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-ad2b373eb003ca2a39a62ea5671acd90a05832fde91d87c13e19379dd59dc02a | Using Green's Theorem, find the area between the curves $ y^2=4x $ and $ x^2=4y $. | [
"\\(\\frac{16}{3}\\)"
] | [
"Using Green's Theorem, find the area between the curves $ y^2=4x $ and $ x^2=4y $."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-c96be48d2fb24dd29d26c919020caf6df8024047a479c9fff0747999b5daad6d | Using Green's Theorem, find the outward flux of the vector field \( \mathbf{F} = (x^2 + y^2) \mathbf{i} + (x - y) \mathbf{j} \) across the closed curve \( C \), where \( C \) is the rectangle with vertices at \( (0, 0) \), \( (2, 0) \), \( (2, 7) \), and \( (0, 2) \). | [
"14"
] | [
"Using Green's Theorem, find the outward flux of the vector field \\( \\mathbf{F} = (x^2 + y^2) \\mathbf{i} + (x - y) \\mathbf{j} \\) across the closed curve \\( C \\), where \\( C \\) is the rectangle with vertices at \\( (0, 0) \\), \\( (2, 0) \\), \\( (2, 7) \\), and \\( (0, 2) \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 6
} | 11 | 0 | 0 |
omr-ws-9b576ccd6dfcd2998026c8c8e68dad831a0a5393e3d09afe86e1eca4f69a7d6f | Using Green's theorem and parametrizing the curve, calculate the area enclosed by the curve $\frac{x^{\frac{2}{3}}}{9} + \frac{y^{\frac{2}{3}}}{4} = 1$. | [
"\\( 81\\pi \\)"
] | [
"Using Green's theorem and parametrizing the curve, calculate the area enclosed by the curve $\\frac{x^{\\frac{2}{3}}}{9} + \\frac{y^{\\frac{2}{3}}}{4} = 1$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-92737e7551067ef53c638ea847d0a814dfafa581a066ddd52e767d93350f85d7 | Using Green's theorem, find the outward flux for the vector field \( \mathbf{F} = 2e^{xy} \mathbf{i} + y^3 \mathbf{j} \) over the square bounded by the lines \( x = 1 \), \( x = -1 \), \( y = 1 \), and \( y = -1 \). | [
"4"
] | [
"Using Green's theorem, find the outward flux for the vector field \\( \\mathbf{F} = 2e^{xy} \\mathbf{i} + y^3 \\mathbf{j} \\) over the square bounded by the lines \\( x = 1 \\), \\( x = -1 \\), \\( y = 1 \\), and \\( y = -1 \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 7
} | 19 | 0 | 0 |
omr-ws-86a6982d9e4b7b38bfa9b0145e9e59b40b0fcd1983373ceb34ad963b7dab5fe0 | Using Inclusion-Exclusion, find the number of integers between 1 and 2000 (inclusive) which are divisible by at least one of 2, 3, 5, or 7. | [
"1514"
] | [
"Using Inclusion-Exclusion, find the number of integers between 1 and 2000 (inclusive) which are divisible by at least one of 2, 3, 5, or 7."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f30f7337a31a509c9a86602c08a9f7ff5c2c4a6f00f8576a9f2ed0272a544e9e | Using Ito's formula, find \( df \) for the function \( f(X_t, t) \) given the stochastic process \( X_t = X_0 + \mu S(t) + \sigma B(S(t)) \). | [
"\\( df = f_t dt + \\sigma f_x dB(S(t)) + \\left( \\mu f_x + \\frac{1}{2} \\sigma^2 f_{xx} \\right) dS(t) \\)"
] | [
"Using Ito's formula, find \\( df \\) for the function \\( f(X_t, t) \\) given the stochastic process \\( X_t = X_0 + \\mu S(t) + \\sigma B(S(t)) \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 1
} | 3 | 0 | 0 |
omr-ws-f07153dd67aa4ff21f5e353073bc39f21ea6120b1cf4d253b13c9c76068f85bf | Using Jensen's inequality, find the minimum value of \(2^a + 2^{2b} + 2^{4c}\) given that \(a + b + c = 7\). | [
"\\(7 \\left(2^{\\frac{18}{7}}\\right)\\)"
] | [
"Using Jensen's inequality, find the minimum value of \\(2^a + 2^{2b} + 2^{4c}\\) given that \\(a + b + c = 7\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 16,
"genselect": 6,
"tir": 15
} | 37 | 0 | 0 |
omr-ws-92514eee48c13ff15c7694513d1abb7a97a87fe8e32aa4584b25f398705022a0 | Using L'Hôpital's rule, find the limit $$\lim\limits_{x\to 0}\frac{x^2-\sin^2x}{x^2\sin^2x}.$$ | [
"\\(\\frac{1}{3}\\)"
] | [
"Using L'Hôpital's rule, find the limit $$\\lim\\limits_{x\\to 0}\\frac{x^2-\\sin^2x}{x^2\\sin^2x}.$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-5896de0f083d6d79fac964115feb0a7440354d9b7de076e33d29ea5eb6869e44 | Using Lagrange Multipliers, find the minimum cost to build an open box with base dimensions \(x \times y\) cm and height \(z\) cm, given that the volume of the box must be 48 cubic centimeters. The cost is \$3 per square centimeter for the base and \$1 per square centimeter for the sides. | [
"48"
] | [
"Using Lagrange Multipliers, find the minimum cost to build an open box with base dimensions \\(x \\times y\\) cm and height \\(z\\) cm, given that the volume of the box must be 48 cubic centimeters. The cost is \\$3 per square centimeter for the base and \\$1 per square centimeter for the sides."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f2a7a9d526145502c7e5317ae53c0f2c3a3ed6f058bb3a62b7f479671e608351 | Using Lagrange multipliers, find the maximum and minimum of the function \( f(x,y) = x^2 + y^2 - xy + x + y \) on the domain \( D = \{(x,y) : x \leq 0, y \leq 0, x + y \geq -3\} \). | [
"\\( f_{MIN} = -1, f_{MAX} = 6 \\)"
] | [
"Using Lagrange multipliers, find the maximum and minimum of the function \\( f(x,y) = x^2 + y^2 - xy + x + y \\) on the domain \\( D = \\{(x,y) : x \\leq 0, y \\leq 0, x + y \\geq -3\\} \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 27
} | 28 | 0 | 0 |
omr-ws-3ad64e265ea31a7edd6d61df78783e529fd3a8cc9bbd96df545b07082be17660 | Using Lagrange's Mean Value Theorem (LMVT), find all positive solutions to the system of equations where \( U \) and \( V \) are fixed positive numbers: \[ X^2 + Y^2 = U^2 + V^2 \] \[ X^3 + Y^3 = U^3 + V^3 \] | [
"\\((X, Y) = (U, V)\\) or \\((X, Y) = (V, U)\\)"
] | [
"Using Lagrange's Mean Value Theorem (LMVT), find all positive solutions to the system of equations where \\( U \\) and \\( V \\) are fixed positive numbers:\n\\[ X^2 + Y^2 = U^2 + V^2 \\]\n\\[ X^3 + Y^3 = U^3 + V^3 \\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 21
} | 35 | 0 | 0 |
omr-ws-7053e6164c930fd5c3be69b7802c92cb026b74c3f4a4eacf445c69f20c1691a1 | Using Maclaurin's Theorem, obtain the power series for \( e^{-kh} \) up to and including the term \( h^3 \). | [
"\\( 1 - kh + \\frac{k^2 h^2}{2} - \\frac{k^3 h^3}{6} \\)"
] | [
"Using Maclaurin's Theorem, obtain the power series for \\( e^{-kh} \\) up to and including the term \\( h^3 \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-87c52d6da69b6ad87a1e8572ed8812924ecc239f2feb3d47659066e26b5aaf3f | Using Mary's secret code, where each letter is translated to its position in the alphabet and then multiplied by her favorite number and added to her mother's favorite number, F becomes 65 and T becomes 177. Determine Mary's favorite number and her mother's favorite number. | [
"Mary's favourite number is 8 and her mother's favourite number is 17."
