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-521c51594659f0b8ed8eff341ebf83e92afab9a9cee6f9385194d281c67404a9 | Assume that both the attacking player and the defending player have exactly \( m \ge 4 \) pieces on each of their territories. Calculate which player has the greater chance of winning a single round (i.e., each player throws their dice once. Consider that a tie (each player loses one piece) does not mean a victory for ... | [
"\\text{Attacking player}"
] | [
"Assume that both the attacking player and the defending player have exactly \\( m \\ge 4 \\) pieces on each of their territories. Calculate which player has the greater chance of winning a single round (i.e., each player throws their dice once. Consider that a tie (each player loses one piece) does not mean a vict... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 1
} | 16 | 0 | 0 |
omr-ws-d908e7709db55f231db8cb5744edcffe57083570f48bda23537721e7fe6632da | Assume that desserts have an equal probability of being a cake or ice cream and a uniform probability of being any of the following 7 flavors: chocolate, vanilla, strawberry, coconut, cookies & cream, fudge, and raspberry. If Andrew has two desserts and at least one is chocolate flavored ice cream, what is the probabil... | [
"\\(\\frac{7}{105}\\)"
] | [
"Assume that desserts have an equal probability of being a cake or ice cream and a uniform probability of being any of the following 7 flavors: chocolate, vanilla, strawberry, coconut, cookies & cream, fudge, and raspberry. If Andrew has two desserts and at least one is chocolate flavored ice cream, what is the pro... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ecf6853fb9a59d1de779a1bb3292b8d4a2cc447bdb9dde955dc3864f36e692bb | Assume that earthquakes strike a certain region at random times that are exponentially distributed with mean 1 year. Volcanic eruptions take place at random times that are exponentially distributed with mean 2 years. What is the probability that there will be two earthquakes before the next volcanic eruption? | [
"\\dfrac{4}{9}"
] | [
"Assume that earthquakes strike a certain region at random times that are exponentially distributed with mean 1 year. Volcanic eruptions take place at random times that are exponentially distributed with mean 2 years. What is the probability that there will be two earthquakes before the next volcanic eruption?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b56b57d312f17f5117208ab1357ae8d74154c127019e70c00e99821f54e3bfff | Assume that for \( i=1,2,\ldots,n \), \( a_i > 0 \) and \( b_i > 0 \), and \(\sum_{i=1}^{n} a_i = a\) and \(\sum_{i=1}^{n} b_i = b\). Given \(a = 5\) and \(b = 7\), find the maximum possible value of \(\sum_{i=1}^{n} \frac{a_i b_i}{a_i + b_i}\). | [
"\\dfrac{35}{12}"
] | [
"Assume that for \\( i=1,2,\\ldots,n \\), \\( a_i > 0 \\) and \\( b_i > 0 \\), and \\(\\sum_{i=1}^{n} a_i = a\\) and \\(\\sum_{i=1}^{n} b_i = b\\). Given \\(a = 5\\) and \\(b = 7\\), find the maximum possible value of \\(\\sum_{i=1}^{n} \\frac{a_i b_i}{a_i + b_i}\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-98366e770453f0e852e37e7bb54eb264250319f4876c382f418fac9c2e0162a0 | Assume that for pairwise distinct real numbers \(a, b, c, d\) the following holds: \[ (a^2 + b^2 - 1)(a + b) = (b^2 + c^2 - 1)(b + c) = (c^2 + d^2 - 1)(c + d). \] Calculate the value of \(a + b + c + d\) given that \(a = -d\) and \(b = -c\). | [
"0"
] | [
"Assume that for pairwise distinct real numbers \\(a, b, c, d\\) the following holds:\n\\[ (a^2 + b^2 - 1)(a + b) = (b^2 + c^2 - 1)(b + c) = (c^2 + d^2 - 1)(c + d). \\]\nCalculate the value of \\(a + b + c + d\\) given that \\(a = -d\\) and \\(b = -c\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9ca3e42658310a5fcb8adbd221ad770fa02d0a0b407df0b96bd1748a037b51fb | Assume that from the set \( X = \{ x \in \mathbb{N} \mid 0 < x \leq 2013 \} \), we choose 673 numbers. Let \( S \) be the set of all possible values of \( |a - b| \) for any two chosen numbers \( a \) and \( b \). Calculate the sum of all elements in \( S \) that satisfy \( 671 \leq |a - b| \leq 1342 \). | [
"676368"
] | [
"Assume that from the set \\( X = \\{ x \\in \\mathbb{N} \\mid 0 < x \\leq 2013 \\} \\), we choose 673 numbers. Let \\( S \\) be the set of all possible values of \\( |a - b| \\) for any two chosen numbers \\( a \\) and \\( b \\). Calculate the sum of all elements in \\( S \\) that satisfy \\( 671 \\leq |a - b| \\l... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-951d59cdffec84f5cc38cdcd8d1b8e18f638f636a3fabbb6fbc955c246c790f9 | Assume that integers \(a\), \(b\), \(c\), and \(d\) satisfy \(a > b > c > d > 0\) and \(a^2 + ac - c^2 = b^2 + bd - d^2\). Find the smallest possible value of \(ab + cd\). | [
"377"
] | [
"Assume that integers \\(a\\), \\(b\\), \\(c\\), and \\(d\\) satisfy \\(a > b > c > d > 0\\) and \\(a^2 + ac - c^2 = b^2 + bd - d^2\\). Find the smallest possible value of \\(ab + cd\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-405c1cdc9bda528a8c4fa10632c166c9df9d28243413c8844ae77d25ed5bf05c | Assume that oil spilled from a ruptured tanker spreads in a circular pattern whose radius increases at a constant rate of 2 ft/s. How fast is the area increasing when the radius of the spill is 60 ft? | [
"240\\pi"
] | [
"Assume that oil spilled from a ruptured tanker spreads in a circular pattern whose radius increases at a constant rate of 2 ft/s. How fast is the area increasing when the radius of the spill is 60 ft?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8082c2102e4243b7c0325d6c920a6e543e77125b13a6304ba88ab7613e918eca | Assume that pentagon $AKHIL$ can be inscribed into a circle. Let $U$ be the intersection of diagonals $AH$ and $KI$, and let $M$ be the intersection of $UL$ and $AI$. Furthermore, assume that $|HU|=4\cdot|KU|$, $|UM|=2\cdot|LM|$, and $[AUK]+[HUI]=5$, where $|\cdot|$ denotes the length of the corresponding segment, and ... | [
"\\frac{10\\sqrt{6}}{17}"
] | [
"Assume that pentagon $AKHIL$ can be inscribed into a circle. Let $U$ be the intersection of diagonals $AH$ and $KI$, and let $M$ be the intersection of $UL$ and $AI$. Furthermore, assume that $|HU|=4\\cdot|KU|$, $|UM|=2\\cdot|LM|$, and $[AUK]+[HUI]=5$, where $|\\cdot|$ denotes the length of the corresponding segme... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 3,
"tir": 9
} | 15 | 0 | 0 |
omr-ws-05db3aaa1ae7883d9c0f16334f2d221af4b0deda18cf51476c3ef8df3cc429cc | Assume that positive numbers \(a, b, c, x, y, z\) satisfy \(cy + bz = a\), \(az + cx = b\), and \(bx + ay = c\). Find the minimum value of the function \[ f(x, y, z) = \frac{x^2}{x+1} + \frac{y^2}{y+1} + \frac{z^2}{z+1}. \] | [
"\\(\\frac{1}{2}\\)"
] | [
"Assume that positive numbers \\(a, b, c, x, y, z\\) satisfy \\(cy + bz = a\\), \\(az + cx = b\\), and \\(bx + ay = c\\). Find the minimum value of the function \\[ f(x, y, z) = \\frac{x^2}{x+1} + \\frac{y^2}{y+1} + \\frac{z^2}{z+1}. \\]"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-3b08bbf3e8f0f9232a42715f31fc806d91808deed249d6a4af640b6f2051477e | Assume that random variables \( X_{n} \) converge in distribution to random variable \( X \) and \( \sup_{n}\mathbb{E}|X_{n}|^{2}<\infty \). Let \( p \in (0,2) \). If \( \mathbb{E}|X|^{p} = 5 \), find the limit \( \lim_{n \rightarrow \infty} \mathbb{E}|X_{n}|^{p} \). | [
"5"
] | [
"Assume that random variables \\( X_{n} \\) converge in distribution to random variable \\( X \\) and \\( \\sup_{n}\\mathbb{E}|X_{n}|^{2}<\\infty \\). Let \\( p \\in (0,2) \\). If \\( \\mathbb{E}|X|^{p} = 5 \\), find the limit \\( \\lim_{n \\rightarrow \\infty} \\mathbb{E}|X_{n}|^{p} \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0c70bd27cee48f33454ed3dc1ba06f4592724a792114deca62f3eb966ec4cda5 | Assume that the $A$-mixtilinear excircle touches $AB$, $AC$, and $\odot(ABC)$ at $D$, $E$, and $T$, respectively. Let $AT \cap \odot(TDE) = S$, $\odot(SBC) \cap \odot(TDE) = M$, and $\odot(MBD) \cap \odot(MCE) = P$. The symmetric point of $P$ about $T$ is $Q$. If $I_A$ is the $A$-excenter of $\triangle ABC$, calculate ... | [
"1"
] | [
"Assume that the $A$-mixtilinear excircle touches $AB$, $AC$, and $\\odot(ABC)$ at $D$, $E$, and $T$, respectively. Let $AT \\cap \\odot(TDE) = S$, $\\odot(SBC) \\cap \\odot(TDE) = M$, and $\\odot(MBD) \\cap \\odot(MCE) = P$. The symmetric point of $P$ about $T$ is $Q$. If $I_A$ is the $A$-excenter of $\\triangle A... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f458a07fe4c2ee61a4f83b7b4a4e0c250f84d082c2a587d510e7448137e07653 | Assume that the Earth is a perfect sphere. A point \( P \) on its surface has the property that starting from \( P \), traveling 1 mile south, 1 mile west, and finally 1 mile north returns you to the same point \( P \). Given that \( P \) is not the north pole, find the locus of \( P \). | [
"The locus of \\( P \\) is an infinite number of circles, each 1 mile north of a circle centered above the South Pole with a circumference of \\( \\frac{1}{n} \\) miles, for all positive integers \\( n \\)."
