,0)$. The frog can move to $(x+1,y,z)$, $(x,y+1,z)$, or $(x,y,z+1)$ from its current position $(x,y,z)$. How many ways can the frog move to $(5,5,5)$ if it is bounded by the planes $y = x$ and $y = x + 1$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-18e7c99bc1d5424a65997a98008b65deca7ba75b0277a948d71bb17a1d7db4e6 | A frog is placed at the origin on a number line and moves according to the following rule: in a given move, the frog advances to either the closest integer point with a greater integer coordinate that is a multiple of 3, or to the closest integer point with a greater integer coordinate that is a multiple of 13. A move ... | [
"169"
] | [
"A frog is placed at the origin on a number line and moves according to the following rule: in a given move, the frog advances to either the closest integer point with a greater integer coordinate that is a multiple of 3, or to the closest integer point with a greater integer coordinate that is a multiple of 13. A ... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 29,
"genselect": 2,
"tir": 15
} | 46 | 0 | 0 |
omr-ws-c442615eaf59b886c292b62955c4fbb6175ad36f3aaf4e89172f9561f7b18a9d | A frog is placed at the origin on the number line and moves according to the following rule: in a given move, the frog advances to either the closest point with a greater integer coordinate that is a multiple of $m$, or to the closest point with a greater integer coordinate that is a multiple of $n$, with $m$ and $n$ r... | [
"\\frac{(m + n - 1)!}{m!n!}"
] | [
"A frog is placed at the origin on the number line and moves according to the following rule: in a given move, the frog advances to either the closest point with a greater integer coordinate that is a multiple of $m$, or to the closest point with a greater integer coordinate that is a multiple of $n$, with $m$ and ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d4f0ce585736699893eeda3a9ba635a00f57b9a20d570d1d66d567dec774965e | A frog is placed at the origin on the number line and moves according to the following rule: in a given move, the frog advances to either the closest point with a greater integer coordinate that is a multiple of 3, or to the closest point with a greater integer coordinate that is a multiple of 13. A move sequence is a ... | [
"169"
] | [
"A frog is placed at the origin on the number line and moves according to the following rule: in a given move, the frog advances to either the closest point with a greater integer coordinate that is a multiple of 3, or to the closest point with a greater integer coordinate that is a multiple of 13. A move sequence ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 25,
"genselect": 4
} | 29 | 0 | 0 |
omr-ws-d4abb864fdc03330c3161b75d8ef83640184b613522379c7ff817efa9c5ed901 | A frog is placed at the origin on the number line, and moves according to the following rule: in a given move, the frog advances to either the closest point with a greater integer coordinate that is a multiple of 3, or to the closest point with a greater integer coordinate that is a multiple of 13. A move sequence is a... | [
"169"
] | [
"A frog is placed at the origin on the number line, and moves according to the following rule: in a given move, the frog advances to either the closest point with a greater integer coordinate that is a multiple of 3, or to the closest point with a greater integer coordinate that is a multiple of 13. A move seq... | [
"MATH_training_set"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 34,
"genselect": 1,
"tir": 20
} | 55 | 0 | 0 |
omr-ws-1ab780195dcbe5476ea6280179acd258f2a893c4237f394936077db10477d51e | A frog is placed on each cell of an $n \times n$ square inside an infinite chessboard (so initially there are a total of $n \times n$ frogs). Each move consists of a frog $A$ jumping over an adjacent frog $B$ and landing in the next cell, with $B$ disappearing. Let $M(n)$ be the minimum number of moves required to reac... | [
"18"
] | [
"A frog is placed on each cell of an $n \\times n$ square inside an infinite chessboard (so initially there are a total of $n \\times n$ frogs). Each move consists of a frog $A$ jumping over an adjacent frog $B$ and landing in the next cell, with $B$ disappearing. Let $M(n)$ be the minimum number of moves required ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7a8d0f8cf5b6ec87cd176c1346db78ecb982e4d57ef422543233b2bf29080ab4 | A frog is placed on each grid of an $n \times n$ square inside an infinite chessboard (initially there are $n^2$ frogs). Each move consists of a frog $A$ jumping over an adjacent frog $B$ and landing in the next grid, with $B$ disappearing. Calculate the minimum number of moves required to reach a configuration where n... | [
"18"
] | [
