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-85719061dbe2907e86c03f247b6583378c4e971ca3145404ce33332f6b57e59a | Using the classification of finite abelian groups, determine all finite abelian groups up to isomorphism of order 450. | [
"$\\mathbb{Z}/450\\mathbb{Z}, \\mathbb{Z}/3\\mathbb{Z} \\times \\mathbb{Z}/3\\mathbb{Z} \\times \\mathbb{Z}/25\\mathbb{Z} \\times \\mathbb{Z}/2\\mathbb{Z}, \\mathbb{Z}/3\\mathbb{Z} \\times \\mathbb{Z}/3\\mathbb{Z} \\times \\mathbb{Z}/5\\mathbb{Z} \\times \\mathbb{Z}/5\\mathbb{Z} \\times \\mathbb{Z}/2\\mathbb{Z}, \\... | [
"Using the classification of finite abelian groups, determine all finite abelian groups up to isomorphism of order 450."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-1415308828f8858a3d17c12ea87be3a77c06b44d43033db9841c23e32359c980 | Using the coordinates of points \( A \) and \( B \), and the angles \( P \) and \( K \) (which are equal), derive a formula to calculate all the points along the arc connecting \( A \) and \( B \) that passes through \( H \). | [
"\\( (146.67, 214.14) \\)"
] | [
"Using the coordinates of points \\( A \\) and \\( B \\), and the angles \\( P \\) and \\( K \\) (which are equal), derive a formula to calculate all the points along the arc connecting \\( A \\) and \\( B \\) that passes through \\( H \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c1f7d2f10e2e2d89dd69ccf973519e2daab160474626276a1cc717f989553c63 | Using the definition $\ln x = \sum_{k \geq 0} \frac{2}{2k+1} \left( \frac{x-1}{x+1} \right)^{2k+1}$ for $x > 0$, calculate the value of $\ln(8) - \ln(2) - \ln(4)$. | [
"0"
] | [
"Using the definition $\\ln x = \\sum_{k \\geq 0} \\frac{2}{2k+1} \\left( \\frac{x-1}{x+1} \\right)^{2k+1}$ for $x > 0$, calculate the value of $\\ln(8) - \\ln(2) - \\ln(4)$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0270df488edc9febb77596f0076c60c9be0b7c99b61583f3b0a139b16a21e434 | Using the definition of the Gamma function given by \[ \Gamma(x) = \int_{0}^\infty e^{-y} \cdot y^{x-1} \, dy, \] calculate the value of \( \Gamma'(3) \) in terms of \( \gamma \) and \( H_3 \), where \( H_3 \) is the 3rd harmonic number and \( \gamma \) is Euler's constant. Express your answer as a simplified fraction ... | [
"2 H_3 - 2 \\gamma - \\dfrac{2}{3}"
] | [
"Using the definition of the Gamma function given by\n\n\\[ \\Gamma(x) = \\int_{0}^\\infty e^{-y} \\cdot y^{x-1} \\, dy, \\]\n\ncalculate the value of \\( \\Gamma'(3) \\) in terms of \\( \\gamma \\) and \\( H_3 \\), where \\( H_3 \\) is the 3rd harmonic number and \\( \\gamma \\) is Euler's constant. Express your a... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-872e11628127c31d341c5aa39a49f23a1a3bed42eb71d4c5b59f25a4a30731c1 | Using the definition of the derivative, find the derivative of the function \( f(z) = \arg(z) \) and determine where this derivative is defined. | [
"The derivative of \\( f(z) = \\arg(z) \\) is not defined anywhere."
] | [
"Using the definition of the derivative, find the derivative of the function \\( f(z) = \\arg(z) \\) and determine where this derivative is defined."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 26
} | 48 | 0 | 0 |
omr-ws-e0c94d776e1af1587604ccd68889b05626c53646b0c44033276b5b399895521c | Using the definition of the derivative, find the derivative of the function \( g(z) = x + y \) (where \( z = x + iy \)) and determine where this derivative is defined. | [
"The derivative is not defined anywhere."
] | [
"Using the definition of the derivative, find the derivative of the function \\( g(z) = x + y \\) (where \\( z = x + iy \\)) and determine where this derivative is defined."
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-320364a400676a1025d6218a60464cc37bf14ff3d044ce5cef8ccfcb07568794 | Using the definitions \(\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}\) and \(\Gamma(s) = \int_{0}^\infty x^{s-1} e^{-x} \, dx\), calculate the value of \( \int_{0}^\infty \frac{y^{3}}{e^y - 1} \, dy \). | [
"\\dfrac{\\pi^4}{15}"
] | [
"Using the definitions \\(\\zeta(s) = \\sum_{n=1}^\\infty \\frac{1}{n^s}\\) and \\(\\Gamma(s) = \\int_{0}^\\infty x^{s-1} e^{-x} \\, dx\\), calculate the value of \\( \\int_{0}^\\infty \\frac{y^{3}}{e^y - 1} \\, dy \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5b442e738f787b6980946f334e446084601874022cc5549471afd25d733ac877 | Using the definitions of \( \sinh x \) and \( \cosh x \) in terms of \( e^{x} \) and \( e^{-x} \), calculate the value of \( \cosh x \cosh y - \sinh x \sinh y \) when \( x = \ln(3) \) and \( y = \ln(2) \). Express your answer as a simplified fraction. | [
"\\dfrac{13}{12}"
] | [
"Using the definitions of \\( \\sinh x \\) and \\( \\cosh x \\) in terms of \\( e^{x} \\) and \\( e^{-x} \\), calculate the value of \\( \\cosh x \\cosh y - \\sinh x \\sinh y \\) when \\( x = \\ln(3) \\) and \\( y = \\ln(2) \\). Express your answer as a simplified fraction."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-192c12e25a52d91dec919063c47b0e25f93df109b19e67efd1cab380f7134acf | Using the diagonalization of matrix \( A \), find the fundamental matrix \( \Phi(t) \) with \( \Phi(0) = I \). If \( x_0 = \begin{bmatrix} -1 \\ 0 \end{bmatrix} \), find \( x(t) \) for \( t > 0 \). | [
"\\( \\begin{bmatrix} -\\cos(\\omega t) \\\\ \\omega \\sin(\\omega t) \\end{bmatrix} \\)"
] | [
"Using the diagonalization of matrix \\( A \\), find the fundamental matrix \\( \\Phi(t) \\) with \\( \\Phi(0) = I \\). If \\( x_0 = \\begin{bmatrix} -1 \\\\ 0 \\end{bmatrix} \\), find \\( x(t) \\) for \\( t > 0 \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-219d6e79d318253cec95228016d991d11db86750e9e2a905d628ca620e4bcc31 | Using the digit 2 exactly three times and mathematical symbols, express all numbers from 1 to 12. No English alphabets, words like $\sin$ or $\lim$, or other digits are allowed. | [
"\\( 1 = 2^{2-2}, 2 = 2 + 2 - 2, 3 = 2 + \\frac{2}{2}, 4 = \\sqrt{2 \\times 2}^2, 5 = 2 \\times 2 + \\lfloor \\sqrt{2} \\rfloor, 6 = 2 + 2 + 2, 7 = \\lceil \\sqrt{(2+2)!} \\rceil + 2, 8 = 2 \\times 2 \\times 2, 9 = (2 + \\lfloor \\sqrt{2} \\rfloor)^2, 10 = 2 \\times \\lceil \\sqrt{(2+2)!} \\rceil, 11 = \\frac{22}{2... | [
"Using the digit 2 exactly three times and mathematical symbols, express all numbers from 1 to 12. No English alphabets, words like $\\sin$ or $\\lim$, or other digits are allowed."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c3272cf8890fbc225523b840f029b8dfebba5748ebc26c41c0c8135362b2bbc6 | Using the digit 3 three times and any combination of operators, create the number 8. You may not append digits (e.g., 33 + 3 is not allowed). (Note: Square root is considered not to use the digit 2.) | [
"\\(\\left(\\frac{3!}{3}\\right)^3\\)"
] | [
"Using the digit 3 three times and any combination of operators, create the number 8. You may not append digits (e.g., 33 + 3 is not allowed). (Note: Square root is considered not to use the digit 2.)"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 19
} | 40 | 0 | 0 |
omr-ws-7eb7cd167e847cfde4de0f3560eb1fbb992e0cb248d3cc94826427343c219436 | Using the digit 9 and the four basic operations (+, −, *, /), parentheses, exponentiation, factorial, and square root, construct the number 2023 using as few 9's as possible. | [
"6"
] | [
"Using the digit 9 and the four basic operations (+, −, *, /), parentheses, exponentiation, factorial, and square root, construct the number 2023 using as few 9's as possible."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 1
} | 2 | 0 | 0 |
omr-ws-546b52c1c94542fa616fa69a2a0a7e8cd7c97e07491ba05d050011388cace72b | Using the digits $(0,1,2,3,4,5,6,7,8,9)$, the four arithmetic operations $(+,-, \times, /)$, exponentiation $(^)$, factorial $(!)$, decimal point $(.)$, and one pair of parentheses exactly one time, form an expression with the maximum possible result. The order of operations is: $1$. (Factorial), $2$. (Exponentiation),... | [
