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-9dbd8591f0db85bd6b34189b6da822281bdf9f8b6ec80d2dec2c28ecc3140186 | Using only complex calculus, calculate the integral \[ \int_{\mathbb{R}} \frac{\cos(2 \pi t x) - i \sin(2 \pi t x)}{\alpha_{1} x^{2} + \alpha_{2} x + \alpha_{3}} \, dx \] where $\alpha_{1} \in \mathbb{R}^{+}$, $t \in \mathbb{R}$, and $\alpha_{2}^{2} - 4 \alpha_{1} \alpha_{3} < 0$. | [
" \\dfrac{2\\pi}{\\sqrt{4\\alpha_{1}\\alpha_{3} - \\alpha_{2}^{2}}}"
] | [
"Using only complex calculus, calculate the integral\n\\[\n\\int_{\\mathbb{R}} \\frac{\\cos(2 \\pi t x) - i \\sin(2 \\pi t x)}{\\alpha_{1} x^{2} + \\alpha_{2} x + \\alpha_{3}} \\, dx\n\\]\nwhere $\\alpha_{1} \\in \\mathbb{R}^{+}$, $t \\in \\mathbb{R}$, and $\\alpha_{2}^{2} - 4 \\alpha_{1} \\alpha_{3} < 0$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 2
} | 9 | 0 | 0 |
omr-ws-c8cbf404ba1763210dbda07c68751ad33a769ce6fcc3982e2163fa2625f2b069 | Using only digits from the set \(\{2, 0, 1, 7\}\), how many six-digit numbers can be formed such that the sum of any three consecutive digits of that number is odd? | [
"224"
] | [
"Using only digits from the set \\(\\{2, 0, 1, 7\\}\\), how many six-digit numbers can be formed such that the sum of any three consecutive digits of that number is odd?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0a40b95a1ae7f25210d625786f7566b554b1c74f98293c31fa36076fc1a6188c | Using only four fives and the operations of addition, subtraction, multiplication, division, order of operations, parentheses, and grouping (but no exponents), find expressions that evaluate to each of the numbers from 1 to 20. | [
"5 + 5 + 5 + 5"
] | [
"Using only four fives and the operations of addition, subtraction, multiplication, division, order of operations, parentheses, and grouping (but no exponents), find expressions that evaluate to each of the numbers from 1 to 20."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-ba9f23aaff3b6417692b6ca09a3f3534903a786979d9fa0e7583976238645587 | Using only mathematical signs (and no additional digits), how can you make the expression $1\ 1\ 1$ equal to 6? | [
"\\((1+1+1)! = 6\\)"
] | [
"Using only mathematical signs (and no additional digits), how can you make the expression $1\\ 1\\ 1$ equal to 6?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-330db283051db0dc41ad3f46a2abcc932453b80575604e608a78e9117b6464a7 | Using only once each of the digits $1, 2, 3, 4, 5, 6, 7,$ and $8$, write the square and the cube of a positive integer. Determine what that number can be. | [
"24"
] | [
"Using only once each of the digits $1, 2, 3, 4, 5, 6, 7,$ and $8$, write the square and the cube of a positive integer. Determine what that number can be."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-d514c92f5917201a10ab752027de963ae59482ab02d428fd74547c33f85e3d79 | Using only pennies, nickels, dimes, and quarters, what is the smallest number of coins Freddie would need so he could pay any amount of money less than a dollar? | [
"10"
] | [
"Using only pennies, nickels, dimes, and quarters, what is the smallest number of coins Freddie would need so he could pay any amount of money less than a dollar?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-194f14d789895b0995ab20e15b5b4ec8739b532224d20bfff38883460f422c20 | Using only single-variable calculus, find the value of $\zeta(4) = \sum_{n=1}^{\infty} \frac{1}{n^4}$. Express your answer in terms of $\pi$. | [
"\\dfrac{\\pi^4}{90}"
] | [
"Using only single-variable calculus, find the value of $\\zeta(4) = \\sum_{n=1}^{\\infty} \\frac{1}{n^4}$. Express your answer in terms of $\\pi$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-58f1d1888d31a9b528b4a4eb560c054d827290d62f0590ae0950ea91d79329ea | Using only the digit 2 and the addition symbol, what is the minimum number of 2’s needed to create an expression with a value of 2020? | [
"29"
] | [
"Using only the digit 2 and the addition symbol, what is the minimum number of 2’s needed to create an expression with a value of 2020?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 4
} | 19 | 0 | 0 |
omr-ws-1b3ed106273de81669f6be82af2cdd7db836475d0e152aba573a6d0ad1107da6 | Using only the digits $1$ and $2$, write down some numbers with $2^n$ digits such that the digits of any two of these numbers differ in at least $2^{n-1}$ places. If $N(n)$ is the maximum number of such numbers that can be written, find $N(3)$. | [
"16"
] | [
"Using only the digits $1$ and $2$, write down some numbers with $2^n$ digits such that the digits of any two of these numbers differ in at least $2^{n-1}$ places. If $N(n)$ is the maximum number of such numbers that can be written, find $N(3)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e03b7dc910fadbb652c9be7ed8da94e2a2eda3130b4ca8d79f0f2d57b85f5dc7 | Using only the digits $3$ and $a$, we form the number $3a33a333a3333a\cdots33\cdots3a$ with a total of 90 digits. If the number is a multiple of $18$, what are the possible values of $a$? | [
"\\(0, 6\\)"
] | [
"Using only the digits $3$ and $a$, we form the number $3a33a333a3333a\\cdots33\\cdots3a$ with a total of 90 digits. If the number is a multiple of $18$, what are the possible values of $a$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 16
} | 19 | 0 | 0 |
omr-ws-deffe794411bb52b033c005ad8a565e3138a33a6d70526bf679ec5f6658869ca | Using only the digits 1, 2, 3, 4, and 5, two players A and B compose a 2005-digit number N by selecting one digit at a time, with A starting first. The last player to play wins if and only if N is divisible by 9. Who will win if both players play optimally? | [
"B wins.",
"Player B wins."
