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-122e1488772610c48fc45f491379d4ef45edf89397bee398d13fdeabafce92f9 | A pair of dice is rolled. The player wins immediately if the total is 7 or 11, and loses immediately if the total is 2, 3, or 12. Any other total is a "carry-over." If the first throw is a carry-over, the player continues rolling until they either win by rolling the same carry-over again or lose by rolling a 7. What is... | [
"\\( \\frac{244}{495} \\)"
] | [
"A pair of dice is rolled. The player wins immediately if the total is 7 or 11, and loses immediately if the total is 2, 3, or 12. Any other total is a \"carry-over.\" If the first throw is a carry-over, the player continues rolling until they either win by rolling the same carry-over again or lose by rolling a 7. ... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-93fcd5aa69deb56d5073b28a644962e1f17129c98966919be4fc1e43075042ca | A pair of dice is thrown. If it is known that one die shows a four, what is the probability that the other die shows a 5? | [
"\\( \\frac{1}{6} \\)"
] | [
"A pair of dice is thrown. If it is known that one die shows a four, what is the probability that the other die shows a 5?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-32384438cde3ab31959131e31f51d5668fdcd418a37f2a8164f720db1bd4a730 | A pair of dice is thrown. If it is known that one die shows a four, what is the probability that the total of both dice is greater than 7? | [
"\\( \\frac{1}{2} \\)"
] | [
"A pair of dice is thrown. If it is known that one die shows a four, what is the probability that the total of both dice is greater than 7?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-41422caa79d743d48907e697071a8ea90d4817a449542e36e4bbe98025487e9e | A pair of integers $(a, b)$ is called 2010-good if $a^2 + b^2$ is divisible by 2010. A pair is called very good if it is 2010-good and $a^3 + b^3$ is also divisible by 2010. Find the number of very good pairs $(a, b)$ such that $0 \leq a, b < 2010$. | [
"2"
] | [
"A pair of integers $(a, b)$ is called 2010-good if $a^2 + b^2$ is divisible by 2010. A pair is called very good if it is 2010-good and $a^3 + b^3$ is also divisible by 2010. Find the number of very good pairs $(a, b)$ such that $0 \\leq a, b < 2010$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4a8c1246be725eb38c864f21cb08a607f3bff26c199a4b72d7d41cb8dc61234a | A pair of integers $(m,n)$ satisfies $m|n^2+1$ and $n|m^2+1$. Describe all such pairs. | [
"\\((F_{2k}, F_{2k+2})\\) and \\((F_{2k+2}, F_{2k})\\) for \\(k \\geq 0\\)."
] | [
"A pair of integers $(m,n)$ satisfies $m|n^2+1$ and $n|m^2+1$. Describe all such pairs."
] | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 4
} | 4 | 0 | 0 |
omr-ws-b099e40cf9804e28ac8159f6e13c8ed68de59b3ec74238532eec9b8d306a6497 | A pair of integers $(m,n)$ satisfies $m|n^2+1$ and $n|m^2+1$. If the number of such pairs is finite, find all pairs; otherwise, describe a set of such pairs. | [
"\\((F_{2k-1}, F_{2k+1})\\) for \\(k \\geq 1\\) where \\(F_k\\) denotes the \\(k\\)-th Fibonacci number."
] | [
"A pair of integers $(m,n)$ satisfies $m|n^2+1$ and $n|m^2+1$. If the number of such pairs is finite, find all pairs; otherwise, describe a set of such pairs."
] | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 5
} | 5 | 0 | 0 |
omr-ws-2a75d3f6c137d6c317ba5558bbc7e1829767b3c560c2d44f9f5faeddb7b366af | A pair of integers \( (m, n) \) is called good if \( m \mid n^2 + n \) and \( n \mid m^2 + m \). | [
"(m, n) \\text{ where } m = n \\text{ or } m + n = -1"
] | [
"A pair of integers \\( (m, n) \\) is called good if \\( m \\mid n^2 + n \\) and \\( n \\mid m^2 + m \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 1
} | 9 | 0 | 0 |
omr-ws-93026f06ef7c0e5b1d71074d4d16ec5f6902a26ad761580a3d82ef2f6164b334 | A pair of natural numbers $(m, n)$ is said to be $good$ if there are natural numbers $k > 1$ that are divisible by $3^{2m} + 3^m + 1$ and $3^n - 1$. Find the number of $good$ pairs $(m, n)$ with $m, n \in \{1, 2, \ldots, 99, 100\}$. | [
"2211"
] | [
"A pair of natural numbers $(m, n)$ is said to be $good$ if there are natural numbers $k > 1$ that are divisible by $3^{2m} + 3^m + 1$ and $3^n - 1$. Find the number of $good$ pairs $(m, n)$ with $m, n \\in \\{1, 2, \\ldots, 99, 100\\}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 1
} | 11 | 0 | 0 |
omr-ws-67d217a2f73a8a511b6217b022a702688db2cfa72910c0371f7fa8b230655c53 | A pair of opposite vertices and the midpoints of a pair of opposite edges of a cube are connected to form a quadrilateral. If each edge of the cube has length $k$, find the area of the quadrilateral. | [
"\\(\\frac{k^2\\sqrt6}{2}\\)"
] | [
"A pair of opposite vertices and the midpoints of a pair of opposite edges of a cube are connected to form a quadrilateral. If each edge of the cube has length $k$, find the area of the quadrilateral."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 2,
"tir": 4
} | 8 | 0 | 0 |
omr-ws-244d3a4170a54658fade0b6572bf1bd4a365dd0b91a4fdf3e197690a6e4d85ec | A pair of positive integers $(a, b)$ is called special if there is an integer $k$ where $\frac{6a - 3b}{ab + 3} = k$ and $a + b \leq 2021$. Find the number of special pairs. | [
"680"
] | [
"A pair of positive integers $(a, b)$ is called special if there is an integer $k$ where $\\frac{6a - 3b}{ab + 3} = k$ and $a + b \\leq 2021$. Find the number of special pairs."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-e289d24615dd5f86d297fdac436c817184476e9dc56370288991490ea0e6f00c | A pair of positive integers $(x, y)$, where $x < y$, is a $k$-tangent pair if $\arctan \frac{1}{k} = \arctan \frac{1}{x} + \arctan \frac{1}{y}$. Compute the second largest integer that appears in a $2012$-tangent pair. | [
"811641"
] | [
"A pair of positive integers $(x, y)$, where $x < y$, is a $k$-tangent pair if $\\arctan \\frac{1}{k} = \\arctan \\frac{1}{x} + \\arctan \\frac{1}{y}$. Compute the second largest integer that appears in a $2012$-tangent pair."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b397044733786ad17a0a0087fcf6c591a169f141d51cd9413b6e8feda46795f3 | A pair of positive integers \( m, n \) is called [i]guerrera[/i] if there exist positive integers \( a, b, c, d \) such that \( m = ab \), \( n = cd \), and \( a + b = c + d \). We start by painting the numbers 3 and 5. If a positive integer \( x \) is not painted and there exists a painted positive integer \( y \) suc... | [
"\\(\\mathbb{Z}^{+} \\setminus \\{1,2\\}\\)"
] | [
"A pair of positive integers \\( m, n \\) is called [i]guerrera[/i] if there exist positive integers \\( a, b, c, d \\) such that \\( m = ab \\), \\( n = cd \\), and \\( a + b = c + d \\). We start by painting the numbers 3 and 5. If a positive integer \\( x \\) is not painted and there exists a painted positive in... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-461c3eb0d2ad20181ee88e9b38a424aae81829f06340ddb39fb873a8dd752f39 | A pair of positive integers \((a, b)\) is called an average couple if there exist positive integers \(k\) and \(c_1, c_2, \ldots, c_k\) such that \[ \frac{c_1 + c_2 + \cdots + c_k}{k} = a \quad \text{and} \quad \frac{s(c_1) + s(c_2) + \cdots + s(c_k)}{k} = b, \] where \(s(n)\) denotes the sum of the digits of \(n\) in ... | [
"11111111108888888889"
] | [
"A pair of positive integers \\((a, b)\\) is called an average couple if there exist positive integers \\(k\\) and \\(c_1, c_2, \\ldots, c_k\\) such that\n\\[\n\\frac{c_1 + c_2 + \\cdots + c_k}{k} = a \\quad \\text{and} \\quad \\frac{s(c_1) + s(c_2) + \\cdots + s(c_k)}{k} = b,\n\\]\nwhere \\(s(n)\\) denotes the sum... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 2
} | 7 | 0 | 0 |
omr-ws-46d034a041960f37a86844ff6a3a07e32a8fd5a3c6c44042e04cf54f435d1e3a | A pair of positive integers \((a, b)\) with \(a < 391\) is called "papusa" if \(\text{lcm}(a, b) > \text{lcm}(a, 391)\). Determine the smallest possible value of \(b\) among all "papusa" pairs. | [