"(9.8)^{76543210!}"
] | [
"Using the digits $(0,1,2,3,4,5,6,7,8,9)$, the four arithmetic operations $(+,-, \\times, /)$, exponentiation $(^)$, factorial $(!)$, decimal point $(.)$, and one pair of parentheses exactly one time, form an expression with the maximum possible result. The order of operations is: $1$. (Factorial), $2$. (Exponentia... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 2
} | 4 | 0 | 0 |
omr-ws-9c1af9b095ca6da169b304a80d3dffc5523a51fff310ee8153d34df6321aefab | Using the digits $1$ to $9$, construct two numbers such that their product is maximal. | [
"\\(9642 \\times 87531\\)"
] | [
"Using the digits $1$ to $9$, construct two numbers such that their product is maximal."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 5,
"tir": 23
} | 34 | 0 | 0 |
omr-ws-4fc2d683ab6bf45ed8de78e20dba68e6659d3541bd1442b683578d594902b0e0 | Using the digits $1$ to $9$, construct two numbers such that their product is minimal. | [
"23456789"
] | [
"Using the digits $1$ to $9$, construct two numbers such that their product is minimal."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 6,
"tir": 3
} | 21 | 0 | 0 |
omr-ws-ddaff4d0bd3ee1b6478ef9cbfa77648f3829e4b51766b127dbdc631441bc50e6 | Using the digits $1, 2, 3, 4, 5$, players $A$ and $B$ compose a $2005$-digit number $N$ by selecting one digit at a time, with $A$ selecting the first digit, $B$ the second, $A$ the third, and so on. Player $A$ wins if and only if $N$ is divisible by $9$. Who will win if both players play optimally? | [
"B wins."
] | [
"Using the digits $1, 2, 3, 4, 5$, players $A$ and $B$ compose a $2005$-digit number $N$ by selecting one digit at a time, with $A$ selecting the first digit, $B$ the second, $A$ the third, and so on. Player $A$ 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": 16,
"genselect": 7,
"tir": 3
} | 26 | 0 | 0 |
omr-ws-7d53bff60d6e82abd9da38a1c324ddf032156b70b8ae32e81d37b80402fbe4f6 | Using the digits $1, 2, 3, 4, 5, 6$ without repetition, three two-digit numbers are formed. The numbers are then added together. How many different sums can be obtained through this procedure? | [
"10"
] | [
"Using the digits $1, 2, 3, 4, 5, 6$ without repetition, three two-digit numbers are formed. The numbers are then added together. How many different sums can be obtained through this procedure?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-9c35e47290964a2db945e46770652376ffb5c5781987afced4aaafc047a54637 | Using the digits $1, 2, 3, 4, 5, 6, 7, 8, 9$ exactly once and the mathematical operations of multiplication (*) and power (^, as in $a^{b}$), how close can you come to the number $123456789$? You can use parentheses as required. | [
"\\(1 \\cdot 2^{9} \\cdot 4 \\cdot 5 \\cdot 6^{3} \\cdot 7 \\cdot 8 = 123,863,040\\)"
] | [
"Using the digits $1, 2, 3, 4, 5, 6, 7, 8, 9$ exactly once and the mathematical operations of multiplication (*) and power (^, as in $a^{b}$), how close can you come to the number $123456789$? You can use parentheses as required."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ba589bd266020471d29b8feafc85554cfcd80dca85e9003afe7ec964e4a47815 | Using the digits $1, 2, 3, 4,$ and $5$ without repetition, the number of $5$-digit numbers is $5! = 120$. If these numbers are listed in increasing order, what is the number that comes in the $60$th place? | [
"32541"
] | [
"Using the digits $1, 2, 3, 4,$ and $5$ without repetition, the number of $5$-digit numbers is $5! = 120$. If these numbers are listed in increasing order, what is the number that comes in the $60$th place?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4289beea163e835995f52bc74dd86dc37c9d86fe2244ad4bf96e8ad51d30f6f5 | Using the digits $1, 3, 5, 7, 9$, how many different numbers less than 130 can be formed? | [
"9"
] | [
"Using the digits $1, 3, 5, 7, 9$, how many different numbers less than 130 can be formed?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6f6763d338a3772efa4a9fb04e2ef1b8d059b81100aacdef03dddd82094f70c7 | Using the digits ${1,3,5,7,9}$ exactly once each, form two integers such that their product is maximized. What are these two integers? | [
"\\(751\\) and \\(93\\)"
] | [
"Using the digits ${1,3,5,7,9}$ exactly once each, form two integers such that their product is maximized. What are these two integers?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 23
} | 23 | 0 | 0 |
omr-ws-985408f789e6ab8c301f26c3bc80a82a4a06b405fbb796a7fcc25c3e8ad0437f | Using the digits 0, 1, 2, 3, 4, 5, 6, how many five-digit numbers greater than 24000 can be made? | [
"10633"
] | [
"Using the digits 0, 1, 2, 3, 4, 5, 6, how many five-digit numbers greater than 24000 can be made?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4988c199400ac70b9bdad4a21b1b869a63ba33113c9762971eedb951be2db94d | Using the digits 0, 1, 2, 3, 4, and 5, find the number of 10-digit numbers such that the difference between consecutive digits is always 1. | [
"990"
] | [
"Using the digits 0, 1, 2, 3, 4, and 5, find the number of 10-digit numbers such that the difference between consecutive digits is always 1."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 7
} | 20 | 0 | 0 |
omr-ws-24e4bcea307bf7b1d640aba58bbce96f0cff306aaba627aa527a45a8543fe9e4 | Using the digits 0, 1, 2, 3, and 4, find the number of 13-digit sequences such that the difference between any two consecutive digits is 1. Examples of such sequences include 0123432123432, 2323432321234, and 3210101234323. | [
"3402"
] | [
"Using the digits 0, 1, 2, 3, and 4, find the number of 13-digit sequences such that the difference between any two consecutive digits is 1. Examples of such sequences include 0123432123432, 2323432321234, and 3210101234323."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b3c0fbcd709b6f599a88ad3faf5f8e72ec154a69f9af88f8552ac3d802c36e7a | Using the digits 0, 1, 3, 5, and 7 without repetition, how many 5-digit natural numbers can be formed which are divisible by 5? | [
"42"
] | [
"Using the digits 0, 1, 3, 5, and 7 without repetition, how many 5-digit natural numbers can be formed which are divisible by 5?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-b6093a8938800ae6a9097b668c91f93bfb12ba717aa2a6d6779311940866a363 | Using the digits 1 through 7 once each, how many numbers are there where the digits alternate between odd and even? | [
"144"
] | [
"Using the digits 1 through 7 once each, how many numbers are there where the digits alternate between odd and even?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-5104256d4bd494ad615b76713df1cdff6a99b4d47d7084f2759271b7146a0113 | Using the digits 1 through 9 exactly once, construct two numbers such that their product is maximized. | [
"9642 \\times 87531"
] | [
"Using the digits 1 through 9 exactly once, construct two numbers such that their product is maximized."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 7
} | 18 | 0 | 0 |
omr-ws-8e99d2e7fa485593b5186099715d2033a782f20d61bcd63862577aaf8844d244 | Using the digits 1 through 9 exactly once, construct two numbers such that their product is minimized. | [
"23456789"
] | [
"Using the digits 1 through 9 exactly once, construct two numbers such that their product is minimized."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 6
} | 17 | 0 | 0 |
omr-ws-dbf6bd012b0f51d462de0a6bde688e7f335e775ab850df601ac1390d29dbc9c5 | Using the digits 1 through 9 exactly once, form three 3-digit numbers such that their product is maximized. Determine the three 3-digit factors and their product. | [
"763, 852, 941; product = 611,721,516"
] | [
"Using the digits 1 through 9 exactly once, form three 3-digit numbers such that their product is maximized. Determine the three 3-digit factors and their product."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 30
} | 51 | 0 | 0 |
omr-ws-7cf7c7e1e1d884ad47e70b3261845c8d54f1d99dc933bc04dde2e5d96474be74 | Using the digits 1 through 9 exactly once, form three 3-digit numbers. What is the smallest possible product of these three numbers? | [