] | [
"Using only the digits 1, 2, 3, 4, and 5, two players A and B compose a 2005-digit number N by selecting one digit at a time, with A starting first. The last player to play wins if and only if N is divisible by 9. Who will win if both players play optimally?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 14
} | 36 | 0 | 0 |
omr-ws-4ef2bc36f7c4d1c80530748fc03845bc81e203fe5c588f45441c6804dd7bc9d6 | Using only the digits 1, 3, and 4, in how many ways can one sum to any positive integer $n$, if the order of summands matters? For example, $1+1+3$, $1+3+1$, and $3+1+1$ are considered different. | [
"\\( a_{2n} = F_{n+1}^2 \\) and \\( a_{2n+1} = F_{n+1}F_{n+2} \\)"
] | [
"Using only the digits 1, 3, and 4, in how many ways can one sum to any positive integer $n$, if the order of summands matters? For example, $1+1+3$, $1+3+1$, and $3+1+1$ are considered different."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a75c1114943d3c031861a2eba74aa0e16c83caede79d920bea6581c43d59fb32 | Using only the digits 7, 8 and 9, how many positive seven-digit integers can be made that are palindromes? | [
"81"
] | [
"Using only the digits 7, 8 and 9, how many positive seven-digit integers can be made that are palindromes?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cea86b0f8850b05140b13244d790fd427483a4be9d4c26b99c8bc6759daa028a | Using only the digits from the number $2010$ and the keys on a calculator, make each of the numbers $1$ to $20$. You don't need to use all the digits, but you can't use them more than once. | [
"16 = \\(\\left( 2 + (0! + 1) \\right)^2\\), 17 = \\(\\left( 2 + (0! + 1) \\right)^2 + 0!\\)"
] | [
"Using only the digits from the number $2010$ and the keys on a calculator, make each of the numbers $1$ to $20$. You don't need to use all the digits, but you can't use them more than once."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-83b5d1e456538f3e9117519473204713a45832076f5f1e0ed0860109f00ee5f4 | Using only the functions factorial (N --> N), square root (R --> R), and integer part (R --> N), starting with the input 3, determine the minimum number of steps required to generate the integer 100. | [
"7"
] | [
"Using only the functions factorial (N --> N), square root (R --> R), and integer part (R --> N), starting with the input 3, determine the minimum number of steps required to generate the integer 100."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 2
} | 5 | 0 | 0 |
omr-ws-f0cd16ab68fae566a004a3128583371c4623f9b668710320f87d1e14a27c5807 | Using only the information $10^{3}=1000$, $10^{4}=10000$, $2^{10}=1024$, $2^{11}=2048$, $2^{12}=4096$, $2^{13}=8192$, find the largest $a$ and $b$ such that one can prove $a < \log_{10}2 < b$. | [
"\\( a = 0.300 \\), \\( b = \\frac{4}{13} \\)"
] | [
"Using only the information $10^{3}=1000$, $10^{4}=10000$, $2^{10}=1024$, $2^{11}=2048$, $2^{12}=4096$, $2^{13}=8192$, find the largest $a$ and $b$ such that one can prove $a < \\log_{10}2 < b$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 12
} | 14 | 0 | 0 |
omr-ws-164193618e391ffa30a7f5018b4c27f256431378ce76d1a3cd887773ec6f683a | Using only the information $10^{3}=1000$, $10^{4}=10000$, $2^{10}=1024$, $2^{11}=2048$, $2^{12}=4096$, $2^{13}=8192$, find the largest $a$ and smallest $b$ such that one can prove $a < \log_{10}2 < b$. | [
"\\( a = 0.3 \\), \\( b = 0.4 \\)"
] | [
"Using only the information $10^{3}=1000$, $10^{4}=10000$, $2^{10}=1024$, $2^{11}=2048$, $2^{12}=4096$, $2^{13}=8192$, find the largest $a$ and smallest $b$ such that one can prove $a < \\log_{10}2 < b$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 1,
"tir": 1
} | 4 | 0 | 0 |
omr-ws-35a17c73ce4584217ca325c8fb22b4b37e3cb7ebd53dcf7647edb95a50b2f3a6 | Using only the numbers 1, 2, 3, 4, and 5 (none more than once) and the operations +, -, *, and / and brackets, how many of the following numbers can be made: 76, 79, 86, 92, 97, and 98? | [
"0"
] | [
"Using only the numbers 1, 2, 3, 4, and 5 (none more than once) and the operations +, -, *, and / and brackets, how many of the following numbers can be made: 76, 79, 86, 92, 97, and 98?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-286e19c7e8301ec29458f6969fd36176611715b931153c2b011e02e41b0ddb17 | Using only the recurrence relation \( I_{n+1} = \frac{e^3}{3} - \frac{n+1}{3} I_n \), can you calculate an approximation (for instance with three digits) of \( I_{100} \)? Explain precisely why there are difficulties in doing so. | [
"115 digits of accuracy"
] | [
"Using only the recurrence relation \\( I_{n+1} = \\frac{e^3}{3} - \\frac{n+1}{3} I_n \\), can you calculate an approximation (for instance with three digits) of \\( I_{100} \\)? Explain precisely why there are difficulties in doing so."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f1ee88e801f7bf3fbaf78b1891200b9ca0a140ee9f1086730cefa44cd325564a | Using only the same digit (from $0$ to $9$) and standard mathematical functions, find the expression that gives $20000$ using the smallest number of this digit. Example solutions provided: - With nine digits $1$: $(1+1)(11-1)^{1+1+1+1}$ - With seven digits $2$: $2\left(\frac{22-2}2\right)^{2+2}$ - With four digits $5$:... | [
"$\\lfloor\\sqrt 5\\rfloor\\lfloor\\sqrt{5!}\\rfloor^{\\lceil\\sqrt{\\lfloor\\sqrt{5!}\\rfloor}\\rceil}$"
] | [
"Using only the same digit (from $0$ to $9$) and standard mathematical functions, find the expression that gives $20000$ using the smallest number of this digit.\n\nExample solutions provided:\n- With nine digits $1$: $(1+1)(11-1)^{1+1+1+1}$\n- With seven digits $2$: $2\\left(\\frac{22-2}2\\right)^{2+2}$\n- With fo... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 3
} | 5 | 0 | 0 |
omr-ws-cf0bf5c34bd5580adcfd6051e16ba672d0db1a72b3f0b07bb0c715e88746d392 | Using only two different digits $2$ and $d$, the following $90$-digit number is formed: $m = 2d22d222d...$. If $m$ is a multiple of $9$, determine all possible values of the digit $d$. | [
"5, 8"
] | [
"Using only two different digits $2$ and $d$, the following $90$-digit number is formed: $m = 2d22d222d...$. If $m$ is a multiple of $9$, determine all possible values of the digit $d$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 13
} | 16 | 0 | 0 |
omr-ws-835416fde1fbf215d66a482a7298fede0639edba4592475313ae14e6d9f3baaa | Using operations a) –– b) and a $10$-stencil, construct the center of a regular $10$-gon with a side of $1$ cm. If the distance from a vertex of the $10$-gon to the center is $r$, find the value of $r^2$. | [
"\\dfrac{3 + \\sqrt{5}}{2}"
] | [
"Using operations a) –– b) and a $10$-stencil, construct the center of a regular $10$-gon with a side of $1$ cm. If the distance from a vertex of the $10$-gon to the center is $r$, find the value of $r^2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-f04571bfb00530bcdb1703e0827e02c625ff8d049e03819aee4b7cfd807ef4d2 | Using operations a) –– b) and a $14$-stencil, construct the center of a regular $14$-gon with a side of $1$ cm. If the distance from the center to one of the vertices is $d$ cm, find the exact value of $d$. | [
"\\dfrac{1}{2 \\sin\\left(\\dfrac{\\pi}{14}\\right)}"
] | [
"Using operations a) –– b) and a $14$-stencil, construct the center of a regular $14$-gon with a side of $1$ cm. If the distance from the center to one of the vertices is $d$ cm, find the exact value of $d$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c06ada470122530dd31ed05fbf5137cc75809b7efb4d9040c2d2422f606d2124 | Using operations a) –– c) and a $12$-stencil, construct the center of a regular $12$-gon with a side of $1$ cm. If the coordinates of the constructed center are $(x, y)$, find the value of $100x + y$. | [
"\\dfrac{102 + \\sqrt{3}}{2}"
] | [
"Using operations a) –– c) and a $12$-stencil, construct the center of a regular $12$-gon with a side of $1$ cm. If the coordinates of the constructed center are $(x, y)$, find the value of $100x + y$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-04b9fc594485fc5a1e24a1d8a21e1392fb2d9a2ab049733b1253a7de036ee7e3 | Using operations a) –– d) and a $10$-stencil, construct the center of a regular $10$-gon with a side of $1$ cm. If the distance from a vertex of the $10$-gon to the center is $d$, find the value of $d^2$. | [