"18"
] | [
"A pair of positive integers \\((a, b)\\) with \\(a < 391\\) is called \"papusa\" if \\(\\text{lcm}(a, b) > \\text{lcm}(a, 391)\\). Determine the smallest possible value of \\(b\\) among all \"papusa\" pairs."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-044d5be5c6f8c662e1c07a4f976d491ba24b6d2789e4a327be9f28ce215cfe34 | A pair of positive integers \((a, b)\) with \(a < 391\) is called "pupusa" if \(\text{lcm}(a, b) > \text{lcm}(a, 391)\). Find the minimum value of \(b\). | [
"18"
] | [
"A pair of positive integers \\((a, b)\\) with \\(a < 391\\) is called \"pupusa\" if \\(\\text{lcm}(a, b) > \\text{lcm}(a, 391)\\). Find the minimum value of \\(b\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2b1e45be0f56c2264cbf29667ecc47fed20d1bf73e5f73038b9810f8ffc27ca7 | A pair of positive integers \((m, n)\) is called a $skena$ pair if it satisfies \((m-n)^2 = pmn - 1\) for some prime number \(p\). Determine how many $skena$ pairs exist. | [
"0"
] | [
"A pair of positive integers \\((m, n)\\) is called a $skena$ pair if it satisfies \\((m-n)^2 = pmn - 1\\) for some prime number \\(p\\). Determine how many $skena$ pairs exist."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-53e70ce7db8b2b8a2731bf0d585a05887fcbf340eb3de2eb561d43b44b265fd4 | A pair of positive integers \((u, v)\) is called "wow" if \(u > v\) and \(u^t - v^t \equiv 1 \pmod{uv + 1}\) for all \(t \equiv \pm 1 \pmod{6}\). Find all "wow" pairs. | [
"\\((u, v) = (n+1, n)\\) for any positive integer \\(n\\)."
] | [
"A pair of positive integers \\((u, v)\\) is called \"wow\" if \\(u > v\\) and \\(u^t - v^t \\equiv 1 \\pmod{uv + 1}\\) for all \\(t \\equiv \\pm 1 \\pmod{6}\\). Find all \"wow\" pairs."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-3d323c7f28f3971d350e712f39a5ec47c97cfabc6a35e7f9be4001be96603553 | A pair of positive integers is called a 'good' pair if their quotient is either 2 or 3. What is the maximum number of good pairs that can be formed among \( n \) arbitrary different positive integers? | [
"\\left\\lfloor \\dfrac{3n - 4}{2} \\right\\rfloor"
] | [
"A pair of positive integers is called a 'good' pair if their quotient is either 2 or 3. What is the maximum number of good pairs that can be formed among \\( n \\) arbitrary different positive integers?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 4
} | 8 | 0 | 0 |
omr-ws-4a9515be12a8e6a58cbdc028540170f69970aded0bbf2bcd72be1337bf8b82de | A pair of positive integers is golden if they end in the same two digits. For example, $(139, 2739)$ and $(350, 850)$ are golden pairs. What is the sum of all two-digit integers $n$ for which $(n^2, n^3)$ is golden? | [
"772"
] | [
"A pair of positive integers is golden if they end in the same two digits. For example, $(139, 2739)$ and $(350, 850)$ are golden pairs. What is the sum of all two-digit integers $n$ for which $(n^2, n^3)$ is golden?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5a692d8d71bff2f297d991d924830a5dcb148b5193fa69284927947f6007ded7 | A pair of six-sided fair dice are labeled so that one die has only even numbers (two each of $2$, $4$, and $6$), and the other die has only odd numbers (two each of $1$, $3$, and $5$). What is the probability that the sum of the numbers on top of the two dice is $7$? | [
"\\(\\frac{1}{3}\\)"
] | [
"A pair of six-sided fair dice are labeled so that one die has only even numbers (two each of $2$, $4$, and $6$), and the other die has only odd numbers (two each of $1$, $3$, and $5$). What is the probability that the sum of the numbers on top of the two dice is $7$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 32
} | 47 | 0 | 0 |
omr-ws-1ac7661e11bb3dad60b69680f1d30a4bcf5ac2d8b70b2c7cbaa6cac6eca13357 | A pair of standard 6-sided fair dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference? | [
"\\(\\frac{1}{12}\\)"
] | [
"A pair of standard 6-sided fair dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 62
} | 94 | 0 | 0 |
omr-ws-59385bb0a411c39856345703be0f31da1aa542ec717b86ac962a6889ff07b646 | A pair of twin primes is two primes that differ by 2. Denote a cornie as a number that is part of more than one pair of twin primes. Find the absolute difference between the number of cornies and the sum of all cornies. | [
"4"
] | [
"A pair of twin primes is two primes that differ by 2. Denote a cornie as a number that is part of more than one pair of twin primes. Find the absolute difference between the number of cornies and the sum of all cornies."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2358219624f1d0051800cde13daf639e4daa3606ed2120aee9ef54bed33cfa6d | A pair of unbiased dice are rolled together until a sum of either 5 or 7 is obtained. What is the probability that a sum of 5 comes before a sum of 7? | [
"\\(\\frac{2}{5}\\)"
] | [
"A pair of unbiased dice are rolled together until a sum of either 5 or 7 is obtained. What is the probability that a sum of 5 comes before a sum of 7?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-69962b4166e868bd57cf79086142254f32db853955dfabc1803b6e1557332512 | A pair of whole numbers $(a,b)$ is called complementary if $a+b$ is a power of $2$ $(1,2,4,8,\text{etc.})$. John partitions the integers from $0$ to $2011$, inclusive, into two sets $X$ and $Y$ such that no two elements in $X$ are complementary, and no two elements in $Y$ are complementary. Given that $0$ is in $X$, ho... | [
"1006"
] | [
"A pair of whole numbers $(a,b)$ is called complementary if $a+b$ is a power of $2$ $(1,2,4,8,\\text{etc.})$. John partitions the integers from $0$ to $2011$, inclusive, into two sets $X$ and $Y$ such that no two elements in $X$ are complementary, and no two elements in $Y$ are complementary. Given that $0$ is in $... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 4
} | 13 | 0 | 0 |
omr-ws-d6083dd321a657d1a5932a4be398b8bf18162fa5a63d6658216d93e6095780b0 | A pair of words consisting only of the letters $a$ and $b$ (with repetitions) is defined as [i]good[/i] if it is $(a,b)$ or of one of the forms $(uv, v)$, $(u, uv)$, where $(u,v)$ is a good pair. For a good pair $(\alpha, \beta)$, let $\gamma$ be the shortest palindrome such that $\alpha\beta = a\gamma b$. Find the len... | [
"40"
] | [
"A pair of words consisting only of the letters $a$ and $b$ (with repetitions) is defined as [i]good[/i] if it is $(a,b)$ or of one of the forms $(uv, v)$, $(u, uv)$, where $(u,v)$ is a good pair. For a good pair $(\\alpha, \\beta)$, let $\\gamma$ be the shortest palindrome such that $\\alpha\\beta = a\\gamma b$. F... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-0e500d01f4ddf008e55806891710df1aad7f478c5fa666d1a9df921cc3693c50 | A palace is divided into a $2003 \times 2003$ grid of square rooms. Someone enters the palace from the northwest corner, visits each room except the southeast corner 100 times, and returns to the northwest corner for the first time before leaving. How many times did this person visit the room at the southeast corner? | [
"99"
] | [
"A palace is divided into a $2003 \\times 2003$ grid of square rooms. Someone enters the palace from the northwest corner, visits each room except the southeast corner 100 times, and returns to the northwest corner for the first time before leaving. How many times did this person visit the room at the southeast cor... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 5,
"tir": 6
} | 11 | 0 | 0 |
omr-ws-1f1ade4106bb015c3f83766b0b9b09f91a1deb7e03dcd4e892680e818f78918f | A palindrome between $1000$ and $10,000$ is chosen at random. What is the probability that it is divisible by $7$? | [
"\\(\\frac{1}{5}\\)"
] | [
"A palindrome between $1000$ and $10,000$ is chosen at random. What is the probability that it is divisible by $7$?"
] | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-76477346a4a7eb77f53b1631a275528bff0d61d42696ad4a8a10836003e7bead | A palindrome between 1000 and 10,000 is chosen at random. What is the probability that it is divisible by 7? | [
"\\dfrac{1}{5}"
] | [
"A palindrome between 1000 and 10,000 is chosen at random. What is the probability that it is divisible by 7?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e8ac34e9fed3ac9d80bdac0b4e040dabeb01bbd86b8e2a748fbbf7ae1254948a | A palindrome is a number that reads the same forward and backward. How many three-digit perfect squares are palindromes? | [
"3"
] | [
"A palindrome is a number that reads the same forward and backward. How many three-digit perfect squares are palindromes?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5bbf5471cc9ee63e04392c027cc61f67a795a617bb19d1917237a2650ec677ea | A palindrome is a number that reads the same forward and backward. What is the smallest 5-digit palindrome in base 2 that can be expressed as a 3-digit palindrome in a different base? Give your response in base 2. | [
"10001_2"
] | [
"A palindrome is a number that reads the same forward and backward. What is the smallest 5-digit palindrome in base 2 that can be expressed as a 3-digit palindrome in a different base? Give your response in base 2."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-aa3dec436087d558f955eed6f5f844d4651169bad633d225e19c543111918080 | A palindrome is a number that reads the same forward as backwards. Suppose that \( N \) is the maximal possible difference between two consecutive three-digit palindromes. Find the number of pairs of consecutive palindromes \( (A, B) \) satisfying \( A < B \) and \( B - A = N \). | [
"8"
] | [
"A palindrome is a number that reads the same forward as backwards. Suppose that \\( N \\) is the maximal possible difference between two consecutive three-digit palindromes. Find the number of pairs of consecutive palindromes \\( (A, B) \\) satisfying \\( A < B \\) and \\( B - A = N \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cdb1608dd8128c8de4db0e63231fc13fa1e4016b3af07f6e12838cfdb1b00f7e | A palindrome is a number that reads the same forwards and backwards, such as 23432. If \(0 < n < 100,000\) and \(n\) is an integer, compute the probability \(p/q\) in simplest form that \(n\) is a palindrome. | [
"\\(\\frac{122}{11111}\\)"
] | [
"A palindrome is a number that reads the same forwards and backwards, such as 23432. If \\(0 < n < 100,000\\) and \\(n\\) is an integer, compute the probability \\(p/q\\) in simplest form that \\(n\\) is a palindrome."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-9b0c52cdc6557843126266c5d3dbb40de0318a2968cb95100c162e628c19c9d0 | A palindrome is a number that reads the same forwards and backwards, such as 3003. How many positive four-digit integers are palindromes? | [
"90"
] | [
"A palindrome is a number that reads the same forwards and backwards, such as 3003. How many positive four-digit integers are palindromes?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b8038f46013b9f663f02ee12be0b65fde9d9a0710a26ec29543fc7e95c29cc0b | A palindrome is a number that reads the same forwards and backwards. For example, $1$, $11$, and $141$ are all palindromes. How many palindromes between $1$ and $1000$ are divisible by $11$? | [
"107"
] | [
"A palindrome is a number that reads the same forwards and backwards. For example, $1$, $11$, and $141$ are all palindromes. How many palindromes between $1$ and $1000$ are divisible by $11$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fa2009891c6c3744ad4e9f1ca8cc0c41fd9154a99c60df0ff17a6beec81ba227 | A palindrome is a positive integer which is the same when read forwards or backwards. Find the sum of the two smallest palindromes that are multiples of $137$. | [
"10960"
] | [
"A palindrome is a positive integer which is the same when read forwards or backwards. Find the sum of the two smallest palindromes that are multiples of $137$."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-b12456556e30eec3a7a09d5c740d2a179f68514f6ddc9402f95be58146a6aed3 | A palindrome is an integer that reads the same forward and backward, such as 3663. What percent of the palindromes between 100 and 500 contain at least one 5? | [
"10\\%"
] | [
"A palindrome is an integer that reads the same forward and backward, such as 3663. What percent of the palindromes between 100 and 500 contain at least one 5?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-50801bb4214bf3b024b88ce80ae3108edbabbd922b28e14b44e032069c37256c | A pan with dimensions 25 cm by 20 cm by 15 cm is filled with water to a depth of 3 cm. Lead cubes with an edge length of 4 cm are placed flat on the bottom. Determine the value of \( n \) such that when the \( n \)-th cube is placed in the pan, all cubes are completely covered by water for the first time. | [
"8"
] | [
"A pan with dimensions 25 cm by 20 cm by 15 cm is filled with water to a depth of 3 cm. Lead cubes with an edge length of 4 cm are placed flat on the bottom. Determine the value of \\( n \\) such that when the \\( n \\)-th cube is placed in the pan, all cubes are completely covered by water for the first time."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 2
} | 10 | 0 | 0 |
omr-ws-d358cc559f2ebb31e5c3194d1368eb20190a6f1f4c7b872805d21cd59c82c82f | A paniolo is 4 miles south of the Wailuku River, which flows due east. He is also 8 miles west and 7 miles north of his cabin. If he wants to water his horse at the river and return home, what is the shortest distance he can travel? | [
"17 miles"
] | [
"A paniolo is 4 miles south of the Wailuku River, which flows due east. He is also 8 miles west and 7 miles north of his cabin. If he wants to water his horse at the river and return home, what is the shortest distance he can travel?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-16bac9230098e9faa159e3ec30bc34ea561139f32915fd32219576be943ed94c | A paper drinking cup in the shape of a cone with height \( h \) and semivertical angle \( \theta \) is filled with water. A ball is placed carefully in the cup, causing water to overflow. What is the radius of the ball that causes the greatest volume of water to spill out of the cup? | [
"\\( \\frac{h \\sin(\\theta)}{3} \\)"
] | [
"A paper drinking cup in the shape of a cone with height \\( h \\) and semivertical angle \\( \\theta \\) is filled with water. A ball is placed carefully in the cup, causing water to overflow. What is the radius of the ball that causes the greatest volume of water to spill out of the cup?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-277eed98d7d6f8835695e2721cdbe905f012bbb96bb0140c47771cd0eb786630 | A paper rectangle $ABCD$ has dimensions $AB = 3$ and $BC = 9$. The rectangle is folded such that vertices $A$ and $C$ coincide. What is the area of the resulting pentagon? | [
"\\(\\frac{39}{2}\\)"
] | [
"A paper rectangle $ABCD$ has dimensions $AB = 3$ and $BC = 9$. The rectangle is folded such that vertices $A$ and $C$ coincide. What is the area of the resulting pentagon?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 7,
