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-af03422e0d81175c8b51232ee0b01d026f31d463507cbc37f120f956ec3c807d | Using an echelon form matrix, solve the system of equations: \[ 3x - 3y - 4z = -2 \] \[ x + 4y + 2z = 6 \] | [
"\\(\\left( \\frac{2}{3} + \\frac{2}{3} t, \\frac{4}{3} - \\frac{2}{3} t, t \\right)\\)"
] | [
"Using an echelon form matrix, solve the system of equations:\n\\[ 3x - 3y - 4z = -2 \\]\n\\[ x + 4y + 2z = 6 \\]"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-a0aa108eefeca8a9521998daf4ebd614cd14ac9f6648b942d49965ec4a9cc147 | Using an infinite supply of equilateral triangles and squares, each with a side length of one inch, what is the convex polygon with the largest number of sides that can be constructed without any pieces overlapping? | [
"12"
] | [
"Using an infinite supply of equilateral triangles and squares, each with a side length of one inch, what is the convex polygon with the largest number of sides that can be constructed without any pieces overlapping?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-1b5a6d1dca2ee99d780d4941c044543b4a8d74c487fa65a159aefa734ee48cb7 | Using an iterative method $x_{r+1} = F(x_{r})$, find each of the roots of the equation $\sin x = \ln x$ to 6 decimal places. | [
"2.2191071"
] | [
"Using an iterative method $x_{r+1} = F(x_{r})$, find each of the roots of the equation $\\sin x = \\ln x$ to 6 decimal places."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 6,
"tir": 12
} | 32 | 0 | 0 |
omr-ws-1d86fc9c802857129d5a5911686351c6df95682a745c02201f0308e8717ad2cd | Using an iterative method $x_{r+1} = F(x_{r})$, find each of the roots of the equation $x^{3} + 100 \cos x = 0$ to 6 decimal places. | [
"\\( x \\approx -1.534645, 1.612756, 4.010988 \\)"
] | [
"Using an iterative method $x_{r+1} = F(x_{r})$, find each of the roots of the equation $x^{3} + 100 \\cos x = 0$ to 6 decimal places."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 9
} | 9 | 0 | 0 |
omr-ws-bd98046062cc7e05ec4f827a060c8663464ce4fce17e6cda2f0da05a36a5e7e0 | Using an iterative process based on the equation \( x = \sqrt[3]{\frac{16}{3}x + 1} \), find the coordinates of the stationary point \(P\), giving each coordinate correct to 3 decimal places. | [
"(2.396, -1.494)"
] | [
"Using an iterative process based on the equation \\( x = \\sqrt[3]{\\frac{16}{3}x + 1} \\), find the coordinates of the stationary point \\(P\\), giving each coordinate correct to 3 decimal places."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2aba700e998c40279bf4a00d3c44d77a5b40823ae6b4eb2ccdb7f74cc9092404 | Using at most four 5's and four operations (either exponentiation or factorial), create the largest possible number or prove that no such number can be defined. | [
"\\left(5^{5^{5^5}}\\right)!"
] | [
"Using at most four 5's and four operations (either exponentiation or factorial), create the largest possible number or prove that no such number can be defined."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 3
} | 16 | 0 | 0 |
omr-ws-b457da2db4d1dc926c979ce0a19cae9052f2e35ef33d2f95048ceb4fafdbb485 | Using calculus, sketch the graph of the function \( f(x) = \frac{5 - 3x^2}{1 - x^2} \). | [
"\\text{Graph sketched with described features}"
] | [
"Using calculus, sketch the graph of the function \\( f(x) = \\frac{5 - 3x^2}{1 - x^2} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-f6064186f456bca0874c5677e38735ba41120690778577b320f4518e5e71169d | Using calculus, sketch the graph of the function \( f(x) = x^2 + x + \frac{1}{x} + \frac{1}{x^2} \) for \( x \neq 0 \). | [
"4"
] | [
"Using calculus, sketch the graph of the function \\( f(x) = x^2 + x + \\frac{1}{x} + \\frac{1}{x^2} \\) for \\( x \\neq 0 \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0af8e93075336e238c48003003c860bee74cdbd1b9c08f4a26bbb3878133c02e | Using coins with values 1 cent, 5 cents, 10 cents, and 25 cents, what is the smallest number of coins needed to be able to make exactly every value from 1 cent to 100 cents? | [
"10"
] | [
"Using coins with values 1 cent, 5 cents, 10 cents, and 25 cents, what is the smallest number of coins needed to be able to make exactly every value from 1 cent to 100 cents?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-14290331c4d6cbad8ddb20e3dc8b7760557a930cbc2b0a62d71313332142e7ff | Using coins with values of 1 cent, 5 cent, 10 cent, and 25 cent, what is the smallest number of coins needed to be able to exactly make each total from 1 cent to a dollar? | [
"10"
] | [
"Using coins with values of 1 cent, 5 cent, 10 cent, and 25 cent, what is the smallest number of coins needed to be able to exactly make each total from 1 cent to a dollar?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 6
} | 14 | 0 | 0 |
omr-ws-bebaf95129a2568beaf6ee1000e07f41c0319364f0c30799d92eeca0ea52bd87 | Using combinations of the positive integers \(a^2\), \(ab\), and \(b^2\), where the number of each element is non-negative, the largest number that cannot be made by adding them is 47. Find the value of \(a\) and \(b\). | [
"4"
] | [
"Using combinations of the positive integers \\(a^2\\), \\(ab\\), and \\(b^2\\), where the number of each element is non-negative, the largest number that cannot be made by adding them is 47. Find the value of \\(a\\) and \\(b\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-4d97d004fedd3d4a3208ed08c240a3d13d62f16a5327cda27124b6efe18b6cd4 | Using commutativity and associativity to quickly find $3^4$. Using commutativity and associativity to quickly find $2^6$. | [
"\\(3^4 = 81\\) and \\(2^6 = 64\\)"
] | [
"Using commutativity and associativity to quickly find $3^4$. Using commutativity and associativity to quickly find $2^6$."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 20
} | 20 | 0 | 0 |
omr-ws-87df3d5d73c5bf043f490b88adcbeee14dbdd41e60bd9ba043ffaa3f0f0d4095 | Using complementary counting with PIE, determine the probability of a specific arrangement of 9 distinguishable people in numbered chairs, given the expression $3*3!*7! - 3*3!*3!*5! + (3!)^3*3!$ for the numerator and $9!$ for the denominator. | [
"\\(\\frac{61}{280}\\)"
] | [
"Using complementary counting with PIE, determine the probability of a specific arrangement of 9 distinguishable people in numbered chairs, given the expression $3*3!*7! - 3*3!*3!*5! + (3!)^3*3!$ for the numerator and $9!$ for the denominator."
] | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 16,
"genselect": 8,
"tir": 31
} | 55 | 0 | 0 |
omr-ws-de8c29f280c96b968a7e0894adf94846fcb830deb79fb6cb4c4e6a58cfe3ec49 | Using complex analysis, calculate the sum $$\sum_{n=0}^{\infty}\frac{1}{n^2-w^2},$$ where $w \in \mathbb{C} \setminus \mathbb{Z}$. | [
"\\(-\\frac{3}{2w^2} + \\frac{\\pi \\cot \\pi w}{2w}\\)"
] | [
"Using complex analysis, calculate the sum $$\\sum_{n=0}^{\\infty}\\frac{1}{n^2-w^2},$$ where $w \\in \\mathbb{C} \\setminus \\mathbb{Z}$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 3,
"tir": 1
} | 5 | 0 | 0 |
omr-ws-517c33a837239ca5ead095b44dec57ef362c4aa2aca7c0fafe2fdc0d061f5169 | Using complex analysis, find \( I_{1} = \int_{- \infty}^{\infty} \frac{1}{x^{2} + 2ax + b}\ dx \) where \( a \) and \( b \) are constants with \( b \leq a^{2} \). | [
"\\dfrac{\\pi}{\\sqrt{a^2 - b}}"
] | [
"Using complex analysis, find \\( I_{1} = \\int_{- \\infty}^{\\infty} \\frac{1}{x^{2} + 2ax + b}\\ dx \\) where \\( a \\) and \\( b \\) are constants with \\( b \\leq a^{2} \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 2
} | 9 | 0 | 0 |
omr-ws-9a8d8d475d19305dfb59f8a3bc7db9ef83097c3f9f9b0bff6375a522636f1a08 | Using complex analysis, find \( I_{n} = \int_{- \infty}^{\infty} \frac{1}{(x^{2} + 2ax + b)^{n}}\ dx \) in the case where \( b < a^{2} \). | [
" \\dfrac{\\sqrt{\\pi} \\, \\Gamma\\left(n - \\dfrac{1}{2}\\right)}{(a^2 - b)^{n - \\frac{1}{2}} \\, \\Gamma(n)} "
] | [
"Using complex analysis, find \\( I_{n} = \\int_{- \\infty}^{\\infty} \\frac{1}{(x^{2} + 2ax + b)^{n}}\\ dx \\) in the case where \\( b < a^{2} \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 6
} | 15 | 0 | 0 |
omr-ws-e5b0c3a05918ef4b50215be053c726edfee5d6261a005ef3dad8d513d3ac0b6f | Using complex analysis, find \( I_{n} = \int_{- \infty}^{\infty} \frac{1}{(x^{2} + 2ax + b)^{n}}\ dx \) in the case where \( b = a^{2} \). | [
"\\infty"
] | [
"Using complex analysis, find \\( I_{n} = \\int_{- \\infty}^{\\infty} \\frac{1}{(x^{2} + 2ax + b)^{n}}\\ dx \\) in the case where \\( b = a^{2} \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0217049d6f89a5accabe0eedd438409e35ca78b6a74f8b807bc25bac5008f65b | Using complex number nth roots of unity, find the value of $\prod_{k=1}^{n-1} \cot\left(\frac{k\pi}{2n}\right)$. | [
"1"
] | [
"Using complex number nth roots of unity, find the value of $\\prod_{k=1}^{n-1} \\cot\\left(\\frac{k\\pi}{2n}\\right)$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-9878595e6099b5459635aca516681ede0abbec29ae70221a4ecf973b5e28fb78 | Using complex numbers, express the condition for three points $A$, $B$, and $C$ on a line in terms of their complex coordinates $a$, $b$, and $c$. Then, apply this condition to the vertices of a triangle $ABC$ and a transversal intersecting the sides (or their extensions) at points $D$, $E$, and $F$. Find the expressio... | [
"-1"
] | [
"Using complex numbers, express the condition for three points $A$, $B$, and $C$ on a line in terms of their complex coordinates $a$, $b$, and $c$. Then, apply this condition to the vertices of a triangle $ABC$ and a transversal intersecting the sides (or their extensions) at points $D$, $E$, and $F$. Find the expr... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 5
} | 9 | 0 | 0 |
omr-ws-29bc33fa2ecb1409f35c00776f63f47c71530cb3ce453aa902b6a509b02c6ca8 | Using contour integration, find the value of the expression $2z^2 \sum_{k=1}^{10} \frac{1}{z^2 - k^2}$ when $z = \frac{3}{2}$. | [
"-\\dfrac{1740}{3059}"
] | [
"Using contour integration, find the value of the expression $2z^2 \\sum_{k=1}^{10} \\frac{1}{z^2 - k^2}$ when $z = \\frac{3}{2}$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-3fd06bf706229051aa4dfe03a9a83bc808b1ecc045f3a6f0e718c2c067be9394 | Using correct units, explain the meaning of \(\int_{10}^{70} v(t) \, dt\) in terms of the rocket's flight. Use a midpoint Riemann's sum with 3 subintervals of equal length to approximate \(\int_{10}^{70} v(t) \, dt\). | [
"2020 feet"
] | [
"Using correct units, explain the meaning of \\(\\int_{10}^{70} v(t) \\, dt\\) in terms of the rocket's flight. Use a midpoint Riemann's sum with 3 subintervals of equal length to approximate \\(\\int_{10}^{70} v(t) \\, dt\\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-303a7319e80d818602457825d12d90be577028b70abd5dd891f3c6092d22ec24 | Using cylindrical coordinates, calculate the volume of the solid bounded above by the sphere \( r^2 + z^2 = a^2 \) and below by the cone \( z = r \cot \phi \), where \( 0 < \phi < \pi / 2 \), and express your answer in terms of \( a \) and \( \phi \). | [
"\\dfrac{2\\pi a^3}{3}(1 - \\cos \\phi)"
] | [
"Using cylindrical coordinates, calculate the volume of the solid bounded above by the sphere \\( r^2 + z^2 = a^2 \\) and below by the cone \\( z = r \\cot \\phi \\), where \\( 0 < \\phi < \\pi / 2 \\), and express your answer in terms of \\( a \\) and \\( \\phi \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-355bbbc1d2617f40381407e812a50a05e5be09157e7294127e68c9512a796563 | Using de Moivre's theorem, express \( \cos(x) \) in terms of \( e^{ix} \) and \( e^{-ix} \). Simplify the expression and find the value of \( A \) in the equation \( \cos(x) = \frac{Ae^{ix} + Ae^{-ix}}{2} \). | [
"1"
] | [
"Using de Moivre's theorem, express \\( \\cos(x) \\) in terms of \\( e^{ix} \\) and \\( e^{-ix} \\). Simplify the expression and find the value of \\( A \\) in the equation \\( \\cos(x) = \\frac{Ae^{ix} + Ae^{-ix}}{2} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e3527d43fa2c9347aaf909b9c565490b9683d36698c1add378fb67664c3d63b4 | Using each digit from 0 to 9 exactly once, write three numbers that fit the ratio 1:3:5. | [
"269 : 807 : 1345"
] | [
"Using each digit from 0 to 9 exactly once, write three numbers that fit the ratio 1:3:5."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 29
} | 29 | 0 | 0 |
omr-ws-21e92b9d74f6fa1eb8498246699a51a70efc0a999eb97023ca3f575f70247301 | Using each of the digits $1, 2, 3, \ldots, 8, 9$ exactly once, we form nine, not necessarily distinct, nine-digit numbers. Determine the maximum possible value of $n$ such that their sum ends in $n$ zeroes. | [