"A frog is placed on each grid of an $n \\times n$ square inside an infinite chessboard (initially there are $n^2$ frogs). Each move consists of a frog $A$ jumping over an adjacent frog $B$ and landing in the next grid, with $B$ disappearing. Calculate the minimum number of moves required to reach a configuration w... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-98e8217d9eff3dbd1902f726a32ddbe95b5a6cee00b9f8f33d3285bde9779584 | A frog is standing at point (0, 0) in the coordinate plane and can jump to points (0, 0), (1, 0), (0, 1), or (1, 1). Each step, it must jump one unit to one of the two possible points. The probability of each type of jump is equal, except if one of the choices is to jump to the left, then the probability of jumping to ... | [
"691"
] | [
"A frog is standing at point (0, 0) in the coordinate plane and can jump to points (0, 0), (1, 0), (0, 1), or (1, 1). Each step, it must jump one unit to one of the two possible points. The probability of each type of jump is equal, except if one of the choices is to jump to the left, then the probability of jumpin... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2f760aaea1152089ede8db700204c75faa4aa03daff0786c214ce34d281650a0 | A frog is standing in the center of a $3 \times 3$ grid of lilypads. Each minute, the frog chooses a square that shares exactly one side with its current square uniformly at random and jumps onto the chosen lilypad. The frog stops jumping once it reaches a lilypad on a corner of the grid. What is the expected number of... | [
"3"
] | [
"A frog is standing in the center of a $3 \\times 3$ grid of lilypads. Each minute, the frog chooses a square that shares exactly one side with its current square uniformly at random and jumps onto the chosen lilypad. The frog stops jumping once it reaches a lilypad on a corner of the grid. What is the expected num... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a299aee15ad9722a6d2ee64ceac7bb672e99088ce11e9a087b338b2f68e6f01d | A frog jumps clockwise around a circle of $n (>1)$ lotus leaves according to the following rule: it skips one leaf and jumps to the next, then skips two leaves and jumps to the following, and so on. If the frog reaches all leaves at least once, find the remainder when the smallest possible even value of $n$ greater tha... | [
"16"
] | [
"A frog jumps clockwise around a circle of $n (>1)$ lotus leaves according to the following rule: it skips one leaf and jumps to the next, then skips two leaves and jumps to the following, and so on. If the frog reaches all leaves at least once, find the remainder when the smallest possible even value of $n$ greate... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-98710c55c573e04bb908301e3e96f58ce9989de0ec4c82ada822a3b5d647a8bf | A frog jumps on five points numbered 1, 2, 3, 4, 5 in clockwise order on a circle. The jumping rules are: - If the frog is at an odd-numbered point, it moves one place clockwise. - If the frog is at an even-numbered point, it moves two places clockwise. If the frog starts at point 5, where will it be after 2010 jumps? | [
"4"
] | [
"A frog jumps on five points numbered 1, 2, 3, 4, 5 in clockwise order on a circle. The jumping rules are:\n- If the frog is at an odd-numbered point, it moves one place clockwise.\n- If the frog is at an even-numbered point, it moves two places clockwise.\nIf the frog starts at point 5, where will it be after 2010... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-65b508ff36c28a08d528986cd18cc34e9cf93466d3990dcf60b4192ed7f47c4f | A frog located at $(x,y)$, with both $x$ and $y$ integers, makes successive jumps of length $5$ and always lands on points with integer coordinates. If the frog starts at $(0,0)$ and ends at $(1,0)$, what is the smallest possible number of jumps the frog makes? | [
"3"
] | [
"A frog located at $(x,y)$, with both $x$ and $y$ integers, makes successive jumps of length $5$ and always lands on points with integer coordinates. If the frog starts at $(0,0)$ and ends at $(1,0)$, what is the smallest possible number of jumps the frog makes?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-4b617dd7ad33549307f64ae74d6f589733a92b4f8932852a2a0abdbf64be3197 | A frog located at $(x,y)$, with both $x$ and $y$ integers, makes successive jumps of length $5$ and always lands on points with integer coordinates. Suppose that the frog starts at $(0,0)$ and ends at $(1,0)$. What is the smallest possible number of jumps the frog makes? | [
"3"
] | [
"A frog located at $(x,y)$, with both $x$ and $y$ integers, makes successive jumps of length $5$ and always lands on points with integer coordinates. Suppose that the frog starts at $(0,0)$ and ends at $(1,0)$. What is the smallest possible number of jumps the frog makes?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 7
} | 30 | 0 | 0 |
omr-ws-9671fd9589feaefcf42ddaa9f4331e5c0ab42340d0b083f6f7f7f813756e967b | A frog makes 3 jumps, each exactly 1 meter long. The directions of the jumps are chosen independently at random. What is the probability that the frog's final position is no more than 1 meter from its starting position? | [
"\\(\\frac{1}{4}\\)",
"\\dfrac{1}{4}"
] | [
"A frog makes 3 jumps, each exactly 1 meter long. The directions of the jumps are chosen independently at random. What is the probability that the frog's final position is no more than 1 meter from its starting position?"
] | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 55,
"genselect": 6,
"tir": 54
} | 115 | 0 | 0 |
omr-ws-d03568f470d878bc6da134462121c343aec6f4204595937a8b95ba5f91450e68 | A frog named Leap is standing at the center of ten concentric circles. The first circle has radius $1$, the second has radius $2$, and each larger circle has a radius equal to the sum of the radii of the two circles before it. Leap jumps around on the circles, taking progressively larger jumps each time, until he lands... | [
"1"
] | [
"A frog named Leap is standing at the center of ten concentric circles. The first circle has radius $1$, the second has radius $2$, and each larger circle has a radius equal to the sum of the radii of the two circles before it. Leap jumps around on the circles, taking progressively larger jumps each time, until he ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f19971a82548985a465399c7e4c64df227d8a998f5d5e474b36b31bcf73e5330 | A frog needs to jump on 10 lily pads in a specific order to get to the other side of the river. If he gets the wrong order, he is sent back to the beginning. What is the minimum number of attempts needed to ensure the frog reaches the other side of the river? | [
"46"
] | [
"A frog needs to jump on 10 lily pads in a specific order to get to the other side of the river. If he gets the wrong order, he is sent back to the beginning. What is the minimum number of attempts needed to ensure the frog reaches the other side of the river?"
] | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 1
} | 1 | 0 | 0 |
omr-ws-1f3843d323338934487323b4f5bd4e3c7fa7cc4f7397c220c9164daa1b4be49f | A frog performs a random walk on the integers, starting from zero. At each step, it jumps one integer forward or one integer backward with equal probability. Find the probability that the frog reaches $+10$ before it reaches $-20$. | [
"\\(\\frac{2}{3}\\)"
] | [
"A frog performs a random walk on the integers, starting from zero. At each step, it jumps one integer forward or one integer backward with equal probability. Find the probability that the frog reaches $+10$ before it reaches $-20$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-5b82f8ed11a9845d7a240765450ad7008c292de1e33f0f1d7be26a94834674bb | A frog sits at the origin $(0,0)$ in the Cartesian plane. In each jump, the frog moves from a lattice point $(x, y)$ to another lattice point $(x', y')$ such that $x \le x'$, $y \le y'$, and the distance between $(x', y')$ and $(x, y)$ is exactly 5. After 42 jumps, the frog is at a distance $d$ from the origin. Find th... | [
"1050"
] | [
"A frog sits at the origin $(0,0)$ in the Cartesian plane. In each jump, the frog moves from a lattice point $(x, y)$ to another lattice point $(x', y')$ such that $x \\le x'$, $y \\le y'$, and the distance between $(x', y')$ and $(x, y)$ is exactly 5. After 42 jumps, the frog is at a distance $d$ from the origin. ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c98f12c3aa17901a071bf9a243c6cf5e14bdc73cd7addbb01d7a0b975270c0ac | A frog sits on the fourth lilypad from the left (third from the right) among six lilypads. Each turn, the frog has a 1/3 probability of moving right and a 2/3 probability of moving left. If the frog can hop off the lilypads when it reaches either the leftmost or the rightmost lilypad, what is the probability that the f... | [
"\\(\\frac{15}{127}\\)"
] | [
"A frog sits on the fourth lilypad from the left (third from the right) among six lilypads. Each turn, the frog has a 1/3 probability of moving right and a 2/3 probability of moving left. If the frog can hop off the lilypads when it reaches either the leftmost or the rightmost lilypad, what is the probability that ... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 8
} | 8 | 0 | 0 |
omr-ws-817c3d393c016852b20bff7c94e12175e5378bb57e448f773b47741d61502159 | A frog sitting at the point $(1, 2)$ begins a sequence of jumps, where each jump is parallel to one of the coordinate axes and has length $1$, and the direction of each jump (up, down, right, or left) is chosen independently at random. The sequence ends when the frog reaches a side of the square with vertices $(0,0), (... | [
"\\(\\frac{5}{8}\\)"
] | [