] | [
"Assume that the Earth is a perfect sphere. A point \\( P \\) on its surface has the property that starting from \\( P \\), traveling 1 mile south, 1 mile west, and finally 1 mile north returns you to the same point \\( P \\). Given that \\( P \\) is not the north pole, find the locus of \\( P \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 23
} | 46 | 0 | 0 |
omr-ws-3e3aa3853b3a1930b5bdc60f1c1aa270db0ef8b7c08919b50ec4a57004df18de | Assume that the calculator approximates every number \( b \in [-2, 2] \) by a number \( \overline{b} \) having \( p \) decimal digits after the decimal point. Perform the following sequence of operations on the calculator: 1. Set \( a = 0.72 \). 2. Calculate \( \delta(a) = \overline{f(a)} - a \). 3. If \( |\delta(a)| >... | [
"\\epsilon"
] | [
"Assume that the calculator approximates every number \\( b \\in [-2, 2] \\) by a number \\( \\overline{b} \\) having \\( p \\) decimal digits after the decimal point. Perform the following sequence of operations on the calculator:\n1. Set \\( a = 0.72 \\).\n2. Calculate \\( \\delta(a) = \\overline{f(a)} - a \\).\n... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-fa5a3b3b9d45572d8987c338164a28a98b898eb3f9d7dce1ea61e0400ae3bfbf | Assume that the function $f$ is defined for all $x \in \mathbb{R}$ and has a continuous derivative. Given $f(0) = 0$ and $f'(x) \in (0,1)$ for all $x \in \mathbb{R}$, calculate the value of $k$ such that the inequality \[ \left\{\int_0^1 f(x) \, dx\right\}^2 = k \int_0^1 \left\{f(x)\right\}^3 \, dx \] holds for the fun... | [
"1"
] | [
"Assume that the function $f$ is defined for all $x \\in \\mathbb{R}$ and has a continuous derivative. Given $f(0) = 0$ and $f'(x) \\in (0,1)$ for all $x \\in \\mathbb{R}$, calculate the value of $k$ such that the inequality\n\\[\n\\left\\{\\int_0^1 f(x) \\, dx\\right\\}^2 = k \\int_0^1 \\left\\{f(x)\\right\\}^3 \\... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b5b49bd967bd3161ac194a44e4c4e12dc5b39bf550a2015576c0a9c5ed8f7035 | Assume that the function \( f \) is defined on \( (-\infty, +\infty) \), measurable, and satisfies the condition \( \int_{-\infty}^{+\infty} |f(x)|^2 \, dx < +\infty \). Let \( g(\alpha) = \alpha^2 \mu\{x \in \mathbb{R} : |f(x)| > \alpha\} \). Find \( \lim_{\alpha \to +\infty} g(\alpha) \). | [
"0"
] | [
"Assume that the function \\( f \\) is defined on \\( (-\\infty, +\\infty) \\), measurable, and satisfies the condition \\( \\int_{-\\infty}^{+\\infty} |f(x)|^2 \\, dx < +\\infty \\). Let \\( g(\\alpha) = \\alpha^2 \\mu\\{x \\in \\mathbb{R} : |f(x)| > \\alpha\\} \\). Find \\( \\lim_{\\alpha \\to +\\infty} g(\\alpha... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-80b2314619b23959d0a36b3c7cd84c074a04c913cfaf8c7b8e304324440adfba | Assume that the function \( f(x) \) has the property that for every positive number \( c \), there exists a polynomial \( P_c(x) \in \mathbb{R}[x] \) such that \( |f(x) - P_c(x)| \leq cx^{1998} \) for all \( x \in \mathbb{R} \). Determine the degree of the polynomial \( f(x) \). | [
"1998"
] | [
"Assume that the function \\( f(x) \\) has the property that for every positive number \\( c \\), there exists a polynomial \\( P_c(x) \\in \\mathbb{R}[x] \\) such that \\( |f(x) - P_c(x)| \\leq cx^{1998} \\) for all \\( x \\in \\mathbb{R} \\). Determine the degree of the polynomial \\( f(x) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f9e3ce2d32b52a529ce6fef21ee4b970038135c67a65e27477a92fc3436e1019 | Assume that the heights of 18-year-old males are normally distributed with a mean of 69 inches and a standard deviation of 6 inches. a) What percentage of 18-year-old males are between 69 and 75 inches tall? b) If 1000 18-year-old males are selected at random, how many will be less than 72 inches tall? | [
"34.13% for part (a) and 692 for part (b)."
] | [
"Assume that the heights of 18-year-old males are normally distributed with a mean of 69 inches and a standard deviation of 6 inches.\na) What percentage of 18-year-old males are between 69 and 75 inches tall?\nb) If 1000 18-year-old males are selected at random, how many will be less than 72 inches tall?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 17
} | 17 | 0 | 0 |
omr-ws-b9d4088e50ed2c7400a72fdae89747ea93eba4eea753fe1bd10213b3a5a2d42b | Assume that the infinite series $\sum_{n=0}^{\infty} a_n$ converges and that $|b_n - 2| \leq a_n^2$ for all $n \in \mathbb{N}$. Let $L$ be the limit of the sequence $(b_n)$. Calculate the value of $L$. | [
"2"
] | [
"Assume that the infinite series $\\sum_{n=0}^{\\infty} a_n$ converges and that $|b_n - 2| \\leq a_n^2$ for all $n \\in \\mathbb{N}$. Let $L$ be the limit of the sequence $(b_n)$. Calculate the value of $L$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4671d6f739c0157df4633867e740144e51d35953b241a0e61fb4bf67ee923af2 | Assume that the length of time \( T \) it takes to do a major construction project varies inversely with the 0.7 power of the number of workers \( W \). If 100 workers can construct an office building in 250 days, write the particular equation expressing \( T \) in terms of \( W \), and use it to predict the number of ... | [
"141"
] | [
"Assume that the length of time \\( T \\) it takes to do a major construction project varies inversely with the 0.7 power of the number of workers \\( W \\). If 100 workers can construct an office building in 250 days, write the particular equation expressing \\( T \\) in terms of \\( W \\), and use it to predict t... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-06422e8e7c45b1b64c4ac74c04f1e06c18a4c9f26d8d9accc2d330f685443ec0 | Assume that the probability that a man has at least one son is 0.8. Form a Markov chain with four states: professional, skilled laborer, unskilled laborer, and a fourth state representing families whose male line has died out. The transition probabilities for having a son are the same as in Problem 1. If there is no so... | [
"\\(\\frac{6}{25}\\) or 30%"
] | [
"Assume that the probability that a man has at least one son is 0.8. Form a Markov chain with four states: professional, skilled laborer, unskilled laborer, and a fourth state representing families whose male line has died out. The transition probabilities for having a son are the same as in Problem 1. If there is ... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-eba3937e596c2dc914fe6660f78c087b42e35ae7b6c85e26742a7d96c88ab230 | Assume that the rate of change in the number of miles $s$ of road cleared per hour by a snowplow is inversely proportional to the height $h$ of the snow. Find the particular solution if $s = 25$ miles when $h = 2$ inches and $s = 12$ miles when $h = 10$ inches. | [
"\\( s = 30.6 - 8.07 \\ln h \\)"
] | [
"Assume that the rate of change in the number of miles $s$ of road cleared per hour by a snowplow is inversely proportional to the height $h$ of the snow. Find the particular solution if $s = 25$ miles when $h = 2$ inches and $s = 12$ miles when $h = 10$ inches."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-4266ca00d602b95866b5c80e030a0e788b0a9e6f2071296ae9bcd686ef3df920 | Assume that the rate of change in the number of miles $s$ of road cleared per hour by a snowplow is inversely proportional to the height $h$ of the snow. Write and solve the differential equation to find $s$ as a function of $h$. | [
"\\( S = 30.6 - \\frac{13}{\\ln 5} \\ln h \\)"
] | [
"Assume that the rate of change in the number of miles $s$ of road cleared per hour by a snowplow is inversely proportional to the height $h$ of the snow. Write and solve the differential equation to find $s$ as a function of $h$."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 3
} | 3 | 0 | 0 |
omr-ws-66f39cba23b6f17b45fec96cc13ca917fc39040233b5e196afa1d5c29d381104 | Assume that the sequence $1, 3, 9, 27, \ldots$ is given by a minimal degree polynomial $P$ such that $P(1) = 1$, $P(2) = 3$, $P(3) = 9$, and $P(4) = 27$. What are the next two numbers in the pattern, i.e., $P(5)$ and $P(6)$? | [
"131"
] | [