] | [
"Using Mary's secret code, where each letter is translated to its position in the alphabet and then multiplied by her favorite number and added to her mother's favorite number, F becomes 65 and T becomes 177. Determine Mary's favorite number and her mother's favorite number."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 15
} | 17 | 0 | 0 |
omr-ws-a5289bcb5bc8db5d102259fc7cee640f683807202521e35c06f6243b21298b47 | Using Muirhead's inequality or any other method, find the smallest positive integer $k$ such that the inequality $k[4,1,0] + k[2,2,1] \geq k[3,2,0] + k[3,1,1]$ holds for all positive real numbers $a$, $b$, and $c$. | [
""
] | [
"Using Muirhead's inequality or any other method, find the smallest positive integer $k$ such that the inequality $k[4,1,0] + k[2,2,1] \\geq k[3,2,0] + k[3,1,1]$ holds for all positive real numbers $a$, $b$, and $c$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 1 |
omr-ws-faccd29815a277233e2e587ad2a8d0efea8987abab010e5f7606a652e35ce29a | Using Newton's cooling law, determine the time of death of a body given the following information: - The body's temperature was 80 degrees Fahrenheit at noon. - The body's temperature was 75 degrees Fahrenheit at 1pm. - The ambient temperature is 70 degrees Fahrenheit. - The body's temperature before death was 98.6 deg... | [
"10:29:02 am"
] | [
"Using Newton's cooling law, determine the time of death of a body given the following information:\n- The body's temperature was 80 degrees Fahrenheit at noon.\n- The body's temperature was 75 degrees Fahrenheit at 1pm.\n- The ambient temperature is 70 degrees Fahrenheit.\n- The body's temperature before death was... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-e9a16a62a42b2a05fb48dc835c602a00e1cd4635dbd80cff9178ff3933fd673c | Using Newton’s method to solve the equation \(\cos x = 0\) to estimate \(\pi/2\) to five decimal places, does the choice of the starting value matter? Provide reasons for your answer. | [
"Yes"
] | [
"Using Newton’s method to solve the equation \\(\\cos x = 0\\) to estimate \\(\\pi/2\\) to five decimal places, does the choice of the starting value matter? Provide reasons for your answer."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-11b9b57ea5c1c32a949a6dfe62a2a43f329189c94a275a498620f272b6cf8d05 | Using Parseval's theorem, find the value of the integral \[ I = \int_{-\infty}^{+\infty} \frac{\sin^4 y}{y^4} \, dy. \] | [
"\\(\\frac{2\\pi}{3}\\)"
] | [
"Using Parseval's theorem, find the value of the integral\n\\[ I = \\int_{-\\infty}^{+\\infty} \\frac{\\sin^4 y}{y^4} \\, dy. \\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 12
} | 25 | 0 | 0 |
omr-ws-47049be136003882d875c6fe1711b35abfd1d9e11c7d868d306b914cc3760fb7 | Using Poisson's summation formula, what function \( f \) should be chosen to obtain the result \( \sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6} \)? | [
"\\(\\frac{\\pi^2}{6}\\)"
] | [
"Using Poisson's summation formula, what function \\( f \\) should be chosen to obtain the result \\( \\sum_{n=1}^{\\infty} \\frac{1}{n^2} = \\frac{\\pi^2}{6} \\)?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ecaf3bcd132b1753a5b7b60fe87b709de604115591394cdc6cbd3127f6e7e902 | Using Pollard's rho algorithm with the function \( f(x) = x^2 + 2 \) and the seed \( x_0 = 13 \), after how many iterations will the algorithm stop when factoring \( N = 1659629823585814114799165699914477904363 \)? Note that no calculators can be used. | [
"3"
] | [
"Using Pollard's rho algorithm with the function \\( f(x) = x^2 + 2 \\) and the seed \\( x_0 = 13 \\), after how many iterations will the algorithm stop when factoring \\( N = 1659629823585814114799165699914477904363 \\)? Note that no calculators can be used."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 2
} | 3 | 0 | 0 |
omr-ws-cdb4ad8ddbf059004fa16492774d19ff0a892d41e9665fbe39009ebf5953e2c6 | Using Riemann Sum, calculate the area bounded by the curve \( y = \frac{1}{x^2} \) and the x-axis from \( x = 1 \) to \( x = 2 \). | [
"\\dfrac{1}{2}"
] | [
"Using Riemann Sum, calculate the area bounded by the curve \\( y = \\frac{1}{x^2} \\) and the x-axis from \\( x = 1 \\) to \\( x = 2 \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-acfdffb464c72ef51c90e7db89a401a9d0e1748d9e57de02878c9ec0dd3d396f | Using Riemann sums, find \[ \lim_{n\to\infty}\sum_{k=1}^{n}\sin\left(\frac{k}{n}\right)\cdot\sin\left(\frac{k}{n^2}\right) \] | [
"\\(\\sin 1 - \\cos 1\\)"
] | [
"Using Riemann sums, find\n\\[\n\\lim_{n\\to\\infty}\\sum_{k=1}^{n}\\sin\\left(\\frac{k}{n}\\right)\\cdot\\sin\\left(\\frac{k}{n^2}\\right)\n\\]"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 4,
"tir": 27
} | 41 | 0 | 0 |
omr-ws-6b320f48d89729d96c235fc41de6ed4309ef08f0543deae83e514964bf9fb04f | Using Rouché's theorem, determine that the equation \( z^8 - 5z^5 - 2z + 1 = 0 \) has 5 roots within the region \( |z| < 1 \). How can these 5 roots be found without using a computer? | [
"5"
] | [
"Using Rouché's theorem, determine that the equation \\( z^8 - 5z^5 - 2z + 1 = 0 \\) has 5 roots within the region \\( |z| < 1 \\). How can these 5 roots be found without using a computer?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-21f3c3416d9e4582fa5ad21e28cb73ff94b6ca303947500a493df4583d0fc5ef | Using Rouché's theorem, find the number of zeros of \( z^4 - 4z^3 + 6z^2 - 4z + 3 \) in the domain \( 0 < |z| < 2 \). | [
"2"
] | [
"Using Rouché's theorem, find the number of zeros of \\( z^4 - 4z^3 + 6z^2 - 4z + 3 \\) in the domain \\( 0 < |z| < 2 \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-b9c3be94a4aba5488a40c157d0f4a1138e130063acf7cf72dfcec842f439db76 | Using Stirling's Approximation, let \( d(n) = \binom{2n}{n} \frac{\sqrt{\pi n}}{4^n} - 1 \). Calculate the value of \( \lim_{n \to \infty} d(n) n^{3/2} \). | [
"-\\infty"
] | [
"Using Stirling's Approximation, let \\( d(n) = \\binom{2n}{n} \\frac{\\sqrt{\\pi n}}{4^n} - 1 \\). Calculate the value of \\( \\lim_{n \\to \\infty} d(n) n^{3/2} \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-a0e9a540b88957586e3a1abbcc8c92abf9f05f2de8300ac4fc0e88b23bbfb0d3 | Using Stirling's approximation for $n!$, which is $\exp(-n)n^{n}\sqrt{2\pi n}$, calculate the value of the expression $$\frac{{2n \choose n} \sqrt{\pi n}}{2^{2n}}$$ for $n = 100$. Express your answer as a decimal rounded to the nearest thousandth. | [
"1.000"
] | [
"Using Stirling's approximation for $n!$, which is $\\exp(-n)n^{n}\\sqrt{2\\pi n}$, calculate the value of the expression $$\\frac{{2n \\choose n} \\sqrt{\\pi n}}{2^{2n}}$$ for $n = 100$. Express your answer as a decimal rounded to the nearest thousandth."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e4332791181541056e7e2cb0388a774d1a8bb73aba5d658c5634bf7a2753764f | Using Stirling's formula $\lim \frac{n^{n}}{n!e^{n}}\sqrt{2n \pi}=1$, evaluate $\lim \frac{n!}{\left(1+\frac{1}{n}\right)^{n^2}}$. | [