"tir": 5
} | 25 | 0 | 0 |
omr-ws-eb4c5a0e3e7c357d2225767c66310fd7e1a19bc029bf41479029d31b78200915 | A paper square is folded such that one vertex comes to a side not containing this vertex, forming three triangles. Three circles are inscribed in these triangles. If the radii of the inscribed circles are $r_1$, $r_2$, and $r_3$, find the value of $\frac{r_1 + r_2 + r_3}{r_1}$ given that $r_1 = r_2 + r_3$. | [
"2"
] | [
"A paper square is folded such that one vertex comes to a side not containing this vertex, forming three triangles. Three circles are inscribed in these triangles. If the radii of the inscribed circles are $r_1$, $r_2$, and $r_3$, find the value of $\\frac{r_1 + r_2 + r_3}{r_1}$ given that $r_1 = r_2 + r_3$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0ee4c497461750638413b98d553fd6ceaea7a1308ba9f8c77945fdb1607df02f | A paper triangle with sides \(a\), \(b\), and \(c\) is folded such that the vertex opposite side \(c\) touches side \(c\). In the resulting quadrilateral, two angles are equal to the fold line. Find the lengths of the segments into which the vertex divides side \(c\). | [
"\\(\\frac{ac}{a+b}\\) and \\(\\frac{bc}{a+b}\\)"
] | [
"A paper triangle with sides \\(a\\), \\(b\\), and \\(c\\) is folded such that the vertex opposite side \\(c\\) touches side \\(c\\). In the resulting quadrilateral, two angles are equal to the fold line. Find the lengths of the segments into which the vertex divides side \\(c\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 2
} | 4 | 0 | 0 |
omr-ws-ca4ec019fd476abd15890609c14c653ed3e133b5c88529229eaa9e63714d8c0c | A paper triangle with sides of lengths 3, 4, and 5 inches is folded so that point \( A \) falls on point \( B \). What is the length of the crease? | [
"\\(\\frac{15}{8}\\)",
"\\dfrac{15}{8}"
] | [
"A paper triangle with sides of lengths 3, 4, and 5 inches is folded so that point \\( A \\) falls on point \\( B \\). What is the length of the crease?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 58,
"genselect": 4,
"tir": 25
} | 87 | 0 | 0 |
omr-ws-fbb24b4ae35b2478670c7f1fbeb2dd29b483bac11ff0faf762571fe288fc6d3e | A paper triangle with sides of lengths 3, 4, and 5 inches is folded so that point \(A\) falls on point \(B\). What is the length in inches of the crease? | [
"\\(\\frac{15}{8}\\)"
] | [
"A paper triangle with sides of lengths 3, 4, and 5 inches is folded so that point \\(A\\) falls on point \\(B\\). What is the length in inches of the crease?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 41,
"genselect": 5,
"tir": 22
} | 68 | 0 | 0 |
omr-ws-e2472d4e8e33a24f4d23d172306db0ddc5cc7450b97f143d7133f2581e528ce0 | A paper with dimensions $15 \times 17$ is to be cut into 7 unique squares with different areas, leaving the smallest possible remaining piece. Determine the side lengths of the 7 unique squares. | [
"10, 8, 7, 5, 3, 2, 1"
] | [
"A paper with dimensions $15 \\times 17$ is to be cut into 7 unique squares with different areas, leaving the smallest possible remaining piece. Determine the side lengths of the 7 unique squares."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 4
} | 7 | 0 | 0 |
omr-ws-eac962e7e49e82992fb5eb47ee63262e247e1add5efe63fc13bcbf862ae5da96 | A paperboy delivers newspapers to 10 houses along Main Street. He never misses three consecutive houses. Compute the number of ways the paperboy could deliver papers in this manner. | [
"274"
] | [
"A paperboy delivers newspapers to 10 houses along Main Street. He never misses three consecutive houses. Compute the number of ways the paperboy could deliver papers in this manner."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-27ae5fda5e92763c82e3fa6204f7f8abb1449870cccbc45aa616223b80df346d | A paperboy delivers newspapers to 10 houses along Main Street. Wishing to save effort, he doesn't always deliver to every house, but to avoid being fired he never misses three consecutive houses. Compute the number of ways the paperboy could deliver papers in this manner. | [
"504"
] | [
"A paperboy delivers newspapers to 10 houses along Main Street. Wishing to save effort, he doesn't always deliver to every house, but to avoid being fired he never misses three consecutive houses. Compute the number of ways the paperboy could deliver papers in this manner.",
"A paperboy delivers newspapers to 10 ... | [
"MATH_training_set",
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 32
} | 32 | 0 | 0 |
omr-ws-95fe5badc6e1569ec0ce4bca0216343f945024f42618a7204b16b26fc3cc49a9 | A paperboy’s route covers 8 consecutive houses along a road. He does not necessarily deliver to all the houses every day, but he always traverses the road in the same direction, and he takes care never to skip over 2 consecutive houses. How many possible routes can he take? | [
"55"
] | [
"A paperboy’s route covers 8 consecutive houses along a road. He does not necessarily deliver to all the houses every day, but he always traverses the road in the same direction, and he takes care never to skip over 2 consecutive houses. How many possible routes can he take?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0ad88e4077a370082b329917ec7c004df5043756d2335135deb6041f0fa4c6eb | A parabola $C$ is given parametrically by $x = at^2$ and $y = 2at$ $(a > 0)$. Find the equation in terms of $t$ which must be satisfied by points on $C$ at which the normal passes through the point $(h, k)$. Let $N(h, k)$ be the number of normals to $C$ passing through $(h, k)$. Find the value of $N(h, k)$ when $h = 2a... | [
"1"
] | [
"A parabola $C$ is given parametrically by $x = at^2$ and $y = 2at$ $(a > 0)$. Find the equation in terms of $t$ which must be satisfied by points on $C$ at which the normal passes through the point $(h, k)$. Let $N(h, k)$ be the number of normals to $C$ passing through $(h, k)$. Find the value of $N(h, k)$ when $h... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7aceef47b8258c91e88efdcce3510118cefe173adb2a0f6196b48c97521fb85f | A parabola $P: y=x^2$ is rotated $135$ degrees clockwise about the origin to $P'$. This image is translated upward $\frac{8+\sqrt{2}}{2}$ to $P''$. Point $A: (0,0)$, $B: (256,0)$, and $C$ is in Quadrant I, on $P''$. If the area of $\triangle ABC$ is at a maximum, it is $a\sqrt{b}+c$ where $a$, $b$, and $c$ are integers... | [
"594"
] | [
"A parabola $P: y=x^2$ is rotated $135$ degrees clockwise about the origin to $P'$. This image is translated upward $\\frac{8+\\sqrt{2}}{2}$ to $P''$. Point $A: (0,0)$, $B: (256,0)$, and $C$ is in Quadrant I, on $P''$. If the area of $\\triangle ABC$ is at a maximum, it is $a\\sqrt{b}+c$ where $a$, $b$, and $c$ are... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-76d09a6668bbfceee45ef97afca90fa66a7f734184a3d1b1f08b8c5fa062fab0 | A parabola $P_1$ with equation $x^2 - 2py = 0$ and a parabola $P_2$ with equation $x^2 + 2py = 0$ are given, where $p > 0$. A line $t$ is tangent to $P_2$. Find the locus of the pole $M$ of the line $t$ with respect to $P_1$. | [