"8"
] | [
"Using each of the digits $1, 2, 3, \\ldots, 8, 9$ exactly once, we form nine, not necessarily distinct, nine-digit numbers. Determine the maximum possible value of $n$ such that their sum ends in $n$ zeroes."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 6
} | 9 | 0 | 0 |
omr-ws-57d8832ebd7742d87690628159b7329cb0d66716483ef9c72cdcbb6af066717e | Using each of the eight digits 1, 3, 4, 5, 6, 7, 8, 9 exactly once, form a three-digit number \( A \), two two-digit numbers \( B \) and \( C \) such that \( B < C \), and a one-digit number \( D \). The numbers must satisfy \( A + D = B + C = 143 \). In how many ways can this be done? | [
"12"
] | [
"Using each of the eight digits 1, 3, 4, 5, 6, 7, 8, 9 exactly once, form a three-digit number \\( A \\), two two-digit numbers \\( B \\) and \\( C \\) such that \\( B < C \\), and a one-digit number \\( D \\). The numbers must satisfy \\( A + D = B + C = 143 \\). In how many ways can this be done?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-b4753811071a1dcea5bcde2d1184d6795cf49758d46d43920d7e25e3f88de914 | Using each of the eight digits 1, 3, 4, 5, 6, 7, 8, and 9 exactly once, form a three-digit number \( A \), two two-digit numbers \( B \) and \( C \) (with \( B < C \)), and a one-digit number \( D \) such that \( A + D = B + C = 143 \). In how many ways can this be done? | [
"48"
] | [
"Using each of the eight digits 1, 3, 4, 5, 6, 7, 8, and 9 exactly once, form a three-digit number \\( A \\), two two-digit numbers \\( B \\) and \\( C \\) (with \\( B < C \\)), and a one-digit number \\( D \\) such that \\( A + D = B + C = 143 \\). In how many ways can this be done?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9d24d42b285da3a326b35b07e89b9a4801cdb1439894bf534916338cbf2e60a0 | Using each of the numbers $1, 2, \ldots, 26$ exactly once, form 13 fractions. What is the maximum number of these fractions that can be integers? | [
"12"
] | [
"Using each of the numbers $1, 2, \\ldots, 26$ exactly once, form 13 fractions. What is the maximum number of these fractions that can be integers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 8,
"tir": 1
} | 14 | 0 | 0 |
omr-ws-89d19ea02175764c46c806f4130806aa828d131a24a9d28dffa261a0a0c00a78 | Using each of the numbers 1-9 exactly once, find a way to make the following three fractions equal: \[ \frac{a}{b} = \frac{c}{de} = \frac{fg}{hi} \] Note that if \(d=3\) and \(e=4\), then \(de=34\) and not \(12\). | [
"\\(\\frac{3}{6} = \\frac{9}{18} = \\frac{27}{54}\\)"
] | [
"Using each of the numbers 1-9 exactly once, find a way to make the following three fractions equal:\n\\[ \\frac{a}{b} = \\frac{c}{de} = \\frac{fg}{hi} \\]\nNote that if \\(d=3\\) and \\(e=4\\), then \\(de=34\\) and not \\(12\\)."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 1,
"tir": 10
} | 15 | 0 | 0 |
omr-ws-d20b9d1b09961fb48659c9cc966d70cd51fb6861fd818bffda79068a5e42db28 | Using each of the ten digits exactly once, construct two five-digit numbers such that their difference is minimized. Determine this minimal difference. | [
"247"
] | [
"Using each of the ten digits exactly once, construct two five-digit numbers such that their difference is minimized. Determine this minimal difference."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-fb3a0d055c9287e363c8285edf9c3e5ec38fddc4578399cd331fe2b7481fdecf | Using elementary means, can one improve the result that if there is a simple group \( G \) of order \( n \), then there are altogether at least 7 groups of order \( n \), at least 4 of which are non-abelian? | [
"Yes"
] | [
"Using elementary means, can one improve the result that if there is a simple group \\( G \\) of order \\( n \\), then there are altogether at least 7 groups of order \\( n \\), at least 4 of which are non-abelian?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 1
} | 16 | 0 | 0 |
omr-ws-7be9a256d37afc69a52c3f60934a83c2693e7f52c754eb7a55c5489710c0db95 | Using elementary row operations, find the inverse of the matrix \(\begin{pmatrix} 1 & 1 & 2 \\ 2 & 1 & 3 \\ -1 & 2 & 2 \end{pmatrix}\). | [
"\\(\\begin{pmatrix} 4 & -2 & -1 \\\\ 7 & -4 & -1 \\\\ -5 & 3 & 1 \\end{pmatrix}\\)"
] | [
"Using elementary row operations, find the inverse of the matrix \\(\\begin{pmatrix} 1 & 1 & 2 \\\\ 2 & 1 & 3 \\\\ -1 & 2 & 2 \\end{pmatrix}\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 29
} | 43 | 0 | 0 |
omr-ws-f7ae6e62988b98081ed6e243426d49eb450344b74259265ce59f7c6a0c9f69de | Using exactly four 4's and any mathematical symbols, write an expression that equals 33. | [
"\\( 4! + 4 + \\sqrt{4} / (0.4) \\)"
] | [
"Using exactly four 4's and any mathematical symbols, write an expression that equals 33."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 3,
"tir": 2
} | 15 | 0 | 0 |
omr-ws-5f2ae9556f3c3919c7a592e17027450bca8a430ead987e6088fd12ffefa49738 | Using five 5s and any mathematical operations, form the number 2023. | [
"2023"
] | [
"Using five 5s and any mathematical operations, form the number 2023."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 3
} | 9 | 0 | 0 |
omr-ws-144e79886a6c6f742f566e9b8a06e35fbb0b805b16ef3a7b7410406cbc6bba82 | Using four 4's and any mathematical operations, express the number 1. | [
"\\( \\frac{4}{4} + 4 - 4 \\)"
] | [
"Using four 4's and any mathematical operations, express the number 1."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 2,
"tir": 23
} | 40 | 0 | 0 |
omr-ws-01946f98d0ba01635ce882cff9475e0200da1afd00ba1c792b9a7947d0c9c081 | Using four 4's and any mathematical operations, express the number 4. | [
"\\(4+(4-4)*4=4\\)"
] | [
"Using four 4's and any mathematical operations, express the number 4."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 25
} | 40 | 0 | 0 |
omr-ws-d4d8db2ee940da28891fc8cb1198abdcef7998d512a9ba4b3651932690c4c8f0 | Using four 4's and any mathematical operations, express the number 5. | [
"\\((4*4+4)/4\\)"
] | [