"Assume that the sequence $1, 3, 9, 27, \\ldots$ is given by a minimal degree polynomial $P$ such that $P(1) = 1$, $P(2) = 3$, $P(3) = 9$, and $P(4) = 27$. What are the next two numbers in the pattern, i.e., $P(5)$ and $P(6)$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9c9b2e90acac64f06d433c6c17d3bd40d5cb5b5919b7c612950e1c0c6584ce48 | Assume that the sequence \( \{a_n\} \) satisfies \( a_1 = 0.72 \) and \( a_{n+1} = f(a_n) \) for \( n \in \mathbb{N} \). Find the smallest \( n_0 \in \mathbb{N} \) such that for every \( n \ge n_0 \) we have \( |a_n - c| < 10^{-6} \). | [
"21"
] | [
"Assume that the sequence \\( \\{a_n\\} \\) satisfies \\( a_1 = 0.72 \\) and \\( a_{n+1} = f(a_n) \\) for \\( n \\in \\mathbb{N} \\). Find the smallest \\( n_0 \\in \\mathbb{N} \\) such that for every \\( n \\ge n_0 \\) we have \\( |a_n - c| < 10^{-6} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 5
} | 16 | 0 | 0 |
omr-ws-afea40e9c1d8fc62d6420f4217911c777cb341f1cb268b15e11c46d47fdf3282 | Assume that the series \( \sum_{n=1}^{\infty} a_n \) is convergent and \( a_n > 0 \). Find the smallest possible value of \( b_{100} \) where \( b_1 \leq b_2 \leq \ldots \leq b_n \leq \ldots \) is a sequence such that \( \lim_{n \to \infty} b_n = \infty \) and the series \( \sum_{n=1}^{\infty} a_n b_n \) is convergent,... | [
"0"
] | [
"Assume that the series \\( \\sum_{n=1}^{\\infty} a_n \\) is convergent and \\( a_n > 0 \\). Find the smallest possible value of \\( b_{100} \\) where \\( b_1 \\leq b_2 \\leq \\ldots \\leq b_n \\leq \\ldots \\) is a sequence such that \\( \\lim_{n \\to \\infty} b_n = \\infty \\) and the series \\( \\sum_{n=1}^{\\in... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 8
} | 13 | 0 | 0 |
omr-ws-9c8a61fed18f3a9acca475d36e16c1238c8e12603bc8ea81eee7bffba7af47f3 | Assume that the ten variables in the expression $(ab + cd)(pqr + stu)$ represent the ten integers from 1 to 10, in some order. What is the greatest possible odd value of the expression? | [
"35145"
] | [
"Assume that the ten variables in the expression $(ab + cd)(pqr + stu)$ represent the ten integers from 1 to 10, in some order. What is the greatest possible odd value of the expression?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 8,
"tir": 12
} | 31 | 0 | 0 |
omr-ws-18895599dc12ddc18ffb6eab7a7c2d1913af4d484b7064b8a565827ce244e517 | Assume that the transformation $t \to U_{t} \in L(\mathbb{R}^3,\mathbb{R}^3)$ is of class $C^1$ and $\forall t, U_{t}^{-1}=U_{t}^\mathbb{T}$. For a given $t$, let $\omega_{t} \in \mathbb{R}^3$ be such that $\forall v \in \mathbb{R}^3, (U_{t})'U_{t}v = \omega_{t} \times v$. Calculate the value of $\omega_{t} \cdot \omeg... | [
"4"
] | [
"Assume that the transformation $t \\to U_{t} \\in L(\\mathbb{R}^3,\\mathbb{R}^3)$ is of class $C^1$ and $\\forall t, U_{t}^{-1}=U_{t}^\\mathbb{T}$. For a given $t$, let $\\omega_{t} \\in \\mathbb{R}^3$ be such that $\\forall v \\in \\mathbb{R}^3, (U_{t})'U_{t}v = \\omega_{t} \\times v$. Calculate the value of $\\o... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-167de0e17941c2992beb39f0d18fbd89f99693a33db1415794e6fe091523b97f | Assume that the zeroes of the polynomial \( P(x) = \left( \sum_{i=0}^{2n-1} a_i x^i \right) + x^{2n} \) are positive integers and that \( \sum_{i=0}^{n-1} a_{2i} = -\sum_{i=1}^n a_{2i-1} \). Determine the zeroes of \( P(x) \). | [
"2 (with multiplicity \\( 2n \\))"
] | [
"Assume that the zeroes of the polynomial \\( P(x) = \\left( \\sum_{i=0}^{2n-1} a_i x^i \\right) + x^{2n} \\) are positive integers and that \\( \\sum_{i=0}^{n-1} a_{2i} = -\\sum_{i=1}^n a_{2i-1} \\). Determine the zeroes of \\( P(x) \\)."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 13
} | 13 | 0 | 0 |
omr-ws-5886c9e8c2f9a791d0c55b659c839d199de76e62fb6c65b10d79ff956184147b | Assume that there exists a perfect power of \(a\) that is congruent to \(-1\) modulo \(m\), and let \(s\) be the least such integer. Given that \(ord_m(a) = d\), find the value of \(d\) in terms of \(s\) and verify that \(a^{2s-d} \equiv -1 \pmod{m}\) does not violate the minimality assumption on \(s\). Express your an... | [
"14"
] | [
"Assume that there exists a perfect power of \\(a\\) that is congruent to \\(-1\\) modulo \\(m\\), and let \\(s\\) be the least such integer. Given that \\(ord_m(a) = d\\), find the value of \\(d\\) in terms of \\(s\\) and verify that \\(a^{2s-d} \\equiv -1 \\pmod{m}\\) does not violate the minimality assumption on... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9f07fca774cda5ba1078a5901fac501c64e4088f321fbb5bd29730dbd1a2341a | Assume that two circles $(O_1)$ and $(O_2)$ intersect at points $A$ and $B$. A line through $B$ intersects $(O_1)$ at $C$ and $(O_2)$ at $D$. Let $M$ be the midpoint of $CD$. Line $AM$ intersects $(O_1)$ at a second point $K$ and $(O_2)$ at a second point $P$. A line $d$ perpendicular to $O_1M$ at $M$ intersects $AC$ a... | [
"\\dfrac{\\sqrt{17}}{6}"
] | [
"Assume that two circles $(O_1)$ and $(O_2)$ intersect at points $A$ and $B$. A line through $B$ intersects $(O_1)$ at $C$ and $(O_2)$ at $D$. Let $M$ be the midpoint of $CD$. Line $AM$ intersects $(O_1)$ at a second point $K$ and $(O_2)$ at a second point $P$. A line $d$ perpendicular to $O_1M$ at $M$ intersects $... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-a47d1e0452d3993e6146a5fc1f80c6b489db0c5b8265981f3f5bbaf43f732cab | Assume that two circles $(O_1)$ and $(O_2)$ intersect at points $A$ and $B$. A line through $B$ intersects $(O_1)$ at $C$ and $(O_2)$ at $D$. Let $M$ be the midpoint of $CD$. Line $AM$ intersects $(O_1)$ at a second point $K$ and $(O_2)$ at a second point $P$. A line $d$ perpendicular to $O_1M$ at $M$ intersects $AC$ a... | [
"\\dfrac{2}{7}"
] | [
"Assume that two circles $(O_1)$ and $(O_2)$ intersect at points $A$ and $B$. A line through $B$ intersects $(O_1)$ at $C$ and $(O_2)$ at $D$. Let $M$ be the midpoint of $CD$. Line $AM$ intersects $(O_1)$ at a second point $K$ and $(O_2)$ at a second point $P$. A line $d$ perpendicular to $O_1M$ at $M$ intersects $... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-516fa087df97f3a74465b8cfef102006144d885d1c046d58c1597c2369ac84ca | Assume that two functions \( f(x) \) and \( g(x) \) are defined on the interval \([0, 2]\), where \( f(x) \) is continuously differentiable and \( g(x) \) is continuously differentiable in the second order and convex. They satisfy the following conditions: 1. \( 0 \leq f(x) \leq g(x) \), \( f(0) = g(0) = f(2) = g(2) \)... | [
"\\dfrac{1}{4}"
] | [
"Assume that two functions \\( f(x) \\) and \\( g(x) \\) are defined on the interval \\([0, 2]\\), where \\( f(x) \\) is continuously differentiable and \\( g(x) \\) is continuously differentiable in the second order and convex. They satisfy the following conditions:\n1. \\( 0 \\leq f(x) \\leq g(x) \\), \\( f(0) = ... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-8f435b4f1cdb27786b019b895bfc41559a896ba94e182ccb9833ff86e9931f82 | Assume that two parallelograms P and Q have equal areas with sides a, b and c, d respectively, where \( c \leq a \leq b \leq d \), and a segment of length d can be placed inside P. If P and Q can be partitioned into four pairwise congruent parts, find the ratio of the area of one of these parts to the area of parallelo... | [
"\\dfrac{1}{4}"
] | [
"Assume that two parallelograms P and Q have equal areas with sides a, b and c, d respectively, where \\( c \\leq a \\leq b \\leq d \\), and a segment of length d can be placed inside P. If P and Q can be partitioned into four pairwise congruent parts, find the ratio of the area of one of these parts to the area of... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2cad35754b4bd117d05c0081f6d9766fcd792f063e3ed6b025c41a6fc3ccbbca | Assume the $n$ sets $A_1, A_2, \ldots, A_n$ are a partition of the set $A=\{1,2,\ldots,29\}$, and the sum of any elements in $A_i$ (for $i=1,2,\ldots,n$) is not equal to $30$. Find the smallest possible value of $n$. | [
"3"
] | [