"\\dfrac{3 + \\sqrt{5}}{2}"
] | [
"Using operations a) –– d) and a $10$-stencil, construct the center of a regular $10$-gon with a side of $1$ cm. If the distance from a vertex of the $10$-gon to the center is $d$, find the value of $d^2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b693fcca61dd725cb15bf19b398aee8d23efa4c5022dad89334981973c00a136 | Using operations a) –– d) and an $n$-stencil, construct the center of a regular $n$-gon with a side of $1$ cm for $n = 7$. In addition to operations a) –– d), it is allowed to connect by a segment any two points, the distance between which is not more than $1$ cm. Calculate the number of distinct segments you need to d... | [
"6"
] | [
"Using operations a) –– d) and an $n$-stencil, construct the center of a regular $n$-gon with a side of $1$ cm for $n = 7$. In addition to operations a) –– d), it is allowed to connect by a segment any two points, the distance between which is not more than $1$ cm. Calculate the number of distinct segments you need... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 3
} | 7 | 0 | 0 |
omr-ws-23df650e23b8895ebb860cb5552520970d84865156e1dc100b838a8fc7f08fbc | Using optimal strategy, what is the expected number of correct guesses when guessing whether each card in a shuffled standard deck is red or black? | [
"\\dfrac{1404}{53}"
] | [
"Using optimal strategy, what is the expected number of correct guesses when guessing whether each card in a shuffled standard deck is red or black?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 6
} | 11 | 0 | 0 |
omr-ws-9b27d5b859da5cf618fa9cc861ef677768fba12d176835bb377075740e5093a6 | Using ordered pairs, describe the partial order $B$ on the power set of $A = \{x, y, z\}$, defined by the subset relation. | [
"B = \\{(S, T) \\mid S \\subseteq T \\subseteq \\{x, y, z\\}\\}"
] | [
"Using ordered pairs, describe the partial order $B$ on the power set of $A = \\{x, y, z\\}$, defined by the subset relation."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4dad8c84bb0d65d8c1bc3f5066f6a63a7f96ae8b1c76420fb32f6bd45c1287df | Using part (a) or otherwise, draw the graphs of \( y = x - 1 \), \( y = x \), \( y = x + 1 \), and \( y = x e^{-\frac{1}{|x|}} \) for \( -\infty < x < \infty \) using the same \( X \) and \( Y \) axes. | [
""
] | [
"Using part (a) or otherwise, draw the graphs of \\( y = x - 1 \\), \\( y = x \\), \\( y = x + 1 \\), and \\( y = x e^{-\\frac{1}{|x|}} \\) for \\( -\\infty < x < \\infty \\) using the same \\( X \\) and \\( Y \\) axes."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 16 |
omr-ws-16b40d8fddf88a54f613799b0ff9775afd6fb9e2f35c5bbdf0bafddec464d575 | Using part (a), solve the equation \(12x^2 - 7x - 10 = 0\). | [
"\\(\\frac{5}{4}, -\\frac{2}{3}\\)"
] | [
"Using part (a), solve the equation \\(12x^2 - 7x - 10 = 0\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 10,
"tir": 27
} | 37 | 0 | 0 |
omr-ws-e648bfe1d20afdedcd0fcc6d4ed17ca24c64da73c8da9678828ece0b178e5ca2 | Using partial fractions, find $f(s)$ if $\mathcal{L}(f) = \frac{1}{s^2+2s+5}$. | [
"\\(\\frac{1}{2}e^{-t}\\sin(2t)\\)"
] | [
"Using partial fractions, find $f(s)$ if $\\mathcal{L}(f) = \\frac{1}{s^2+2s+5}$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 31
} | 45 | 0 | 0 |
omr-ws-c6cc8acea5437de003bada5c7b016dd854ee850b8cc32bc66fa998740032e939 | Using partial fractions, find $f(s)$ if $\mathcal{L}(f) = \frac{3s-2}{s^3(s^2+4)}$. | [
"\\( \\frac{1}{8}(-2t^2 + 6t + 1 - \\cos(2t) - 3\\sin(2t)) \\)"
] | [
"Using partial fractions, find $f(s)$ if $\\mathcal{L}(f) = \\frac{3s-2}{s^3(s^2+4)}$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 30
} | 44 | 0 | 0 |
omr-ws-1d3c1e181f7776e7d15680a133f0b7cda4644afc2120ead1dd9e127fbbe76b83 | Using pennies (1 cent), nickels (5 cents), and dimes (10 cents), how many ways can we make 35 cents using generating functions? | [
"24"
] | [
"Using pennies (1 cent), nickels (5 cents), and dimes (10 cents), how many ways can we make 35 cents using generating functions?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f22b76534b9f861b3c1356fe0a3a77199ceca49405edf3babe1508bc3618e243 | Using pennies (1 cent), nickels (5 cents), dimes (10 cents), quarters (25 cents), and half-dollars (50 cents), how many ways are there to make $50n$ cents, where $n$ is a fixed integer? | [
"\\(\\binom{n+4}{4}+45 \\binom{n+3}{4}+52 \\binom{n+2}{4}+2 \\binom{n+1}{4}\\)"
] | [
"Using pennies (1 cent), nickels (5 cents), dimes (10 cents), quarters (25 cents), and half-dollars (50 cents), how many ways are there to make $50n$ cents, where $n$ is a fixed integer?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5de3d853144181c8e2c5e52443f1cfd1127fbcfb7f4f4c79b8cfc17b412d873f | Using power series around \( t_0 = 0 \), determine the general solution of the ordinary differential equation \[ ty''(t) + (3 + 2t)y'(t) + 6y(t) = 0. \] In the event that any of the series that form the general solution converges to an elementary function, identify that function. | [
"\\( y(t) = Ae^{-2t} + B\\frac{(-4t^2 + 8t^3 - \\dots)\\ln t + 1 + 2t - 4t^2 + \\dots}{t^2} \\)\n\nThe elementary function identified in the solution is \\( e^{-2t} \\)."
] | [
"Using power series around \\( t_0 = 0 \\), determine the general solution of the ordinary differential equation\n\\[ ty''(t) + (3 + 2t)y'(t) + 6y(t) = 0. \\]\nIn the event that any of the series that form the general solution converges to an elementary function, identify that function."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 8,
"tir": 1
} | 14 | 0 | 0 |
omr-ws-9977b0496372a95b3feb2c1dd01215081a80c96b272b36b409b32d587dd0cafc | Using properties of the definite integral, calculate $\int_{a}^{b}\frac{\left| x\right|}{x}dx$, where $a < b$. | [
"\\(|b| - |a|\\)"
] | [
"Using properties of the definite integral, calculate $\\int_{a}^{b}\\frac{\\left| x\\right|}{x}dx$, where $a < b$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 14
} | 28 | 0 | 0 |
omr-ws-fa68c7e5eb7e7ecb494a1a271e7f6daba172b9cd25073c578c53fe71347dfe21 | Using purely Euclidean geometry (straight edge and collapsible compass), determine the length of CD in relation to AB. Additional constructions are allowed, but the answer must be expressed as being equal to other lines, not in numerical form. | [
"\\( CD = AB \\sqrt{3} \\)"
] | [
"Using purely Euclidean geometry (straight edge and collapsible compass), determine the length of CD in relation to AB. Additional constructions are allowed, but the answer must be expressed as being equal to other lines, not in numerical form."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-60aedbf9ba5c36bc2d434e6d0633f835277b8f094b32c98d7f7fdbf04d661fe9 | Using red, blue, and yellow colored toothpicks and marshmallows, how many ways are there to construct distinctly colored regular hexagons? (Note that two colored hexagons are the same if we can either rotate one of the hexagons and obtain the other or flip one of the hexagons about some line and obtain the other.) | [
"92"
] | [
"Using red, blue, and yellow colored toothpicks and marshmallows, how many ways are there to construct distinctly colored regular hexagons? (Note that two colored hexagons are the same if we can either rotate one of the hexagons and obtain the other or flip one of the hexagons about some line and obtain the other.)... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 5
} | 16 | 0 | 0 |
omr-ws-98e678dcae0d15dc9e52b6d451b3315be232fc8aa50736e2e4522f21335b8614 | Using rook polynomials, compute the number of triples $(a, b, c)$ where $0 \leq a \leq 5$, $0 \leq b \leq 6$, $0 \leq c \leq 7$, and $a + b + c = 16$. | [
"6"
] | [
"Using rook polynomials, compute the number of triples $(a, b, c)$ where $0 \\leq a \\leq 5$, $0 \\leq b \\leq 6$, $0 \\leq c \\leq 7$, and $a + b + c = 16$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-29f49e8fb1f13fc167e3284693932dc94f5643e2560a57b714f66ad5fb989d38 | Using seven $7$s and the operations $+,-,\times,\div$, along with brackets, can you form the number 500? | [
"7 \\times 7 \\times 7 + 7 \\times 7 + 7 \\times 7 + 7 + 7 + 7"
] | [
"Using seven $7$s and the operations $+,-,\\times,\\div$, along with brackets, can you form the number 500?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 3