"13994694"
] | [
"Using the digits 1 through 9 exactly once, form three 3-digit numbers. What is the smallest possible product of these three numbers?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-a6b301bab6051ba3a705087982ea83e12d678a4e66772163b9d7fc34cf31de53 | Using the digits 1 through 9, form two numbers such that their product is as large as possible. Each digit must be used exactly once. Which are the two numbers? | [
"96421 and 8753"
] | [
"Using the digits 1 through 9, form two numbers such that their product is as large as possible. Each digit must be used exactly once. Which are the two numbers?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 4
} | 4 | 0 | 0 |
omr-ws-0a307ea899b0581898d676f623a5f8a76f71c620db6d91a820ccbaca4fed9957 | Using the digits 1 through 9, how many different 9-digit numbers can be formed so that all two adjacent digits are consecutive numbers? Repetitions of digits are allowed. | [
"1370"
] | [
"Using the digits 1 through 9, how many different 9-digit numbers can be formed so that all two adjacent digits are consecutive numbers? Repetitions of digits are allowed."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 23
} | 44 | 0 | 0 |
omr-ws-7d35564912c631b3dd09ddb8b169db5011b01e14669594ca1a90036025bddd64 | Using the digits 1 to 9 exactly once, construct two numbers such that their product is maximal. | [
"9642 \\times 87531"
] | [
"Using the digits 1 to 9 exactly once, construct two numbers such that their product is maximal."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-5bfe3b8febf2c9be5cf199d938a28367fede8856fd304c16e821367e1926f4ae | Using the digits 1 to 9 exactly once, construct two numbers such that their product is minimal. | [
"23456789"
] | [
"Using the digits 1 to 9 exactly once, construct two numbers such that their product is minimal."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 5
} | 17 | 0 | 0 |
omr-ws-9eae953cd5e3c2e3fbb68cf0c6dad062067e45f4ecdbfb0b3d4ba5bf8dd77bc9 | Using the digits 1, 2, 3, 4, 5 and the mathematical operations +, -, *, /, ^, and !, along with concatenation, how close can you come to the number 67890? | [
"\\(67718.7\\)"
] | [
"Using the digits 1, 2, 3, 4, 5 and the mathematical operations +, -, *, /, ^, and !, along with concatenation, how close can you come to the number 67890?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-76cc6991b5f58c2f775be49490ee59eee36e6e3472ccbfe4c4fd21dee7d0d758 | Using the digits 1, 2, 3, 4, 5, how many even three-digit numbers less than 500 can be formed if each digit can be used more than once? | [
"40"
] | [
"Using the digits 1, 2, 3, 4, 5, how many even three-digit numbers less than 500 can be formed if each digit can be used more than once?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ae823d9f4c2be25b6210d23d3dc3e5e38a6e0c1c305fa0c5453c54604389834a | Using the digits 1, 2, 3, 4, and 5 exactly once, form all possible five-digit numbers. Calculate the sum of the squares of the numbers in one of the two groups into which these numbers can be divided such that the sum of the squares of the numbers in each group is equal. Find this sum. | [
"78113218860"
] | [
"Using the digits 1, 2, 3, 4, and 5 exactly once, form all possible five-digit numbers. Calculate the sum of the squares of the numbers in one of the two groups into which these numbers can be divided such that the sum of the squares of the numbers in each group is equal. Find this sum."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-57d00ec8e128be031580534ca2973c76e618db06a69704fa5cb19c20f4b7c254 | Using the digits 1, 2, 3, 4, and 5, how many different 16-digit numbers can be formed such that for every adjacent two digits, either both are the same or at least one of them is 1? | [
"64570081"
] | [
"Using the digits 1, 2, 3, 4, and 5, how many different 16-digit numbers can be formed such that for every adjacent two digits, either both are the same or at least one of them is 1?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bed42b17a627748425ecc9754b963137c004398b5d1f01c672b5986511f99302 | Using the digits 1, 2, 3, 5, and 7, how many 5-digit numbers greater than 50000 can be formed? | [
"48"
] | [
"Using the digits 1, 2, 3, 5, and 7, how many 5-digit numbers greater than 50000 can be formed?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-53d33b1c46cfd8be754d33f3cde514b433b6a532d8b3e7bd9fba6e48abdffa62 | Using the digits 1, 2, 3, and 4 exactly once, along with the operations +, -, ×, /, parentheses, and powers, what is the largest number you can make? | [
"\\((.3)^{-\\left((.2)^{-\\left((.1)^{(-4)}\\right)}\\right)}\\)"
] | [
"Using the digits 1, 2, 3, and 4 exactly once, along with the operations +, -, ×, /, parentheses, and powers, what is the largest number you can make?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5a1978ded773ccc382f6b0846417ae6ebd083b0efc2cd914db1f895d0dd5e4af | Using the digits 1963 and the operations +, -, /, and *, along with bracketing, form mathematical expressions. At least one operation must be placed between two digits, and the minus sign cannot be used as a negative sign. Find the smallest natural number that cannot be expressed in this way. Calculate this number. | [
"14"
] | [
"Using the digits 1963 and the operations +, -, /, and *, along with bracketing, form mathematical expressions. At least one operation must be placed between two digits, and the minus sign cannot be used as a negative sign. Find the smallest natural number that cannot be expressed in this way. Calculate this number... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-a2510e2a21241a61c27407b2df2d614325d65d275b412d90f5b59e7af80c3c24 | Using the digits 2, 3, 5, 6, and 7, form a two-digit number and a three-digit number that give the least product. | [
"9175"
] | [
"Using the digits 2, 3, 5, 6, and 7, form a two-digit number and a three-digit number that give the least product."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 2,
"tir": 20
} | 32 | 0 | 0 |
omr-ws-078ed198d21ec61b4ae3170b13bafe59b931a7c10cf98881c83e2efa31eb9dbe | Using the digits 5, 5, 5, and 1 exactly once, along with the operators +, -, *, and /, form the number 24. Note that expressions like $5*5-1^{5}$ and concatenations like $55$ are not allowed. | [
"\\( 5 \\cdot \\left(5 - \\frac{1}{5}\\right) \\)"
] | [
"Using the digits 5, 5, 5, and 1 exactly once, along with the operators +, -, *, and /, form the number 24. Note that expressions like $5*5-1^{5}$ and concatenations like $55$ are not allowed."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-a75c7b5875e7953fce47089726f097c3891a3d43f5921e7b87c26603deb21fd5 | Using the digits \(a, b, c, d, e, f\) (all distinct from each other), Carlos wrote the three-digit number \(\overline{def}\) and Emilio wrote the three-digit number \(\overline{abc}\). If Carlos' number is greater by 9 than Emilio's number, find the least possible value of \(a + b + c\). | [
"14"
] | [
"Using the digits \\(a, b, c, d, e, f\\) (all distinct from each other), Carlos wrote the three-digit number \\(\\overline{def}\\) and Emilio wrote the three-digit number \\(\\overline{abc}\\). If Carlos' number is greater by 9 than Emilio's number, find the least possible value of \\(a + b + c\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 18
} | 41 | 0 | 0 |
omr-ws-da4b13a9ff8e8f447f951d0d6d605e81a528886cd319f757ca2f85e229049a0b | Using the digits from 0 to 9 without repetition, create a ten-digit number such that the first digit is divisible by 1, the first two digits form a number divisible by 2, the first three digits form a number divisible by 3, and so on, up to the entire ten-digit number being divisible by 10. (Optional) Prove that there ... | [
"3816547290"
] | [