"Assume the $n$ sets $A_1, A_2, \\ldots, A_n$ are a partition of the set $A=\\{1,2,\\ldots,29\\}$, and the sum of any elements in $A_i$ (for $i=1,2,\\ldots,n$) is not equal to $30$. Find the smallest possible value of $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e6c2d1c216732273bf66347dea46faef718c4be9da52595c74bd2e50c90d50c0 | Assume the Earth is a perfect sphere with a radius of 4000 miles. Starting at a latitude of 30 degrees North, how many miles will you travel if you move east until you return to your original position? | [
"\\( 4000\\sqrt{3}\\pi \\)"
] | [
"Assume the Earth is a perfect sphere with a radius of 4000 miles. Starting at a latitude of 30 degrees North, how many miles will you travel if you move east until you return to your original position?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 19
} | 34 | 0 | 0 |
omr-ws-ac354e6e7c0793c7032cdd28acbaf3a1eb5349ec5ab7062625efd0158e862ef4 | Assume the constant of integration is $0$. Find a set of necessary conditions for $\int^{d}_{c}{\frac{\ln{ax}}{x^{b}}}\,dx<0$ where $a,b,c,d\in\Re$ and $d>c$. | [
" \n\\begin{cases} \na > 0,\\ c > 0,\\ ad \\leq 1,\\ \\text{and}\\ (b \\neq 1\\ \\text{or}\\ ad < 1) \\\\\n\\text{or} \\\\\na < 0,\\ d < 0,\\ ac \\leq 1,\\ \\text{and}\\ (b \\neq 1\\ \\text{or}\\ ac > 1)\n\\end{cases} \n"
] | [
"Assume the constant of integration is $0$. Find a set of necessary conditions for $\\int^{d}_{c}{\\frac{\\ln{ax}}{x^{b}}}\\,dx<0$ where $a,b,c,d\\in\\Re$ and $d>c$."
] | [
"aops_c7_college_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 4
} | 5 | 0 | 0 |
omr-ws-37aba64c4882a3a156d35a7049b0f11339ba9f6def9c3096d179902585b7016b | Assume the game begins with 10 players, each of whom independently and randomly choose between rock, paper, and scissors with equal probability. Let \( R \), \( P \), and \( S \) count the number of players who play rock, paper, and scissors, respectively. How would you compute the joint probability mass function (PMF)... | [
"\\( P(R = r, P = p, S = s) = \\frac{10!}{r! p! s!} \\left(\\frac{1}{3}\\right)^{10} \\) for \\( r + p + s = 10 \\)."
] | [
"Assume the game begins with 10 players, each of whom independently and randomly choose between rock, paper, and scissors with equal probability. Let \\( R \\), \\( P \\), and \\( S \\) count the number of players who play rock, paper, and scissors, respectively. How would you compute the joint probability mass fun... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-9469a677ed7bd3b30f8bc395e2337e9b08cb4b788723cd1a8d0927c8e66f639e | Assume the highest ranking prospect is the $(m + 1)th$ person to be interviewed. What is the probability that the highest rank candidate among the first $m$ candidates is one of the first $k$ candidates who were rejected? | [
"\\dfrac{k}{m}"
] | [
"Assume the highest ranking prospect is the $(m + 1)th$ person to be interviewed. What is the probability that the highest rank candidate among the first $m$ candidates is one of the first $k$ candidates who were rejected?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-e3189b3441a2936b8b82da476400b5ca26e31afa941285207cf793416ce2e860 | Assume the highest ranking prospect is the (m+1)th person to be interviewed. What is the probability that the highest rank candidate among the first m candidates is one of the first k candidates who were rejected? | [
"\\(\\frac{k}{m}\\)"
] | [
"Assume the highest ranking prospect is the (m+1)th person to be interviewed. What is the probability that the highest rank candidate among the first m candidates is one of the first k candidates who were rejected?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-5f153d270200d6d8ca4743cf2c188f5fc8d01925f422643e98c59fc09bb3996c | Assume the prior \(\theta_i \sim N(0, A)\) independently for \( i = 1, \ldots, k \) and a given \( A > 0 \). Provide an estimator \(\hat{A}\) of \( A \) and derive the empirical Bayes estimator of \(\tau\), denoted as \(\hat{\tau}_B\), where \(\hat{\tau}_B = E(\tau \mid X_1, \ldots, X_k, \hat{A})\). | [
"\\hat{\\tau}_B = \\frac{\\hat{A}}{\\hat{A} + 1} X_i"
] | [
"Assume the prior \\(\\theta_i \\sim N(0, A)\\) independently for \\( i = 1, \\ldots, k \\) and a given \\( A > 0 \\). Provide an estimator \\(\\hat{A}\\) of \\( A \\) and derive the empirical Bayes estimator of \\(\\tau\\), denoted as \\(\\hat{\\tau}_B\\), where \\(\\hat{\\tau}_B = E(\\tau \\mid X_1, \\ldots, X_k,... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 3
} | 11 | 0 | 0 |
omr-ws-be5827393c5a252e5d7ffb635f5a7aa5bdbe6bb6a18949391b5a2a27d9af3f57 | Assume the property Q is possessed by at least one of \(a_1, b_1, c_1\), at least one of \(a_2, b_2, c_2\), at least one of \(a_3, b_3, c_3\), at least one of \(a_4, b_4, c_4\), and at least one of \(a_5, b_5, c_5\). Find the largest value of \(m\) for which it can be said that at least \(m\) of the \(a\)'s, \(b\)'s, o... | [
"5"
] | [
"Assume the property Q is possessed by at least one of \\(a_1, b_1, c_1\\), at least one of \\(a_2, b_2, c_2\\), at least one of \\(a_3, b_3, c_3\\), at least one of \\(a_4, b_4, c_4\\), and at least one of \\(a_5, b_5, c_5\\). Find the largest value of \\(m\\) for which it can be said that at least \\(m\\) of the ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-5391d47426fa429a68c25b510bf6adaa4b5923f13cc3877a363c44e3891ea744 | Assume the speed of sound is $343$ m/s. When Anastasia and Bananastasia both yell at the same time, you hear Anastasia’s yell $5$ seconds before Bananastasia’s yell. If Bananastasia yells first, and then Anastasia yells when she hears Bananastasia yell, you hear Anastasia’s yell $5$ seconds after Bananastasia’s yell. W... | [
"3430 meters"
] | [
"Assume the speed of sound is $343$ m/s. When Anastasia and Bananastasia both yell at the same time, you hear Anastasia’s yell $5$ seconds before Bananastasia’s yell. If Bananastasia yells first, and then Anastasia yells when she hears Bananastasia yell, you hear Anastasia’s yell $5$ seconds after Bananastasia’s ye... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-7271dd159af965c2ce59b194b6837d90f7eb2a844ac5b6b155619a49c718f01f | Assume the weight (in ounces) of a major league baseball is a random variable, and a carton contains 144 baseballs. The weights of individual baseballs are independent and normally distributed with mean $\mu=5$ and standard deviation $\sigma = 2/5$. Let $T$ represent the total weight of all baseballs in the carton. Fin... | [
"0.8508"
] | [
"Assume the weight (in ounces) of a major league baseball is a random variable, and a carton contains 144 baseballs. The weights of individual baseballs are independent and normally distributed with mean $\\mu=5$ and standard deviation $\\sigma = 2/5$. Let $T$ represent the total weight of all baseballs in the cart... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 1
} | 8 | 0 | 0 |
omr-ws-5baea4b337a9dd66e2f2f96edf162b9c7991132de97037c95b66207b826ef5da | Assume there are 18 board members: 13 females and 5 males, including Carl. There are 4 tasks to be assigned, and assigning the same people different tasks constitutes a different assignment. Find the probability that Carl and at least one female are given tasks. | [
"\\(\\frac{15}{68}\\)"
] | [
"Assume there are 18 board members: 13 females and 5 males, including Carl. There are 4 tasks to be assigned, and assigning the same people different tasks constitutes a different assignment. Find the probability that Carl and at least one female are given tasks."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-ba6e89ed6cca28a32be252c72dacea5d969eb0dc4ad6bcf4acb38aa33ea22082 | Assume there are finitely many primes \( p_1, p_2, \ldots, p_k \). Let \( n = p_1 \cdot p_2 \cdot \ldots \cdot p_k \). For any arbitrary number \( m \) such that \( 1 < m < n \), what can we say about \(\gcd(m, n)\)? | [
"\\(\\gcd(m, n) > 1\\)"
] | [