} | 4 | 0 | 0 |
omr-ws-b628846fc31f759d431dba3644b79bb3d3f168dd5583931838f5c1d278577642 | Using several white edge cubes of side $1$, Guille builds a large cube. He then paints $4$ faces of the large cube red. After disassembling the large cube, he finds that there are $431$ smaller cubes with at least one face painted red. Determine the total number of smaller cubes used to assemble the large cube. | [
"729"
] | [
"Using several white edge cubes of side $1$, Guille builds a large cube. He then paints $4$ faces of the large cube red. After disassembling the large cube, he finds that there are $431$ smaller cubes with at least one face painted red. Determine the total number of smaller cubes used to assemble the large cube."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-35f0ca28d2a05b7591757955add9e59a7f14d31c8a52a3388132577fc596fb3c | Using sigma notation, find the number and the sum of its digits for a number in base \( q \) represented by \( a_{n}a_{n-1}a_{n-2}...a_{3}a_{2}a_{1}a_{0} \). | [
"\\( N = \\sum_{i=0}^{n} a_i q^i \\) and the sum of its digits is \\( \\sum_{i=0}^{n} a_i \\)"
] | [
"Using sigma notation, find the number and the sum of its digits for a number in base \\( q \\) represented by \\( a_{n}a_{n-1}a_{n-2}...a_{3}a_{2}a_{1}a_{0} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 21
} | 34 | 0 | 0 |
omr-ws-4d6e92a3e10cc3c0ab06a812d6653c4498370f6484012d22d41f18d71cf4b700 | Using similar methods, determine the approximate probability that a randomly chosen number from 1 to N is prime (note: This formula will have a significant error). | [
"\\dfrac{1}{\\ln N}"
] | [
"Using similar methods, determine the approximate probability that a randomly chosen number from 1 to N is prime (note: This formula will have a significant error)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-480522d7b1034f60f47ad4818b4b5911c7a93a1ebfb3e4d963fe6e3364ca191e | Using single variable calculus, maximize the length of a piece of wood that can be turned around a 90-degree corner in a corridor, where the first section of the corridor is 2 meters wide and the next section is 3 meters wide. | [
"\\left(2^{2/3} + 3^{2/3}\\right)^{3/2}"
] | [
"Using single variable calculus, maximize the length of a piece of wood that can be turned around a 90-degree corner in a corridor, where the first section of the corridor is 2 meters wide and the next section is 3 meters wide."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-7dd23778431b66ca6b41457fad5468e8c84325cc97b0f63d1347ab9537dc0472 | Using some squares of size $2 \times 2$, some $P$-rectangles (a $2 \times 3$ or $3 \times 2$ rectangle without one cell in the corner), and some $S$-rectangles (a $2 \times 3$ or $3 \times 2$ rectangle without two cells in opposite corners), one forms a rectangle of size $1993 \times 2000$. Let $s$ denote the sum of th... | [
"996500"
] | [
"Using some squares of size $2 \\times 2$, some $P$-rectangles (a $2 \\times 3$ or $3 \\times 2$ rectangle without one cell in the corner), and some $S$-rectangles (a $2 \\times 3$ or $3 \\times 2$ rectangle without two cells in opposite corners), one forms a rectangle of size $1993 \\times 2000$. Let $s$ denote th... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 5
} | 16 | 0 | 0 |
omr-ws-afed096ad45383bbe0ceebb9036fd63ccaa56944c7b9e1f06f2e59b019d14e55 | Using spherical coordinates, compute the integral $\int \int \int \left(x^2+y^2+z^2 \right) dv$ where the region is inside $\rho = 2\cos \phi$ and outside $\rho = \cos \phi$. | [
"\\(\\frac{1984}{75}\\)"
] | [
"Using spherical coordinates, compute the integral $\\int \\int \\int \\left(x^2+y^2+z^2 \\right) dv$ where the region is inside $\\rho = 2\\cos \\phi$ and outside $\\rho = \\cos \\phi$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5bf10327379f2064e216f9d16a14c054967fec2dedfb57504f90926eb7f45809 | Using squares of side 1, a stair-like figure is formed in stages. The first stage uses 1 square, the second uses 5 squares, and so on. Determine the last stage for which the corresponding figure uses fewer than 2014 squares. | [
"32"
] | [
"Using squares of side 1, a stair-like figure is formed in stages. The first stage uses 1 square, the second uses 5 squares, and so on. Determine the last stage for which the corresponding figure uses fewer than 2014 squares."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 7,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-67649c86f93b4626792536255504d257accdf0a840f78c8e7ba818545ac73845 | Using the "key-shaped" (Hankel) contour described in (ii), solve the integral \(\int_{0}^\infty \frac{x^{p}}{1+x^{n}} \, dx\) where \(|p| < 1\) and \(n \in \mathbb{N}^{+}\). | [
"\\dfrac{\\pi}{n \\sin\\left( \\dfrac{\\pi (p + 1)}{n} \\right)}"
] | [
"Using the \"key-shaped\" (Hankel) contour described in (ii), solve the integral \\(\\int_{0}^\\infty \\frac{x^{p}}{1+x^{n}} \\, dx\\) where \\(|p| < 1\\) and \\(n \\in \\mathbb{N}^{+}\\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5e3fe662a06be74267d819107673d51fda51a2cb17ce1332fbf28d92ef0fa5ca | Using the 9 letters A, B, C, D, E, F, G, H, I, in how many different ways can a group of 5 be selected if F must be one of the letters? | [
"70"
] | [
"Using the 9 letters A, B, C, D, E, F, G, H, I, in how many different ways can a group of 5 be selected if F must be one of the letters?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-4b14aec9272a1255833b76ee1956e75a09a179aee0f6c3f01c9d38ceaaaa1ec4 | Using the 9 letters A, B, C, D, E, F, G, H, I, in how many different ways can a group of 6 be selected if exactly 3 of the 6 must be consonants? | [
"10"
] | [
"Using the 9 letters A, B, C, D, E, F, G, H, I, in how many different ways can a group of 6 be selected if exactly 3 of the 6 must be consonants?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3d19be70b89a2441a7e2662377a17cf39b8cb56e3a47eaaaedb2a6e5af4f7214 | Using the AM-GM inequality, find the maximum value of \( \sin{A} \sin{B} \sin{C} \) for any triangle \( \triangle ABC \). | [
"\\dfrac{3\\sqrt{3}}{8}"
] | [
"Using the AM-GM inequality, find the maximum value of \\( \\sin{A} \\sin{B} \\sin{C} \\) for any triangle \\( \\triangle ABC \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ab10a1f2d51f2520a135175368e3d55889644a21be9a0ce14d7e3c4c116216ab | Using the Binomial Theorem, calculate the value of the following expression for \( n = 10 \): \[ \dbinom{10}{1} + 2\dbinom{10}{2} + 3\dbinom{10}{3} + \cdots + 10\dbinom{10}{10} \] | [
"5120"
] | [
"Using the Binomial Theorem, calculate the value of the following expression for \\( n = 10 \\):\n\n\\[\n\\dbinom{10}{1} + 2\\dbinom{10}{2} + 3\\dbinom{10}{3} + \\cdots + 10\\dbinom{10}{10}\n\\]"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-31ac3a5e1ab0cab1abdfc1f4625d4f150f4d935db80d75fb92bf3abeb6a2413a | Using the Cauchy integral formula, evaluate the integral $\int_{0}^{\pi} \frac{d \theta}{ (a+\cos \theta )^2},$ where $a > 1.$ | [
"\\(\\frac{\\pi a}{(a^2-1)^{3/2}}\\)"
] | [
"Using the Cauchy integral formula, evaluate the integral $\\int_{0}^{\\pi} \\frac{d \\theta}{ (a+\\cos \\theta )^2},$ where $a > 1.$"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 26
} | 41 | 0 | 0 |
omr-ws-b44ef6e142260c8bbb09580dedd865b1c6e91d090580e40b319a1e4ba500ac44 | Using the Cauchy-Riemann equations, construct an analytic function \( f(z) = u(x, y) + iv(x, y) \) given that \( v(x, y) = \frac{y}{x^2 + y^2} \). Find \( u(x, y) \). | [
"\\( u(x, y) = -\\frac{x}{x^2 + y^2} + c \\)"
] | [
"Using the Cauchy-Riemann equations, construct an analytic function \\( f(z) = u(x, y) + iv(x, y) \\) given that \\( v(x, y) = \\frac{y}{x^2 + y^2} \\). Find \\( u(x, y) \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 5,
"tir": 8
} | 18 | 0 | 0 |
omr-ws-407b0ea6a7f434329663093cfd5d8b21b51fd77b9650ca9f0e5af432ccca1bf3 | Using the Cayley table of \( S_3 \) (the group of all permutations of \( A = \{1, 2, 3\} \)), find all the proper and improper subgroups of \( S_3 \) and elaborate the diagram. | [
"The proper and improper subgroups of \\( S_3 \\) are \\( S_3 \\), \\(\\{e\\}\\), \\(\\{e, (12)\\}\\), \\(\\{e, (13)\\}\\), \\(\\{e, (23)\\}\\), and \\(\\{e, (123), (132)\\}\\)."