"\\(+\\infty\\)"
] | [
"Using Stirling's formula $\\lim \\frac{n^{n}}{n!e^{n}}\\sqrt{2n \\pi}=1$, evaluate $\\lim \\frac{n!}{\\left(1+\\frac{1}{n}\\right)^{n^2}}$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-7fdfe0e2317c1d97ebd39771655a8eb9a55e86e89b0eff679f04bd481647583f | Using Stirling's formula, calculate the value of $k$ in the expression $\frac{1000!}{(500!)^2} \approx 2^{1000} \sqrt{\frac{2}{\pi \cdot 1000}} \cdot k$. | [
"1"
] | [
"Using Stirling's formula, calculate the value of $k$ in the expression $\\frac{1000!}{(500!)^2} \\approx 2^{1000} \\sqrt{\\frac{2}{\\pi \\cdot 1000}} \\cdot k$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3a30b530bae8591660dc33297e03fc7652e1819d376cabc7369f4b61c3f3e57d | Using Stirling's formula, calculate the value of \( k \) in the expression \[ \binom{n}{\alpha n} \approx \frac{k}{\sqrt{2 \pi \alpha (1-\alpha) n}} 2^{n H(\alpha)}, \] where \( H(\alpha) = - \alpha \log_2{\alpha} - (1-\alpha) \log_2{(1-\alpha)} \), \( 0 < \alpha < 1 \), and \( \alpha n \in \mathbb{N} \). Find \( k \). | [
"1"
] | [
"Using Stirling's formula, calculate the value of \\( k \\) in the expression\n\\[\n\\binom{n}{\\alpha n} \\approx \\frac{k}{\\sqrt{2 \\pi \\alpha (1-\\alpha) n}} 2^{n H(\\alpha)},\n\\]\nwhere \\( H(\\alpha) = - \\alpha \\log_2{\\alpha} - (1-\\alpha) \\log_2{(1-\\alpha)} \\), \\( 0 < \\alpha < 1 \\), and \\( \\alph... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b3bef934b3241cbae2b5e7568a3f752875a9f436d49e1d70cc6c2869334218e6 | Using Stirling's formula, calculate the value of the constant $C$ such that: $$\binom{2n}{n} \leq C \cdot \frac{4^n}{\sqrt{\pi n}}$$ for all $n \geq 1$. Express your answer as a simplified fraction. | [
"1"
] | [
"Using Stirling's formula, calculate the value of the constant $C$ such that:\n$$\\binom{2n}{n} \\leq C \\cdot \\frac{4^n}{\\sqrt{\\pi n}}$$\nfor all $n \\geq 1$. Express your answer as a simplified fraction."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-46622e9e55bef60210d45490201cb28ce9d3a35d1c8494185cd0c3f5ac3f2d95 | Using Stokes' Theorem, compute the line integral $\int _{C} 3y \, dx - xz \, dy + yz^2 \, dz$, where $C$ is the curve defined by the intersection of the plane $z=2$ with the parabola $x^2 + y^2 = 2z$, and $C$ is oriented counterclockwise with respect to the vector $(0,0,1)$. | [
"-20\\pi"
] | [
"Using Stokes' Theorem, compute the line integral $\\int _{C} 3y \\, dx - xz \\, dy + yz^2 \\, dz$, where $C$ is the curve defined by the intersection of the plane $z=2$ with the parabola $x^2 + y^2 = 2z$, and $C$ is oriented counterclockwise with respect to the vector $(0,0,1)$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-aec51d91792265fc4c3e3b9c3c2dfb1666b6b82aa593d6a437a67d13afd9092b | Using Stokes' Theorem, rewrite \( I \) as a line integral and verify that it yields the same result as the surface integral. Specifically, evaluate the line integral along the bottom curve \( z = \frac{1}{4} \) with \( \rho = \frac{1}{2} \). | [
"\\(\\frac{3\\pi}{32}\\)"
] | [
"Using Stokes' Theorem, rewrite \\( I \\) as a line integral and verify that it yields the same result as the surface integral. Specifically, evaluate the line integral along the bottom curve \\( z = \\frac{1}{4} \\) with \\( \\rho = \\frac{1}{2} \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b2eb55ddab106f011e778468a7bfbbfb2bebfb1afafd6dd73435fccd1d8efad8 | Using Stokes' theorem over the surface \( S \) with boundary \( C1 \cup C2 \) and the result from Problem 2, calculate the line integral of \( F \) over the curve \( C1 \). Show the orientation chosen for the curve. | [
"\\oint_{C1} \\mathbf{F} \\cdot d\\mathbf{r} = \\iint_{S} (\\nabla \\times \\mathbf{F}) \\cdot d\\mathbf{S} + \\oint_{C2} \\mathbf{F} \\cdot d\\mathbf{r}"
] | [
"Using Stokes' theorem over the surface \\( S \\) with boundary \\( C1 \\cup C2 \\) and the result from Problem 2, calculate the line integral of \\( F \\) over the curve \\( C1 \\). Show the orientation chosen for the curve."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 5
} | 10 | 0 | 0 |
omr-ws-45b3bef6ebc89f54a20cc3cb4211a7a490a5bee78bb3b41bf4b9e49278636675 | Using Taylor series, calculate the expansion of $(\sin(x^3))^{\frac{1}{3}}$ up to $o(x^{13})$. | [
"\\( x - \\frac{x^7}{18} - \\frac{x^{13}}{3240} + O(x^{19}) \\)"
] | [
"Using Taylor series, calculate the expansion of $(\\sin(x^3))^{\\frac{1}{3}}$ up to $o(x^{13})$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 3,
"tir": 20
} | 29 | 0 | 0 |
omr-ws-effc85d9f652cc30872ef26bce5dcac3e7da37b4fff00cd6231c088e4c41b964 | Using Taylor's theorem, find \(\lim_{x \to 0} \left[x - x^2 \ln\left(1 + \frac{1}{x}\right)\right]\). | [
"0"
] | [
"Using Taylor's theorem, find \\(\\lim_{x \\to 0} \\left[x - x^2 \\ln\\left(1 + \\frac{1}{x}\\right)\\right]\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-62dc3ec4f1aaf4d305b6a098a0c8b4de7905c5faaa79921df49626be77da5d91 | Using \( k \) different colors to paint \( n \) blocks of a positive \( n \)-sided shape, where areas with common edges must be painted different colors, find the total number of coloring methods. (Assume \( n \geq 4 \) and \( k \geq 2 \)). | [
"(k-1)^n + (-1)^n (k-1)"
] | [
"Using \\( k \\) different colors to paint \\( n \\) blocks of a positive \\( n \\)-sided shape, where areas with common edges must be painted different colors, find the total number of coloring methods. (Assume \\( n \\geq 4 \\) and \\( k \\geq 2 \\))."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0bbfde7642fde6e79b1af429355b9226db11b316f7477caeee1c36df228f6e10 | Using a 16-pint container and a 9-pint container, how can the man measure exactly 1 pint of water? | [
"Post 7 provides a detailed step-by-step solution to measure exactly 1 pint of water using a 16-pint container and a 9-pint container. The steps are as follows:\n\n1. Fill the 16-pint container.\n2. Pour water from the 16-pint container into the 9-pint container until it is full. There are now 7 pints in the 16-pin... | [
"Using a 16-pint container and a 9-pint container, how can the man measure exactly 1 pint of water?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 17
} | 65 | 0 | 0 |
omr-ws-80bbed7924ce538e199f47ff6caabfdb988cbbdd3cbf83ee04330f624993f5d4 | Using a 3-litre jug and a 5-litre jug, how can you measure exactly 1 litre of water? | [
"1 litre can be measured by first filling the 3-litre jug, pouring it into the 5-litre jug, filling the 3-litre jug again, and then pouring from the 3-litre jug into the 5-litre jug until it's full, leaving exactly 1 litre in the 3-litre jug."