"Assume there are finitely many primes \\( p_1, p_2, \\ldots, p_k \\). Let \\( n = p_1 \\cdot p_2 \\cdot \\ldots \\cdot p_k \\). For any arbitrary number \\( m \\) such that \\( 1 < m < n \\), what can we say about \\(\\gcd(m, n)\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-07d2a01e29e9d06423ca582f304e33627427b5653f2505f15d85b75f147973a9 | Assume there exists a language with 25 characters, and words consist of 17 characters each. A ring made of paper has $5^{18}$ characters printed around it. For any word $\omega$, let $\mathrm{count}(\omega)$ be the maximum number of paper slices that can be obtained containing exactly the word $\omega$ by cutting the r... | [
"24 \\times 5^{16}"
] | [
"Assume there exists a language with 25 characters, and words consist of 17 characters each. A ring made of paper has $5^{18}$ characters printed around it. For any word $\\omega$, let $\\mathrm{count}(\\omega)$ be the maximum number of paper slices that can be obtained containing exactly the word $\\omega$ by cutt... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-885d3fbce9781e113ab9e81aefa1b4bcf58ca14155ed8e999f0462993bb4cabe | Assume there exists a permutation $\{a_1, a_2, a_3, \ldots, a_n\}$ of $\{1, 2, \ldots, n\}$ such that $\{a_1+1, a_2+2, \ldots, a_n+n\}$ and $\{a_1-1, a_2-2, \ldots, a_n-n\}$ are complete residue systems modulo $n$. Find the sum of all possible values of $n$ less than $100$ for which this condition holds. | [
"1633"
] | [
"Assume there exists a permutation $\\{a_1, a_2, a_3, \\ldots, a_n\\}$ of $\\{1, 2, \\ldots, n\\}$ such that $\\{a_1+1, a_2+2, \\ldots, a_n+n\\}$ and $\\{a_1-1, a_2-2, \\ldots, a_n-n\\}$ are complete residue systems modulo $n$. Find the sum of all possible values of $n$ less than $100$ for which this condition hold... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-00436f7a4c69a8849edd920a43048a692d84bd4c83e54bf5d4d67b38eb76a0c7 | Assume two non-intersecting circles \(O_1\) and \(O_2\) are internally tangent to a larger circle \(O\) at points \(A\) and \(B\) respectively. Two internal common tangents of circles \(O_1\) and \(O_2\) intersect the larger circle \(O\) at points \(C, D, E,\) and \(F\) respectively. Lines \(CE\) and \(FD\) lie on oppo... | [
"\\frac{(a_x b_y - a_y b_x)(c_x - e_x) - (c_x e_y - c_y e_x)(a_x - b_x)}{(a_x - b_x)(c_y - e_y) - (a_y - b_y)(c_x - e_x)}"
] | [
"Assume two non-intersecting circles \\(O_1\\) and \\(O_2\\) are internally tangent to a larger circle \\(O\\) at points \\(A\\) and \\(B\\) respectively. Two internal common tangents of circles \\(O_1\\) and \\(O_2\\) intersect the larger circle \\(O\\) at points \\(C, D, E,\\) and \\(F\\) respectively. Lines \\(C... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 4
} | 5 | 0 | 0 |
omr-ws-36a7e1b90b90704e8f3069fd520857d4d5b69f0d00c30d950522c62888a7b943 | Assume two random subsets \( A \) and \( B \) of the set \(\{1,2,3,4,5,6\}\). Find the probability that \( B \) is a subset of \( A \) or \( A \) and \( B \) are disjoint. | [
"\\(\\frac{729}{2^{11}}\\)"
] | [
"Assume two random subsets \\( A \\) and \\( B \\) of the set \\(\\{1,2,3,4,5,6\\}\\). Find the probability that \\( B \\) is a subset of \\( A \\) or \\( A \\) and \\( B \\) are disjoint."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8a6e8406f7eb6c41b3971a01941fdceff8e9a58c8faf8690200662599796e54b | Assume we can find positive integers \( R, k, l \) such that for every positive integer \( M \) the number \( Mp^kq^l + R \) is captivating. Let \( D \) be the smallest constant such that every integer \( n \ge D \) is captivating. If \( p = 2 \) and \( q = 3 \), and \( R = 1 \), find the smallest possible value of \( ... | [
"37"
] | [
"Assume we can find positive integers \\( R, k, l \\) such that for every positive integer \\( M \\) the number \\( Mp^kq^l + R \\) is captivating. Let \\( D \\) be the smallest constant such that every integer \\( n \\ge D \\) is captivating. If \\( p = 2 \\) and \\( q = 3 \\), and \\( R = 1 \\), find the smallest... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 5
} | 7 | 0 | 0 |
omr-ws-56b496efb95be717669d34980872ea9a647fd8b603f210b96c361395b3c95c52 | Assume we have a calendrical system in which leap years happen every four years, no matter what. In a 150-year period, what is the maximum possible number of leap years? | [
"38"
] | [
"Assume we have a calendrical system in which leap years happen every four years, no matter what. In a 150-year period, what is the maximum possible number of leap years?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-53a510bd3a9027810aea15f9db7a29ac51e1c90153d8aae2a30e20e5711d842f | Assume we have a cylinder lying on its side with a diameter of 20 units. If the cylinder is filled to 25% of its volume with gas, at what height on the stick (placed down the middle of the cylinder) should the gas leave? | [
"6"
] | [
"Assume we have a cylinder lying on its side with a diameter of 20 units. If the cylinder is filled to 25% of its volume with gas, at what height on the stick (placed down the middle of the cylinder) should the gas leave?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 3
} | 10 | 0 | 0 |
omr-ws-8ec00ba1e4e66d681cc7d5324f44b77e51f900633979d457493c891042bfd6a8 | Assume you have a die where the probability to get "1" is 1/2, and the probability to get 2, 3, 4, 5, or 6 is equal. Let \( X \) be the number of eyes that you get when the die is tossed. a) Find the probability distribution of \( X \). b) Find the expected value \( E(X) \). c) Find the variance \( Var(X) \). | [
"\\( E(X) = 2.5 \\) and \\( Var(X) = 3.25 \\)"
] | [
"Assume you have a die where the probability to get \"1\" is 1/2, and the probability to get 2, 3, 4, 5, or 6 is equal. Let \\( X \\) be the number of eyes that you get when the die is tossed.\na) Find the probability distribution of \\( X \\).\nb) Find the expected value \\( E(X) \\).\nc) Find the variance \\( Var... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 21
} | 21 | 0 | 0 |
omr-ws-3fcfa7254aef2ade11e0cf34e3d027ee3e6e1dd70597d90d42d906ab682060e3 | Assume you start with 16 dollars. A fair coin is flipped; if it comes up heads, 8 dollars are added to that amount, otherwise half the money is lost. Find the probability that you will have less than 16 dollars after the fourth flip. | [
"\\dfrac{9}{16}"
] | [
"Assume you start with 16 dollars. A fair coin is flipped; if it comes up heads, 8 dollars are added to that amount, otherwise half the money is lost. Find the probability that you will have less than 16 dollars after the fourth flip."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a676c2d2b51f2102c14012e37a2dc1ba54e74d812ac8d79aa592f6f23fe371de | Assuming $0 < y < 1$, find the maximum value of \[ \int_0^y \sqrt{x^4 + (y - y^2)^2} \, dx \] | [
"\\dfrac{1}{3}"
] | [
"Assuming $0 < y < 1$, find the maximum value of\n\\[ \\int_0^y \\sqrt{x^4 + (y - y^2)^2} \\, dx \\]"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-8ff27ee9cdf7ac96f1c62f2584a32760b4cfa1e1c9af4f6e65d7c37cf445271d | Assuming 5 people each produce 13,146,337,252,759,428 dollars per year (5 of each denomination of the third Zimbabwe dollar * 73), how many years will it take to distribute an equal amount of money to 101 numbers? How much money will each number have after those years? Note: The denominations of the Third Zimbabwe Doll... | [
"65731686263797140"
] | [
"Assuming 5 people each produce 13,146,337,252,759,428 dollars per year (5 of each denomination of the third Zimbabwe dollar * 73), how many years will it take to distribute an equal amount of money to 101 numbers? How much money will each number have after those years?\n\nNote: The denominations of the Third Zimba... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-82467ee78298612ed6195f7f14e487c4fbc9aa1362a8b7d8127e18823a1c958d | Assuming Earth's orbit around the Sun is circular with a mean distance of 9.29 x 10^7 miles and that one revolution takes 365 days, find the linear speed of the Earth in miles per hour. | [
"2292 mph"
] | [