] | [
"Using the Cayley table of \\( S_3 \\) (the group of all permutations of \\( A = \\{1, 2, 3\\} \\)), find all the proper and improper subgroups of \\( S_3 \\) and elaborate the diagram."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 7,
"tir": 22
} | 29 | 0 | 0 |
omr-ws-1049ba95d7e723e4d7c90343fcad461b0a7ded5c4b0a8ffda664f95e571ae900 | Using the Cayley-Hamilton theorem, find the value of \( A^{64} + 2A^{37} - 58I \) where \( A = \begin{pmatrix} -3 & -4 & -4 \\ 2 & 2 & 1 \\ 0 & 1 & 2 \end{pmatrix} \). | [
"\\( 68A^2 + 2A - 125I \\)"
] | [
"Using the Cayley-Hamilton theorem, find the value of \\( A^{64} + 2A^{37} - 58I \\) where \\( A = \\begin{pmatrix} -3 & -4 & -4 \\\\ 2 & 2 & 1 \\\\ 0 & 1 & 2 \\end{pmatrix} \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a16b4c21889bd7c8a5c3456f0f3d9f52b447209a2c3254499aef53f691b70369 | Using the Cesaro sum, someone claims that \(\sum_{n=1}^\infty n = -\frac{1}{12}\). Evaluate the validity of this claim. | [
"\\(-\\frac{1}{12}\\)"
] | [
"Using the Cesaro sum, someone claims that \\(\\sum_{n=1}^\\infty n = -\\frac{1}{12}\\). Evaluate the validity of this claim."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-845870cd01c071441fe77c173f37491b89a2831095c348ce7ac8f725e04da7f2 | Using the Euclidean algorithm, find the gcd of $3645$ and $2357$. Also, express the gcd as a linear combination of $3645$ and $2357$. | [
"\\(1 = 635 \\cdot 3645 - 982 \\cdot 2357\\)"
] | [
"Using the Euclidean algorithm, find the gcd of $3645$ and $2357$. Also, express the gcd as a linear combination of $3645$ and $2357$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 27
} | 28 | 0 | 0 |
omr-ws-51ba9e9f5af12b6e8b5b52d107dca2389d335a92735a66206d414d3d787c3e83 | Using the Euclidean algorithm, find the gcd of $7645$ and $2872$. Also, express the gcd as a linear combination of $7645$ and $2872$. | [
"\\(1 = 3543 \\cdot 2872 - 1331 \\cdot 7645\\)"
] | [
"Using the Euclidean algorithm, find the gcd of $7645$ and $2872$. Also, express the gcd as a linear combination of $7645$ and $2872$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 28
} | 29 | 0 | 0 |
omr-ws-274a32f84a387ad4b5bea44c19efbe30ad2bf412090351dabc22f8f2d3ac0186 | Using the Ferro/Tartaglia/Cardan method to solve the cubic equation \(x^3 + 3x^2 + 6x + 8 = 0\), after completing the cube and letting \(X = u - v\), the solutions are given by \(u^3 = -2 \pm \sqrt{5}\) and \(v = \frac{1}{u}\). Determine the three solutions for \((u, v)\). | [
"\\left( \\omega^2 \\sqrt[3]{-2 + \\sqrt{5}}, \\omega \\sqrt[3]{2 + \\sqrt{5}} \\right)"
] | [
"Using the Ferro/Tartaglia/Cardan method to solve the cubic equation \\(x^3 + 3x^2 + 6x + 8 = 0\\), after completing the cube and letting \\(X = u - v\\), the solutions are given by \\(u^3 = -2 \\pm \\sqrt{5}\\) and \\(v = \\frac{1}{u}\\). Determine the three solutions for \\((u, v)\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-e176058a2c520822cf162df2631126c8bde5ed0ccb123a8858502e4cc4e2e6d3 | Using the Fourier series of \( f(x) = (\pi - |x|)^2 \) on \([- \pi, \pi]\), calculate the value of \[ \sum_{n=1}^{\infty} \frac{1}{n^2}. \] | [
"\\dfrac{\\pi^2}{6}"
] | [
"Using the Fourier series of \\( f(x) = (\\pi - |x|)^2 \\) on \\([- \\pi, \\pi]\\), calculate the value of \n\n\\[ \\sum_{n=1}^{\\infty} \\frac{1}{n^2}. \\]"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-edc6888e18c92d9c2516f8dc0ace2115b3b9a5e3a4299206b8d2ff5756710a6c | Using the Fourier series of \( f(x) = (\pi - |x|)^2 \) on \([- \pi, \pi]\), calculate the value of \[ \sum_{n=1}^{\infty} \frac{1}{n^4}. \] | [
"\\dfrac{\\pi^4}{90}"
] | [
"Using the Fourier series of \\( f(x) = (\\pi - |x|)^2 \\) on \\([- \\pi, \\pi]\\), calculate the value of \n\n\\[ \\sum_{n=1}^{\\infty} \\frac{1}{n^4}. \\]"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-619f885a092f47e9cf29a171c4e775e17f29a64623cfab90ced5c30a2389013d | Using the Fourier series of \( f(x) = x(\pi - x) \) for \( 0 < x < \pi \), given by \[ f(x) = \sum \frac{4}{\pi n^3}[1 - (-1)^n] \sin(nx), \] calculate the value of the series \(\sum_{n=1}^{\infty} \frac{1}{n^4}\). | [
"\\dfrac{\\pi^4}{90}"
] | [
"Using the Fourier series of \\( f(x) = x(\\pi - x) \\) for \\( 0 < x < \\pi \\), given by\n\\[ f(x) = \\sum \\frac{4}{\\pi n^3}[1 - (-1)^n] \\sin(nx), \\]\ncalculate the value of the series \\(\\sum_{n=1}^{\\infty} \\frac{1}{n^4}\\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a3d10f64e150e2c0c8b2665ad39d6c5779989070b4c7256592c47e4caad270df | Using the Gamma function, calculate the value of \( s \) for which the Laplace transform of \( t^4 \) equals \( \frac{24}{s^5} \). | [
"s > 0"
] | [
"Using the Gamma function, calculate the value of \\( s \\) for which the Laplace transform of \\( t^4 \\) equals \\( \\frac{24}{s^5} \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-16e365d9bc6539631d783818c85ba679ebf866799cf8a3190cb95b8201e7689f | Using the Gamma function, calculate the value of \(\Gamma(1)\). | [
"1"
] | [
"Using the Gamma function, calculate the value of \\(\\Gamma(1)\\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8b48dec90602a9ecc6c7edecc6efb238a7d328256e96659d89ab5476d8d94394 | Using the Gauss-Laguerre method, approximate the integral \[ \int_{0}^{\infty}\frac{e^{-x}}{x+2}dx \] and express your answer in the form \(\frac{m}{n} + \frac{4}{\theta^5}\), where \(m\) and \(n\) are integers, and \(0 < \theta < 1\). Find the value of \(m+n\). | [
"19"
] | [
"Using the Gauss-Laguerre method, approximate the integral\n\n\\[\n\\int_{0}^{\\infty}\\frac{e^{-x}}{x+2}dx\n\\]\n\nand express your answer in the form \\(\\frac{m}{n} + \\frac{4}{\\theta^5}\\), where \\(m\\) and \\(n\\) are integers, and \\(0 < \\theta < 1\\). Find the value of \\(m+n\\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a5f52e31b562991b34d5034b270911eae106d92c0c3cc4e151211c8a130503cb | Using the Gauss-Ostrogradsky theorem, calculate the integral $$\int\limits_{S}(1+2x)dy\wedge dz+(2x+3y)dz\wedge dx+(3x+4z)dx\wedge dy,$$ where $S$ is the outer side of the pyramid surface defined by $$\frac{x}{a}+\frac{y}{b}+\frac{z}{c}\leq 1; \ \ x,y,z\geq 0.$$ | [
"\\(\\frac{3}{2}abc\\)"
] | [
"Using the Gauss-Ostrogradsky theorem, calculate the integral\n$$\\int\\limits_{S}(1+2x)dy\\wedge dz+(2x+3y)dz\\wedge dx+(3x+4z)dx\\wedge dy,$$\nwhere $S$ is the outer side of the pyramid surface defined by\n$$\\frac{x}{a}+\\frac{y}{b}+\\frac{z}{c}\\leq 1; \\ \\ x,y,z\\geq 0.$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-5ad77d81bc32dc01e99c7882986b84edc6f72d9f95d5493c8305d879234cccaa | Using the Intermediate Value Theorem (IVT) and Rolle's Theorem (RT), estimate the smallest distance between the point \((0,0)\) and the curve \(y = e^x\). | [