"\\( x^2 + 2py = 0 \\)"
] | [
"A parabola $P_1$ with equation $x^2 - 2py = 0$ and a parabola $P_2$ with equation $x^2 + 2py = 0$ are given, where $p > 0$. A line $t$ is tangent to $P_2$. Find the locus of the pole $M$ of the line $t$ with respect to $P_1$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 46,
"genselect": 8,
"tir": 16
} | 70 | 0 | 0 |
omr-ws-1a5374e4c0fe7fb355f231c433793557fd89df7c0c8e2cbff5835b048ed98031 | A parabola $P_1$ with equation $x^2 - 2py = 0$ and parabola $P_2$ with equation $x^2 + 2py = 0, p > 0$, are given. A line $t$ is tangent to $P_2.$ Find the locus of pole $M$ of the line $t$ with respect to $P_1.$ | [
"x^2 + 2py = 0"
] | [
"A parabola $P_1$ with equation $x^2 - 2py = 0$ and parabola $P_2$ with equation $x^2 + 2py = 0, p > 0$, are given. A line $t$ is tangent to $P_2.$ Find the locus of pole $M$ of the line $t$ with respect to $P_1.$"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-516fc338defacdf61b0b0dd716692de40d8c2eea00c559682945b6a27ffd5a0a | A parabola $\mathcal{P}$ with focus $F,$ vertex $V,$ and directrix $l$ is defined on the plane. Points $X$ and $Y$ are defined on $\mathcal{P}$ such that $X, F,$ and $Y$ are collinear. Circle $\omega$ is centered on $l$ and is tangent to $XF$ at $X.$ Let $Z$ be the antipode of $X$ with respect to $\omega,$ and let $Q$ ... | [
"5"
] | [
"A parabola $\\mathcal{P}$ with focus $F,$ vertex $V,$ and directrix $l$ is defined on the plane. Points $X$ and $Y$ are defined on $\\mathcal{P}$ such that $X, F,$ and $Y$ are collinear. Circle $\\omega$ is centered on $l$ and is tangent to $XF$ at $X.$ Let $Z$ be the antipode of $X$ with respect to $\\omega,$ a... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 5
} | 13 | 0 | 0 |
omr-ws-1c8924e0e83b391b155ea11137f4030da1342f58f9f90166124eb50e3f27ae17 | A parabola $y = x^2 - a$ meets the right branch of the hyperbola $y = 1/x$ at point $A$, and meets its left branch at points $B$ and $C$. Find all possible values of $a$ if the triangle $ABC$ is a right triangle. | [
"-2"
] | [
"A parabola $y = x^2 - a$ meets the right branch of the hyperbola $y = 1/x$ at point $A$, and meets its left branch at points $B$ and $C$. Find all possible values of $a$ if the triangle $ABC$ is a right triangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6cc0ec495b346d5c22c5abdd882cea5aa11f33e9cb8df5893f5becb0a44d3f87 | A parabola $y = x^2 - a$ meets the right branch of the hyperbola $y = 1/x$ at point $A$, and meets its left branch at points $B$ and $C$. Find the area of the right triangle $ABC$ for all possible values of $a$. | [
"\\(\\frac{\\sqrt{5}}{2}\\)"
] | [
"A parabola $y = x^2 - a$ meets the right branch of the hyperbola $y = 1/x$ at point $A$, and meets its left branch at points $B$ and $C$. Find the area of the right triangle $ABC$ for all possible values of $a$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 5,
"tir": 5
} | 14 | 0 | 0 |
omr-ws-9824e562b44b256e4023584a13c0408b08148dcbbe65c00ec6e79b13f640f02c | A parabola $y = x^2$ in the $xy$-plane is rotated $180^{\circ}$ about a point $(a, b)$. The resulting parabola has roots at $x = 40$ and $x = 48$. Compute $a + b$. | [
"30"
] | [
"A parabola $y = x^2$ in the $xy$-plane is rotated $180^{\\circ}$ about a point $(a, b)$. The resulting parabola has roots at $x = 40$ and $x = 48$. Compute $a + b$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-1c5974e2037cbe88b80f424c1185a711a19e18cd39dc7c844ae2519679c2531a | A parabola \( P: y = x^2 \) is rotated 135 degrees clockwise about the origin to \( P' \), then translated upward by \( \frac{8+\sqrt{2}}{2} \) to \( P'' \). Point \( A: (0,0) \), \( B: (256,0) \), and \( C \) is in Quadrant I on \( P'' \). If the area of \( \triangle ABC \) is at a maximum, it is \( a\sqrt{b} + c \) w... | [
"594"
] | [
"A parabola \\( P: y = x^2 \\) is rotated 135 degrees clockwise about the origin to \\( P' \\), then translated upward by \\( \\frac{8+\\sqrt{2}}{2} \\) to \\( P'' \\). Point \\( A: (0,0) \\), \\( B: (256,0) \\), and \\( C \\) is in Quadrant I on \\( P'' \\). If the area of \\( \\triangle ABC \\) is at a maximum, i... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 46,
"genselect": 8
} | 54 | 0 | 0 |
omr-ws-637e188ae321e5fb45146a0a7ade51e314373a0e54b77653548e7a1e2bfe4a84 | A parabola \( p \) is given by the equation \( y = -x^2 \). A point \( A \) with coordinates \( (2, 3) \) is marked that does not lie on \( p \). All possible parabolas \( q \) of the form \( y = x^2 + ax + b \) are drawn through point \( A \), intersecting \( p \) at two points \( X \) and \( Y \). Find the coordinate... | [
"(2, -\\dfrac{1}{2})"
] | [
"A parabola \\( p \\) is given by the equation \\( y = -x^2 \\). A point \\( A \\) with coordinates \\( (2, 3) \\) is marked that does not lie on \\( p \\). All possible parabolas \\( q \\) of the form \\( y = x^2 + ax + b \\) are drawn through point \\( A \\), intersecting \\( p \\) at two points \\( X \\) and \\(... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-56a085390658e1130d8d1faec120687943c04242c15f28a56497213b9c6efa57 | A parabola \( y^2 = -12x \) rolls without slipping around the parabola \( y^2 = 12x \). Find the locus of the vertex of the rolling parabola. | [
"\\( x^3 = (x+6)y^2 \\)"
] | [
"A parabola \\( y^2 = -12x \\) rolls without slipping around the parabola \\( y^2 = 12x \\). Find the locus of the vertex of the rolling parabola."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 4,
"tir": 2
} | 8 | 0 | 0 |
omr-ws-43b8b7a5a88ebcf5679450fd3698435da5604772ed80e9b977d6fbcb3fb2f162 | A parabola and a hyperbola intersect at three points \( A, B, C \). The axis of the parabola is the asymptote of the hyperbola. Let \( G \) be the intersection point of the medians of the triangle \( ABC \). If the equation of the axis of the parabola is \( x = k \), find the value of \( k \) given that the coordinates... | [
"3"
] | [
"A parabola and a hyperbola intersect at three points \\( A, B, C \\). The axis of the parabola is the asymptote of the hyperbola. Let \\( G \\) be the intersection point of the medians of the triangle \\( ABC \\). If the equation of the axis of the parabola is \\( x = k \\), find the value of \\( k \\) given that ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0ed998248a639decc2af41236cd390c5aee9f3315e081d8cda947b95804d41e8 | A parabola and an ellipse share a focus, and the directrix of the parabola is the line containing the minor axis of the ellipse. The parabola and ellipse intersect at two points. Given that the equation of the ellipse is $\frac{x^2}{25} + \frac{y^2}{9} = 1,$ find the distance between those two points. | [
"\\tfrac{4\\sqrt{14}}{3}"
] | [
"A parabola and an ellipse share a focus, and the directrix of the parabola is the line containing the minor axis of the ellipse. The parabola and ellipse intersect at two points. Given that the equation of the ellipse is $\\frac{x^2}{25} + \\frac{y^2}{9} = 1,$ find the distance between those two points."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-89a6e9488975814c688e75ed5f1712c6cf27cc708f30a5c4476c521afcaa7e9b | A parabola given by $y = x^2$ is rotated around the $y$-axis to form a paraboloid. A cut at the plane $y = a$ leaves a section of the paraboloid with a volume of $2016\pi$ below the plane. What is the value of $a$? Express your answer in simplest form. | [