"Assuming Earth's orbit around the Sun is circular with a mean distance of 9.29 x 10^7 miles and that one revolution takes 365 days, find the linear speed of the Earth in miles per hour."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ea8e29b9a5ecfca5b3671837dc88f3b0f0f56caa4e8efe0abd2bb9c95dba87a9 | Assuming Schanuel's conjecture, let $\alpha$ be an algebraic number such that $i\alpha$ is not a rational number. Calculate the degree of the field extension $\mathbb{Q}(\pi, e^{\alpha \pi})$ over $\mathbb{Q}$. | [
"\\infty"
] | [
"Assuming Schanuel's conjecture, let $\\alpha$ be an algebraic number such that $i\\alpha$ is not a rational number. Calculate the degree of the field extension $\\mathbb{Q}(\\pi, e^{\\alpha \\pi})$ over $\\mathbb{Q}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a74b66a549afb83e3ebe311fc6bf3a502e1fd5f042143ad39ed5fd744c1ba439 | Assuming \( A = \mathbb{Q} \), find all functions \( f \) which satisfy the functional equation \( f(x + y) = f(x) f(y) - f(xy) + 1 \) and \( f(1997) \neq f(1998) \). | [
"\\( f(x) = x + 1 \\)",
"f(x) = x + 1"
] | [
"Assuming \\( A = \\mathbb{Q} \\), find all functions \\( f \\) which satisfy the functional equation \\( f(x + y) = f(x) f(y) - f(xy) + 1 \\) and \\( f(1997) \\neq f(1998) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 8
} | 39 | 0 | 0 |
omr-ws-1e6422f476b165f14068e71d693fed6451f4f7589cfcdc911f320ab871b12b81 | Assuming \( f(n) = \sqrt{1 + n \sqrt{1 + (n+1) \sqrt{1 + \cdots}}} \) is a polynomial in \( n \), calculate the value of \( f(3) = \sqrt{1 + 3 \sqrt{1 + 4 \sqrt{1 + 5 \sqrt{1 + \cdots}}}} \). | [
"4"
] | [
"Assuming \\( f(n) = \\sqrt{1 + n \\sqrt{1 + (n+1) \\sqrt{1 + \\cdots}}} \\) is a polynomial in \\( n \\), calculate the value of \\( f(3) = \\sqrt{1 + 3 \\sqrt{1 + 4 \\sqrt{1 + 5 \\sqrt{1 + \\cdots}}}} \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 3
} | 18 | 0 | 0 |
omr-ws-9d916ba0d5b7f3dd2f6022ef14048c4650ea5a3e3df0a4705c2162cab1b50fe6 | Assuming \( f(x) \) is twice differentiable, calculate the value of \( \lim_{h \to 0} \frac{f(3+h) + f(3-h) - 2f(3)}{h^2} \) given that \( f''(3) = 5 \). | [
"5"
] | [
"Assuming \\( f(x) \\) is twice differentiable, calculate the value of \\( \\lim_{h \\to 0} \\frac{f(3+h) + f(3-h) - 2f(3)}{h^2} \\) given that \\( f''(3) = 5 \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3083b1e7945e62007b4ab2139c7565ab82059e65eb1c2b6122466a9e331c8414 | Assuming \( u_0 > -1 \), use \( w_n = \frac{1}{u_{n+1}} - \frac{1}{u_n} \) to find an equivalent of the sequence \( (u_n) \). | [
"\\frac{1}{\\sum_{k=0}^{n-1} w_k}"
] | [
"Assuming \\( u_0 > -1 \\), use \\( w_n = \\frac{1}{u_{n+1}} - \\frac{1}{u_n} \\) to find an equivalent of the sequence \\( (u_n) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 7
} | 16 | 0 | 0 |
omr-ws-bddfa3b00410e9fd1d985b2946d0d76e5005c885d30245a43d8421d0d3d4a8ff | Assuming non-zero denominators, the equation $\frac{k}{x-2}+\frac{w}{(x-2)^{2}}+\frac{bx+f}{x^{2}+1}= \frac{3x^{2}+3x+7}{(x-2)^{2}(x^{2}+1)}$ holds for all values of $x$. Find the value of $k+w+b+f$. | [
"5"
] | [
"Assuming non-zero denominators, the equation $\\frac{k}{x-2}+\\frac{w}{(x-2)^{2}}+\\frac{bx+f}{x^{2}+1}= \\frac{3x^{2}+3x+7}{(x-2)^{2}(x^{2}+1)}$ holds for all values of $x$. Find the value of $k+w+b+f$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-145269ae206dbbb0ff34ba75886d37f031d371538a050956f71b765b3c5595af | Assuming part (a), we can define $$g(s) = f(5^{a_1} + 5^{a_2} + \ldots + 5^{a_k}),$$ where $s = a_1 + a_2 + \ldots + a_k$. Find the least positive integer $p$ for which $$g(s) = g(s + p), \quad \text{for all } s \ge 1,$$ or show that no such $p$ exists. | [
"4"
] | [
"Assuming part (a), we can define\n$$g(s) = f(5^{a_1} + 5^{a_2} + \\ldots + 5^{a_k}),$$\nwhere $s = a_1 + a_2 + \\ldots + a_k$. Find the least positive integer $p$ for which\n$$g(s) = g(s + p), \\quad \\text{for all } s \\ge 1,$$\nor show that no such $p$ exists."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-2fb3dc280648b8b0686abe5cfed9b5b5c27a5dd4254cb101206febf057b0e355 | Assuming real values for $p$, $q$, $r$, and $s$, the equation $$x^4 + px^3 + qx^2 + rx + s$$ has four non-real roots. The sum of two of these roots is $4 + 7i$, and the product of the other two roots is $3 - 4i$. Find $q$. | [
"71"
] | [
"Assuming real values for $p$, $q$, $r$, and $s$, the equation $$x^4 + px^3 + qx^2 + rx + s$$ has four non-real roots. The sum of two of these roots is $4 + 7i$, and the product of the other two roots is $3 - 4i$. Find $q$."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 11
} | 26 | 0 | 0 |
omr-ws-a393865f9126d136b90e8bef8c3a736d27f1219cfa333c9a1a3ecf49d3b0d020 | Assuming real values for $p, q, r$, and $s$, the equation $$x^4 + px^3 + qx^2 + rx + s$$ has four non-real roots. The sum of two of these roots is $q + 6i$, and the product of the other two roots is $3 - 4i$. Find the smallest value of $q$. | [
"-3"
] | [
"Assuming real values for $p, q, r$, and $s$, the equation $$x^4 + px^3 + qx^2 + rx + s$$ has four non-real roots. The sum of two of these roots is $q + 6i$, and the product of the other two roots is $3 - 4i$. Find the smallest value of $q$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-aa7393ed26d069bcae75c2b29f2e3cce2ffdd5b7ef48aab301c6ca4ebae22a52 | Assuming that $p^3$ does not divide $2^{p-1}-1$ for large enough $p$, find the density of the set $\{n \mid 2^n-1 \text{ is cubefree}\}$. | [
"0.7"
] | [
"Assuming that $p^3$ does not divide $2^{p-1}-1$ for large enough $p$, find the density of the set $\\{n \\mid 2^n-1 \\text{ is cubefree}\\}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6e5d5b42e094faf085ae140641d2e5ea852d0b2938d2118ab6b73c3a7212f22e | Assuming that ProtestanT picks an answer for every question, what are the chances that he gets exactly 3 out of the 10 questions right? Express your answer as a common fraction. | [
"\\(\\frac{15}{128}\\)"
] | [
"Assuming that ProtestanT picks an answer for every question, what are the chances that he gets exactly 3 out of the 10 questions right? Express your answer as a common fraction."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 10
} | 25 | 0 | 0 |
omr-ws-f2f00c1fe2ec20365f644726f47c290993b9d0f79efa04bb37c55e0f93ab76c5 | Assuming that \( f(x) \) is continuous in the interval \((0,1)\), calculate the value of \[ \int_{x=0}^{x=1} \int_{y=x}^{y=1} \int_{z=x}^{z=y} f(x)f(y)f(z)\;dz dy dx \] given that \(\int_{0}^{1} f(t)\; dt = 2\). | [
"\\dfrac{4}{3}"
] | [
"Assuming that \\( f(x) \\) is continuous in the interval \\((0,1)\\), calculate the value of\n\\[\n\\int_{x=0}^{x=1} \\int_{y=x}^{y=1} \\int_{z=x}^{z=y} f(x)f(y)f(z)\\;dz dy dx\n\\]\ngiven that \\(\\int_{0}^{1} f(t)\\; dt = 2\\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c23725dc5610f60609b51667fd6cbf13ee62f423e33c5313cce87de3cdd105cc | Assuming that any 5 aircrafts include 3 that are arranged in a straight line, and there are exactly 4 crafts arranged in a straight line, what is the minimum number of planes required? | [
"5"
] | [
"Assuming that any 5 aircrafts include 3 that are arranged in a straight line, and there are exactly 4 crafts arranged in a straight line, what is the minimum number of planes required?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 8
} | 18 | 0 | 0 |
omr-ws-1a4b6c4fe2481f1a4c92eb746c309e42239cc8a3dde6f38738249890749a9079 | Assuming that each distinct arrangement of the letters in "DISCUSSIONS" is equally likely to occur, what is the probability that a random arrangement of the letters in "DISCUSSIONS" has all the \(S\)’s together? | [
"\\(\\frac{4}{165}\\)"
] | [
"Assuming that each distinct arrangement of the letters in \"DISCUSSIONS\" is equally likely to occur, what is the probability that a random arrangement of the letters in \"DISCUSSIONS\" has all the \\(S\\)’s together?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-e3d7a31520dbec84219ae826e26f4af627285f5cf8c1a7fff669aaaeae076946 | Assuming that every prime has primitive roots, find all primes that have exactly \( k \) primitive roots. (a) \( k = 4 \) (b) \( k = 12 \). | [
"\\( p = 11, 13 \\) for \\( k = 4 \\) and \\( p = 29, 37 \\) for \\( k = 12 \\)."