"0.78"
] | [
"Using the Intermediate Value Theorem (IVT) and Rolle's Theorem (RT), estimate the smallest distance between the point \\((0,0)\\) and the curve \\(y = e^x\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-10c00bc6f23017b7eb8a19e951ad6144464f32caf94fe4bf2f086a96f110d014 | Using the Interpolational Lagrange Formula, evaluate the expression: \[A = \dfrac{a^4}{(a-b)(a-c)} + \dfrac{b^4}{(b-a)(b-c)} + \dfrac{c^4}{(c-a)(c-b)}\] (Hint: $A$ is equal to $B = a^2 + b^2 + c^2 + ab + ac + bc$) | [
"a^2 + b^2 + c^2 + ab + ac + bc"
] | [
"Using the Interpolational Lagrange Formula, evaluate the expression:\n\\[A = \\dfrac{a^4}{(a-b)(a-c)} + \\dfrac{b^4}{(b-a)(b-c)} + \\dfrac{c^4}{(c-a)(c-b)}\\]\n(Hint: $A$ is equal to $B = a^2 + b^2 + c^2 + ab + ac + bc$)"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-de7cf173292ee9f3dc40b913c78c8900dfd3b616aa08c94e74dc9d3a5f1c443d | Using the Maclaurin expansion of $\cos x$ and neglecting all terms of degree greater than 4, approximate $\pi$ by substituting $x = \frac{1}{2}\pi$ and solving the resulting quartic polynomial equation. To how many significant figures is this approximation accurate? | [
"2"
] | [
"Using the Maclaurin expansion of $\\cos x$ and neglecting all terms of degree greater than 4, approximate $\\pi$ by substituting $x = \\frac{1}{2}\\pi$ and solving the resulting quartic polynomial equation. To how many significant figures is this approximation accurate?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bae7d46efe76fd8d11fca618114a6e79418d873ef227c39aab4e636f86b4621b | Using the Maclaurin expansion of $\cos x$ and neglecting all terms of degree greater than 4, approximate $\pi$ by substituting $x = \frac{1}{3}\pi$ and solving the resulting quartic polynomial equation. To how many significant figures is this approximation accurate? | [
"3"
] | [
"Using the Maclaurin expansion of $\\cos x$ and neglecting all terms of degree greater than 4, approximate $\\pi$ by substituting $x = \\frac{1}{3}\\pi$ and solving the resulting quartic polynomial equation. To how many significant figures is this approximation accurate?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-162cd2778e882b65fe545d16ddc00ea0692b91fa504676e2ed81b0a24a776ad0 | Using the Maclaurin expansion of $\cos x$ and neglecting terms of degree greater than 4, find an approximate value for $\pi$ by substituting $x = \frac{1}{2} \pi$ and solving the resulting quartic polynomial equation. Determine the accuracy of this approximation in terms of significant figures. | [
"3.185"
] | [
"Using the Maclaurin expansion of $\\cos x$ and neglecting terms of degree greater than 4, find an approximate value for $\\pi$ by substituting $x = \\frac{1}{2} \\pi$ and solving the resulting quartic polynomial equation. Determine the accuracy of this approximation in terms of significant figures."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 7
} | 18 | 0 | 0 |
omr-ws-b15ccb2f7b2bf9683171b9ae0ca7d135333d2f794da6dbee9acc6a0c0fd30421 | Using the Maclaurin expansion of $\cos x$ and neglecting terms of degree greater than 4, find an approximate value for $\pi$ by substituting $x = \frac{1}{3} \pi$ and solving the resulting quartic polynomial equation. Determine the accuracy of this approximation in terms of significant figures. | [
"3.1479"
] | [
"Using the Maclaurin expansion of $\\cos x$ and neglecting terms of degree greater than 4, find an approximate value for $\\pi$ by substituting $x = \\frac{1}{3} \\pi$ and solving the resulting quartic polynomial equation. Determine the accuracy of this approximation in terms of significant figures."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-febbc338a9d13b58fe759ffc7a1027dbaa51f2badef36bca9ae023b84c6af52a | Using the Maclaurin series for $\sinh x$, find the sum of the infinite series \[ \frac{1}{2} + \frac{9}{2 \times 4 \times 6} + \frac{9^2}{2 \times 4 \times 6 \times 8 \times 10} + \frac{9^3}{2 \times 4 \times 6 \times 8 \times 10 \times 12 \times 14} + \cdots \] | [
"\\(\\frac{1}{3} \\sinh \\left(\\frac{3}{2}\\right)\\)"
] | [
"Using the Maclaurin series for $\\sinh x$, find the sum of the infinite series\n\n\\[\n\\frac{1}{2} + \\frac{9}{2 \\times 4 \\times 6} + \\frac{9^2}{2 \\times 4 \\times 6 \\times 8 \\times 10} + \\frac{9^3}{2 \\times 4 \\times 6 \\times 8 \\times 10 \\times 12 \\times 14} + \\cdots\n\\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-b7de50680fa36277315c511f60d8729441c2ba098b4091e4e9bfcd4fb57ad14d | Using the Mean Value Theorem for \( f(x) = \ln x \) on the interval \( x \in [e, \pi] \), find the value of \( k \) such that \( e^e \cdot \pi^\pi = e^{2\pi + k} \). | [
"e + \\pi \\ln \\pi - 2\\pi"
] | [
"Using the Mean Value Theorem for \\( f(x) = \\ln x \\) on the interval \\( x \\in [e, \\pi] \\), find the value of \\( k \\) such that \\( e^e \\cdot \\pi^\\pi = e^{2\\pi + k} \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-21c7e5794ef7bfae503ae2bbb62af9c06226aeaf45c57bd4e4cccadefe00255a | Using the Newton-polygon algorithm for computing Puiseux series of the algebraic curve given by the equation \( f(x,y) = y^3 - 5y^2 + 8y - x - 4 = 0 \), how can one handle the situation where the Newton polygon lies along the axes, and how to obtain the correct slopes of one and one-half for the Puiseux series expansio... | [
"y = 1 + x + \\cdots \\quad \\text{and} \\quad y = 2 \\pm x^{1/2} + \\cdots"
] | [
"Using the Newton-polygon algorithm for computing Puiseux series of the algebraic curve given by the equation \\( f(x,y) = y^3 - 5y^2 + 8y - x - 4 = 0 \\), how can one handle the situation where the Newton polygon lies along the axes, and how to obtain the correct slopes of one and one-half for the Puiseux series e... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 5
} | 11 | 0 | 0 |
omr-ws-31c0603e4a6c6156ee115e8b9652cdb504e1e7e351443c25d46118403c7b8929 | Using the Peano Postulates and the definition of addition, find the value of \(a + b\) where \(a\) is the successor of \(0\) and \(b\) is also the successor of \(0\). | [
"2"
] | [
"Using the Peano Postulates and the definition of addition, find the value of \\(a + b\\) where \\(a\\) is the successor of \\(0\\) and \\(b\\) is also the successor of \\(0\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3bdf9f88084ade5964cda8cfe84246258c185dea658dbd3e603699d6cb1e3a86 | Using the Principle of Inclusion-Exclusion (PIE), determine the number of 4-letter words where at least two consecutive letters are the same, and explain why the given calculation is incorrect. | [