"Using four 4's and any mathematical operations, express the number 5."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-e364926f2546c449a33827b110b2b092ea559528766668303ddf6a3e7c4ae6b8 | Using four 4's and any mathematical operations, express the number 6. | [
"\\(4 + \\frac{4 + 4}{4}\\)"
] | [
"Using four 4's and any mathematical operations, express the number 6."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-cb4347ac670f793d4f0c6da49c1c1a66411f67ef5e677db96d254e9527df48ee | Using four 4s and any mathematical operations (excluding square roots), generate the number 224. If using exponentiation or logarithms, the bases and exponents must also be generated using the four 4s. | [
"(4! + 4) \\times (4 + 4)"
] | [
"Using four 4s and any mathematical operations (excluding square roots), generate the number 224. If using exponentiation or logarithms, the bases and exponents must also be generated using the four 4s."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-580f9af59ff2c8f9643627c1d61dc74794db21b45571bde42c601f50bd7a1e38 | Using four fours and mathematical signs, form the number 24. | [
"\\(4! \\times (4 - 4)^4 = 24\\)\n\n(Note: The expression \\(4! \\times (4 - 4)^4\\) simplifies to \\(24 \\times 0^4 = 24 \\times 0 = 0\\), which is incorrect. The correct expression from the forum is \\(4! \\times (4 - 4 + 4) = 24\\).)\n\nCorrected"
] | [
"Using four fours and mathematical signs, form the number 24."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 2,
"tir": 3
} | 9 | 0 | 0 |
omr-ws-36ce87913806424f3022f883fbcea1497d18736b3e7622309d489881f7a75286 | Using four fours and mathematical signs, form the number 25. | [
"${(4+\\frac{4}{4})^ {\\sqrt{4}} }$"
] | [
"Using four fours and mathematical signs, form the number 25."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-f91a1aad95a2bf45e901d72472dce68f17e9983768bea0d1ebf572f48a13e8ea | Using four fours and mathematical signs, form the number 97. | [
"\\( 4 \\cdot 4! + \\frac{4}{4} \\)"
] | [
"Using four fours and mathematical signs, form the number 97."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-fceecb3fcd18a48b4a4ab3e113f9aae3c21d1433df250f3d3fdcb4c1047be4e8 | Using four planes, determine the maximum number of distinct regions into which space can be divided. Calculate the sum of the digits of this maximum number. | [
"6"
] | [
"Using four planes, determine the maximum number of distinct regions into which space can be divided. Calculate the sum of the digits of this maximum number."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d293dcc3ec7fa59c4317ba4290400d8ed958a78b8209c406a884c5f4e333cac6 | Using four zeros (0, 0, 0, 0), can you perform mathematical operations to obtain the number 24? | [
"\\( (0! + 0! + 0! + 0!)! = 24 \\)"
] | [
"Using four zeros (0, 0, 0, 0), can you perform mathematical operations to obtain the number 24?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 7,
"tir": 11
} | 29 | 0 | 0 |
omr-ws-7a066d2718d5cc2e903baa647195d52933b45d6ed515e2703d38e90b33ae98dd | Using generating functions, determine the number of ways Alex can make 16¢ using an infinite number of 1¢, 2¢, 5¢, and 10¢ coins. | [
"25"
] | [
"Using generating functions, determine the number of ways Alex can make 16¢ using an infinite number of 1¢, 2¢, 5¢, and 10¢ coins."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-c9643a3bb9490f5c0ef1162b47e4b3482bc1ab2ad0234c2c0b60c5b5053e2dbc | Using generating functions, determine the number of ways to get a sum of 27 when rolling five dice. | [
"35"
] | [
"Using generating functions, determine the number of ways to get a sum of 27 when rolling five dice."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-eab55efb82d3ee7742a145cf0089a5d2193940bccdd0f6d469ec69d20169f013 | Using generating functions, find the coefficient of \(x^{10}\) in the expression \((x^2 + x^3 + x^4)^4\) and explain why this does not give the correct answer for the problem of forming a sequence of 10 letters from 4 'a's, 4 'b's, 4 'c's, and 4 'd's, where each letter must appear at least twice. | [
"10"
] | [
"Using generating functions, find the coefficient of \\(x^{10}\\) in the expression \\((x^2 + x^3 + x^4)^4\\) and explain why this does not give the correct answer for the problem of forming a sequence of 10 letters from 4 'a's, 4 'b's, 4 'c's, and 4 'd's, where each letter must appear at least twice."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-aa68508966029d7bcc6b6f33e8d05c8f5913bf6a4423a65e4aeb800fbb4d292d | Using generating functions, find the number of ordered pairs (A, B) of subsets of \(\{1, 2, \ldots, 10\}\) such that each element of A is larger than \(|B|\) and each element of B is larger than \(|A|\). | [
"17711"
] | [
"Using generating functions, find the number of ordered pairs (A, B) of subsets of \\(\\{1, 2, \\ldots, 10\\}\\) such that each element of A is larger than \\(|B|\\) and each element of B is larger than \\(|A|\\)."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 8,
"tir": 26
} | 36 | 0 | 0 |
omr-ws-317f7b87b2fc2f2c7910a9b793db09ee72bdf5690c65f970a8ec2ba643f6f1d2 | Using generating functions, find the number of ordered triplets \((x, y, z)\) that satisfy the inequality \(2x + 3y + z < 12\). | [
"84"
] | [
"Using generating functions, find the number of ordered triplets \\((x, y, z)\\) that satisfy the inequality \\(2x + 3y + z < 12\\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-387507d62a49f07692708882dadd9ce55b792551b262a870541bc1d4cc17d9d4 | Using generating functions, find the number of ways to divide $n$ coins into $5$ boxes such that the number of coins in boxes $1$ and $2$ is an even number that does not exceed $10$, and the remaining boxes contain $3$ to $5$ coins. The generating function for this problem is given by $(1+x^2+x^4+x^6+x^8+x^{10})^2(x^3+... | [
"[x^n] \\left( (1 + x^2 + x^4 + x^6 + x^8 + x^{10})^2 (x^3 + x^4 + x^5)^3 \\right)"
] | [
"Using generating functions, find the number of ways to divide $n$ coins into $5$ boxes such that the number of coins in boxes $1$ and $2$ is an even number that does not exceed $10$, and the remaining boxes contain $3$ to $5$ coins. The generating function for this problem is given by $(1+x^2+x^4+x^6+x^8+x^{10})^2... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-8f4d0b920c10d52294d40dfa4d484c13c30540dbac52d3ab2d62ed38e224ac71 | Using generating functions, find the probability that the sum of the numbers on 11 fair dice is divisible by 4. | [