] | [
"Assuming that every prime has primitive roots, find all primes that have exactly \\( k \\) primitive roots. (a) \\( k = 4 \\) (b) \\( k = 12 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 7
} | 7 | 0 | 0 |
omr-ws-007388ce2e8a2b577c3ed1e1c1ffcc15abae10bae4b29f226829af9c2e4602d2 | Assuming that for \( n \) odd no 3 diagonals of a regular \( n \)-gon are concurrent, generalize the number of compartments into which the \( n \)-gon is dissected by the diagonals. | [
"\\(\\binom{n}{4} + \\binom{n-1}{2} + 1\\)"
] | [
"Assuming that for \\( n \\) odd no 3 diagonals of a regular \\( n \\)-gon are concurrent, generalize the number of compartments into which the \\( n \\)-gon is dissected by the diagonals."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-b190c6feab4a2d8c88e2cc6b2d4afd08874509b227aa24c4592d825353fbac2d | Assuming that in the multiplication of bacteria by fission (division into two bacteria) the probability of a bacterium dividing during a time interval Δ(t) is equal to aΔ(t) + o(Δt) and is not dependent on the number of previous divisions or on the number of bacteria present, find the probability that if at time 0 ther... | [
"e^{-a t} \\left(1 - e^{-a t}\\right)^{i - 1}"
] | [
"Assuming that in the multiplication of bacteria by fission (division into two bacteria) the probability of a bacterium dividing during a time interval Δ(t) is equal to aΔ(t) + o(Δt) and is not dependent on the number of previous divisions or on the number of bacteria present, find the probability that if at time 0... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-5143f844f0de884849640e50ac21b428ff88e0492e98cbbbf1b321e608198907 | Assuming that one in \( n \) letters is lost in the mail, find the probability that Superman received Joseph's letter, given that Superman would have answered if he had received it. | [
"\\(\\frac{n-1}{2n-1}\\)"
] | [
"Assuming that one in \\( n \\) letters is lost in the mail, find the probability that Superman received Joseph's letter, given that Superman would have answered if he had received it."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 6,
"tir": 12
} | 20 | 0 | 0 |
omr-ws-a6895e5a969b339520dcaee1cbfd1f37c14bf6d0257a9da700266e91f72015b8 | Assuming that one in \( n \) letters is lost in the mail, find the probability that Superman received the letter, given that he would have answered if he had received it. | [
"\\(\\frac{n-1}{2n-1}\\)"
] | [
"Assuming that one in \\( n \\) letters is lost in the mail, find the probability that Superman received the letter, given that he would have answered if he had received it."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 1
} | 2 | 0 | 0 |
omr-ws-e9d5c360f34edd13390e3f5010430a240e071d13676f598c0a2afd2b0e1b7323 | Assuming that real numbers $x$ and $y$ satisfy the equation $y\left(1+x^2\right)=x\left(\sqrt{1-4y^2}-1\right)$, find the maximum value of $xy$. | [
"\\(\\frac{\\frac{\\sqrt{2}}{\\sqrt[4]{3}} + \\sqrt{1 + \\frac{2}{\\sqrt{3}}}}{2 + \\frac{2}{\\sqrt{3}}}\\)"
] | [
"Assuming that real numbers $x$ and $y$ satisfy the equation $y\\left(1+x^2\\right)=x\\left(\\sqrt{1-4y^2}-1\\right)$, find the maximum value of $xy$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3af5558daaed0f83af3ae22f9f10a221704bde06a5e02887bc9a10cf7c11eb90 | Assuming that the answers to 25 questions are distributed as 5 As, 5 Bs, 5 Cs, 5 Ds, and 5 Es, if you pick five questions at random, what should you guess on them in order to maximize your chance of getting at least two right? | [
"AAAAA"
] | [
"Assuming that the answers to 25 questions are distributed as 5 As, 5 Bs, 5 Cs, 5 Ds, and 5 Es, if you pick five questions at random, what should you guess on them in order to maximize your chance of getting at least two right?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 2
} | 11 | 0 | 0 |
omr-ws-94dec503f599239a362fc0b9f2ccf7a3564a87ac0d4ba2b02ab083f6b307219d | Assuming that the birth of a boy or a girl is equally likely, what is the probability that the three children in a family include at least one boy and one girl? Express your answer as a common fraction. | [
"\\frac{3}{4}"
] | [
"Assuming that the birth of a boy or a girl is equally likely, what is the probability that the three children in a family include at least one boy and one girl? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-82df22310410a99003a1a5679d937a31bd4839e23f5672c80a7ce1014eca1ef9 | Assuming that the computational cost of adding two integers is one unit, estimate the total computational cost of the recursive function $fibonacci(N)$ and the non-recursive function $fibonacci(N)$ in terms of $N$. | [
"Recursive: $O(\\phi^n)$, Non-recursive: $O(n)$"
] | [
"Assuming that the computational cost of adding two integers is one unit, estimate the total computational cost of the recursive function $fibonacci(N)$ and the non-recursive function $fibonacci(N)$ in terms of $N$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 14
} | 16 | 0 | 0 |
omr-ws-952852dd792735c4906b880ec91b3322610743e280e5be9facc5cfbc21ded4aa | Assuming that the quarter earned 10% annual interest for the entire 103 year period, would it be worth more or less than $40,000? | [
"less than $40,000$"
] | [
"Assuming that the quarter earned 10% annual interest for the entire 103 year period, would it be worth more or less than $40,000?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-3f8caaf429ba11fb912cd8d586e48edce8e5a95dc92d35f043f2d304596ab274 | Assuming that the size of the fish population satisfies the logistic equation \( \frac{dP}{dt} = kP(1 - \frac{P}{K}) \), where the initial population \( P(0) = 400 \), the carrying capacity \( K = 8400 \), and the population doubles in the first year, determine the constant \( k \) and solve the equation to find an exp... | [
"P(t) = \\frac{8400}{1 + 20 \\left( \\frac{19}{40} \\right)^t}"
] | [
"Assuming that the size of the fish population satisfies the logistic equation \\( \\frac{dP}{dt} = kP(1 - \\frac{P}{K}) \\), where the initial population \\( P(0) = 400 \\), the carrying capacity \\( K = 8400 \\), and the population doubles in the first year, determine the constant \\( k \\) and solve the equation... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-0ec25d482ad2864e85f05044a23a9ca43303500ddae79d2a947c8f809a80c5b1 | Assuming that the top of the can is 3 times thicker than the sides, find the dimensions (radius \( r \), height \( h \)) that will minimize the surface area of this can. Imagine the top has 3 layers to it. | [
"h = 4r"
] | [
"Assuming that the top of the can is 3 times thicker than the sides, find the dimensions (radius \\( r \\), height \\( h \\)) that will minimize the surface area of this can. Imagine the top has 3 layers to it."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d01126db07044c8a74c4074d90a84933e890ef9ed5fc63ec4016fe1f2bc5f082 | Assuming that the top of the can is 3 times thicker than the sides, find the dimensions (radius \( r \), height \( h \)) that will minimize the surface area of this can. The top has 3 layers. | [
"h = 4r"
] | [
"Assuming that the top of the can is 3 times thicker than the sides, find the dimensions (radius \\( r \\), height \\( h \\)) that will minimize the surface area of this can. The top has 3 layers."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ae58a9ca88fe68b80d5c8ce58972d391c2f41e30269c50026ceb27cc08a03646 | Assuming the 14 goals scored in a match between Germany and Brazil could be scored in any order, what is the probability that the score was never tied after the first goal? | [
"$\\frac{1}{7}$"
] | [
"Assuming the 14 goals scored in a match between Germany and Brazil could be scored in any order, what is the probability that the score was never tied after the first goal?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b67c501020eaba191f2f6ec4ab68461e33e872a5416ca57de6979ecf638930b3 | Assuming the axiom of choice does not hold, does there exist a Riemann-Integrable bijection \( f : [0, 1] \to [0, 1] \) whose inverse is not Riemann-Integrable? | [
"No"
] | [
"Assuming the axiom of choice does not hold, does there exist a Riemann-Integrable bijection \\( f : [0, 1] \\to [0, 1] \\) whose inverse is not Riemann-Integrable?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 1
} | 14 | 0 | 0 |