] | [
"Using a 3-litre jug and a 5-litre jug, how can you measure exactly 1 litre of water?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 19
} | 19 | 0 | 0 |
omr-ws-c6a1c9e78fb70448d0bf3942f84a663abbf9b344a2cbd1df2b5508b9cf3e6fa3 | Using a 3-litre jug and a 5-litre jug, how can you measure exactly 4 litres of water? | [
"1. Fill the 5-liter jug.\n2. Pour 3 liters from the 5-liter jug into the 3-liter jug.\n3. Empty the 3-liter jug.\n4. Pour the remaining 2 liters from the 5-liter jug into the 3-liter jug.\n5. Refill the 5-liter jug.\n6. Pour 1 liter from the 5-liter jug into the 3-liter jug to fill it.\n7. You now have exactly 4 l... | [
"Using a 3-litre jug and a 5-litre jug, how can you measure exactly 4 litres of water?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 15
} | 15 | 0 | 0 |
omr-ws-76c2a17daef96fc7c68d4ee4007e5b0d2ff3563534b6d21e630d36ca4058b3f8 | Using a 5-liter container and a 9-liter container, how can Mr. Fat measure out exactly 2 liters of milk from a milk tank? | [
"2 liters"
] | [
"Using a 5-liter container and a 9-liter container, how can Mr. Fat measure out exactly 2 liters of milk from a milk tank?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 4
} | 19 | 0 | 0 |
omr-ws-83f039c5dc2dd72dee5911c50bbcb8d1d491f92c6418bb435a516a1d3b0d5df0 | Using a 9-pint container and a 16-pint container, how can a man measure exactly 1 pint of water? | [
"1 pint"
] | [
"Using a 9-pint container and a 16-pint container, how can a man measure exactly 1 pint of water?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 4
} | 19 | 0 | 0 |
omr-ws-b912d5405dc103499204039a28aa2ece035d38a5b983d043696945ee50aa5db3 | Using a 99 ml glass and a 105 ml glass, is it possible to measure exactly 1000 ml of water? If so, describe the process. | [
"\\text{No}"
] | [
"Using a 99 ml glass and a 105 ml glass, is it possible to measure exactly 1000 ml of water? If so, describe the process."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e67f7b9f616a664e2af80dfb5ff2766f31d3e83c87bfe6943faffe7c14379061 | Using a 99 ml glass and a 105 ml glass, is it possible to measure exactly 1500 ml of water? If so, describe the process. | [
"19 times with the 105 ml glass to add water and 5 times with the 99 ml glass to remove water."
] | [
"Using a 99 ml glass and a 105 ml glass, is it possible to measure exactly 1500 ml of water? If so, describe the process."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 30,
"genselect": 6,
"tir": 8
} | 44 | 0 | 0 |
omr-ws-5b5e0d81adedd3dea26226fcc83fe8595c18937ecd1ee443f4eb0afebca5c6b3 | Using a Markov chain, compute the expected number of rolls for Alice to get two consecutive rolls that sum to 9 when rolling a 6-sided die repeatedly. | [
"24"
] | [
"Using a Markov chain, compute the expected number of rolls for Alice to get two consecutive rolls that sum to 9 when rolling a 6-sided die repeatedly."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-62e1d36de012ad61a4c9f7fe3d3c2bc47a909854a810022061ffb5a7ebaf2906 | Using a balance scale, what is the minimum number of weighings necessary to identify two fake coins out of 60 coins and determine whether they are heavier or lighter than the genuine ones? All the coins look identical, all the genuine coins have the same weight, and the two fake coins have the same weight. You can comp... | [
"10"
] | [
"Using a balance scale, what is the minimum number of weighings necessary to identify two fake coins out of 60 coins and determine whether they are heavier or lighter than the genuine ones? All the coins look identical, all the genuine coins have the same weight, and the two fake coins have the same weight. You can... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e826f763f399541e1e307de193d6464a165580d1bb0bc2a8e2972f165d5fa6dc | Using a calculator with only the $\sin$, $\cos$, $\tan$, $\arcsin$, $\arccos$, and $\arctan$ buttons working, starting from an initial display of $0$, determine the minimum number of button presses required to display the value $\frac{1}{2}$. | [
"no_valid_answer_found"
] | [
"Using a calculator with only the $\\sin$, $\\cos$, $\\tan$, $\\arcsin$, $\\arccos$, and $\\arctan$ buttons working, starting from an initial display of $0$, determine the minimum number of button presses required to display the value $\\frac{1}{2}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-eca7eb83a584cad21db1172b27027b14da611123f7a2d6d430660edba469f8af | Using a combinatorial method, express $\binom{2n+2}{n+1}$ in terms of $\binom{2n}{n}$ and $\binom{2n}{n-1}$. Find the value of $A$ in the equation $\binom{2n+2}{n+1} = A\left(\binom{2n}{n} + \binom{2n}{n-1}\right)$. | [
"2"
] | [
"Using a combinatorial method, express $\\binom{2n+2}{n+1}$ in terms of $\\binom{2n}{n}$ and $\\binom{2n}{n-1}$. Find the value of $A$ in the equation $\\binom{2n+2}{n+1} = A\\left(\\binom{2n}{n} + \\binom{2n}{n-1}\\right)$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-339b21713df391cf9c20979319dd1d3235018f25b0e58497896212988c35b963 | Using a compass and a ruler, construct a point $K$ inside an acute-angled triangle $ABC$ such that $\angle KBA = 2\angle KAB$ and $\angle KBC = 2\angle KCB$. | [
"K"
] | [
"Using a compass and a ruler, construct a point $K$ inside an acute-angled triangle $ABC$ such that $\\angle KBA = 2\\angle KAB$ and $\\angle KBC = 2\\angle KCB$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9472127b760bdc8510c31bc16a36c748a0550ee7bf04d32d5721ce84a36b640e | Using a compass and a ruler, construct a triangle $ABC$ given the lengths of sides $b$ and $c$, and the length of the segment $AI$, where $I$ is the incenter of the triangle. | [
"\\triangle ABC"
] | [
"Using a compass and a ruler, construct a triangle $ABC$ given the lengths of sides $b$ and $c$, and the length of the segment $AI$, where $I$ is the incenter of the triangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 2
} | 15 | 0 | 0 |
omr-ws-5b2e35cff6130c5ee0fa77874a5c9685f0a50cec21f7a50344cf7e779a29b6ef | Using a compass and a ruler, restore the triangle $ABC$ given three points: $D$, $E$, and $M$, where $D$ and $E$ are the midpoints of the altitudes $AH$ and $CP$ respectively, and $M$ is the midpoint of the side $AC$. | [
"\\triangle ABC"
] | [