"\\(\\frac{1}{4}\\)"
] | [
"Using generating functions, find the probability that the sum of the numbers on 11 fair dice is divisible by 4."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7abe4b27847e8645b0ba67d9781f3c4a4c46285e83da66850c48b820f7c6964e | Using generating functions, find the value of $\sum_{k \geq 1} \frac{\phi (k)}{k^6}$. | [
"\\dfrac{\\zeta(5)}{\\zeta(6)}"
] | [
"Using generating functions, find the value of $\\sum_{k \\geq 1} \\frac{\\phi (k)}{k^6}$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-718f8e6129497fad6fc41b030037455bcbc866e59ee3a72a2e01100d592bc297 | Using inequalities, find the range of \( V = x(12-2x)(20-2x) \), where \( V \) models the volume of a box made from a \( 12 \times 20 \) rectangle with squares of side length \( x \) cut out from each corner. | [
"\\( 0 \\leq V \\leq \\frac{64}{27} \\cdot \\left(28 + 19\\sqrt{19}\\right) \\)"
] | [
"Using inequalities, find the range of \\( V = x(12-2x)(20-2x) \\), where \\( V \\) models the volume of a box made from a \\( 12 \\times 20 \\) rectangle with squares of side length \\( x \\) cut out from each corner."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 27,
"genselect": 5,
"tir": 26
} | 58 | 0 | 0 |
omr-ws-4239012444bb280f3ae611dd42c816d4ce0d080954133529e28c32041b081ba2 | Using integrals, find the area of the region bounded by the graph of \( y = x^3 + 3x^2 - 2 \) and the tangent lines of that graph that pass through the point \((0, -3)\). | [
"\\(\\frac{27}{64}\\)"
] | [
"Using integrals, find the area of the region bounded by the graph of \\( y = x^3 + 3x^2 - 2 \\) and the tangent lines of that graph that pass through the point \\((0, -3)\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 1,
"tir": 21
} | 26 | 0 | 0 |
omr-ws-dbc6b737818fc94a803f473dca380f2e3de08f930d266cd3e55e0573355b1583 | Using integration, find the area in the first quadrant enclosed by the curve $\frac{x^2}{16}+\frac{y^2}{9}=1$. | [
"} 6\\pi \\]"
] | [
"Using integration, find the area in the first quadrant enclosed by the curve $\\frac{x^2}{16}+\\frac{y^2}{9}=1$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5a10054b833f486531fd2bed849753ba46ff71d95f8f6af1a6634a07438750ae | Using laws of logarithms, change the form of the equation \( y = \frac{1}{5} \log_3 (9x - 36)^{15} - 13 \). Graph the function by first stating the BASIC FUNCTION and then describing each transformation in order. Specifically describe what happens to the domain, range, asymptotes, s-intercept, and vertical compression/... | [
"\\( y = 3 \\log_3 (x - 4) - 7 \\)"
] | [
"Using laws of logarithms, change the form of the equation \\( y = \\frac{1}{5} \\log_3 (9x - 36)^{15} - 13 \\). Graph the function by first stating the BASIC FUNCTION and then describing each transformation in order. Specifically describe what happens to the domain, range, asymptotes, s-intercept, and vertical com... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 17
} | 32 | 0 | 0 |
omr-ws-6ec56e86ecf678318cab50baa2dd4f6a11932250040bdc37e3dd704bf8dfdd42 | Using less than 10 twos (and only twos), create as many numbers as possible from 1 to 1000. You can concatenate twos, use powers, factorials, and take the 2nd root, etc. For example, the number 458 can be expressed as $222 + 2^{2^2}$. | [
"1000"
] | [
"Using less than 10 twos (and only twos), create as many numbers as possible from 1 to 1000. You can concatenate twos, use powers, factorials, and take the 2nd root, etc. For example, the number 458 can be expressed as $222 + 2^{2^2}$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-97ea6a363aa9bfde27ad550399a20b894e7f81b63d628633c069594f79ab5f19 | Using logs to base ten, find all real values of \( x \) for which \( x^{\log x} = 100x \). | [
"\\( x = 100 \\) or \\( x = \\frac{1}{10} \\)"
] | [
"Using logs to base ten, find all real values of \\( x \\) for which \\( x^{\\log x} = 100x \\)."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 24
} | 24 | 0 | 0 |
omr-ws-f0c8393df2d7a8886f3a78c3f436d6efaf92195de2e91c93346669330b29203c | Using mass points, determine the ratio of $CP$ to $CM$, where $M$ is the midpoint of hypotenuse $AB$ in the right triangle $\triangle ABC$ with an inscribed square $WXYZ$ of side length $4$ and hypotenuse $AB$ of length $60$. | [
"\\(\\frac{1}{8}\\)"
] | [
"Using mass points, determine the ratio of $CP$ to $CM$, where $M$ is the midpoint of hypotenuse $AB$ in the right triangle $\\triangle ABC$ with an inscribed square $WXYZ$ of side length $4$ and hypotenuse $AB$ of length $60$."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-401d47cb75fcbc74bd5226a8413e00eb9443a9e1e8855a4c33dd2fe3d42165dc | Using mass points, determine the ratio of the area of the triangle formed by the cevians dividing the sides of a triangle in the ratio 2:1 to the area of the original triangle. | [
"\\dfrac{1}{70}"
] | [
"Using mass points, determine the ratio of the area of the triangle formed by the cevians dividing the sides of a triangle in the ratio 2:1 to the area of the original triangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-5b5d5f0af6012056c3ed16afb2111e14b0d0a373940c0a0dec75ec234261d2e4 | Using mathematical relations, state the assertion that the function $y=3x-x^2$ increases when the argument varies in the interval from $-1$ to $+1$. | [
"If arbitrary $x_1$ and $x_2$ are such that $-1\\le x_1<x_2\\le 1$, then $3x_1-x_1^2<3x_2-x_2^2.$"
] | [
"Using mathematical relations, state the assertion that the function $y=3x-x^2$ increases when the argument varies in the interval from $-1$ to $+1$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8eacb58b0618ada80fb011cfda14aadfa52e0cc55dafde3a075d8f9fcafba771 | Using methods other than a computer, find the 7 roots of the equation $e^z = 7z^7$ within the region $|z| < 1$. | [
"7"
] | [