"\\( 24\\sqrt{7} \\)"
] | [
"A parabola given by $y = x^2$ is rotated around the $y$-axis to form a paraboloid. A cut at the plane $y = a$ leaves a section of the paraboloid with a volume of $2016\\pi$ below the plane. What is the value of $a$? Express your answer in simplest form."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-dde3206f83df66e0ef947e80c2db3dba17e1a935a82b0be92744c0b9e36e522d | A parabola goes through the points $(1,5)$, $(10,0)$, and $(a,b)$. If $(a,b)$ is the vertex, find $a$ as a function of $b$. | [
"\\( a = \\frac{-9b+50\\pm9\\sqrt{b^2-5b}}{5} \\)"
] | [
"A parabola goes through the points $(1,5)$, $(10,0)$, and $(a,b)$. If $(a,b)$ is the vertex, find $a$ as a function of $b$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-13ab412cae49eb04a5dfb6675a22dddd18d5f44b1dfa1fba7309666092da092d | A parabola has a latus rectum of length $4a$ and is tangential to both the coordinate axes. Find the locus of its focus. | [
"\\( y^2 = \\frac{a^2 x^2}{x^2 - a^2} \\)"
] | [
"A parabola has a latus rectum of length $4a$ and is tangential to both the coordinate axes. Find the locus of its focus."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-592e621e3817ed206190394238289687c3b47be9fd1d5ecc57d7c1f321b7c3d7 | A parabola has a turning point at \( (z, -8) \), intersects the y-axis at \( y = 10 \), and has one x-intercept at \( x = 5 \). Find the value of \( z \) and the equation of the parabola. | [
"y = \\frac{2}{25}(x - 15)^2 - 8"
] | [
"A parabola has a turning point at \\( (z, -8) \\), intersects the y-axis at \\( y = 10 \\), and has one x-intercept at \\( x = 5 \\). Find the value of \\( z \\) and the equation of the parabola."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-d9a47dbef8f14300996b931a74d05a5c562149419a83efa1b3c57f6f7916bcbb | A parabola has a vertex at \((4, -5)\) and the equation \(y = ax^2 + bx + c\). If \(a\) is positive, what is the sign of \(c\)? | [
"positive"
] | [
"A parabola has a vertex at \\((4, -5)\\) and the equation \\(y = ax^2 + bx + c\\). If \\(a\\) is positive, what is the sign of \\(c\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 3
} | 16 | 0 | 0 |
omr-ws-f3fa0a366fdc2417e9727f7e91aad5269b80db0f43816c4d06f6f5e606b554b7 | A parabola has a vertex at \((a, b)\) and passes through another point \((c, d)\). Find the equation of the parabola. | [
"\\( y = \\frac{d - b}{(c - a)^2} x^2 - \\frac{2a(d - b)}{(c - a)^2} x + \\frac{a^2(d - b)}{(c - a)^2} + b \\)"
] | [
"A parabola has a vertex at \\((a, b)\\) and passes through another point \\((c, d)\\). Find the equation of the parabola."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-45f99cec5055a4fe4049c25ebc1b6e5fccdc8e2f37017399387d2c1c701be219 | A parabola has focus $(3,3)$ and directrix $3x + 7y = 21.$ Express the equation of the parabola in the form \[ax^2 + bxy + cy^2 + dx + ey + f = 0,\]where $a,$ $b,$ $c,$ $d,$ $e,$ $f$ are integers, $a$ is a positive integer, and $\gcd(|a|,|b|,|c|,|d|,|e|,|f|) = 1.$ | [
"49x^2 - 42xy + 9y^2 - 222x - 54y + 603 = 0"
] | [
"A parabola has focus $(3,3)$ and directrix $3x + 7y = 21.$ Express the equation of the parabola in the form\n\\[ax^2 + bxy + cy^2 + dx + ey + f = 0,\\]where $a,$ $b,$ $c,$ $d,$ $e,$ $f$ are integers, $a$ is a positive integer, and $\\gcd(|a|,|b|,|c|,|d|,|e|,|f|) = 1.$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e72f388b624ac6a84e4e7781b3f49aad48ea988e57da6738a388963138f5671c | A parabola has focus $F$ and vertex $V$ with $VF = 10$. Let $AB$ be a chord of length $100$ that passes through $F$. Determine the area of $\triangle VAB$. | [
"$100\\sqrt{10}$"
] | [
"A parabola has focus $F$ and vertex $V$ with $VF = 10$. Let $AB$ be a chord of length $100$ that passes through $F$. Determine the area of $\\triangle VAB$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-6462886c8e743ee048a7283e9301ac3e8d1529cc435a6ef81f1781f7db5c2912 | A parabola has focus $F$ and vertex $V$. There are two equilateral triangles that have two vertices on the parabola and the remaining vertex on $F$. The ratio of the areas of these two triangles can be expressed as $m+n\sqrt{p}$, where $m$ and $n$ are positive integers and $p$ is a prime. Compute $m+n+p$. | [
"156"
] | [
"A parabola has focus $F$ and vertex $V$. There are two equilateral triangles that have two vertices on the parabola and the remaining vertex on $F$. The ratio of the areas of these two triangles can be expressed as $m+n\\sqrt{p}$, where $m$ and $n$ are positive integers and $p$ is a prime. Compute $m+n+p$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-e00c7134616e8591e4354fae04e139f5563a893a8ee954e22eea595bb58b01bf | A parabola has its focus at the point $(4,0)$ and its directrix is the line $y=-4$. A beam of light travels right to left, parallel to the $x$-axis, and strikes the parabola where $x=8$. At what point will the reflected beam contact the parabola again? | [
"(0, 0)"
] | [
"A parabola has its focus at the point $(4,0)$ and its directrix is the line $y=-4$. A beam of light travels right to left, parallel to the $x$-axis, and strikes the parabola where $x=8$. At what point will the reflected beam contact the parabola again?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 4
} | 14 | 0 | 0 |
omr-ws-36a3126d1bcd4213ef40aa9d485ff0fdffe6814d2fba963c1f91b1236ba9c2c3 | A parabola has its minimum at (0, 0) and passes through the point A(2, 4). An equilateral triangle is inscribed inside this parabola. Find the area of the equilateral triangle. | [
"$\\sim 8.055948494$ or $\\sim 6.550345684$"
] | [
"A parabola has its minimum at (0, 0) and passes through the point A(2, 4). An equilateral triangle is inscribed inside this parabola. Find the area of the equilateral triangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-82e84e0db129fb23923e58fc5a27668042d589695cbf657f1eedc51c9453c824 | A parabola has vertex $V = (0,0)$ and focus $F = (0,1).$ Let $P$ be a point in the first quadrant, lying on the parabola, so that $PF = 101.$ Find $P.$ | [
"(20,100)"
] | [
"A parabola has vertex $V = (0,0)$ and focus $F = (0,1).$ Let $P$ be a point in the first quadrant, lying on the parabola, so that $PF = 101.$ Find $P.$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3fc5acc785fcdedc65812f692972442ec86a0d46c37b60bb07db1a330ab14094 | A parabola in the Cartesian plane is tangent to the $x$-axis at $(1,0)$ and to the $y$-axis at $(0,3)$. The sum of the coordinates of the vertex of the parabola can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$. | [
"46"
] | [
"A parabola in the Cartesian plane is tangent to the $x$-axis at $(1,0)$ and to the $y$-axis at $(0,3)$. The sum of the coordinates of the vertex of the parabola can be written as $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 29,
"genselect": 5,
"tir": 14