"Using a compass and a ruler, restore the triangle $ABC$ given three points: $D$, $E$, and $M$, where $D$ and $E$ are the midpoints of the altitudes $AH$ and $CP$ respectively, and $M$ is the midpoint of the side $AC$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f81b8058907147e8a0e1ce27336fdbe2002fae21406e2bf577b66b9be23deec3 | Using a compass and a ruler, split a triangle into two smaller triangles such that the sum of the squares of the sides of each smaller triangle is the same. | [
"AD"
] | [
"Using a compass and a ruler, split a triangle into two smaller triangles such that the sum of the squares of the sides of each smaller triangle is the same."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 2
} | 5 | 0 | 0 |
omr-ws-ba9e0edaa98824c9fed408bcabafef28c9df4ee33c3ed7f7a138e56b523c69bb | Using a compass and a straight edge, construct a triangle given the point of intersection of its medians, the center of the circumscribed circle, and the point of intersection with one of the angle bisectors. | [
"\\triangle ABC"
] | [
"Using a compass and a straight edge, construct a triangle given the point of intersection of its medians, the center of the circumscribed circle, and the point of intersection with one of the angle bisectors."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 3
} | 7 | 0 | 0 |
omr-ws-d00d27755504543d2bcf251964e39e98284eb6b13027b14cbe3c363117bd4236 | Using a compass and ruler, construct a line that is tangent to the graph of $y = \sin x$ at a point above the x-axis and at a point below the x-axis. The line can also intersect the graph at other points. | [
"y = \\cos(a) x"
] | [
"Using a compass and ruler, construct a line that is tangent to the graph of $y = \\sin x$ at a point above the x-axis and at a point below the x-axis. The line can also intersect the graph at other points."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 6
} | 9 | 0 | 0 |
omr-ws-c06e08c7901414f6680a995bb3e724a4349106a969cbfd5ffc8163396d5782c4 | Using a compass and ruler, draw a line that divides the area of a fixed triangle \(ABC\) into two equal parts. | [
"\\text{Draw a median of the triangle}"
] | [
"Using a compass and ruler, draw a line that divides the area of a fixed triangle \\(ABC\\) into two equal parts."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 3
} | 16 | 0 | 0 |
omr-ws-bd9cece3ba3471d94cb02039ea81d85dfb466f7caa95e903f9af8ee5e2999c82 | Using a compass and straightedge, construct a regular heptadecagon (17-gon) inscribed in a circle of radius 1. Let $A_1, A_2, \ldots, A_{17}$ be the vertices of this heptadecagon in order. Calculate the value of $\sum_{k=1}^{8} A_kA_{k+1}^2$, where $A_kA_{k+1}$ denotes the length of the side between vertices $A_k$ and ... | [
"16\\left(1 - \\cos\\frac{2\\pi}{17}\\right)"
] | [
"Using a compass and straightedge, construct a regular heptadecagon (17-gon) inscribed in a circle of radius 1. Let $A_1, A_2, \\ldots, A_{17}$ be the vertices of this heptadecagon in order. Calculate the value of $\\sum_{k=1}^{8} A_kA_{k+1}^2$, where $A_kA_{k+1}$ denotes the length of the side between vertices $A_... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-4d5f82beee5fa5a3c33e0dcf8e4f443165dd3ffcdcc4253cc7e4d1507e772e34 | Using a compass and straightedge, construct an acute-angled triangle given the feet of its three altitudes. | [
"\\triangle ABC"
] | [
"Using a compass and straightedge, construct an acute-angled triangle given the feet of its three altitudes."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 3
} | 7 | 0 | 0 |
omr-ws-8d4551e9da85316f95156215984ff0b967ce42f077544a64db0b2a4857e3f774 | Using a compass and straightedge, construct an angle of 15 degrees by trisecting an angle of 45 degrees. If the length of one side of the constructed angle is 1 unit, find the length of the other side adjacent to the 15-degree angle, expressed in simplest radical form. | [
"\\sqrt{6} - \\sqrt{2}"
] | [
"Using a compass and straightedge, construct an angle of 15 degrees by trisecting an angle of 45 degrees. If the length of one side of the constructed angle is 1 unit, find the length of the other side adjacent to the 15-degree angle, expressed in simplest radical form."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-f60f76878f909f04a8f20e3af80abfac173c91d1d0bb1bece2d0f3a596875555 | Using a compass and straightedge, how do you divide a circle into 9 equal parts? | [
"\\text{It is impossible to divide a circle into 9 equal parts using only a compass and straightedge.}"
] | [
"Using a compass and straightedge, how do you divide a circle into 9 equal parts?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-9eef4efc4a65cfc61e382982e3fa7a5b4e6c548b2814c4e4acb287ce68cbef27 | Using a compass, a straight-edge, and a trisector, determine the number of distinct ways to construct a regular 7-gon by first constructing an angle of $2\pi/7$ radians. | [
"3"
] | [
"Using a compass, a straight-edge, and a trisector, determine the number of distinct ways to construct a regular 7-gon by first constructing an angle of $2\\pi/7$ radians."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 2
} | 9 | 0 | 0 |
omr-ws-c58a950d6588cdfd22097901f7bc85444b42d8c6803213be532cfe545c3d9677 | Using a computer, verify that the sequence of rationals \( (E_n)_{n=0}^{\infty} \), defined by \( E_n = \frac{A_n}{B_n} \), converges to the Napier constant \( e \). Find \( E_{10} \), \( E_{100} \), and (if possible) \( E_{1000} \). Compare the results with the decimals of \( e \). | [
"E_{1000} \\approx e"
] | [
"Using a computer, verify that the sequence of rationals \\( (E_n)_{n=0}^{\\infty} \\), defined by \\( E_n = \\frac{A_n}{B_n} \\), converges to the Napier constant \\( e \\). Find \\( E_{10} \\), \\( E_{100} \\), and (if possible) \\( E_{1000} \\). Compare the results with the decimals of \\( e \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-666d785e24aba57b77f7eda85ec094c7c03efe0cef5df8167b25b91ad0fddc97 | Using a double-sided ruler that allows drawing a straight line through two given points and a straight line parallel to this one at a distance of 3 cm, divide a given segment into three equal parts without using a compass. | [
"The segment is divided into three equal parts using the described method."