"Using methods other than a computer, find the 7 roots of the equation $e^z = 7z^7$ within the region $|z| < 1$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 6
} | 15 | 0 | 0 |
omr-ws-4a1088132306c06356b9f021afd15fb4139745669c65e104a3f7297a87a72dfd | Using multi-index notation, let \( f(x) = e^{x_1} \) and \( g(x) = e^{x_2} \). Calculate the value of \( \partial^{\alpha}(fg) \) at \( x = (0,0) \) for \( \alpha = (2,2) \). | [
"1"
] | [
"Using multi-index notation, let \\( f(x) = e^{x_1} \\) and \\( g(x) = e^{x_2} \\). Calculate the value of \\( \\partial^{\\alpha}(fg) \\) at \\( x = (0,0) \\) for \\( \\alpha = (2,2) \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ffe4e8ef560cb6e014c6d35b96ec05b7da835e8e51e34003b5c04eb95285b152 | Using one or more of 20 3 cent stamps and 20 5 cent stamps, how many different amounts of postage can be made? | [
"146"
] | [
"Using one or more of 20 3 cent stamps and 20 5 cent stamps, how many different amounts of postage can be made?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7822c40fb0433a6feec2ac40f256b225dbc932159d61fd2024e5bc2bc846466b | Using only $+, -, \times, \div$, and parentheses, form the following lists of integers into arithmetic expressions equal to $24$: (i) $2, 5, 10, 13$, (ii) $5, 7, 10, 11$, (iii) $1, 7, 13, 13$, (iv) $3, 4, 8, 13$. Every integer in a list must be used the same number of times as it appears in the list. | [
"\\frac{(13 - 4) \\times 8}{3}"
] | [
"Using only $+, -, \\times, \\div$, and parentheses, form the following lists of integers into arithmetic expressions equal to $24$:\n(i) $2, 5, 10, 13$,\n(ii) $5, 7, 10, 11$,\n(iii) $1, 7, 13, 13$,\n(iv) $3, 4, 8, 13$.\nEvery integer in a list must be used the same number of times as it appears in the list."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-c37a8d8eb4c25ec9fa71a6e057560383d982ad5656032e1415c6fce48da5685a | Using only 32 cent and 20 cent stamps, Charlie put 3.36 dollars of postage on a package. He used twice as many 32 cent stamps as 20 cent stamps. Determine how many of each type of stamp he used. | [
"4 (20 cent stamps) and 8 (32 cent stamps)"
] | [
"Using only 32 cent and 20 cent stamps, Charlie put 3.36 dollars of postage on a package. He used twice as many 32 cent stamps as 20 cent stamps. Determine how many of each type of stamp he used."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 12,
"tir": 24
} | 36 | 0 | 0 |
omr-ws-dc6401e743d8fa046c8134e5985fb832618cc3b5285b8196ee18ba029beffeda | Using only 33 gram nuts and 63 gram bolts, determine the minimum possible total weight of nuts and bolts needed to balance a 432 gram motor in the left pan. Calculate the total weight of nuts and bolts used in your solution. | [
"558"
] | [
"Using only 33 gram nuts and 63 gram bolts, determine the minimum possible total weight of nuts and bolts needed to balance a 432 gram motor in the left pan. Calculate the total weight of nuts and bolts used in your solution."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-48b50be8fb977c6aebf919d3a513758f8fc65011453dd38f909feb0265680f19 | Using only 33 gram nuts and 63 gram bolts, level a balance with a 432 gram motor in the left pan. Specify how many nuts and how many bolts go in each pan and justify your answer. | [
"One 63 gram bolt in the left pan and 15 33 gram nuts in the right pan."
] | [
"Using only 33 gram nuts and 63 gram bolts, level a balance with a 432 gram motor in the left pan. Specify how many nuts and how many bolts go in each pan and justify your answer."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 1,
"tir": 13
} | 28 | 0 | 0 |
omr-ws-68053cda79f8e591c3b86ae9cf17dc7ff669a5c58f3f65330ff438a9cc79a3ca | Using only 5 zeroes and any mathematical operations, how can you get to 120? | [
"120"
] | [
"Using only 5 zeroes and any mathematical operations, how can you get to 120?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-61440c92b9939ed7379410825990fa65d65e7ef3ef56ecf4ca5e0a8ee43fbdb6 | Using only Lagrange interpolation polynomial, find the inverse \( C^{-1} \) of the Cauchy matrix \( C = ||c(i,j)|| \) where \( c(i,j) = \frac{1}{i+j} \). | [
"C^{-1}(i,j) = (-1)^{i+j} \\frac{\\prod_{k=1}^n (i+k)(k+j)}{(i + j) \\prod_{\\substack{k=1 \\\\ k \\neq i}}^n (i - k) \\prod_{\\substack{k=1 \\\\ k \\neq j}}^n (j - k)}"
] | [
"Using only Lagrange interpolation polynomial, find the inverse \\( C^{-1} \\) of the Cauchy matrix \\( C = ||c(i,j)|| \\) where \\( c(i,j) = \\frac{1}{i+j} \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 8
} | 9 | 0 | 0 |
omr-ws-2deae3658879d6a76ec13d1fb3901d16cb2a1f56b0fdc6c5a0f6df434fed2174 | Using only Lagrange interpolation polynomial, find the inverse \( V^{-1} \) of the Vandermonde matrix \( V = V(x_1, x_2, \ldots, x_n) \) where \( a(i,j) = x_i^{j-1} \) and \( x_i \ne x_j \) for \( i \ne j \). | [
"V^{-1}_{j,i} = \\dfrac{(-1)^{n-j} \\displaystyle\\sum_{1 \\leq m_1 < \\cdots < m_{n-j} \\leq n \\atop m_k \\neq i} x_{m_1} \\cdots x_{m_{n-j}}}"
] | [
"Using only Lagrange interpolation polynomial, find the inverse \\( V^{-1} \\) of the Vandermonde matrix \\( V = V(x_1, x_2, \\ldots, x_n) \\) where \\( a(i,j) = x_i^{j-1} \\) and \\( x_i \\ne x_j \\) for \\( i \\ne j \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 6
} | 7 | 0 | 0 |
omr-ws-77d40e8706351420b9ce7b3a417993e1c14e1142f6fa7ee8bac2804863c55ea2 | Using only a 3-liter bucket and a 5-liter bucket, how can you measure exactly 4 liters of water? | [
"Fill the 5-liter bucket and pour it into the 3-liter bucket, leaving 2 liters in the 5-liter bucket. Empty the 3-liter bucket and pour the 2 liters from the 5-liter bucket into the 3-liter bucket. Fill the 5-liter bucket again and top up the 3-liter bucket from it, which will require 1 liter, leaving 4 liters in t... | [
"Using only a 3-liter bucket and a 5-liter bucket, how can you measure exactly 4 liters of water?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 16
} | 17 | 0 | 0 |
omr-ws-4e2b6b5964f03675414790a81104320736791077337031ea848aa7bf41e87327 | Using only a 9-minute timer and a 13-minute timer, which positive integer values of $m$ minutes cannot be measured? For a particular value of $m$ that can be measured, how many different ways can we measure $m$ minutes, and what are all these different ways? | [