"406250"
] | [
"Using the Principle of Inclusion-Exclusion (PIE), determine the number of 4-letter words where at least two consecutive letters are the same, and explain why the given calculation is incorrect."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3f357ebe873cb08938ad463f434cd97f9f11e15c883e77f6392cdfb331c9c44b | Using the Principle of Inclusion-Exclusion, calculate the exact number of ways to roll five six-sided dice such that at least three dice show a six, and subtract from it the number of ways exactly three dice show a six. What is the resulting number? | [
"26"
] | [
"Using the Principle of Inclusion-Exclusion, calculate the exact number of ways to roll five six-sided dice such that at least three dice show a six, and subtract from it the number of ways exactly three dice show a six. What is the resulting number?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f8ac78457bda24ad10e55007f4b35c63bdfcd1e7e5f2e3f26d7e05c50a4891c7 | Using the Principle of Inclusion-Exclusion, find the number of primes less than $100$. | [
"25"
] | [
"Using the Principle of Inclusion-Exclusion, find the number of primes less than $100$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3dfd098ecece77c2df8c8334b2272bf80af5295cd354342a7258e7bc58121ed3 | Using the Pythagorean theorem, a triple integral, and infinitesimal calculus, calculate the volume \( V \) of a sphere with radius \( R = 5 \). Express your answer in terms of \( \pi \). | [
"\\dfrac{500}{3}\\pi"
] | [
"Using the Pythagorean theorem, a triple integral, and infinitesimal calculus, calculate the volume \\( V \\) of a sphere with radius \\( R = 5 \\). Express your answer in terms of \\( \\pi \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-afc55606304d75827e2629fb0320dd4fc3b84e6edf3a0e2470d6d59e9eeb9784 | Using the Rational Root Theorem, find all possible rational roots of the polynomial \( f(x) = 4x^4 - 5x^3 + 10x^2 - 20x - 24 \). | [
"\\(\\pm 1, \\pm 2, \\pm 3, \\pm 4, \\pm 6, \\pm 8, \\pm 12, \\pm 24, \\pm \\frac{1}{2}, \\pm \\frac{3}{2}, \\pm \\frac{1}{4}, \\pm \\frac{3}{4}\\)"
] | [
"Using the Rational Root Theorem, find all possible rational roots of the polynomial \\( f(x) = 4x^4 - 5x^3 + 10x^2 - 20x - 24 \\)."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-97cc740bee81b6422115a7a45304f0ae9f6f7c6e7fe9caf1fd1541b2fe104989 | Using the Residue Theorem and an appropriate Laurent series, obtain an exact expression (not involving integrals) for the following integral: \[ \mathop\oint\limits_{C_k}\sqrt{(z - 1)(z - 2)(z - 3)}\,dz \] where \( C_k(t): [0, 2\pi] \to re^{it} \) with \( 2 < r < 3 \) and tracking for each value of the integral, entire... | [
"0"
] | [
"Using the Residue Theorem and an appropriate Laurent series, obtain an exact expression (not involving integrals) for the following integral:\n\\[ \\mathop\\oint\\limits_{C_k}\\sqrt{(z - 1)(z - 2)(z - 3)}\\,dz \\]\nwhere \\( C_k(t): [0, 2\\pi] \\to re^{it} \\) with \\( 2 < r < 3 \\) and tracking for each value of ... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-aeae0a26206194b79be78dbdeb7ae53f62e18129d536c526c2bdb86a69fb2901 | Using the Residue Theorem, calculate the integral $\int_{0}^{2\pi}e^{\cos x}\cos (\sin x)dx$. | [
"\\(2\\pi\\)"
] | [
"Using the Residue Theorem, calculate the integral $\\int_{0}^{2\\pi}e^{\\cos x}\\cos (\\sin x)dx$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-4e285924615a6f9a33fed31af7f7acae2e016dd5e11fa55e218ad96e35e6203e | Using the Residue theorem, compute the integral $\int_{0}^{+\infty}\frac{\sqrt{x}\log(x)}{x^3-1}dx$. | [
"\\(\\frac{\\pi^2}{9}\\)"
] | [
"Using the Residue theorem, compute the integral $\\int_{0}^{+\\infty}\\frac{\\sqrt{x}\\log(x)}{x^3-1}dx$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 1,
"tir": 2
} | 10 | 0 | 0 |
omr-ws-e504fb82eeb974c0561460563d4c6baf0798b92f82bc68942125cbfba29b5cad | Using the Riemann summation, find the expression of the integral: \[ I(x) = \int_{0}^{2\pi} \ln(x^2 - 2x \cos(t) + 1) \, dt. \] | [
"\n\\begin{cases}\n0 & \\text{if } |x| \\leq 1, \\\\\n4\\pi \\ln |x| & \\text{if } |x| > 1.\n\\end{cases}\n"
] | [
"Using the Riemann summation, find the expression of the integral:\n\\[ I(x) = \\int_{0}^{2\\pi} \\ln(x^2 - 2x \\cos(t) + 1) \\, dt. \\]"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 4
} | 17 | 0 | 0 |
omr-ws-0d30a1782892dddb45888fc77f3b5b5d1c000fa0cc58455e89c36ed4f4fba9f7 | Using the Schwarz inequality \((\int_{a}^{b} f(x) g(x) \ dx)^{2} \leq (\int_{a}^{b} (f(x)^{2}\ dx) (\int_{a}^{b} (g(x))^{2}\ dx)\), set \( f(x) = 1 \), and choose \( g(x) = e^{x} \), \( a = 0 \), and \( b = t \) for \( t > 0 \). Calculate the value of \( k \) such that: \[ \frac{e^{t} -1}{e^{t} +1} \leq \frac{kt}{2}. \... | [
"1"
] | [
"Using the Schwarz inequality \\((\\int_{a}^{b} f(x) g(x) \\ dx)^{2} \\leq (\\int_{a}^{b} (f(x)^{2}\\ dx) (\\int_{a}^{b} (g(x))^{2}\\ dx)\\), set \\( f(x) = 1 \\), and choose \\( g(x) = e^{x} \\), \\( a = 0 \\), and \\( b = t \\) for \\( t > 0 \\). Calculate the value of \\( k \\) such that:\n\n\\[\n\\frac{e^{t} -1... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bfceaca99bee17320210f41c40fab44705487f69d71ef11ac13051ab10c2a77a | Using the Schwarz inequality \((\int_{a}^{b} f(x) g(x) \ dx)^{2} \leq (\int_{a}^{b} (f(x)^{2}\ dx) (\int_{a}^{b} (g(x))^{2}\ dx)\), with \( f(x) = x \) and a suitable choice of \( g(x) \), find the smallest constant \( C \) such that: \[ \int_{0}^{1} e^{- \frac{1}{2} x^{2}}\ dx \geq C (1- e^{- \frac{1}{4}})^{2}. \] | [
"12"
] | [
"Using the Schwarz inequality \\((\\int_{a}^{b} f(x) g(x) \\ dx)^{2} \\leq (\\int_{a}^{b} (f(x)^{2}\\ dx) (\\int_{a}^{b} (g(x))^{2}\\ dx)\\), with \\( f(x) = x \\) and a suitable choice of \\( g(x) \\), find the smallest constant \\( C \\) such that:\n\n\\[\n\\int_{0}^{1} e^{- \\frac{1}{2} x^{2}}\\ dx \\geq C (1- e... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-165b0a90a8ec9eea9abcb29504f7c8d96104dc3e15fc81b4b789a4fc4c6db522 | Using the Taylor series expansions for $e^{i\cdot x}$, $\cos{x}$, and $\sin{x}$, find the value of the expression $a + bi$ where $a$ and $b$ are the real and imaginary parts, respectively, of $e^{i\cdot \frac{\pi}{4}}$. | [
"\\dfrac{\\sqrt{2}}{2} + \\dfrac{\\sqrt{2}}{2}i"
] | [
