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-238df0252920dfbcacc62a26c6a4e55430e09f79e18063ae458016610483712f | Use the multinomial theorem to expand $(x + y + z)^4$. | [
"\\((x + y + z)^4 = \\sum_{\\substack{a,b,c \\geq 0 \\\\ a + b + c = 4}} \\binom{4}{a,b,c} x^ay^bz^c = \\sum_{sym} (x^4 + 4x^3y + 3x^2y^2 + 6x^2yz)\\)"
] | [
"Use the multinomial theorem to expand $(x + y + z)^4$."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 6,
"tir": 30
} | 43 | 0 | 0 |
omr-ws-35d2c574d6388517a28af1e971c88012f6a0f889928b4d586a233469e0c5dbba | Use the multiplicative inverse of [87] in \(\mathbb{Z}_{150}\) to find all \(x\) in \(\mathbb{Z}_{150}\) such that \([87]x = [11]\). | [
"\\text{No solution}"
] | [
"Use the multiplicative inverse of [87] in \\(\\mathbb{Z}_{150}\\) to find all \\(x\\) in \\(\\mathbb{Z}_{150}\\) such that \\([87]x = [11]\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-31291f76521053e4252734a9951bf91eb198702e1c2575d03cef7a811518c042 | Use the net force equation \( F = ma \) and the given force due to air resistance \( r = pv \) to write a differential equation for the ball’s velocity \( v(t) \). | [
"\\(\\frac{dv}{dt} = \\frac{p}{m}v - g\\)"
] | [
"Use the net force equation \\( F = ma \\) and the given force due to air resistance \\( r = pv \\) to write a differential equation for the ball’s velocity \\( v(t) \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 2
} | 3 | 0 | 0 |
omr-ws-05617a3443c98136900b64bba053acc0a1fbbbd83790ffeaa8d15bb87f585ff4 | Use the normal distribution approximation to find out how likely it would be for a fair coin to give results as unbalanced as 573 heads and 427 tails. | [
"\\(0.00000194939\\)"
] | [
"Use the normal distribution approximation to find out how likely it would be for a fair coin to give results as unbalanced as 573 heads and 427 tails."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4b23b1670ec91a76056df9176ff072790cc78a0b53bcfdb98668ab5a6674a0b3 | Use the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ exactly once to form three different 3-digit numbers such that the third number is three times the first number, and the second number is the difference between the third number and the first number. | [
"192, 384, 576"
] | [
"Use the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ exactly once to form three different 3-digit numbers such that the third number is three times the first number, and the second number is the difference between the third number and the first number."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 28
} | 28 | 0 | 0 |
omr-ws-a6514d28a4e101e36feea4e3afa572dc1b9a97608f54ad62840b1946d5c5a21f | Use the numbers 2, 3, 4, and 5 and the operations of addition, subtraction, multiplication, division, and exponentiation to generate the largest possible result. | [
"2^{3^{4^5}}"
] | [
"Use the numbers 2, 3, 4, and 5 and the operations of addition, subtraction, multiplication, division, and exponentiation to generate the largest possible result."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-3f7fca23f4ead6b51d6ef2b79f7a3c510d41904f53c3e71dabefe878fdd55e17 | Use the numbers 3, 3, 7, 7 with basic operations (+, -, *, /) and braces to get 24 as a result. | [
"\\( 7 \\left(\\frac{3}{7} + 3\\right) \\)"
] | [
"Use the numbers 3, 3, 7, 7 with basic operations (+, -, *, /) and braces to get 24 as a result."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-37be9d3d73a686240324ba614a646beb663a5e774ac84eff8c3e792ae50c7327 | Use the parametrization from Problem 1 to find the area enclosed by the curve \( x^{\frac{2}{3}} + y^{\frac{2}{3}} = a^{\frac{2}{3}} \) using the formula \( A = \frac{1}{2} \int_{0}^{2\pi} \left[ x(t) \frac{dy(t)}{dt} - y(t) \frac{dx(t)}{dt} \right] dt \). | [