"A frog sitting at the point $(1, 2)$ begins a sequence of jumps, where each jump is parallel to one of the coordinate axes and has length $1$, and the direction of each jump (up, down, right, or left) is chosen independently at random. The sequence ends when the frog reaches a side of the square with vertices $(0,... | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 6,
"tir": 26
} | 47 | 0 | 0 |
omr-ws-2a1f0bc5598028576f48b09f2e233b93602c80e6f6d8f458826a3df14abf0191 | A frog sitting at the point (1, 2) begins a sequence of jumps, where each jump is parallel to one of the coordinate axes and has length 1, and the direction of each jump (up, down, right, or left) is chosen independently at random. The sequence ends when the frog reaches a side of the square with vertices (0,0), (0,4),... | [
"\\dfrac{5}{8}"
] | [
"A frog sitting at the point (1, 2) begins a sequence of jumps, where each jump is parallel to one of the coordinate axes and has length 1, and the direction of each jump (up, down, right, or left) is chosen independently at random. The sequence ends when the frog reaches a side of the square with vertices (0,0), (... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d7afb8b6554fe39124c51fbfca5d17b5dd9da3c34b3a1b62987e4fa23a1b016e | A frog started from the origin of the coordinate plane and made 3 jumps. Each time, the frog jumped a distance of 5 units and landed on a point with integer coordinates. How many different end positions are possible for the frog? | [
"72"
] | [
"A frog started from the origin of the coordinate plane and made 3 jumps. Each time, the frog jumped a distance of 5 units and landed on a point with integer coordinates. How many different end positions are possible for the frog?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3498548a3ed2238bdeb039f6c996e3bed04742bfe4817166013df2cca9523eeb | A frog started from the origin of the coordinate plane and made three jumps. Each time the frog jumped a distance of 5 units and landed at a point with integer coordinates. How many different possibilities of the final position of the frog are there? | [
"217"
] | [
"A frog started from the origin of the coordinate plane and made three jumps. Each time the frog jumped a distance of 5 units and landed at a point with integer coordinates. How many different possibilities of the final position of the frog are there?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 5
} | 8 | 0 | 0 |
omr-ws-5172f203a3415b97bae0a845363bcbd19761250f5683ecb144eb74aee026c2eb | A frog starts at $(0,0)$ and every move consists of choosing a direction and jumping with vector length $1$. After $3$ moves, what is the probability the frog is on the unit disk? | [
"\\(\\frac{1}{4}\\)"
] | [
"A frog starts at $(0,0)$ and every move consists of choosing a direction and jumping with vector length $1$. After $3$ moves, what is the probability the frog is on the unit disk?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 18
} | 32 | 0 | 0 |
omr-ws-8a0e1cf8a2530ea23cb667c4d262857e9f6f8fe3f9f4fdcd284c493887b1cf7f | A frog starts at $(0,0)$ in the coordinate plane. In a move, if he is at the square $(x,y)$ he can move to $(x+1,y)$, $(x,y+1)$, or $(x+7,y+11)$. How many possible points can the frog reach in $20$ moves or less? | [
"1761"
] | [
"A frog starts at $(0,0)$ in the coordinate plane. In a move, if he is at the square $(x,y)$ he can move to $(x+1,y)$, $(x,y+1)$, or $(x+7,y+11)$. How many possible points can the frog reach in $20$ moves or less?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 7,
"tir": 23
} | 33 | 0 | 0 |
omr-ws-047f5d1f9a3ed41a2725bd870d25f5f4d84eaa157e711d134dbe052fab83db91 | A frog starts at $1$ along the number line. Every second, he flips a fair coin; if the coin lands heads, he moves $1$ unit forward, and if the coin lands tails, he moves $2$ units forward. What is the probability that he lands on a multiple of $5$ before he lands on a multiple of $10$? | [
"\\dfrac{16}{21}"
] | [
"A frog starts at $1$ along the number line. Every second, he flips a fair coin; if the coin lands heads, he moves $1$ unit forward, and if the coin lands tails, he moves $2$ units forward. What is the probability that he lands on a multiple of $5$ before he lands on a multiple of $10$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2b44955bb8040f35a541deef37ff63fe6efef790c96c147ff835adf44d8a073a | A frog starts at 3 on the number line and jumps over the positive integers. Initially, her jump size is 1. Every time she lands on a power of two, her jump size increases by 1. The 20th power of 2 she visits is $2^N$. Find $N$. | [