} | 48 | 0 | 0 |
omr-ws-9769e47f978024e340d733ab34f49cad8d0a57aedd2fd2417cda1cddc2f894e0 | A parabola is inscribed in a unit square such that it is tangent to two sides. What is the area of the shaded region formed by the parabola and the square? | [
"\\(\\frac{1}{6}\\)"
] | [
"A parabola is inscribed in a unit square such that it is tangent to two sides. What is the area of the shaded region formed by the parabola and the square?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 1,
"tir": 4
} | 6 | 0 | 0 |
omr-ws-e92f5288f3b9be49779b7abd34ac8d92ee63cc83d7950d7a9a15eccb8926a400 | A parabola passes through a circle with center \( O \) and radius \( 1 \) at points \( P \) and \( Q \). What is the form of the parabola so that the area bounded by \( OP \), \( OQ \), and the parabola from \( P \) to \( Q \) is the greatest? | [
"y = -x^2 + \\dfrac{5}{4}"
] | [
"A parabola passes through a circle with center \\( O \\) and radius \\( 1 \\) at points \\( P \\) and \\( Q \\). What is the form of the parabola so that the area bounded by \\( OP \\), \\( OQ \\), and the parabola from \\( P \\) to \\( Q \\) is the greatest?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 6
} | 8 | 0 | 0 |
omr-ws-78ca6826f5416e6af638fe8eb0cb9e101efead58cb10f1ac5ffa8b4d9bd54b91 | A parabola passes through the points (-1,4), (1,0), and (2,7). Find the sum of the x-coordinates associated with y = 64. | [
"\\(\\frac{2}{3}\\)"
] | [
"A parabola passes through the points (-1,4), (1,0), and (2,7). Find the sum of the x-coordinates associated with y = 64."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-6e34688c29482ffdd718d30fc23f5e0692e76621f1798863bdf65667f58c1c7d | A parabola passes through the points (5,1) and (13,-7) and has the y-axis as its directrix. What are the coordinates of all possible vertices of this parabola? | [
"\\((\\tfrac12, -2)\\) or \\((4, 5)\\)"
] | [
"A parabola passes through the points (5,1) and (13,-7) and has the y-axis as its directrix. What are the coordinates of all possible vertices of this parabola?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 17
} | 17 | 0 | 0 |
omr-ws-ab2701efc952e881fb37cdef0bab187c6349488f9207531742bb62d7eaa6c121 | A parabola passes through the vertices B, C, and D of a square ABCD. Given that A(2,1) and C(2,3), find the focus of the parabola. | [
"(2, 13/4)"
] | [
"A parabola passes through the vertices B, C, and D of a square ABCD. Given that A(2,1) and C(2,3), find the focus of the parabola."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-195fdc9d3b17802e99420d4cc5550bcb7551557a22be5d041bd07b73fc3c9cfb | A parabola touches a semicircle and passes through the endpoints of its diameter. What is the probability that a point thrown into the semicircle at random will fall in the domain bounded by the arc of the semicircle and the parabola? | [
"$\\frac{3\\pi -8}{\\pi }$"
] | [
"A parabola touches a semicircle and passes through the endpoints of its diameter. What is the probability that a point thrown into the semicircle at random will fall in the domain bounded by the arc of the semicircle and the parabola?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f4aa24dbdc127eee3fe1a3b61ebbb59f48d26029490fdca1785a5b8e209bb368 | A parabola touches the base of a square and passes through its upper vertices. What is the probability that a point thrown in the square at random will fall in the domain between the upper side of the square and the parabola? | [
"$\\frac{2}{3}$"
] | [
"A parabola touches the base of a square and passes through its upper vertices. What is the probability that a point thrown in the square at random will fall in the domain between the upper side of the square and the parabola?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5096f64e8dca4bc39c01dd5c74b2fd4d01ba8c8406bd40084efb4d5c0cf94242 | A parabola with directrix $x + y + 2 = 0$ touches the line $2x + y - 5 = 0$ at the point $(2, 1)$. Determine the semi-latusrectum of the parabola. | [
"\\(\\frac{9\\sqrt{2}}{2}\\)"
] | [
"A parabola with directrix $x + y + 2 = 0$ touches the line $2x + y - 5 = 0$ at the point $(2, 1)$. Determine the semi-latusrectum of the parabola."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 7,
"tir": 8
} | 30 | 0 | 0 |
omr-ws-ce4480c332bf9dd2effdae3481ba0451dbd827551cfc6f3dba6ea097fca03442 | A parabola with equation $y = x^2 + bx + c$ passes through the points $(2,3)$ and $(4,3)$. What is $c$? | [
"11"
] | [
"A parabola with equation $y = x^2 + bx + c$ passes through the points $(2,3)$ and $(4,3)$. What is $c$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-930795b52dbdbe8427bf8dd318aa66bec43bdc1eaeb21c8b2c3eea53b556d26e | A parabola with equation $y=x^2+bx+c$ passes through the points $(-1,-11)$ and $(3,17)$. What is $c$? | [
"-7"
] | [
"A parabola with equation $y=x^2+bx+c$ passes through the points $(-1,-11)$ and $(3,17)$. What is $c$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a9015889a6d7b50041f742b5d683a048f545a34b985e53c471668a9d7d183101 | A parabola with equation \( y = ax^2 + bx + c \) is reflected about the \( x \)-axis, then both the original and the reflected parabolas are translated horizontally in opposite directions by 5 units to become the graphs \( y = f(x) \) and \( y = g(x) \). What describes the graph \( y = (f + g)(x) \)? | [
"\\( y = k \\) (horizontal line)"
] | [
"A parabola with equation \\( y = ax^2 + bx + c \\) is reflected about the \\( x \\)-axis, then both the original and the reflected parabolas are translated horizontally in opposite directions by 5 units to become the graphs \\( y = f(x) \\) and \\( y = g(x) \\). What describes the graph \\( y = (f + g)(x) \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 5,
"tir": 14
} | 24 | 0 | 0 |
omr-ws-5799eb78f38bc4f5d7f084c1fdbdf72111bdf8e420fa4b6a932f8bd44c568760 | A parabolic arch has a height of 20 meters and a width of 36 meters at the base. At which height above the base is the arch 18 meters wide? | [
"15 meters"
] | [
"A parabolic arch has a height of 20 meters and a width of 36 meters at the base. At which height above the base is the arch 18 meters wide?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-20bad4c73d9d55299352da916033ffe5f36643e0125b707a761a843a72d591c3 | A parabolic arch has a span of 129 feet and a maximum height of 25 feet. Choose a suitable rectangular coordinate system and find the equation of the parabola. Then calculate the height of the arch at points 10 feet, 20 feet, and 40 feet from the center. | [
"24.4, 22.6, 15.4"
] | [
"A parabolic arch has a span of 129 feet and a maximum height of 25 feet. Choose a suitable rectangular coordinate system and find the equation of the parabola. Then calculate the height of the arch at points 10 feet, 20 feet, and 40 feet from the center."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 25
} | 25 | 0 | 0 |
omr-ws-370abc4b1c49c976281fdd5db811559bfedc30b13cf338017b22db82c3121487 | A parabolic dish is 10 units deep and 100 units in diameter. How far away from its vertex is its focus? | [
"62.5"
] | [