"Using the digits from 0 to 9 without repetition, create a ten-digit number such that the first digit is divisible by 1, the first two digits form a number divisible by 2, the first three digits form a number divisible by 3, and so on, up to the entire ten-digit number being divisible by 10.\n\n(Optional) Prove tha... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 21
} | 34 | 0 | 0 |
omr-ws-08fc5513bbea3f4dfb369f982a402eddc6ce70faa74ba193d78077673e0f3944 | Using the digits from 1 to 10, how many ways can the blanks be filled with +, -, *, /, or nothing (to combine digits) to make the expression equal to 100? | [
"0"
] | [
"Using the digits from 1 to 10, how many ways can the blanks be filled with +, -, *, /, or nothing (to combine digits) to make the expression equal to 100?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-b5ff815151d28681d36379d56327961937cc9136056ca2377c67cd11b5898fc4 | Using the digits from 1 to 10, how many ways can the blanks be filled with +, -, or nothing (to combine digits) to make the expression equal to 100? | [
"0"
] | [
"Using the digits from 1 to 10, how many ways can the blanks be filled with +, -, or nothing (to combine digits) to make the expression equal to 100?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-ddb8a573ae01f2924377afbb01440b936d013a1347e0e89a99d4597acb2d806b | Using the digits from the set {2, 0, 1, 5}, how many n-digit numbers can be formed that are divisible by 3? | [
"\\begin{cases} \n4^{n-1} + 1 & \\text{if } n \\equiv 0 \\pmod{3}, \\\\\n4^{n-1} - 1 & \\text{if } n \\equiv 1 \\pmod{3}, \\\\\n4^{n-1} & \\text{if } n \\equiv 2 \\pmod{3}.\n\\end{cases}"
] | [
"Using the digits from the set {2, 0, 1, 5}, how many n-digit numbers can be formed that are divisible by 3?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-150e5a1db6fb8adbfaa60eda6f40ef10e654cd50e7e5fdce156e6aa492c66e13 | Using the digits of the year 2005 in order and the standard symbols $+$, $-$, $\times$, $\div$, $\sqrt{}$, brackets, and $!$, create the longest unbroken list of consecutive integers starting from 1. Find the longest unbroken list possible and a long list with as few breaks as possible. | [
"27"
] | [
"Using the digits of the year 2005 in order and the standard symbols $+$, $-$, $\\times$, $\\div$, $\\sqrt{}$, brackets, and $!$, create the longest unbroken list of consecutive integers starting from 1. Find the longest unbroken list possible and a long list with as few breaks as possible."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-304a231016598cabef17dd681cf687f71c6da3dd051058595f99b76e5b875b7d | Using the distance formula $d = |x_2 - x_1| + |y_2 - y_1|$, find the locus of all points $(x, y)$ satisfying $x \ge 0$, $y \ge 0$ and which are equidistant from the origin $(0,0)$ and a fixed point $(a,b)$ in the plane, where $a > b$. Distinguish the cases according to the signs of $a$ and $b$. | [
"\\( x = \\frac{a - b}{2} \\) or \\( x + y = \\frac{a + b}{2} \\)"
] | [
"Using the distance formula $d = |x_2 - x_1| + |y_2 - y_1|$, find the locus of all points $(x, y)$ satisfying $x \\ge 0$, $y \\ge 0$ and which are equidistant from the origin $(0,0)$ and a fixed point $(a,b)$ in the plane, where $a > b$. Distinguish the cases according to the signs of $a$ and $b$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 2
} | 4 | 0 | 0 |
omr-ws-90f2470c5b970c0365169f561628460d41c55de942e120e1aeadf8f5559bac1a | Using the distribution of \( T_{k+1} - T_k - T \), determine the pdf for \( T_k \). Is there any difference between the pdf of \( T_k \) and \( T_k - T \)? | [
"\\frac{\\lambda^k t^{k-1} e^{-\\lambda t}}{(k-1)!}"
] | [
"Using the distribution of \\( T_{k+1} - T_k - T \\), determine the pdf for \\( T_k \\). Is there any difference between the pdf of \\( T_k \\) and \\( T_k - T \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-f7d4af2a3fc1c27dc5e1846e146417f816017d9933f3c6cac205a5a9730c6965 | Using the dominated convergence theorem (or otherwise), find \(\lim_{n \to \infty} \mathbb{E}\left(\frac{Y_n}{Y_n + 1}\right)^2\) where \(Y_n = \frac{1}{n} \sum_{i=1}^{n} X_i\) and \(X_1, X_2, \ldots\) are i.i.d. with common mean \(0\) and common finite variance \(\sigma^2\). | [
"0"
] | [
"Using the dominated convergence theorem (or otherwise), find \\(\\lim_{n \\to \\infty} \\mathbb{E}\\left(\\frac{Y_n}{Y_n + 1}\\right)^2\\) where \\(Y_n = \\frac{1}{n} \\sum_{i=1}^{n} X_i\\) and \\(X_1, X_2, \\ldots\\) are i.i.d. with common mean \\(0\\) and common finite variance \\(\\sigma^2\\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d040f04462783ca2f34cafc571b203637b0612abe40c29cf73f23f23819cd143 | Using the encoding matrix \( A = \begin{pmatrix} 2 & 5 \\ 1 & 3 \end{pmatrix} \), how would you encode the message "MATHS IS GREAT" and what message would be received? | [
"YMVOYQOKCDRF"
] | [
"Using the encoding matrix \\( A = \\begin{pmatrix} 2 & 5 \\\\ 1 & 3 \\end{pmatrix} \\), how would you encode the message \"MATHS IS GREAT\" and what message would be received?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-5cdf7b524870615b49442796411e51b98473b18fc63f5e4a69d72e6c90915ba7 | Using the encoding matrix \( A = \begin{pmatrix} 2 & 5 \\ 1 & 3 \end{pmatrix} \), how would you encode the word "MATH" and what "word" would be sent using this code? | [
"EPBR"
] | [
"Using the encoding matrix \\( A = \\begin{pmatrix} 2 & 5 \\\\ 1 & 3 \\end{pmatrix} \\), how would you encode the word \"MATH\" and what \"word\" would be sent using this code?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 3
} | 11 | 0 | 0 |
omr-ws-782ea1d25de601c1089753329e88b64bb7b385b87c1cf92926ce6a960926ca87 | Using the equality \(\overrightarrow{OA_0} + \overrightarrow{A_0A_1} + \overrightarrow{A_1A_2} + \cdots + \overrightarrow{A_nA_{n-1}} = \overrightarrow{OA_{n+1}}\), express the sums \(\sum_{k=0}^n \cos k\alpha\) and \(\sum_{k=0}^n \sin k\alpha\) in terms of \(n\) and \(\alpha\). | [
"\\(\\sum_{k=0}^n \\cos k\\alpha = \\frac{\\sin \\left(\\frac{(n+1)\\alpha}{2}\\right) \\cos \\left(\\frac{n\\alpha}{2}\\right)}{\\sin \\left(\\frac{\\alpha}{2}\\right)}\\), \\(\\sum_{k=0}^n \\sin k\\alpha = \\frac{\\sin \\left(\\frac{(n+1)\\alpha}{2}\\right) \\sin \\left(\\frac{n\\alpha}{2}\\right)}{\\sin \\left(\... | [
"Using the equality \\(\\overrightarrow{OA_0} + \\overrightarrow{A_0A_1} + \\overrightarrow{A_1A_2} + \\cdots + \\overrightarrow{A_nA_{n-1}} = \\overrightarrow{OA_{n+1}}\\), express the sums \\(\\sum_{k=0}^n \\cos k\\alpha\\) and \\(\\sum_{k=0}^n \\sin k\\alpha\\) in terms of \\(n\\) and \\(\\alpha\\)."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 8
} | 8 | 0 | 0 |
omr-ws-2ea5817a7f5cf6ef5c7550fd1196952f2c9e234d48dc35da45623c5d7fc06656 | Using the equation $\cos 7 \theta = 0$, calculate the value of \[ \sin^{2} \frac{\pi}{14} + \sin^{2} \frac{3 \pi}{14} + \sin^{2} \frac{5 \pi}{14}. \] | [
"\\dfrac{5}{4}"
] | [
"Using the equation $\\cos 7 \\theta = 0$, calculate the value of \n\n\\[\n\\sin^{2} \\frac{\\pi}{14} + \\sin^{2} \\frac{3 \\pi}{14} + \\sin^{2} \\frac{5 \\pi}{14}.\n\\]"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9fe632f05119969080cafa3a9b275afaffe44a1857a524d79a5cb0a49b1e4c19 | Using the expansion \[ \frac{\sin(\pi z)}{\pi} = z \prod_{k=1}^{\infty} \left(1 - \frac{z^2}{k^2}\right), \] evaluate the product \[ P = \prod_{k=1}^{10} \frac{(2k)^2}{(2k-1)(2k+1)}. \] Express your answer as a simplified fraction. | [
"no_valid_answer_found"
] | [
"Using the expansion\n\\[\n\\frac{\\sin(\\pi z)}{\\pi} = z \\prod_{k=1}^{\\infty} \\left(1 - \\frac{z^2}{k^2}\\right),\n\\]\nevaluate the product\n\\[\nP = \\prod_{k=1}^{10} \\frac{(2k)^2}{(2k-1)(2k+1)}.\n\\]\nExpress your answer as a simplified fraction."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-30ede8ed18ad5f5eff789a2116a3e9e5dfb4ba966eb7af96362b84291607f10b | Using the expansion of a natural number \( n \) in base 8, determine the criterion for divisibility by 7. | [
"The sum of the digits of the number in base 8 must be divisible by 7."