"94"
] | [
"A frog starts at 3 on the number line and jumps over the positive integers. Initially, her jump size is 1. Every time she lands on a power of two, her jump size increases by 1. The 20th power of 2 she visits is $2^N$. Find $N$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-e32381268d0d65bd010295a2e588aaa05537666e4217e8ef3b716eb36d141d23 | A frog starts at \((1, 1)\). Every second, if the frog is at point \((x, y)\), it moves to \((x + 1, y)\) with probability \(\frac{x}{x+y}\) and moves to \((x, y + 1)\) with probability \(\frac{y}{x+y}\). The frog stops moving when its \(y\)-coordinate is 10. Suppose the probability that when the frog stops its \(x\)-c... | [
"13"
] | [
"A frog starts at \\((1, 1)\\). Every second, if the frog is at point \\((x, y)\\), it moves to \\((x + 1, y)\\) with probability \\(\\frac{x}{x+y}\\) and moves to \\((x, y + 1)\\) with probability \\(\\frac{y}{x+y}\\). The frog stops moving when its \\(y\\)-coordinate is 10. Suppose the probability that when the f... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 6,
"tir": 15
} | 25 | 0 | 0 |
omr-ws-476a49baa8e8af4dbe5aaf170ad467987a872b73091b1d494178599761fea7b9 | A frog starts at a vertex of a convex polygon and makes a series of jumps. Each jump is either clockwise, skipping one vertex, or counterclockwise, skipping two vertices. If the frog makes a number of jumps equal to the number of sides of the polygon and visits all vertices exactly once before returning to the starting... | [
"The set of natural numbers which are not multiples of \\( 6 \\) or which are multiples of \\( 30 \\)."
] | [
"A frog starts at a vertex of a convex polygon and makes a series of jumps. Each jump is either clockwise, skipping one vertex, or counterclockwise, skipping two vertices. If the frog makes a number of jumps equal to the number of sides of the polygon and visits all vertices exactly once before returning to the sta... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 1,
"tir": 2
} | 5 | 0 | 0 |
omr-ws-3945b37b3b48dd33be368618ee6a22615a1493e10fd7cfd76329f67d6864b469 | A frog starts at coordinate 1 and makes 100 jumps, passing through all coordinates from 1 to 100 and returning to 1. Each jump is in the opposite direction of the previous jump. What is the shortest possible total distance the frog can travel? | [
"198"
] | [
"A frog starts at coordinate 1 and makes 100 jumps, passing through all coordinates from 1 to 100 and returning to 1. Each jump is in the opposite direction of the previous jump. What is the shortest possible total distance the frog can travel?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-280b4beb95abd3f6a2d5eb78399b1c8d45f53d5bba67d28c8345de34bbd957c8 | A frog starts at lily pad $100$. Each minute, if the frog is at lily pad $n$, it hops to lily pad $n + 1$ with probability $\frac{n-1}{n}$, and hops all the way back to lily pad $1$ with probability $\frac{1}{n}$. Let $N$ be the position of the frog after $1000$ minutes. What is the expected value of $N$? | [
"100"
] | [
"A frog starts at lily pad $100$. Each minute, if the frog is at lily pad $n$, it hops to lily pad $n + 1$ with probability $\\frac{n-1}{n}$, and hops all the way back to lily pad $1$ with probability $\\frac{1}{n}$. Let $N$ be the position of the frog after $1000$ minutes. What is the expected value of $N$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7cdf0f487fa08169dedb91f39230520526e564cb2e3703a61a4e94cd7457934d | A frog starts at point 0 on a number line and jumps $k$ units to the right or left on the $k$-th step for each positive integer $k$. How many steps are required for the frog to reach at least 100 different points? | [
"99"
] | [
"A frog starts at point 0 on a number line and jumps $k$ units to the right or left on the $k$-th step for each positive integer $k$. How many steps are required for the frog to reach at least 100 different points?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 5
} | 13 | 0 | 0 |