"\\(\\frac{3a^2 \\pi}{8}\\)"
] | [
"Use the parametrization from Problem 1 to find the area enclosed by the curve \\( x^{\\frac{2}{3}} + y^{\\frac{2}{3}} = a^{\\frac{2}{3}} \\) using the formula \\( A = \\frac{1}{2} \\int_{0}^{2\\pi} \\left[ x(t) \\frac{dy(t)}{dt} - y(t) \\frac{dx(t)}{dt} \\right] dt \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 4
} | 18 | 0 | 0 |
omr-ws-fed2d115232cb73bd7b506c7b694763bd2c03747324c380ee3147cb0dc823cdc | Use the piecewise linear finite element method to solve the boundary value problem: \[ D^2 u + 2y = \cos x \quad \text{for} \quad 0 < x < 1 \] with boundary conditions: \[ Du(0) = 1, \quad Du(1) = -1 \] and the partition: \[ x_0 = 0, \quad x_1 = \frac{1}{2}, \quad x_2 = 1 \] | [
"\\left[1.2106, 0.8618, 0.7487\\right]"
] | [
"Use the piecewise linear finite element method to solve the boundary value problem:\n\\[ D^2 u + 2y = \\cos x \\quad \\text{for} \\quad 0 < x < 1 \\]\nwith boundary conditions:\n\\[ Du(0) = 1, \\quad Du(1) = -1 \\]\nand the partition:\n\\[ x_0 = 0, \\quad x_1 = \\frac{1}{2}, \\quad x_2 = 1 \\]"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a9b86ad667b13b37fc7eea4308db3dbbf85bd054e7e3e83ac3adfacd2ae2da57 | Use the power series method to solve the differential equation $\frac{d^2y}{dx^2} - y = x$. | [
"\\( y = A\\cosh x + B\\sinh x - x \\)"
] | [
"Use the power series method to solve the differential equation $\\frac{d^2y}{dx^2} - y = x$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-27548d6deadbc9f75d8c7def3b227d7df60cc30d4045ae2fff59a9d99139cbbf | Use the power series method to solve the differential equation \((1+x^2)y'' + 6xy' + 6y = 0\) about the ordinary point \(x=0\). | [
"\\( y = a_0(1 - 3x^2 + 5x^4 - \\ldots) + a_1(x - 2x^3 + 3x^5 - \\ldots) \\)"
] | [
"Use the power series method to solve the differential equation \\((1+x^2)y'' + 6xy' + 6y = 0\\) about the ordinary point \\(x=0\\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-21c1b2d782523826cf0119c87706c2633b8179328ed44c2bf965220a097b132d | Use the previous parts of this problem to place \( \pi \) into a chain of inequalities. | [
"\\( 3\\frac{89}{630} < \\pi < 3\\frac{1}{7} \\)"
] | [
"Use the previous parts of this problem to place \\( \\pi \\) into a chain of inequalities."
] | [
"c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-cbaf22c52b124c7f3c4d75414d71171f78c829774ecb664bb0da7e010dc5ecd4 | Use the principle of inclusion and exclusion to find the number of ways to form a main committee of 10 persons from 12 married couples such that no married couple is included in the committee. | [
"67584"
] | [
"Use the principle of inclusion and exclusion to find the number of ways to form a main committee of 10 persons from 12 married couples such that no married couple is included in the committee."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7e19398de19b0f3be6d0f2627638bfeeb709d832a329aafadde9e70523287c5e | Use the product \((1-x)^{-1}(1-x)^{-\frac{1}{2}}\) to find the value of the series \[1 + \frac{1}{2} + \frac{1}{2} \cdot \frac{3}{4} + \frac{1}{2} \cdot \frac{3}{4} \cdot \frac{5}{6} + \cdots + \frac{1}{2} \cdot \frac{3}{4} \cdot \frac{5}{6} \cdots \frac{2n-3}{2n-2}.\] | [
"\\(\\frac{(2n-1)!!}{2^n \\cdot n!}\\)"
] | [