] | [
"Using a double-sided ruler that allows drawing a straight line through two given points and a straight line parallel to this one at a distance of 3 cm, divide a given segment into three equal parts without using a compass."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 5
} | 19 | 0 | 0 |
omr-ws-56a78b968b2385376f64f2cf6ce27ca685bd09f4e2c86b97808b7bb3d911b1d4 | Using a formal root $\beta$ of $f(x)$ from the above, consider the finite field $F_{16} = \{c_0 + c_1\beta + c_2\beta^2 + c_3\beta^3\}$, where $c_i \in \mathbb{Z}_2$ for $0 \le i \le 3$. Find the multiplicative inverse of $(1 + \beta)$ in $F_{16}$. | [
"\\(1 - \\beta + \\beta^2 - \\beta^3 + \\beta^4\\)"
] | [
"Using a formal root $\\beta$ of $f(x)$ from the above, consider the finite field $F_{16} = \\{c_0 + c_1\\beta + c_2\\beta^2 + c_3\\beta^3\\}$, where $c_i \\in \\mathbb{Z}_2$ for $0 \\le i \\le 3$. Find the multiplicative inverse of $(1 + \\beta)$ in $F_{16}$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-95ac32cdf273847fcbc4d5a3e44a365e7652220cdbb2be7834456ee2476a566c | Using a given one-to-one and onto function \( f: [0,1] \rightarrow (0,1) \), construct a bijective function from \([-1,1]\) to \(\mathbb{R}\). | [
"F(x) = \\begin{cases} \\dfrac{f(x)}{1 - f(x)} & \\text{if } x \\in [0,1], \\\\ -\\dfrac{f(-x)}{1 - f(-x)} & \\text{if } x \\in [-1,0). \\end{cases}"
] | [
"Using a given one-to-one and onto function \\( f: [0,1] \\rightarrow (0,1) \\), construct a bijective function from \\([-1,1]\\) to \\(\\mathbb{R}\\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 1
} | 2 | 0 | 0 |
omr-ws-e9ee7e8c9a682a07113450fecff1c9021dade748e11a3e426fcb2bbb6420754f | Using a hierarchy diagram, represent the partial order $B$ on the power set of $A = \{x, y, z\}$, defined by the subset relation. | [
"\\text{Cube}"
] | [
"Using a hierarchy diagram, represent the partial order $B$ on the power set of $A = \\{x, y, z\\}$, defined by the subset relation."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 7
} | 19 | 0 | 0 |
omr-ws-d85052ebd3da6652b5a3d1197cef6bfee30b0f9c508d9709f2ccb8ba7912c1ce | Using a midpoint plotter that draws the exact midpoint of two previously drawn points, starting with two points 1 unit apart, determine the minimum number of times the midpoint plotter must be used to achieve two points strictly between $\frac{1}{2017}$ and $\frac{1}{2016}$ units apart. Also, describe the strategy to a... | [
"17"
] | [
"Using a midpoint plotter that draws the exact midpoint of two previously drawn points, starting with two points 1 unit apart, determine the minimum number of times the midpoint plotter must be used to achieve two points strictly between $\\frac{1}{2017}$ and $\\frac{1}{2016}$ units apart. Also, describe the strate... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 6
} | 14 | 0 | 0 |
omr-ws-a45efa76c5a9c2e304f81b429ef981967c051e6326e907cdd0f3a96ca2409fba | Using a nozzle to paint each square in a $1 \times n$ stripe, when the nozzle is aiming at the $i$-th square, the square is painted black, and simultaneously, its left and right neighboring square (if exists) each has an independent probability of $\tfrac{1}{2}$ to be painted black. In the optimal strategy (i.e., achie... | [
"\\( T(n) = 2 - 2 \\left( -\\frac{1}{2} \\right)^n \\)"
] | [
"Using a nozzle to paint each square in a $1 \\times n$ stripe, when the nozzle is aiming at the $i$-th square, the square is painted black, and simultaneously, its left and right neighboring square (if exists) each has an independent probability of $\\tfrac{1}{2}$ to be painted black. In the optimal strategy (i.e.... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e2013ff4c1f3f5a987a3a0020ca45700fe60eb0893baa8f4dfff4560a58a22c0 | Using a one-half washer contour in the upper complex plane, which includes the half-circle contour \( \textbf{C} \), compute the value of the integral \[ I = \int_0^{\infty} \frac{\sin^4(x)}{x^4} \, dx. \] Express your answer in terms of \(\pi\). | [
"\\dfrac{\\pi}{3}"
] | [
"Using a one-half washer contour in the upper complex plane, which includes the half-circle contour \\( \\textbf{C} \\), compute the value of the integral\n\\[ I = \\int_0^{\\infty} \\frac{\\sin^4(x)}{x^4} \\, dx. \\]\nExpress your answer in terms of \\(\\pi\\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-2de30a0c6b90bb2144c928a45beb4062256c8c1a3871f34af0df505aaac17d55 | Using a ruler and compass, construct an angle of $3^\circ$. If the angle is constructed correctly, what is the measure of the angle formed by trisecting this $3^\circ$ angle? | [
"1^\\circ"
] | [
"Using a ruler and compass, construct an angle of $3^\\circ$. If the angle is constructed correctly, what is the measure of the angle formed by trisecting this $3^\\circ$ angle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-17d1a5073465d4a6dc705dcc8d717c62df9f8626dbf9f5d5160b3fd60c2a3f72 | Using a ruler and compass, construct the parallel projection of the incenter of triangle $ABC$ given the parallel projection of triangle $ABC$ as $A_1B_1C_1$ and the parallel projections $A_1H_1$ and $C_1L_1$ of the altitude $AH$ and the bisector $CL$ of $\triangle ABC$ respectively. | [
"I_1"
] | [
"Using a ruler and compass, construct the parallel projection of the incenter of triangle $ABC$ given the parallel projection of triangle $ABC$ as $A_1B_1C_1$ and the parallel projections $A_1H_1$ and $C_1L_1$ of the altitude $AH$ and the bisector $CL$ of $\\triangle ABC$ respectively."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4a7c66a392917eb3cc3060434183a4085b3e2e176a0839b64015058ecf8e571a | Using a ruler and compass, construct the parallel projection of the orthocenter of triangle $ABC$ given the parallel projection of triangle $ABC$ as $A_1B_1C_1$ and the parallel projections $A_1H_1$ and $C_1L_1$ of the altitude $AH$ and the bisector $CL$ of $\triangle ABC$ respectively. | [
"H_1"
] | [
"Using a ruler and compass, construct the parallel projection of the orthocenter of triangle $ABC$ given the parallel projection of triangle $ABC$ as $A_1B_1C_1$ and the parallel projections $A_1H_1$ and $C_1L_1$ of the altitude $AH$ and the bisector $CL$ of $\\triangle ABC$ respectively."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 2
} | 15 | 0 | 0 |
omr-ws-a8da9a36bd08573b054cae129056fb1dcfe0ac9cd050725f68cee0cb4390acc4 | Using a ruler with two marks at a distance of 1 unit, construct a perpendicular from a given point to a given line. The ruler allows constructing points $P_1$ and $P_2$ on any line $l$ such that $AP_1 = AP_2 = 1$ for a given point $A$ on $l$. | [
"\\text{Perpendicular constructed from the given point to the line using the marked ruler.}"
] | [
"Using a ruler with two marks at a distance of 1 unit, construct a perpendicular from a given point to a given line. The ruler allows constructing points $P_1$ and $P_2$ on any line $l$ such that $AP_1 = AP_2 = 1$ for a given point $A$ on $l$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 2
} | 13 | 0 | 0 |
omr-ws-fdc88fc1c94dec17524085670a22738abe42abc8e2390f299e159ad869dd8145 | Using a ruler with two parallel edges (where the distance between the edges is equal to the breadth of the ruler), construct a line through a given point parallel to a given line. | [
"Place the ruler so one edge aligns with the given line, draw a parallel line using the other edge. Slide the ruler, keeping the second edge on the drawn line, until the first edge passes through the given point, then draw the desired parallel line."