omr-ws-3dabfaccbe4a2b3c46c0a08da9bf7059bcd4e33905a84b1fbcce521adf497d19 | Assuming the constraint $4x - 3y = 0$ is correct, find a second constraint to determine the specific coordinates $(x, y)$ of the point on segment $AB$ that is closest to $C$. | [
"\\(\\left( \\frac{36}{25}, \\frac{48}{25}, 0 \\right)\\)"
] | [
"Assuming the constraint $4x - 3y = 0$ is correct, find a second constraint to determine the specific coordinates $(x, y)$ of the point on segment $AB$ that is closest to $C$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c4fa8b2fc559166d04ad0c2d0e9b038baa7e6877d803957283fb40a4a936a26c | Assuming the end of the slide is on the ground, what is the tallest a slide can be to the nearest hundredth of a foot if the horizontal distance from the end of the slide to the base of the ladder is 6.5 feet, given that the length of the slide must be at least twice its height? | [
"3.75"
] | [
"Assuming the end of the slide is on the ground, what is the tallest a slide can be to the nearest hundredth of a foot if the horizontal distance from the end of the slide to the base of the ladder is 6.5 feet, given that the length of the slide must be at least twice its height?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1c7506b125c87094edc5207e791560d257130461de23898d40d596d3ac64aa76 | Assuming the number of goals per game is a Poisson variable based on the mean calculated in Problem 1, find the probability that: a) Exactly 3 goals are scored in the final match. b) More than 2 goals are scored in both semi-final matches. c) Exactly 2 goals are scored in two out of four quarter-final matches. d) Exact... | [
"0.144"
] | [
"Assuming the number of goals per game is a Poisson variable based on the mean calculated in Problem 1, find the probability that:\na) Exactly 3 goals are scored in the final match.\nb) More than 2 goals are scored in both semi-final matches.\nc) Exactly 2 goals are scored in two out of four quarter-final matches.\... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-f327351c3adeff42a673ad9e3cbd3c454b58576b6f16dc38137064ea74d042cf | Assuming the orbits of Venus and Earth are circular with the Sun at the center and in the same plane, with radii of 108,200,000 km and 149,600,000 km respectively, what is the maximum possible angle (to the nearest degree) between Venus and the Sun as viewed from Earth? | [
"46"
] | [
"Assuming the orbits of Venus and Earth are circular with the Sun at the center and in the same plane, with radii of 108,200,000 km and 149,600,000 km respectively, what is the maximum possible angle (to the nearest degree) between Venus and the Sun as viewed from Earth?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-27584b3b342449135603631bbfa05e51562e6d587b923497a20a24f8a6a7cc2d | Assuming the problem asks for the y-intercept of the tangent line of a circle centered at the origin with area $\pi$ and slope $2$, what is the difference between this answer and the correct value of $\tan(2\pi)$? | [
"\\(\\sqrt{5}\\)"
] | [
"Assuming the problem asks for the y-intercept of the tangent line of a circle centered at the origin with area $\\pi$ and slope $2$, what is the difference between this answer and the correct value of $\\tan(2\\pi)$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-46b39988a16654fa648cddfe30fa9b87592f1ff7da93aa0134f88e06a8312d77 | Assuming the tank is full at time \( t = 0 \), find the value of \( h(t) \) at \( t = 144\sqrt{6} \) if \( h(t) = \left( -\frac{1}{144} t + \sqrt{6} \right)^2 \). | [
"0"
] | [
"Assuming the tank is full at time \\( t = 0 \\), find the value of \\( h(t) \\) at \\( t = 144\\sqrt{6} \\) if \\( h(t) = \\left( -\\frac{1}{144} t + \\sqrt{6} \\right)^2 \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6d06e8f70efb099da348994ed9e5779ec3c8611a9f92c0053b90ae6fdd0e5295 | At $t = 0$ Tal starts walking in a line from the origin. He continues to walk forever at a rate of $1$ m/s and lives happily ever after. But Michael (who is Tal’s biggest fan) can’t bear to say goodbye. At $t = 10$ s he starts running on Tal’s path at a rate of $n$ such that $n > 1$ m/s. Michael runs to Tal, gives him ... | [
"\\dfrac{n + 1}{n - 1}"
] | [
"At $t = 0$ Tal starts walking in a line from the origin. He continues to walk forever at a rate of $1$ m/s and lives happily ever after. But Michael (who is Tal’s biggest fan) can’t bear to say goodbye. At $t = 10$ s he starts running on Tal’s path at a rate of $n$ such that $n > 1$ m/s. Michael runs to Tal, gives... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d5a83c5aca145bd9d260f92d928aa3a8954092a1da100f36159a867b1f0923e4 | At $t = 0$, there is exactly $1$ $K$-fish in a pond. After each second, each $K$-fish has a $\left( \frac{1}{2} \right)^n$ probability of turning into $n^2$ $K$-fish for integers $n \ge 1$. What is the expected number of $K$-fish in the pond after 5 seconds? | [
"7776"
] | [
"At $t = 0$, there is exactly $1$ $K$-fish in a pond. After each second, each $K$-fish has a $\\left( \\frac{1}{2} \\right)^n$ probability of turning into $n^2$ $K$-fish for integers $n \\ge 1$. What is the expected number of $K$-fish in the pond after 5 seconds?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c7e56f60e01e7997e4c1513ca055f07ddff06c2cd6f013b94d246d045a69ea5b | At 1 minute before 1 o'clock, we put the numbers from 1 to 10 in a vase and take out the number 1. At half a minute before 1 o'clock, we put the numbers from 11 to 20 in the vase and take out the number 2. At 1/3 of a minute before 1 o'clock, we put the numbers from 21 to 30 in the vase and take out the number 3, and s... | [
"0"
] | [
"At 1 minute before 1 o'clock, we put the numbers from 1 to 10 in a vase and take out the number 1. At half a minute before 1 o'clock, we put the numbers from 11 to 20 in the vase and take out the number 2. At 1/3 of a minute before 1 o'clock, we put the numbers from 21 to 30 in the vase and take out the number 3, ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1753c4b2871c3bdcf17357ef02d41812ea35e4c018488c31d582ad1f55580208 | At 11:00 a.m. how many degrees are in the smaller angle formed by the minute hand and the hour hand of the clock? | [
"30"
] | [
"At 11:00 a.m. how many degrees are in the smaller angle formed by the minute hand and the hour hand of the clock?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3a5932e30fdc279848ebde0438b98a51eecb4f14b755e6c6aefd49a93551a1f8 | At 12 o'clock, the three hands of a clock (hour, minute, second) are in one line. What is the first time after noon where the second hand is the angle bisector of the angle formed by the hour and minute hands? | [
"\\dfrac{86400}{1427}"
] | [
"At 12 o'clock, the three hands of a clock (hour, minute, second) are in one line. What is the first time after noon where the second hand is the angle bisector of the angle formed by the hour and minute hands?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-2feafd4449c2ac8e903aaedd9a61a6a9e93d8e09050dd701fa25089c65a7035d | At 12:00 PM, the temperature of an oven and a chicken inside it is 100°F. After 50 minutes, the oven's temperature reaches 200°F. Determine the temperature of the chicken at 12:50 PM using an initial value problem (IVP) differential equation method, excluding Laplace Transforms. | [
"136.8"
] | [
"At 12:00 PM, the temperature of an oven and a chicken inside it is 100°F. After 50 minutes, the oven's temperature reaches 200°F. Determine the temperature of the chicken at 12:50 PM using an initial value problem (IVP) differential equation method, excluding Laplace Transforms."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-6b43c462bdf06c75b978c2a9252da1dd590a37b5a867c7eaf3ab782d71685789 | At 12:00 noon, Anne, Beth, and Carmen begin running laps around a circular track of length 300 meters, all starting from the same point on the track. Each jogger maintains a constant speed in one of the two possible directions for an indefinite period of time. If Anne's speed is different from the other two speeds, cal... | [
"0",
"100"
] | [
"At 12:00 noon, Anne, Beth, and Carmen begin running laps around a circular track of length 300 meters, all starting from the same point on the track. Each jogger maintains a constant speed in one of the two possible directions for an indefinite period of time. If Anne's speed is different from the other two speeds... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 5
} | 10 | 0 | 0 |
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