"Use the product \\((1-x)^{-1}(1-x)^{-\\frac{1}{2}}\\) to find the value of the series\n\n\\[1 + \\frac{1}{2} + \\frac{1}{2} \\cdot \\frac{3}{4} + \\frac{1}{2} \\cdot \\frac{3}{4} \\cdot \\frac{5}{6} + \\cdots + \\frac{1}{2} \\cdot \\frac{3}{4} \\cdot \\frac{5}{6} \\cdots \\frac{2n-3}{2n-2}.\\]"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bcfe5c89da8e276d0d4a45d5b644d7d88d853a4459a8bebdd9a080edd33b575d | Use the property that if \( m \leq f(x) \leq M \) for \( a \leq x \leq b \), where \( m \) is the absolute minimum and \( M \) is the absolute maximum of \( f \) on the interval \([a, b]\), to estimate the value of the integral \(\int_{5\pi}^{6\pi} (5x - 10\sin(x)) \, dx\). Find the smaller value and the larger value. | [
"\\dfrac{85}{3}\\pi^2 + 5\\sqrt{3}\\pi"
] | [
"Use the property that if \\( m \\leq f(x) \\leq M \\) for \\( a \\leq x \\leq b \\), where \\( m \\) is the absolute minimum and \\( M \\) is the absolute maximum of \\( f \\) on the interval \\([a, b]\\), to estimate the value of the integral \\(\\int_{5\\pi}^{6\\pi} (5x - 10\\sin(x)) \\, dx\\). Find the smaller ... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-9d3d1bdb0c2f18c802fb92d54e011541f6613cee797f4a58e19d992b8bee53a5 | Use the property that the product of two numbers of the form $(a + kb)$, where $a, k, b$ are integers, is also of the same form. Given a general solution of $x^{2} - 2y^{2} = 1$ and a particular solution of $x^{2} - 2y^{2} = -1$, obtain several solutions of $x^{2} - 2y^{2} = -1$. | [
"(239, 169)"
] | [
"Use the property that the product of two numbers of the form $(a + kb)$, where $a, k, b$ are integers, is also of the same form. Given a general solution of $x^{2} - 2y^{2} = 1$ and a particular solution of $x^{2} - 2y^{2} = -1$, obtain several solutions of $x^{2} - 2y^{2} = -1$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-f04aea509eb574f40a133b472d56e10c48c094630e2de63e32ab84891a3edeea | Use the quadratic formula to find the roots of the polynomial \( kx^2 + (k+1)x - (k+2) \). | [
"\\( x = \\frac{-k-1 \\pm \\sqrt{5k^2 + 10k + 1}}{2k} \\)"
] | [
"Use the quadratic formula to find the roots of the polynomial \\( kx^2 + (k+1)x - (k+2) \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-a287dde81d56181f6d65b16e382aa03ff9362a732613144ea014fbac201a9f15 | Use the ratio test to analyze whether the given series converges or diverges. If the ratio test is inconclusive, use a different test to analyze the series. $$\sum_{k=1}^{\infty} \frac{1 \cdot 4 \cdot 7 \cdots (3k-2)}{2 \cdot 4 \cdot 6 \cdots (2k)}$$ | [
"\\(\\frac{3}{2}\\)"
] | [
"Use the ratio test to analyze whether the given series converges or diverges. If the ratio test is inconclusive, use a different test to analyze the series.\n\n$$\\sum_{k=1}^{\\infty} \\frac{1 \\cdot 4 \\cdot 7 \\cdots (3k-2)}{2 \\cdot 4 \\cdot 6 \\cdots (2k)}$$"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-844a21a85a5081ace073ea337ee73c451335de49ad6be85e8bf6fb090b322a85 | Use the ratio test to determine the convergence of the series \(\sum_{n=1}^{\infty} \frac{(2x)^n}{n^2}.\) | [
"\\(-\\frac{1}{2} \\le x \\le \\frac{1}{2}\\)"
] | [
"Use the ratio test to determine the convergence of the series \\(\\sum_{n=1}^{\\infty} \\frac{(2x)^n}{n^2}.\\)"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 28
} | 42 | 0 | 0 |
omr-ws-5e4aaf73f583b513f7385e256ed5c359faa703e07ea8b67e1eaf6dd46c185c1f | Use the reduction formula to find \( \int_{0}^{1} (1+x^{2})^{3} dx \). | [
"\\dfrac{96}{35}"
] | [
"Use the reduction formula to find \\( \\int_{0}^{1} (1+x^{2})^{3} dx \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e9926e92c93871416a8b028c35bac6f3a64b2182832a466a4ead531cea3c15da | Use the reduction formula to find \( \int_{0}^{1} (1+x^{2})^{\frac{3}{2}} dx \). | [