] | [
"Using a ruler with two parallel edges (where the distance between the edges is equal to the breadth of the ruler), construct a line through a given point parallel to a given line."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 8
} | 9 | 0 | 0 |
omr-ws-64d4275df2505a4baaa67f03bbdfb9df26ca16e3ac382f58901ae202aaeed387 | Using a ruler with two parallel edges (where the distance between the edges is equal to the breadth of the ruler), construct the bisector of a given angle. | [
"OP"
] | [
"Using a ruler with two parallel edges (where the distance between the edges is equal to the breadth of the ruler), construct the bisector of a given angle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 6
} | 8 | 0 | 0 |
omr-ws-0fa3f9bd8835760b5d899b02a33520b6160642079773f73e6c868adb8b94e3b1 | Using a ruler with two parallel edges (where the distance between the edges is equal to the breadth of the ruler), construct the center of a circle through three given non-collinear points. | [
"O"
] | [
"Using a ruler with two parallel edges (where the distance between the edges is equal to the breadth of the ruler), construct the center of a circle through three given non-collinear points."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 6
} | 17 | 0 | 0 |
omr-ws-1853b7067ba0a8fcfb88ddc34a1cc2ea71293bf6ab50e27ae498e393de24a84c | Using a ruler with two parallel edges (where the distance between the edges is equal to the breadth of the ruler), construct the midpoint of a given rectilinear line segment. | [
"The midpoint of the segment \\(AB\\) can be constructed by the described method in Post 2."
] | [
"Using a ruler with two parallel edges (where the distance between the edges is equal to the breadth of the ruler), construct the midpoint of a given rectilinear line segment."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 6,
"tir": 2
} | 16 | 0 | 0 |
omr-ws-e660eadeec9409b49fe9bc8b0b33463acc028b77b7c9c55e531caa80ec093dbc | Using a similar approach, determine the exact expression for the following integral: \[ \mathop\oint\limits_{C_k}\sqrt{(z - 1)(z + 1)(z - 3)}\,dz \] where \( 1 < r < 3 \). | [
"0"
] | [
"Using a similar approach, determine the exact expression for the following integral:\n\\[ \\mathop\\oint\\limits_{C_k}\\sqrt{(z - 1)(z + 1)(z - 3)}\\,dz \\]\nwhere \\( 1 < r < 3 \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-322348f5b85c3c51424a72d8b2ddc6239457f8e5be812f2e577c6080eeb11f9c | Using a simple calculator that can perform the operations +, -, *, /, and square root, how can you find the value of $3^{\sqrt{2}}$ with an accuracy of 6 digits? | [
"4.728804"
] | [
"Using a simple calculator that can perform the operations +, -, *, /, and square root, how can you find the value of $3^{\\sqrt{2}}$ with an accuracy of 6 digits?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 6
} | 10 | 0 | 0 |
omr-ws-cecf7502b78cc26a438366c7cb483edf2c2be5cc40a1f17ab81781518ea8ae12 | Using a special machine that returns the order of $a$ modulo $n$ for any relatively prime positive integers $a$ and $n$, devise an algorithm of time complexity at most $O(\log (n))$ to factorize natural numbers $n$ of the form $pq$, where $p$ and $q$ are safe primes (i.e., $p$, $q$, $\frac{p-1}{2}$, and $\frac{q-1}{2}$... | [
"d"
] | [
"Using a special machine that returns the order of $a$ modulo $n$ for any relatively prime positive integers $a$ and $n$, devise an algorithm of time complexity at most $O(\\log (n))$ to factorize natural numbers $n$ of the form $pq$, where $p$ and $q$ are safe primes (i.e., $p$, $q$, $\\frac{p-1}{2}$, and $\\frac{... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-4c76a3c6926a009bc8d8dc7f2c1b35fc53fda79f19457d51556765de44cc08a8 | Using a straight edge and compass, divide a straight line of length \( m \) into two parts such that the difference of the squares of the lengths of these parts is equal to their product. | [
"\\(\\frac{3-\\sqrt{5}}{2}m\\)"
] | [
"Using a straight edge and compass, divide a straight line of length \\( m \\) into two parts such that the difference of the squares of the lengths of these parts is equal to their product."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 6,
"tir": 2
} | 8 | 0 | 0 |
omr-ws-3735eb165b68abf0cfe32b50983569056b5acbc73956fe9e91600a8465ad1d86 | Using a straight edge and compasses alone, construct the centre of a regular polygon starting with one side. Also, show how to construct the complete regular polygon. | [
"O"
] | [
"Using a straight edge and compasses alone, construct the centre of a regular polygon starting with one side. Also, show how to construct the complete regular polygon."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 3
} | 11 | 0 | 0 |
omr-ws-d25eb421eff96f603479ce7b9bbc9e84cb3a37a1400e3add19ac4ef82ac41878 | Using a straight wall and three fences each of length 10 meters, what is the largest area Bahman can enclose in meters squared? | [
"\\( 75\\sqrt{3} \\)"
] | [
"Using a straight wall and three fences each of length 10 meters, what is the largest area Bahman can enclose in meters squared?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 7,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-de8994e91f1e70e85d7663b7356cda9623263ff76b36ef4f2935d5d6b43f545e | Using a straightedge and compass, construct a line through a random point outside a triangle that divides the triangle into two regions of equal area. | [
"PD"
] | [
"Using a straightedge and compass, construct a line through a random point outside a triangle that divides the triangle into two regions of equal area."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 5
} | 9 | 0 | 0 |
omr-ws-f57cc4d33301766bea82a82f7329701e495909cb5b58d9ef552b877f4ee433d2 | Using a three-liter container and a five-liter container with an unlimited supply of lemonade, how can you measure out exactly four liters of lemonade? Determine two methods of solving this problem. | [
"First method: Fill the 5-liter container, pour 3 liters into the 3-liter container, empty the 3-liter container, pour the remaining 2 liters into it, fill the 5-liter container again, and top up the 3-liter container to get 4 liters in the 5-liter container. Second method: Fill the 3-liter container, pour it into ... | [
"Using a three-liter container and a five-liter container with an unlimited supply of lemonade, how can you measure out exactly four liters of lemonade? Determine two methods of solving this problem."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 27
} | 27 | 0 | 0 |
omr-ws-1a2509c1f7e90c60b95def03c65cf9a9e2f54bf25e6e3117adb1ee7d07975d84 | Using a triangle, find the ratio of \( p:a \) in terms of \( \sin x \) and \( \sin y \). | [
"\\(\\frac{\\sin{x}}{\\sin{y}}\\)"
] | [
"Using a triangle, find the ratio of \\( p:a \\) in terms of \\( \\sin x \\) and \\( \\sin y \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 1,
"tir": 18
} | 26 | 0 | 0 |
omr-ws-844903a2f176d6393d49b244fe726b46c766890fb7f506af9e4d8ac808013d2c | Using a two-pan balance and a weight of 1 gram, how can we obtain exactly 1 kilogram of sugar in the smallest possible number of weighings? | [
"10"
] | [
"Using a two-pan balance and a weight of 1 gram, how can we obtain exactly 1 kilogram of sugar in the smallest possible number of weighings?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-1e05106dd0806516750b1c1caf05841bb283682c9481c405be7287502b264efc | Using a two-pan balance scale and three 1-gram weights, three 5-gram weights, and three 50-gram weights, find the number of different positive weights of chemicals that can be measured. | [
"63"
] | [