] | [
"Using the expansion of a natural number \\( n \\) in base 8, determine the criterion for divisibility by 7."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-c946791f83b6c2e5264a69ff2de7a48f0d6450c208e2bc0103b99e55811d2f0b | Using the expression $\frac{1}{2} \int(x\frac{dy}{dt} -y\frac{dx}{dt})$, calculate the value of $[A] - [P] - \pi a^{2}$, where $[A]$ is the area swept out by $OA$, and $f=\frac{1}{2}\int_{0}^{2\pi} ((x+\frac{dy}{dt})\cos t + (y-\frac{dx}{dt})\sin t)dt$. Express your answer in terms of $a$ and $f$. | [
"-f"
] | [
"Using the expression $\\frac{1}{2} \\int(x\\frac{dy}{dt} -y\\frac{dx}{dt})$, calculate the value of $[A] - [P] - \\pi a^{2}$, where $[A]$ is the area swept out by $OA$, and $f=\\frac{1}{2}\\int_{0}^{2\\pi} ((x+\\frac{dy}{dt})\\cos t + (y-\\frac{dx}{dt})\\sin t)dt$. Express your answer in terms of $a$ and $f$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 6
} | 10 | 0 | 0 |
omr-ws-ce97ca44aae2575f499f83295a33cbba3ba9f0fc8ac7d258d7daf4f7850d8af2 | Using the expressions \(x^2-1\), \(3\), \(x^2+x+1\), \(\frac{x^2-1}{x}\), \(x^2\), and \(x-2\), combine them using addition, subtraction, multiplication, and division to reach \(x\). Exponents and square roots are not allowed. | [
"\\(\\frac{{x^2-1}}{\\frac{x^2-1}{x}} \\times \\frac{(x^2 + x + 1 - (x^2) - (x-2))}{3}=x\\)"
] | [
"Using the expressions \\(x^2-1\\), \\(3\\), \\(x^2+x+1\\), \\(\\frac{x^2-1}{x}\\), \\(x^2\\), and \\(x-2\\), combine them using addition, subtraction, multiplication, and division to reach \\(x\\). Exponents and square roots are not allowed."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 6,
"tir": 13
} | 32 | 0 | 0 |
omr-ws-95beffa69978361a97dfb640ff5c869c4982267ffb77e3eac0955d7bfd371a99 | Using the expressions found for \( C_{n} \) and \( S_{n} \), find \( \int_{0}^{\pi} x^{5} \sin x\ dx \). | [
"\\(\\pi^{5} - 20 \\pi^{3} + 120 \\pi\\)"
] | [
"Using the expressions found for \\( C_{n} \\) and \\( S_{n} \\), find \\( \\int_{0}^{\\pi} x^{5} \\sin x\\ dx \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-cec94e4f9bdee235a71468f190016155731096018d0563da37eef361b06e1a06 | Using the extended Euclid's algorithm, find the multiplicative inverse of \( 1234 \) modulo \( 4321 \). | [
"3239"
] | [
"Using the extended Euclid's algorithm, find the multiplicative inverse of \\( 1234 \\) modulo \\( 4321 \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 28
} | 42 | 0 | 0 |
omr-ws-a1ca7a1e08fb7eeb838c2d5dc24c18173c5f3c83c3502f9d8ae3272505bacb12 | Using the extended Euclid's algorithm, find the multiplicative inverse of \( 550 \) modulo \( 1769 \). | [
"544"
] | [
"Using the extended Euclid's algorithm, find the multiplicative inverse of \\( 550 \\) modulo \\( 1769 \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-11822ef43a176d7f7fcbf2e86644119ebcb76c05ea454c3ee8c59f77a9a51784 | Using the fact that if \( I - A \) is nonsingular, then \( \sum_{k=0}^{n} A^k = (I - A^{n+1})(I - A)^{-1} \), find some trigonometric summation formulas, where \( A = \begin{pmatrix} t\cos{\varphi} & -t\sin{\varphi} \\ t\sin{\varphi} & t\cos{\varphi} \end{pmatrix} \) and \( A^0 := I \). | [
"\n\\begin{aligned}\n\\sum_{k=0}^{n} t^k \\cos(k\\varphi) &= \\frac{1 - t \\cos \\varphi - t^{n+1} \\cos((n+1)\\varphi) + t^{n+2} \\cos(n\\varphi)}{1 - 2t \\cos \\varphi + t^2}, \\\\\n\\sum_{k=0}^{n} t^k \\sin(k\\varphi) &= \\frac{t \\sin \\varphi - t^{n+1} \\sin((n+1)\\varphi) + t^{n+2} \\sin(n\\varphi)}{1 - 2t \\... | [
"Using the fact that if \\( I - A \\) is nonsingular, then \\( \\sum_{k=0}^{n} A^k = (I - A^{n+1})(I - A)^{-1} \\), find some trigonometric summation formulas, where \\( A = \\begin{pmatrix} t\\cos{\\varphi} & -t\\sin{\\varphi} \\\\ t\\sin{\\varphi} & t\\cos{\\varphi} \\end{pmatrix} \\) and \\( A^0 := I \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-4ec5347c6cf676333b790f53c114a3fb9c07b9023917e12c61c4a2ff7864acfc | Using the formula $A = P(1 + r)^n$, determine how long it will take for $\$250.00$ to increase to $\$500$ at 4% annual interest. | [
"17.67"
] | [
"Using the formula $A = P(1 + r)^n$, determine how long it will take for $\\$250.00$ to increase to $\\$500$ at 4% annual interest."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-082c7ab1070531b65fbbe17dbadc8014b09c29951cceb6a3bad0d56f87ebe817 | Using the formula $\mathcal{L}^{-1} \{\tfrac{\ln s}{s} \} = - \ln t - \gamma$, find the expression for $\mathcal{L} \{\ln (2t) \}$. | [
"\\dfrac{\\ln 2 - \\gamma - \\ln s}{s}"
] | [
"Using the formula $\\mathcal{L}^{-1} \\{\\tfrac{\\ln s}{s} \\} = - \\ln t - \\gamma$, find the expression for $\\mathcal{L} \\{\\ln (2t) \\}$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5b15c2d6fed8a4c3627067c995134e0d24678b3e524d0a73855ec174d813c3cc | Using the formula $\pi \cot(\pi z) = \frac{1}{z} + \sum_{n=1}^{\infty} \frac{2z}{z^2 - n^2}$, calculate the values of $\zeta(2k)$ in terms of Bernoulli numbers and powers of $\pi$. | [
"\\zeta(2k) = \\frac{(-1)^{k+1} B_{2k} (2\\pi)^{2k}}{2 (2k)!}"
] | [
"Using the formula $\\pi \\cot(\\pi z) = \\frac{1}{z} + \\sum_{n=1}^{\\infty} \\frac{2z}{z^2 - n^2}$, calculate the values of $\\zeta(2k)$ in terms of Bernoulli numbers and powers of $\\pi$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-c435cdc6b2553a0f2dd907c46ac836933bb6510b3f5ea5920a870ca5b767eddf | Using the formula \( d = 0.045r^2 + 1.1r \), calculate the stopping distance \( d \) in feet for a car traveling at 40 mph, 55 mph, and 65 mph. | [
"116, 197, 262"
] | [
"Using the formula \\( d = 0.045r^2 + 1.1r \\), calculate the stopping distance \\( d \\) in feet for a car traveling at 40 mph, 55 mph, and 65 mph."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 29
} | 29 | 0 | 0 |
omr-ws-2ede08ed84c2c97f6e7a880aaa80424ada9541143bdb0ef2c79570768f930632 | Using the formula \( n^2 + 3^2 = x^2 \), find two values of \( n \) and two corresponding values of \( x \) that satisfy the formula, excluding \( n = 4 \) and \( x = 5 \). | [
"n = -4, x = -5"
] | [
"Using the formula \\( n^2 + 3^2 = x^2 \\), find two values of \\( n \\) and two corresponding values of \\( x \\) that satisfy the formula, excluding \\( n = 4 \\) and \\( x = 5 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-cd8ca4c2a801b7a3a5157646b689c35569863bd1226c2a9db9f73d7cfddc3ba2 | Using the formula \( y_2(x) = y_1(x) \int_{x_0}^{x} \frac{W(t)}{y_1(t)^2} \, dt \) with \( x_0 = 0 \) and expanding the integrand in powers of \( t \) to order \( t^3 \), find the first three non-zero terms in the power series expansion for a solution \( y_2 \) of the differential equation \( xy'' - (1 - x^2)y' - (1 + ... | [