"A parabolic dish is 10 units deep and 100 units in diameter. How far away from its vertex is its focus?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-516631937b6346e4aa42a23feaa4d639b533321b3823524d4a4c10df6b8db381 | A parallelepiped has a surface area of 216 and a volume of 216. Calculate the sum of all possible values of the length of one of its edges. | [
"6"
] | [
"A parallelepiped has a surface area of 216 and a volume of 216. Calculate the sum of all possible values of the length of one of its edges."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-0e8cd954663c6cf01b1f060b257f966a3b5fe2583e0eadd71aa0acaa609e65ed | A parallelepiped has two opposite parallelogram faces with side lengths $2$ and $3$ and a $60^\circ$ angle, two opposite parallelogram faces with side lengths $2$ and $5$ and a $60^\circ$ angle, and two opposite parallelogram faces with side lengths $3$ and $5$ and a $60^\circ$ angle. Find its volume. | [
"\\(\\frac{15\\sqrt{3}}{2}\\)"
] | [
"A parallelepiped has two opposite parallelogram faces with side lengths $2$ and $3$ and a $60^\\circ$ angle, two opposite parallelogram faces with side lengths $2$ and $5$ and a $60^\\circ$ angle, and two opposite parallelogram faces with side lengths $3$ and $5$ and a $60^\\circ$ angle. Find its volume."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7d314172f30ea66113f8acc3a0295ef11d4524df2b31d6b6367cb50a8405d627 | A parallelepiped is cut by a plane along a 6-gon. Suppose this 6-gon can be completely covered by a rectangle $\pi$ with dimensions $a \times b$. Determine the maximum possible value of $\frac{a}{b}$ if one of the parallelepiped's faces can also be completely covered by the rectangle $\pi$. | [
"\\sqrt{3}"
] | [
"A parallelepiped is cut by a plane along a 6-gon. Suppose this 6-gon can be completely covered by a rectangle $\\pi$ with dimensions $a \\times b$. Determine the maximum possible value of $\\frac{a}{b}$ if one of the parallelepiped's faces can also be completely covered by the rectangle $\\pi$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 2
} | 7 | 0 | 0 |
omr-ws-876fdb80bd6737859af68fd283cfde6476956d03bd57ef8ae23af4b0bf5331d3 | A parallelepiped with side lengths $a$, $b$, and $c$ is cut by a plane through its intersection of diagonals which is perpendicular to one of these diagonals. Calculate the area of the intersection of the plane and the parallelepiped. | [
"\\(\\frac{2abc}{\\sqrt{a^2 + b^2 + c^2}}\\)"
] | [
"A parallelepiped with side lengths $a$, $b$, and $c$ is cut by a plane through its intersection of diagonals which is perpendicular to one of these diagonals. Calculate the area of the intersection of the plane and the parallelepiped."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a9279a6c7f65a2b525cf625b219cd364bf07832a9914435e79a64688532800de | A parallelogram can intersect two internally tangent circles at most $x$ times and at least $y$ times. Compute $y^{x}$. | [
"0"
] | [
"A parallelogram can intersect two internally tangent circles at most $x$ times and at least $y$ times. Compute $y^{x}$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 7
} | 18 | 0 | 0 |
omr-ws-3374aaa313b9dcf08b53fa814aa3676c04fcbaa82a7dbcfca3b1cb296ce1a993 | A parallelogram circumscribes the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and two of its angular points are on the lines $x^2-h^2=0$. If the other two angular points are on the conic $\frac{x^2}{a^2}+\frac{y^2}{b^2}\left(1-\frac{a^2}{h^2}\right)=k$, find the value of $k$. | [
"1"
] | [
"A parallelogram circumscribes the ellipse $\\frac{x^2}{a^2}+\\frac{y^2}{b^2}=1$ and two of its angular points are on the lines $x^2-h^2=0$. If the other two angular points are on the conic $\\frac{x^2}{a^2}+\\frac{y^2}{b^2}\\left(1-\\frac{a^2}{h^2}\\right)=k$, find the value of $k$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-892d728454a758bc0c3bb1741fc4198a4fa4301dca2c5a86890c27a791dc04a8 | A parallelogram circumscribes the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). If two opposite vertices of the parallelogram lie on the curve \(f(x, y) = x^2 + y^2 - 1 = 0\), find the equation of the curve on which the other two vertices lie. Express your answer in the form \(g(x, y) = 0\). | [
"\\dfrac{(b^2 - 1)x^2}{a^2} + \\dfrac{(a^2 - 1)y^2}{b^2} + 1 = 0"
] | [
"A parallelogram circumscribes the ellipse \\(\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1\\). If two opposite vertices of the parallelogram lie on the curve \\(f(x, y) = x^2 + y^2 - 1 = 0\\), find the equation of the curve on which the other two vertices lie. Express your answer in the form \\(g(x, y) = 0\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 3
} | 4 | 0 | 0 |
omr-ws-cc85da8be88b7ede2a988759ce85589206c0ab516224d6125e4d6f32859bad7c | A parallelogram has a height of 5, an area of 50, and angles of 150° and 30°. What is the perimeter of this parallelogram? | [
"40"
] | [
"A parallelogram has a height of 5, an area of 50, and angles of 150° and 30°. What is the perimeter of this parallelogram?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-40bb24da97736037b6007240fcbbdc41d00eb23e892d7c514eb69c70f6b198fb | A parallelogram has adjacent sides of lengths $s$ units and $2s$ units forming a 45-degree angle. The area of the parallelogram is $8\sqrt 2$ square units. What is the value of $s$? Express your answer in simplest radical form. | [
"2\\sqrt{2}"
] | [
"A parallelogram has adjacent sides of lengths $s$ units and $2s$ units forming a 45-degree angle. The area of the parallelogram is $8\\sqrt 2$ square units. What is the value of $s$? Express your answer in simplest radical form."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-96cc2ba0f86c222cdc8acf46b69d5c59f414c7297a9f6518743e87127f87d80b | A parallelogram has one of its diagonals double the length of the other, and the area of the parallelogram is \(50 \, \text{cm}^2\). Find the length of the longest diagonal. | [
"10\\sqrt{2}"
] | [
"A parallelogram has one of its diagonals double the length of the other, and the area of the parallelogram is \\(50 \\, \\text{cm}^2\\). Find the length of the longest diagonal."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 29
} | 29 | 0 | 0 |
omr-ws-a72d73f1e2df874743ea462fa525bd1f6ab7eca3d6a395a5b96b6564b2b914bb | A parallelogram has one side of length 20, and two diagonals of lengths 24 and 10. What is its area? | [
"120"
] | [
"A parallelogram has one side of length 20, and two diagonals of lengths 24 and 10. What is its area?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-50a6fb5741c5d3228ff4d2d7e0b1ed8c25f21f372ba01ee669fa2382e72b8e03 | A parallelogram has sides measuring \(2\) meters and \(6\) meters, with its shorter diagonal also measuring \(6\) meters. Find the length of the longer diagonal. | [
"2\\sqrt{11}"
] | [
"A parallelogram has sides measuring \\(2\\) meters and \\(6\\) meters, with its shorter diagonal also measuring \\(6\\) meters. Find the length of the longer diagonal."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
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