"Using the Taylor series expansions for $e^{i\\cdot x}$, $\\cos{x}$, and $\\sin{x}$, find the value of the expression $a + bi$ where $a$ and $b$ are the real and imaginary parts, respectively, of $e^{i\\cdot \\frac{\\pi}{4}}$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-daf11a6132667c2102b7fd25c76c451752b3100284d2ea57f0b761d20176b679 | Using the Taylor series for $e^x$ up to the term in $x^n$ together with the remainder term, determine the smallest integer $n$ to ensure that the estimated value of $e$ is correct to four decimal places, and find the estimated value. | [
"\\( n = 9 \\), estimated value \\( e \\approx 2.7183 \\)"
] | [
"Using the Taylor series for $e^x$ up to the term in $x^n$ together with the remainder term, determine the smallest integer $n$ to ensure that the estimated value of $e$ is correct to four decimal places, and find the estimated value."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-d4650f94a633709e71cd8009300912d8a13cdebb1c848d6db1928e937c4e5052 | Using the Von Neumann construction of natural numbers, let \( m \) and \( n \) be natural numbers such that \( m \in n \). Define \( f(m, n) \) to be 1 if \( s(m) = n \), and 2 if \( s(m) \in n \). Calculate the sum of all possible values of \( f(m, n) \) for a given \( m \) and \( n \) where \( m \in n \). | [
"3"
] | [
"Using the Von Neumann construction of natural numbers, let \\( m \\) and \\( n \\) be natural numbers such that \\( m \\in n \\). Define \\( f(m, n) \\) to be 1 if \\( s(m) = n \\), and 2 if \\( s(m) \\in n \\). Calculate the sum of all possible values of \\( f(m, n) \\) for a given \\( m \\) and \\( n \\) where \... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-796de8df95b568246adc451f10a82af180e4d62434176e2b57e7e2dd8eee2d35 | Using the \(\delta\)-\(\epsilon\) definition of a limit, find the largest possible value of \(\delta\) such that for any \(\epsilon > 0\), if \(0 < |x - 2| < \delta\), then \(|x^3 - 8| < \epsilon\). Given that for \(1 < x < 3\), it follows that \(|x^3 - 8| \leq 19|x - 2|\), express \(\delta\) in terms of \(\epsilon\). | [
"\\min\\left\\{1, \\dfrac{\\epsilon}{19}\\right\\}"
] | [
"Using the \\(\\delta\\)-\\(\\epsilon\\) definition of a limit, find the largest possible value of \\(\\delta\\) such that for any \\(\\epsilon > 0\\), if \\(0 < |x - 2| < \\delta\\), then \\(|x^3 - 8| < \\epsilon\\). Given that for \\(1 < x < 3\\), it follows that \\(|x^3 - 8| \\leq 19|x - 2|\\), express \\(\\delt... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3d4230edfe5d5d9fe94e50b53e1b5664ef3e27f34583c33ae00496a1ed29d74a | Using the area integral \( \int y \frac{dx}{dt} \, dt \), find the area between the curve described by the parametric equations and the x-axis for \( \pi \geq t \geq t_0 \). | [
"\\(\\frac{7\\pi^{2}+36}{48}\\)"
] | [
"Using the area integral \\( \\int y \\frac{dx}{dt} \\, dt \\), find the area between the curve described by the parametric equations and the x-axis for \\( \\pi \\geq t \\geq t_0 \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-532b9c431623fa7238a762d6597cad8ebd1cae6426465590d6315e0443f61347 | Using the area of a regular pentagon, calculate the value of $4\sin \frac{2\pi}{5} + \tan \frac{2\pi}{5} - 5\cot \frac{\pi}{5}$. | [
"0"
] | [
"Using the area of a regular pentagon, calculate the value of $4\\sin \\frac{2\\pi}{5} + \\tan \\frac{2\\pi}{5} - 5\\cot \\frac{\\pi}{5}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e3c70c5b17b8455ebd7f6cffa6cf454662faaac4b355da0427603cfb338e98d5 | Using the arithmetic operations and the numbers $5, 5, 3, 3$, obtain the number $100$. | [
"\\(\\frac{5 \\cdot 3}{0.5 \\cdot 0.3}\\)"
] | [
"Using the arithmetic operations and the numbers $5, 5, 3, 3$, obtain the number $100$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 2,
"tir": 6
} | 15 | 0 | 0 |
omr-ws-50d78fdc1acbd52e47f0ad3ad1bf10994f6663aa245e2384c09071a5b96d56e2 | Using the axiom of choice, construct a function \( f: \mathbb{R} \to \mathbb{R} \setminus \{0\} \) such that \( f(P) = \mathbb{R} \setminus \{0\} \) for every non-degenerate interval \( P \subset \mathbb{R} \). | [
"f(x) = \\begin{cases} \ny & \\text{if } x \\in A_y \\text{ where } \\{A_y\\}_{y \\in \\mathbb{R} \\setminus \\{0\\}} \\text{ is a partition of } \\mathbb{R} \\text{ into dense subsets} \\\\\n\\end{cases}"
] | [
"Using the axiom of choice, construct a function \\( f: \\mathbb{R} \\to \\mathbb{R} \\setminus \\{0\\} \\) such that \\( f(P) = \\mathbb{R} \\setminus \\{0\\} \\) for every non-degenerate interval \\( P \\subset \\mathbb{R} \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-bb6e06ad41f0704cd6dfa6eecd865e88b03551489c8a14833083cc5cd92a5a40 | Using the balance-scale shown, determine how many ♦ are necessary to replace the ? in the last picture. The given balance equations are: 1. ▼♦ = ○ 2. ▼ = ♦◘ 3. ○○ = ◘◘◘ 4. ▼ = ? Explain your reasoning. | [
"5"
] | [
"Using the balance-scale shown, determine how many ♦ are necessary to replace the ? in the last picture. The given balance equations are:\n1. ▼♦ = ○\n2. ▼ = ♦◘\n3. ○○ = ◘◘◘\n4. ▼ = ?\n\nExplain your reasoning."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cdf168d6c9aee324893617166c56b26fc1477577293b3df31513c96135636b79 | Using the binomial expansion of $\sqrt{(1-x)}$ with $x=\frac{1}{100}$, find the approximate value of $\sqrt{11}$ as a fraction in the form $\frac{a}{b}$, where $a$ and $b$ are integers. Then, calculate the absolute error of this approximation to $\sqrt{11}$, correct to one significant figure. Provide your answer as the... | [
"34533.0000002"
] | [
"Using the binomial expansion of $\\sqrt{(1-x)}$ with $x=\\frac{1}{100}$, find the approximate value of $\\sqrt{11}$ as a fraction in the form $\\frac{a}{b}$, where $a$ and $b$ are integers. Then, calculate the absolute error of this approximation to $\\sqrt{11}$, correct to one significant figure. Provide your ans... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-fcf8e929f48170d82c6596466aafa763cf54120c36779dfe66cd5aff41b38357 | Using the binomial expansion, calculate the exact value of $(1+\sqrt{2})^{5}$ and express it in the form $a+b\sqrt{2}$, where $a$ and $b$ are integers. Find the value of $a+b$. | [
"70"
] | [
"Using the binomial expansion, calculate the exact value of $(1+\\sqrt{2})^{5}$ and express it in the form $a+b\\sqrt{2}$, where $a$ and $b$ are integers. Find the value of $a+b$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e0872c1177354294d688cce534b32ea9aa9982a45726a1046aef98ed46745e96 | Using the binomial theorem, determine the ninth digit from the right in the number \(101^{10}\). | [
"1"
] | [
"Using the binomial theorem, determine the ninth digit from the right in the number \\(101^{10}\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.