"Using a two-pan balance scale and three 1-gram weights, three 5-gram weights, and three 50-gram weights, find the number of different positive weights of chemicals that can be measured."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-66f54cc733e44d3e44291bee218c4c2d8581ae08560195da88e6307daa3c9dea | Using a two-way table, compute the total number of students in the age group 13–18 years who liked adventure or biography, given the survey data. | [
"49"
] | [
"Using a two-way table, compute the total number of students in the age group 13–18 years who liked adventure or biography, given the survey data."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d64472f74102330156986c875c715d9b810192f0d7abd76bea49709133b70cb4 | Using all digits from 1 to 9 (each digit used once and only once), form fractions equal in value to: a) \( \frac{1}{3} \) b) \( \frac{1}{4} \) c) \( \frac{1}{5} \) d) \( \frac{1}{6} \) e) \( \frac{1}{7} \) f) \( \frac{1}{8} \) g) \( \frac{1}{9} \) | [
"a) \\( \\frac{5823}{17469} \\)\nb) \\( \\frac{3942}{15768} \\)\nc) \\( \\frac{2697}{13485} \\)\nd) \\( \\frac{2943}{17658} \\)\ne) \\( \\frac{2394}{16758} \\)\nf) \\( \\frac{3187}{25496} \\)\ng) \\( \\frac{6381}{57429} \\)"
] | [
"Using all digits from 1 to 9 (each digit used once and only once), form fractions equal in value to:\na) \\( \\frac{1}{3} \\)\nb) \\( \\frac{1}{4} \\)\nc) \\( \\frac{1}{5} \\)\nd) \\( \\frac{1}{6} \\)\ne) \\( \\frac{1}{7} \\)\nf) \\( \\frac{1}{8} \\)\ng) \\( \\frac{1}{9} \\)"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 5
} | 5 | 0 | 0 |
omr-ws-036a0071be66433fc30254e0266fe313c5dabb4d981a59d50af104963ae87698 | Using all natural numbers from 0 to 9, create four different natural numbers that are divisible by 9: one with 4 digits, one with 3 digits, one with 2 digits, and one with 1 digit. Given that the sum of these numbers is 2016, find the four numbers. | [
"9"
] | [
"Using all natural numbers from 0 to 9, create four different natural numbers that are divisible by 9: one with 4 digits, one with 3 digits, one with 2 digits, and one with 1 digit. Given that the sum of these numbers is 2016, find the four numbers."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-d26c441df88892d514051e0f524fa08616aa72b6b87ff1a02e61c48569433ac3 | Using all nine cards with digits $1, 2, 3, 4, 5, 6, 7, 8,$ and $9$, form some numbers such that all numbers are composite. Determine the smallest possible value of their sum. | [
"99"
] | [
"Using all nine cards with digits $1, 2, 3, 4, 5, 6, 7, 8,$ and $9$, form some numbers such that all numbers are composite. Determine the smallest possible value of their sum."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 25,
"genselect": 8,
"tir": 3
} | 36 | 0 | 0 |
omr-ws-559f8517dff9ca3ad97983edce0140335503237affd2f02b0a2d34023edc3b1f | Using all nine cards with digits $1, 2, 3, 4, 5, 6, 7, 8,$ and $9$, form some numbers such that all numbers are prime. Determine the smallest possible value of their sum. | [
"207"
] | [
"Using all nine cards with digits $1, 2, 3, 4, 5, 6, 7, 8,$ and $9$, form some numbers such that all numbers are prime. Determine the smallest possible value of their sum."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 8,
"tir": 8
} | 28 | 0 | 0 |
omr-ws-815a272bc56d411000421997977de38b891bf896182b36525e8a38ebb625e293 | Using all of the digits of the number 123 456 789, is it possible to rearrange the digits to produce a prime number? If it exists, how do you find this number? | [
"No\n\nExplanation: The sum of the digits of the number 123 456 789 is 45, which is divisible by 9. Any rearrangement of these digits will also have a sum of 45, meaning the number will always be divisible by 9 and thus cannot be a prime number."
] | [
"Using all of the digits of the number 123 456 789, is it possible to rearrange the digits to produce a prime number? If it exists, how do you find this number?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-4f6225196e7ca50143318d564b62616d841d407f574eb772ba4518b3b31e8ef1 | Using all the digits 3, 4, 5, 6, and 7 exactly once in each numeral, how many 5-digit numerals are divisible by 55? | [
"4"
] | [
"Using all the digits 3, 4, 5, 6, and 7 exactly once in each numeral, how many 5-digit numerals are divisible by 55?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-95cd90a0298859adc2fc642ae8bd2496ca76484fe77b3b4226c4b3369b6d9613 | Using all the digits from 0 to 9, form two 5-digit numbers. What is the smallest possible difference between these two numbers? | [
"247"
] | [
"Using all the digits from 0 to 9, form two 5-digit numbers. What is the smallest possible difference between these two numbers?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-805fd49454d28dc168c48f236739611065a500bc3296be18b14a817bd7710ec8 | Using all the digits from 0 to 9, form two five-digit numbers. What is the smallest possible difference between these two numbers? | [
"247"
] | [
"Using all the digits from 0 to 9, form two five-digit numbers. What is the smallest possible difference between these two numbers?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-0026d2cb17167f39b41f331dca28a363ac907a50bbb99b88b6ae73e8041245fc | Using all the digits from 1 to 5 exactly once, find the largest possible product of two factors. | [
"\\(52 \\times 431\\) \n\nTo provide the numerical value:\n\\[ 52 \\times 431 = 22412 \\]"
] | [
"Using all the digits from 1 to 5 exactly once, find the largest possible product of two factors."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-846a7ff9682f6655c5444cad79c30332090acbb2a7a7ace0706b754a7d10ab31 | Using all the digits from 1 to 9 compose three 3-digit numbers so that their product has the greatest possible value. | [
"147; 258; 369"
] | [
"Using all the digits from 1 to 9 compose three 3-digit numbers so that their product has\nthe greatest possible value."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f533efceb99030a1c370fced3431e44f1f76c978f96b4a8b1918adff6c99ad5b | Using all the digits from 1 to 9 compose three 3-digit numbers which are in the ratio $1 : 2 : 3$. | [
"Find what remainders result from the division of the numbers $A$ and $B$ by 9"
] | [
"Using all the digits from 1 to 9 compose three 3-digit numbers which are in the ratio $1 : 2 : 3$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3c728c7af43395a6c245820e5169a4d726b35a4169a6d1e7a346515ed953ff28 | Using an ample supply of identical $1 \times 1$ and identical $2 \times 2$ tiles, how many ways are there to tile a $3 \times 8$ grid? | [
"171"
] | [
"Using an ample supply of identical $1 \\times 1$ and identical $2 \\times 2$ tiles, how many ways are there to tile a $3 \\times 8$ grid?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 29,
"tir": 9
} | 38 | 0 | 0 |
omr-ws-bdb42386fb403072ac14fd3dff84010bfb9a2f72bace52ad2fe0feb331d74e5d | Using an ample supply of identical 1 × 1 and identical 2 × 2 tiles, how many ways are there to tile a 3 × 8 grid? | [
"171"
] | [
"Using an ample supply of identical 1 × 1 and identical 2 × 2 tiles, how many ways are there to tile a 3 × 8 grid?"
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 60,
"tir": 5
} | 65 | 0 | 0 |
omr-ws-16a47a89a7b0e33c811b4b13b199d8732fce242fcf03f8487ca0ccd1546eb19a | Using an apparatus that can divide any segment into three equal segments, along with a ruler and pencil, how can you find the midpoint of any given segment? | [
"M"
] | [
"Using an apparatus that can divide any segment into three equal segments, along with a ruler and pencil, how can you find the midpoint of any given segment?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 6
} | 15 | 0 | 0 |
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