"\\( x - \\frac{3x^2}{2} + \\frac{5x^3}{6} \\)"
] | [
"Using the formula \\( y_2(x) = y_1(x) \\int_{x_0}^{x} \\frac{W(t)}{y_1(t)^2} \\, dt \\) with \\( x_0 = 0 \\) and expanding the integrand in powers of \\( t \\) to order \\( t^3 \\), find the first three non-zero terms in the power series expansion for a solution \\( y_2 \\) of the differential equation \\( xy'' - ... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-164b8a6bfef2598cb4c0ae9d27a5a71f045bc29564863ec4e882d8624b30d034 | Using the formula \(1 + x + x^2 + \cdots + x^n = \frac{x^{n+1} - 1}{x-1}\), determine the formula for the sum \(1 + 2x + 3x^2 + 4x^3 + 5x^4\) by differentiating and multiplying by \(x\). | [
"\\(\\frac{5 x^6 - 6 x^5 + 1}{(x - 1)^2}\\)"
] | [
"Using the formula \\(1 + x + x^2 + \\cdots + x^n = \\frac{x^{n+1} - 1}{x-1}\\), determine the formula for the sum \\(1 + 2x + 3x^2 + 4x^3 + 5x^4\\) by differentiating and multiplying by \\(x\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-eefac8ebcb1edc2cca838fa3d26653096e2afb09d9623f967b632a6ba49a78d6 | Using the formula \(1 + x + x^2 + \cdots + x^n = \frac{x^{n+1} - 1}{x-1}\), determine the formula for the sum \(1^2x + 2^2x^2 + 3^2x^3 + 4^2x^4\) by differentiating and multiplying by \(x\). | [
"\\(\\frac{25 x^7 - 59 x^6 + 36 x^5 - x - 1}{(x-1)^3}\\)"
] | [
"Using the formula \\(1 + x + x^2 + \\cdots + x^n = \\frac{x^{n+1} - 1}{x-1}\\), determine the formula for the sum \\(1^2x + 2^2x^2 + 3^2x^3 + 4^2x^4\\) by differentiating and multiplying by \\(x\\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7988bdb99cc0d256381d725745b07b9648139468800db26b2360ced98caef361 | Using the formula derived in Problem 2 and the error term from Problem 3, calculate the smallest positive integer $N$ such that the number of primes less than or equal to $N$ differs from $\frac{N}{\ln(N)}$ by less than $100$. | [
"8100"
] | [
"Using the formula derived in Problem 2 and the error term from Problem 3, calculate the smallest positive integer $N$ such that the number of primes less than or equal to $N$ differs from $\\frac{N}{\\ln(N)}$ by less than $100$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 2
} | 5 | 0 | 0 |
omr-ws-ef25f5360a490b36c51ce5f3e3b8ec90182a09d7f75e1a9514cdd4b5844a07d6 | Using the formula for the sum of a geometric progression \( 1 + x + x^2 + \ldots + x^n = \frac{1 - x^{n+1}}{1 - x} \) (where \( x \neq 1 \)), deduce the formula for \( 1 + x + x^2 + \ldots + nx^{n-1} \). | [
"\\frac{1 - (n+1)x^n + nx^{n+1}}{(1 - x)^2}"
] | [
"Using the formula for the sum of a geometric progression \\( 1 + x + x^2 + \\ldots + x^n = \\frac{1 - x^{n+1}}{1 - x} \\) (where \\( x \\neq 1 \\)), deduce the formula for \\( 1 + x + x^2 + \\ldots + nx^{n-1} \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-8c69e2ec51cddf2c6a7fd87055331eccbaaef7a7b8ac3188dc07a3e9053c4a06 | Using the four triangles formed by dividing a regular hexagon $ABCDEF$ with an area of $12$ cm$^2$, a rectangle $PQRS$ is assembled. What is the area of the rectangle $PQRS$? | [
"16"
] | [
"Using the four triangles formed by dividing a regular hexagon $ABCDEF$ with an area of $12$ cm$^2$, a rectangle $PQRS$ is assembled. What is the area of the rectangle $PQRS$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3b1151c8a17d863acfdce36c29681e9a1a01ebd06c1c487bd32cbc6318f6e13c | Using the geometric interpretation indicated above, let us ascertain the validity of the following statements: Find whether the following sequences have limits: (a) $x_n=1+\frac{(-1)^n}{n}$, (b) $x_n=\left[1+\frac{(-1)^n}{n}\right]$. | [
"(a) $\\lim\\limits_{n\\to \\infty }x_n=1$; (b) the sequence has no limit"
] | [
"Using the geometric interpretation indicated above, let us ascertain the validity of the following statements: Find whether the following sequences have limits: (a) $x_n=1+\\frac{(-1)^n}{n}$, (b) $x_n=\\left[1+\\frac{(-1)^n}{n}\\right]$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3f3fc37da568666798de6438e4fb1eb7b976c1d7ae5b36bbd3fd6c05635bb346 | Using the given axioms and properties, find the expression for \(-(xy)\) in terms of \(x\) and \(y\) without using \(-x = (-1)x\). If \(x = 3\) and \(y = 4\), calculate the value of \(-(xy)\) using your derived expression. | [
"-12"
] | [
"Using the given axioms and properties, find the expression for \\(-(xy)\\) in terms of \\(x\\) and \\(y\\) without using \\(-x = (-1)x\\). If \\(x = 3\\) and \\(y = 4\\), calculate the value of \\(-(xy)\\) using your derived expression."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d5dfbece2ad0396fabd22a23e3d2c734b1e8ac0f8314d9376aaff808b6e05294 | Using the given data, find a quadratic function and a cubic function which model the situation. Explain the process used. Data: Gude numbers (from tip): 1, 2, 3, 4, 5, 6, 7, 8 Distance from tip (cm): 10, 23, 38, 55, 74, 96, 120, 149 | [
"y = -0.00291x^3 + 1.299x^2 + 8.144x + 1.325"
] | [
"Using the given data, find a quadratic function and a cubic function which model the situation. Explain the process used.\n\nData:\nGude numbers (from tip): 1, 2, 3, 4, 5, 6, 7, 8\nDistance from tip (cm): 10, 23, 38, 55, 74, 96, 120, 149"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0664d3f0950cb7b9325b0e5bef7e02caa0c1dd636f2dc10a15801ade4d95f398 | Using the given identity, calculate the coefficient of $x^{10}$ in the expansion of \[ \prod_{n \geq 1}\left(1-x^{n}\right)^{6}. \] | [
"18"
] | [
"Using the given identity, calculate the coefficient of $x^{10}$ in the expansion of \n\\[\n\\prod_{n \\geq 1}\\left(1-x^{n}\\right)^{6}.\n\\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 8
} | 10 | 0 | 0 |
omr-ws-a41c4988a8fe6d3c4321364edc7857437a8ddb209a1e3a9369ef4846aaa9e2e3 | Using the given patterns in base 10: \[ 9 - 1 = 2^3, \] \[ 9^2 - 1 = 2^4 \cdot 5, \] \[ 9^4 - 1 = 2^3 \cdot 5 \cdot 41, \] determine the length of the period in the base 9 expansion of each of the fractions \( \frac{1}{5} \), \( \frac{1}{2^5} \), and \( \frac{1}{5 \cdot 14} \). | [
"6"
] | [