"1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 14, 15, 16, 17, 19, 20, 21, 23, 24, 25, 28, 29, 30, 32, 33, 34, 37, 38, 41, 42, 43, 46, 47, 50, 51, 55, 56, 59, 60, 64, 68, 69, 73, 77, 82, 86, 95"
] | [
"Using only a 9-minute timer and a 13-minute timer, which positive integer values of $m$ minutes cannot be measured? For a particular value of $m$ that can be measured, how many different ways can we measure $m$ minutes, and what are all these different ways?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-8fe0c6f2d64b742edbbc0cf53b5cd82a752b2d4376baec357c3bf91eef630f8c | Using only a compass (no ruler), find the center of a given circle. | [
"O"
] | [
"Using only a compass (no ruler), find the center of a given circle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-db125c96174880b610740d7e7190430b1b1fccd516d112d4151c0f09e8546ba9 | Using only a compass and a ruler, construct an angle of $1^\circ$ given an angle of $19^\circ$. | [
"1^\\circ"
] | [
"Using only a compass and a ruler, construct an angle of $1^\\circ$ given an angle of $19^\\circ$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-164bb1a557d6b16bb5c37b712d046def590a8ed9303a65563d09a08a16e1694e | Using only a compass and a ruler, reconstruct triangle $ABC$ given the following three points: point $M$ (the intersection of its medians), point $I$ (the center of its inscribed circle), and point $Q_a$ (the touch point of the inscribed circle to side $BC$). | [
"\\triangle ABC"
] | [
"Using only a compass and a ruler, reconstruct triangle $ABC$ given the following three points: point $M$ (the intersection of its medians), point $I$ (the center of its inscribed circle), and point $Q_a$ (the touch point of the inscribed circle to side $BC$)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-46b9e58e564eb2ef5a070d27113f514fa7f175e1e1d8e57d8646b7e03a55934d | Using only a compass and no straight-edge or any other tools, construct a square and let the coordinates of its vertices be $(0,0)$, $(a,0)$, $(a,a)$, and $(0,a)$. Find the value of $a^2$ if the circle used to start the construction has a radius of $1$. | [
"2"
] | [
"Using only a compass and no straight-edge or any other tools, construct a square and let the coordinates of its vertices be $(0,0)$, $(a,0)$, $(a,a)$, and $(0,a)$. Find the value of $a^2$ if the circle used to start the construction has a radius of $1$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8dc0bae6dceda0b78917242aa9962678f15457ca598f79d7c649647fece03eb5 | Using only a compass and ruler, find the arithmetic mean (AM), geometric mean (GM), and harmonic mean (HM) of the numbers 5 and 10. | [
"AM = 7.5, GM = \\(5\\sqrt{2}\\), HM = \\(\\frac{20}{3}\\)"
] | [
"Using only a compass and ruler, find the arithmetic mean (AM), geometric mean (GM), and harmonic mean (HM) of the numbers 5 and 10."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 23
} | 23 | 0 | 0 |
omr-ws-25a24b2d7d35b09a21a13a63985b286021a638f5df0e704f0cce16a50f04145a | Using only a compass and straightedge, construct a square such that two adjacent vertices lie on a given circle and the other two vertices lie on a given line that does not intersect the circle. | [
"ABCD"
] | [
"Using only a compass and straightedge, construct a square such that two adjacent vertices lie on a given circle and the other two vertices lie on a given line that does not intersect the circle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 2
} | 7 | 0 | 0 |
omr-ws-fd13b6e613963c895be5d8fd21cf8fc85db9c5e6efb947351f7b3e455d133c07 | Using only a compass, construct a 1 degree arc on a circle if a 19 degree arc is given. | [
"A 1 degree arc can be constructed using the given methods."
] | [
"Using only a compass, construct a 1 degree arc on a circle if a 19 degree arc is given."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 1,
"tir": 20
} | 22 | 0 | 0 |
omr-ws-6b16bb3cca044529e3b9a6f000a6d5e66bbdfc9e99c2d266da737593cfdda50c | Using only a compass, construct a figure of area $2$ from a given square of area $1$. | [
"2"
] | [
"Using only a compass, construct a figure of area $2$ from a given square of area $1$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3d85b533b34af97ed662134af98cb35d9c755d31e8784936e8a88313c6b519e3 | Using only a compass, construct two circles that intersect at right angles; that is, the tangents at the two points where they meet are perpendicular. | [
"\\text{Two circles constructed with centers at endpoints of perpendicular diameters of an initial circle intersect at right angles.}"
] | [
"Using only a compass, construct two circles that intersect at right angles; that is, the tangents at the two points where they meet are perpendicular."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 4
} | 5 | 0 | 0 |
omr-ws-a3fc8e7926184f9e16b77222c026bc2d9a4d7223a8536d13383d484e2a4f898f | Using only a compass, determine the minimal number of circles one must draw to ascertain if a given angle in a plane is acute. | [
"1"
] | [
"Using only a compass, determine the minimal number of circles one must draw to ascertain if a given angle in a plane is acute."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ae3e538a49fb2185beaa33527bbdf7d785f858d47b8a7344e87bcad63a2ccf3d | Using only a compass, divide a given square into two parts of equal area. | [
"Draw a circle centered at one corner with radius equal to the square's diagonal, then draw another circle from the opposite corner; their intersection points define a line through the square's center, dividing it into two equal areas."
] | [
"Using only a compass, divide a given square into two parts of equal area."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 4
} | 5 | 0 | 0 |
omr-ws-c0ce9452dbdefa3c3ba0e7cd060cf994efcc4ce601aabc8413bb786e5cdc6958 | Using only a compass, draw a circle that passes through three given points. | [
"The construction process described in the forum discussions."