omr-ws-dcdcf647cd2bbf4d5848b20ea8acadd5602eb7cc9958deb79e89926e6597aad7 | A frog starts at point 1 and jumps 100 times, passing through all the points from 1 to 100 and returning to point 1. There are 99! different ways for the frog to jump. Assuming each way is equally likely, what is the average distance that the frog jumps? (Find the expected value.) | [
"\\(\\frac{100 \\times 101}{3}\\)"
] | [
"A frog starts at point 1 and jumps 100 times, passing through all the points from 1 to 100 and returning to point 1. There are 99! different ways for the frog to jump. Assuming each way is equally likely, what is the average distance that the frog jumps? (Find the expected value.)"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 25
} | 29 | 0 | 0 |
omr-ws-2e2749ec6912fabbb6730025c7a085fd04bf01be511e4a01f0035fa1b5dcac40 | A frog starts at point 1 and jumps 100 times, passing through all the points from 1 to 100 and returning to point 1. There are 99! different ways for the frog to jump. Assuming each way is equally likely, what is the average distance the frog jumps? (Find the expected value.) | [
"\\(\\frac{10100}{3}\\)"
] | [
"A frog starts at point 1 and jumps 100 times, passing through all the points from 1 to 100 and returning to point 1. There are 99! different ways for the frog to jump. Assuming each way is equally likely, what is the average distance the frog jumps? (Find the expected value.)"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 22
} | 26 | 0 | 0 |
omr-ws-128a7d9884bb3e4461d155b71e0b1e15d25564f0b410ba7d46ae22d92a329c5c | A frog starts at position 0 on the x-axis and wants to jump to position \( X \). Each jump can be an integer length between 1 and \( \min(10, X - \text{current position}) \). What is the expected number of jumps the frog needs to make to reach position \( X \)? | [
"H_X"
] | [
"A frog starts at position 0 on the x-axis and wants to jump to position \\( X \\). Each jump can be an integer length between 1 and \\( \\min(10, X - \\text{current position}) \\). What is the expected number of jumps the frog needs to make to reach position \\( X \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 1
} | 6 | 0 | 0 |
omr-ws-f159e6c3ecab64de19e84c4881e5e512fc36b3256b70171a74921e7ef3b38735 | A frog starts at the center of a circular island with radius \( r \). The frog jumps in a sequence of distances: 1/2 meters, 1/4 meters, 1/8 meters, 1/16 meters, and so on, each time turning 90 degrees left or right. After a finite number of jumps, the frog does not reach the water. What is the minimum value of \( r \)... | [
"\\(\\frac{\\sqrt 5}3\\)"
] | [
"A frog starts at the center of a circular island with radius \\( r \\). The frog jumps in a sequence of distances: 1/2 meters, 1/4 meters, 1/8 meters, 1/16 meters, and so on, each time turning 90 degrees left or right. After a finite number of jumps, the frog does not reach the water. What is the minimum value of ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-bb5a88c5d6b8f46b90fba99ca4db695b6b6b3fca56ad43f1495d138306285a65 | A frog starts at the origin (0,0). At each minute, it picks a random integer $x$, turns $x$ degrees counterclockwise, and jumps exactly 1 unit. After 2 minutes, what is the probability that the frog is exactly one unit from the origin? | [
"\\(\\frac{1}{3}\\)"
] | [
"A frog starts at the origin (0,0). At each minute, it picks a random integer $x$, turns $x$ degrees counterclockwise, and jumps exactly 1 unit. After 2 minutes, what is the probability that the frog is exactly one unit from the origin?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-85205efd620d2e9b596b75744a7d891d4f38959e6c4f1019364c24315fde13a7 | A frog starts at the origin \((0,0)\) of the plane. At every second, if the frog is at point \((x,y)\), it can jump to one of the points \((x+1,y)\) or \((x, y+1)\). Let \(P(n)\) be the number of distinct paths the frog can take to reach a point \((n,n)\) such that there are exactly \(n\) collinear points on the path. ... | [
"48"
] | [
"A frog starts at the origin \\((0,0)\\) of the plane. At every second, if the frog is at point \\((x,y)\\), it can jump to one of the points \\((x+1,y)\\) or \\((x, y+1)\\). Let \\(P(n)\\) be the number of distinct paths the frog can take to reach a point \\((n,n)\\) such that there are exactly \\(n\\) collinear p... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 7
} | 11 | 0 | 0 |
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