"Using the given patterns in base 10:\n\\[ 9 - 1 = 2^3, \\]\n\\[ 9^2 - 1 = 2^4 \\cdot 5, \\]\n\\[ 9^4 - 1 = 2^3 \\cdot 5 \\cdot 41, \\]\ndetermine the length of the period in the base 9 expansion of each of the fractions \\( \\frac{1}{5} \\), \\( \\frac{1}{2^5} \\), and \\( \\frac{1}{5 \\cdot 14} \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d760a3226206d9ae54d66e568990004dd8b274fe4a17c9b66ab7fb6bab3f0761 | Using the given values of \( f(x) \), compute \( f'(2.5) \) using the Richardson extrapolation technique. x f(x) 2.10 0.7419 2.30 0.8329 2.40 0.8754 2.50 0.8960 2.60 0.9162 | [
"0.204"
] | [
"Using the given values of \\( f(x) \\), compute \\( f'(2.5) \\) using the Richardson extrapolation technique.\n\n x f(x)\n 2.10 0.7419\n 2.30 0.8329\n 2.40 0.8754\n 2.50 0.8960\n 2.60 0.9162"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-cf1b2289475df85751198bf2901000c2b7e105c9d5760347e87695e1b3793ee2 | Using the height function \( f(t) = 900 - 16t^2 \), evaluate the average speed of the ball between the times \( t = 7.5 - \left(\frac{1}{2}\right)^n \) and \( t = 7.5 - \left(\frac{1}{2}\right)^{n+1} \) and determine if it is possible to evaluate the limit of these average speeds using a geometric series. | [
"240"
] | [
"Using the height function \\( f(t) = 900 - 16t^2 \\), evaluate the average speed of the ball between the times \\( t = 7.5 - \\left(\\frac{1}{2}\\right)^n \\) and \\( t = 7.5 - \\left(\\frac{1}{2}\\right)^{n+1} \\) and determine if it is possible to evaluate the limit of these average speeds using a geometric seri... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 7
} | 8 | 0 | 0 |
omr-ws-c0ac8d155991c8b328771fa5240062ce5b8c735d3880cc733c092854370abbed | Using the hook-length formula, calculate the number of ways to fill a $3 \times 3$ grid with the numbers $1$ through $9$ such that each row and column is increasing. The hook-length formula is given by $\frac{n!}{ \prod_{(i,j) \in Y} \text{hook}\,(i,j)}$, where $n = 9$ and $\text{hook}\,(i,j)$ is the number of cells in... | [
"\\(5 \\cdot 4 \\cdot 3 \\cdot 4 \\cdot 3 \\cdot 2 \\cdot 3 \\cdot 2 \\cdot 1\\)"
] | [
"Using the hook-length formula, calculate the number of ways to fill a $3 \\times 3$ grid with the numbers $1$ through $9$ such that each row and column is increasing. The hook-length formula is given by $\\frac{n!}{ \\prod_{(i,j) \\in Y} \\text{hook}\\,(i,j)}$, where $n = 9$ and $\\text{hook}\\,(i,j)$ is the numbe... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-a89c9409f8b656156ba7b0b4baa9d840be0515e166304aff8e2cb37b4a2f19c2 | Using the identity $$\int_{0}^{\infty}t^{z-1}\sin{t}~dt = \Gamma(z)\sin(\frac{1}{2}\pi z),$$ calculate the value of the integral $$\int_{0}^{\infty}\sin{t^3}~dt.$$ Express your answer in terms of $\Gamma$ and $\pi$. | [
"\\frac{1}{3} \\Gamma\\left(\\frac{1}{3}\\right) \\sin\\left(\\frac{\\pi}{6}\\right)"
] | [
"Using the identity $$\\int_{0}^{\\infty}t^{z-1}\\sin{t}~dt = \\Gamma(z)\\sin(\\frac{1}{2}\\pi z),$$ calculate the value of the integral $$\\int_{0}^{\\infty}\\sin{t^3}~dt.$$ Express your answer in terms of $\\Gamma$ and $\\pi$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a115158b76ab8e4a9fa1eaf52e307a9679d5513f98a997f2676d0df0c635db8e | Using the identity $(1-x)^{-\frac{1}{2}}(1-x)^{-\frac{1}{2}}=(1-x)^{-1}$, find the coefficient of $x^{10}$ in the expansion of $(1-x)^{-1}$. | [
"1"
] | [
"Using the identity $(1-x)^{-\\frac{1}{2}}(1-x)^{-\\frac{1}{2}}=(1-x)^{-1}$, find the coefficient of $x^{10}$ in the expansion of $(1-x)^{-1}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e1530a77660313b05cc091b49c073f990ce03b719ac0f5566b81c576cedc29ac | Using the identity $4\sin \theta \sin \left(\frac{\pi}{3} - \theta\right) \sin \left(\frac{\pi}{3} + \theta\right) = \sin 3\theta$, derive a similar identity for $\cos 3\theta$. Then, using this identity, find the value of $\cot \theta \cot \left(\frac{\pi}{3} - \theta\right) \cot \left(\frac{\pi}{3} + \theta\right)$ w... | [
"\\dfrac{\\sqrt{3}}{3}"
] | [
"Using the identity $4\\sin \\theta \\sin \\left(\\frac{\\pi}{3} - \\theta\\right) \\sin \\left(\\frac{\\pi}{3} + \\theta\\right) = \\sin 3\\theta$, derive a similar identity for $\\cos 3\\theta$. Then, using this identity, find the value of $\\cot \\theta \\cot \\left(\\frac{\\pi}{3} - \\theta\\right) \\cot \\left... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-96b67600095fa71e0e60eb3fdfd986a9a198cf5f3dc35935a6491181107b1635 | Using the identity $\lg 11 = 1.0413\ldots$, find the least positive integer $n$ such that no $n$-digit number is divisible by $11$. | [
"1"
] | [
"Using the identity $\\lg 11 = 1.0413\\ldots$, find the least positive integer $n$ such that no $n$-digit number is divisible by $11$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 4
} | 19 | 0 | 0 |
omr-ws-ab1b3249be09db3652e39a0ecbfe8e46799ed3ad4b2592aca1717a90a32ffeb7 | Using the identity \[ \frac{1}{1+x^{2}}=\int_{0}^{+\infty}\exp(-y(1+x^{2}))dy, \] find the value of the integral \[ \int_{0}^{+\infty}\frac{\cos(ax)}{1+x^{2}}dx. \] | [
"\\dfrac{\\pi}{2} e^{-|a|}"
] | [
"Using the identity\n\\[\n\\frac{1}{1+x^{2}}=\\int_{0}^{+\\infty}\\exp(-y(1+x^{2}))dy,\n\\]\nfind the value of the integral\n\\[\n\\int_{0}^{+\\infty}\\frac{\\cos(ax)}{1+x^{2}}dx.\n\\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 8
} | 19 | 0 | 0 |
omr-ws-40664237a83b9f6c013dbc5f8d247717a3aaf250435eebae35d82f05c9ad4cc9 | Using the inclusion/exclusion principle, find a formula for the cardinality of $ |\bigcup_{i\neq j}^{n}(A_i\bigcap A_j)| $. | [
" \\sum_{k=2}^n (-1)^k (k - 1) \\sum_{1 \\leq i_1 < i_2 < \\cdots < i_k \\leq n} |A_{i_1} \\cap A_{i_2} \\cap \\cdots \\cap A_{i_k}| "
] | [
"Using the inclusion/exclusion principle, find a formula for the cardinality of $ |\\bigcup_{i\\neq j}^{n}(A_i\\bigcap A_j)| $."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7536911e76a6932a80587a142245cf1b8cbaf795d78f35d11c4c42a6eba2ea82 | Using the inequality $ \frac{n \sqrt[n]{e}}{e} < \sqrt[n]{n!} < \frac{n \sqrt[n]{ne}}{e} $, find the value of $L$ where $L = \lim_{n \rightarrow + \infty} \frac{\sqrt[n]{n!}}{n}$. | [
"\\dfrac{1}{e}"
] | [
"Using the inequality $ \\frac{n \\sqrt[n]{e}}{e} < \\sqrt[n]{n!} < \\frac{n \\sqrt[n]{ne}}{e} $, find the value of $L$ where $L = \\lim_{n \\rightarrow + \\infty} \\frac{\\sqrt[n]{n!}}{n}$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-66cc2050d9f831b01ec2b07f31b14d6b7d67859d7a20aa3467a937dcb2dc9e44 | Using the inequality \(\pi(2n) - \pi(n) < \frac{n \log(4)}{\log(n)}\) (where \(n > 1\)), find the smallest integer \(k\) such that \(\pi(2n) \leq \frac{n \log(k)}{\log(n)}\) for all \(n > 1\). | [