] | [
"Using only a compass, draw a circle that passes through three given points."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 35,
"genselect": 3,
"tir": 7
} | 45 | 0 | 0 |
omr-ws-7fa802830e008938bcc8e0d5449a59be780f880658f8011b31681014b8b144e0 | Using only a ruler (without divisions), construct the symedian of an acute-angled unequal triangle \(ABC\) by drawing no more than four lines. The symedian is the line symmetrical to the median relative to the corresponding angle bisector. | [
"AM'"
] | [
"Using only a ruler (without divisions), construct the symedian of an acute-angled unequal triangle \\(ABC\\) by drawing no more than four lines. The symedian is the line symmetrical to the median relative to the corresponding angle bisector."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 6
} | 15 | 0 | 0 |
omr-ws-16195fb321e282d24d80dd813c4f1060066b36e67179654c812eb12febf019d1 | Using only a ruler (without divisions), construct the symedian of an unequal triangle \(ABC\) by drawing no more than six lines. The symedian is the straight line symmetrical to the median relative to the corresponding bisector. The triangle has an incircle \(I\) and a circumcircle, the center of which is not marked. | [
"AD"
] | [
"Using only a ruler (without divisions), construct the symedian of an unequal triangle \\(ABC\\) by drawing no more than six lines. The symedian is the straight line symmetrical to the median relative to the corresponding bisector. The triangle has an incircle \\(I\\) and a circumcircle, the center of which is not ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 3
} | 10 | 0 | 0 |
omr-ws-83d84e7d24664ebd96c615793cb52a83d0737ed612f9758ef23d9a4779082d19 | Using only a ruler and compasses, construct two circles that intersect orthogonally (i.e., the tangents to the circles at a point of their intersection are perpendicular). | [
"\\text{Two circles constructed as described intersect orthogonally.}"
] | [
"Using only a ruler and compasses, construct two circles that intersect orthogonally (i.e., the tangents to the circles at a point of their intersection are perpendicular)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e05a171d50548fac115e67e492fbcd60d0948d406ce1df5fdff0a9bcfa9edb83 | Using only a ruler without divisions and without going beyond a $2 \times 2$ square on grid paper, divide the diagonal of the square into 6 equal parts. | [
"\\text{Draw lines from (0,2) to (1,0) and from (2,0) to (1,2), intersecting the diagonal at points that divide it into six equal parts.}"
] | [
"Using only a ruler without divisions and without going beyond a $2 \\times 2$ square on grid paper, divide the diagonal of the square into 6 equal parts."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b6e4197ec0806f39c190636e4615e16a2ec6c28dd03b2ebdf004dd5f215ee86b | Using only a ruler without divisions, divide the diagonal of a $2 \times 2$ square into 6 equal parts. | [
"\\text{Diagonally divided into six equal parts using ruler-only construction.}"
] | [
"Using only a ruler without divisions, divide the diagonal of a $2 \\times 2$ square into 6 equal parts."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 6
} | 20 | 0 | 0 |
omr-ws-0e424955793ca20bb336e08ae1ec30a5e6e160fc4f774ff838d0169230a3da04 | Using only a ruler without divisions, show how to divide the diagonal of a $1 \times 4$ rectangle into three equal parts. | [
"The first trisection point is the intersection of the line through \\((0,-2)\\) and \\((4,5)\\) and the line between \\((0,5)\\) and \\((2,-2)\\). The second trisection point is the intersection of the line between \\((0,4)\\) and \\((4,-1)\\) and the line between \\((2,-2)\\) and \\((3,2)\\)."
] | [
"Using only a ruler without divisions, show how to divide the diagonal of a $1 \\times 4$ rectangle into three equal parts."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-8f9a7b3a35d5070f469686e00a27b344055b0856513c4465e6ec22b94f201319 | Using only a ruler, construct a triangle whose area is equal to the area of a given trapezoid $ABCD$ with bases $BC = a$ and $AD = 2a$. | [
"\\triangle ABE"
] | [
"Using only a ruler, construct a triangle whose area is equal to the area of a given trapezoid $ABCD$ with bases $BC = a$ and $AD = 2a$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 3
} | 15 | 0 | 0 |
omr-ws-dbf56c6b0d90f3d63712d9f0a5dae0c4bdedbe697396374a4428793e9691f3c6 | Using only a ruler, construct the perpendicular from an exterior point to the extension of a given diameter of a circle. | [
"PQ"
] | [
"Using only a ruler, construct the perpendicular from an exterior point to the extension of a given diameter of a circle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-f9860d07c153afb612efd4537912bc0a47bad3c638e1b9b3de88c1b517020a34 | Using only a straightedge and a compass, construct a line segment of length $\sqrt{2010}$ given a line segment of length 1. If the constructed line segment has length $\sqrt{a}$, find the value of $a$. | [
"2010"
] | [
"Using only a straightedge and a compass, construct a line segment of length $\\sqrt{2010}$ given a line segment of length 1. If the constructed line segment has length $\\sqrt{a}$, find the value of $a$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9dfeabaa5c7960f3f376c2ac949cc24f752ef6836b7a5d72f8236d2680d30ee4 | Using only a straightedge and a compass, determine the exact value of $\cos(20^\circ)$ if it is possible to trisect a $60^\circ$ angle. If it is not possible, provide the value of $\cos(20^\circ)$ in terms of radicals. | [
" \\sqrt[3]{\\dfrac{1 + i\\sqrt{3}}{16}} + \\sqrt[3]{\\dfrac{1 - i\\sqrt{3}}{16}} "
] | [
"Using only a straightedge and a compass, determine the exact value of $\\cos(20^\\circ)$ if it is possible to trisect a $60^\\circ$ angle. If it is not possible, provide the value of $\\cos(20^\\circ)$ in terms of radicals."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-0bd2c6b1e66205442c72553edb847178e9f40923aac4ed580f9d2b935fdc6b69 | Using only a straightedge, divide a given segment into three equal parts. (With a straightedge, you can join two given points and draw a line perpendicular to a given line from a given point.) | [
"AB \\text{ is divided into three equal segments.}"
] | [
"Using only a straightedge, divide a given segment into three equal parts. (With a straightedge, you can join two given points and draw a line perpendicular to a given line from a given point.)"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 4
} | 18 | 0 | 0 |
omr-ws-c09200fdeb7f77715390bdb8b1a6b7ef13b88e0a9d525d7744f79c7dc93f04b8 | Using only an angle of $\frac{\pi}{7}$ and a ruler, construct an angle of $\frac{\pi}{14}$. | [
"\\dfrac{\\pi}{14}"
] | [
"Using only an angle of $\\frac{\\pi}{7}$ and a ruler, construct an angle of $\\frac{\\pi}{14}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-5deb77b905bce85265fd25498687163b92619c89a34a693f14f90678af7d9c65 | Using only an ordinary floating-point device (calculator or spreadsheet), give a reasonable floating-point approximation of \( M = \max_{x \in [0, \infty)} x^{1000} e^{-0.001x} \). Do not use anything that uses arbitrary-precision arithmetic. | [
"\\( 5.07596 \\times 10^{5565} \\)"
] | [
"Using only an ordinary floating-point device (calculator or spreadsheet), give a reasonable floating-point approximation of \\( M = \\max_{x \\in [0, \\infty)} x^{1000} e^{-0.001x} \\). Do not use anything that uses arbitrary-precision arithmetic."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-b9fe8a2acfca48dc4ee3067110cef9e50fbe441ec1d7178dfbf29ffa4056f1ae | Using only an unmarked ruler, construct a square whose vertices lie on a given circle, with the circle's center also given. | [
"\\text{Square constructed with vertices on the given circle using the described method.}"
] | [
"Using only an unmarked ruler, construct a square whose vertices lie on a given circle, with the circle's center also given."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ef701c1490cc037bcc9db1186248f060fa051e8e3e3c9eb6273993a1d06a4c9b | Using only basic mathematical tools such as integration and Taylor series, prove that \( e < \pi \) and find the smallest integer \( n \) such that \( \left(1 + \frac{1}{n}\right)^n < \pi \). | [
"1"
] | [
"Using only basic mathematical tools such as integration and Taylor series, prove that \\( e < \\pi \\) and find the smallest integer \\( n \\) such that \\( \\left(1 + \\frac{1}{n}\\right)^n < \\pi \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
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