"14"
] | [
"Using the inequality \\(\\pi(2n) - \\pi(n) < \\frac{n \\log(4)}{\\log(n)}\\) (where \\(n > 1\\)), find the smallest integer \\(k\\) such that \\(\\pi(2n) \\leq \\frac{n \\log(k)}{\\log(n)}\\) for all \\(n > 1\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 4
} | 6 | 0 | 0 |
omr-ws-549135cf6fdeab2c05fe99d1d489d80413407a61b820cc319a7c9849a52b191f | Using the information $10^3 = 1000$, $10^4 = 10000$, $2^{10} = 1024$, $2^{11} = 2048$, $2^{12} = 4096$, and $2^{13} = 8192$, find the largest $a$ and the smallest $b$ such that $a < \log_{10}2 < b$. | [
"\\( a = \\frac{3}{10} \\) and \\( b = \\frac{4}{13} \\)"
] | [
"Using the information $10^3 = 1000$, $10^4 = 10000$, $2^{10} = 1024$, $2^{11} = 2048$, $2^{12} = 4096$, and $2^{13} = 8192$, find the largest $a$ and the smallest $b$ such that $a < \\log_{10}2 < b$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 18
} | 22 | 0 | 0 |
omr-ws-93916cb20fe14540843e9f6238537d7e2d6ba2e2ecc1472e7c18de04daca57ac | Using the information $10^3 = 1000$, $10^4 = 10000$, $2^{10} = 1024$, $2^{11} = 2048$, $2^{12} = 4096$, and $2^{13} = 8192$, find the largest $a$ and the smallest $b$ such that one can prove $a < \log_{10}2 < b$. | [
"\\( a = \\frac{3}{10} \\) and \\( b = \\frac{4}{13} \\)"
] | [
"Using the information $10^3 = 1000$, $10^4 = 10000$, $2^{10} = 1024$, $2^{11} = 2048$, $2^{12} = 4096$, and $2^{13} = 8192$, find the largest $a$ and the smallest $b$ such that one can prove $a < \\log_{10}2 < b$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 5,
"tir": 13
} | 18 | 0 | 0 |
omr-ws-e6d83fc7191fde1eaea606e770246def2748bb03ba2f2937a7aa1c860b30c80c | Using the information provided, calculate Diane's adjusted gross income: - Gross income: \$38,500 (wages) + \$120 (interest) + \$5,850 (gambling earnings) - Deduction for tax-deferred retirement account: \$2,400 | [
"42070"
] | [
"Using the information provided, calculate Diane's adjusted gross income:\n- Gross income: \\$38,500 (wages) + \\$120 (interest) + \\$5,850 (gambling earnings)\n- Deduction for tax-deferred retirement account: \\$2,400"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ad313d9a755d46ee5c98070afb11487ca4dec9d6f24f38726d6df7e6a3f3c48b | Using the information provided, calculate Diane's taxable income: - Adjusted gross income (from Problem 2) - Personal exemption: \$3,950 - Standard deduction: \$6,200 - Deductions: \$2,250 (state taxes) + \$4,500 (mortgage interest) + \$2,300 (property taxes) | [
"37000"
] | [
"Using the information provided, calculate Diane's taxable income:\n- Adjusted gross income (from Problem 2)\n- Personal exemption: \\$3,950\n- Standard deduction: \\$6,200\n- Deductions: \\$2,250 (state taxes) + \\$4,500 (mortgage interest) + \\$2,300 (property taxes)"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-3452676861440be9d99f988146de380ff6d755c640bf3033d5f9ba19b0161a72 | Using the information that 4652 tosses of a coin yield 1868 heads, find an approximate value of the probability that the number of heads will be more than 960 but less than or equal to 1080 in 2500 tosses of the same coin. | [
"0.9589"
] | [
"Using the information that 4652 tosses of a coin yield 1868 heads, find an approximate value of the probability that the number of heads will be more than 960 but less than or equal to 1080 in 2500 tosses of the same coin."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 4,
"tir": 20
} | 29 | 0 | 0 |
omr-ws-940e72e82a10712747088406f744ed03800e5f4733f9dc3a1a585c716a717c4f | Using the information that \(10^3 = 1000\), \(10^4 = 10000\), \(2^{10} = 1024\), \(2^{11} = 2048\), \(2^{12} = 4096\), and \(2^{13} = 8192\), find the largest \(a\) and smallest \(b\) such that \(a < \log_{10}(2) < b\). | [
"\\( a = \\frac{3}{10}, b = \\frac{4}{13} \\)"
] | [
"Using the information that \\(10^3 = 1000\\), \\(10^4 = 10000\\), \\(2^{10} = 1024\\), \\(2^{11} = 2048\\), \\(2^{12} = 4096\\), and \\(2^{13} = 8192\\), find the largest \\(a\\) and smallest \\(b\\) such that \\(a < \\log_{10}(2) < b\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 13
} | 16 | 0 | 0 |
omr-ws-bd1380ec3023523af95aed085b1b97f27c13a18859d088bc15e4a0d72ee2c421 | Using the integer \( K \) found in Problem 1, determine a method to factor the number 3,142,039 into its prime components. | [
"3103"
] | [
"Using the integer \\( K \\) found in Problem 1, determine a method to factor the number 3,142,039 into its prime components."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-6d4896a6d57ca879dcd2e9c0fc7c6499fdc6e9b19e937dc5b07ad2a7c341fd1b | Using the least cost method, find a starting basic feasible solution for the transportation problem given the following cost matrix and supply/demand constraints: \[ \begin{array}{c|cccc|c} & 1 & 2 & 3 & 4 & \text{Supply} \\ \hline 1 & 1 & 2 & 3 & 2 & 150 \\ 2 & 2 & 4 & 1 & 2 & 200 \\ 3 & 1 & 3 & 5 & 3 & 250 \\ \text{d... | [
"1650"
] | [
"Using the least cost method, find a starting basic feasible solution for the transportation problem given the following cost matrix and supply/demand constraints:\n\n\\[\n\\begin{array}{c|cccc|c}\n & 1 & 2 & 3 & 4 & \\text{Supply} \\\\\n\\hline\n1 & 1 & 2 & 3 & 2 & 150 \\\\\n2 & 2 & 4 & 1 & 2 & 200 \\\\\n3 & 1 & 3... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-5552bc3262171fb6ff1978ba16aa668b04eabbaea78c3b9b25a6fa610efdb1cf | Using the left end of the arch as the origin, find an equation for the height \( y \) (in feet) above the water of a point on the arch as a function of the point's position \( x \) (in feet) in relation to the left endpoint, given that the arch is 24 ft wide at the water level and the peak of the arch is 40 ft above th... | [
"\\( y = -\\dfrac{5}{18}x^2 + \\dfrac{20}{3}x \\)"
] | [
"Using the left end of the arch as the origin, find an equation for the height \\( y \\) (in feet) above the water of a point on the arch as a function of the point's position \\( x \\) (in feet) in relation to the left endpoint, given that the arch is 24 ft wide at the water level and the peak of the arch is 40 ft... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 19
} | 34 | 0 | 0 |
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