"\\dfrac{7\\sqrt{2} + 3\\ln(1+\\sqrt{2})}{8}"
] | [
"Use the reduction formula to find \\( \\int_{0}^{1} (1+x^{2})^{\\frac{3}{2}} dx \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0349c00418ee0448a534c54561d3dc2d9b3280f3385c520548754f18707207d6 | Use the reduction formula to find \( \int_{0}^{1} \frac{1}{(1+x^{2})^{3}} dx \). | [
"\\dfrac{8 + 3\\pi}{32}"
] | [
"Use the reduction formula to find \\( \\int_{0}^{1} \\frac{1}{(1+x^{2})^{3}} dx \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6342aa314188ff20fab39a176642f3e2bed449578ed2c96d629af8652eb2428b | Use the reduction formula to find \(\int_{0}^{1} (1+x^{2})^{3}\ dx\). | [
"\\(\\frac{96}{35}\\)"
] | [
"Use the reduction formula to find \\(\\int_{0}^{1} (1+x^{2})^{3}\\ dx\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-eb71cee4b4e8883bda92642e910a57370cc67bff92e0f1ac8411fb4d31f40545 | Use the reduction formula to find \(\int_{0}^{1} (1+x^{2})^{\frac{3}{2}}\ dx\). | [
"\\(\\frac{1}{8}(3\\ln(1+\\sqrt{2})+7\\sqrt{2})\\)"
] | [
"Use the reduction formula to find \\(\\int_{0}^{1} (1+x^{2})^{\\frac{3}{2}}\\ dx\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-837cdad39b1979c5d70ff79801aad26cccc2378e2ef435e3122719d251597f1a | Use the reduction formula to find \(\int_{0}^{1} \frac{1}{(1+x^{2})^{3}}\ dx\). | [
"\\(\\frac{3}{32}\\pi + \\frac{1}{4}\\)"
] | [
"Use the reduction formula to find \\(\\int_{0}^{1} \\frac{1}{(1+x^{2})^{3}}\\ dx\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 29
} | 52 | 0 | 0 |
omr-ws-1d9aacba1f6066a5ea3acd6961e6726347912e45e2d17f1d74e3c36be2aae413 | Use the remainder theorem to find the remainder when \( 2x^3 + x^2 - 7x - 6 \) is divided by (a) \( x - 2 \) and (b) \( x + 1 \). Hence, factorize the cubic expression. | [
"(x - 2)(x + 1)(2x + 3)"
] | [
"Use the remainder theorem to find the remainder when \\( 2x^3 + x^2 - 7x - 6 \\) is divided by (a) \\( x - 2 \\) and (b) \\( x + 1 \\). Hence, factorize the cubic expression."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5da4b4d45288d01a223f09e86f9f9e0ac45b1ab00711c6a6c708e41bf0de8dce | Use the residue theorem to compute the integral $$ \int_{0}^{2\pi} \cos^k(\theta) d\theta $$ where $k \geq 0$ is a non-negative integer. | [
"\\(\\begin{cases} \n0 & \\text{if } k \\text{ is odd} \\\\\n\\frac{2\\pi}{4^l} \\binom{2l}{l} & \\text{if } k = 2l \\text{ (even)}\n\\end{cases}\\)"
] | [
"Use the residue theorem to compute the integral $$ \\int_{0}^{2\\pi} \\cos^k(\\theta) d\\theta $$ where $k \\geq 0$ is a non-negative integer."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 5,
"tir": 23
} | 30 | 0 | 0 |
omr-ws-aa1b1ac5d966131634ae3c32abfade6bbdd037e922d2673222c5e8ccf4a7fa0a | Use the residue theorem to compute the integral $$ \int_{0}^{\infty} \frac{1}{1+x^3}dx$$ Given Hint: Consider the contour given by the boundary of the region $\{re^{i\theta} \hspace{0.1cm} | \hspace{0.1cm} \theta \in [0, \frac{2\pi}{3}], 0 \leq r \leq R\}$, and take $R$ to infinity. | [
"\\(\\frac{2\\pi}{3\\sqrt{3}}\\)"
] | [
"Use the residue theorem to compute the integral $$ \\int_{0}^{\\infty} \\frac{1}{1+x^3}dx$$\n\nGiven Hint: Consider the contour given by the boundary of the region $\\{re^{i\\theta} \\hspace{0.1cm} | \\hspace{0.1cm} \\theta \\in [0, \\frac{2\\pi}{3}], 0 \\leq r \\leq R\\}$, and take $R$ to infinity."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 29
} | 43 | 0 | 0 |
omr-ws-31ca24ef63b32323666bce27cf5dbf667a44143832e0e48f3241975cb55e1af1 | Use the result $\int_{0}^{\infty} \frac{\ln (a^{2}+x^{2})}{\cosh x} \ dx = 2 \pi \ln \Big[\sqrt{2 \pi} \ \frac{\Gamma (\frac{3}{4} + \frac{a}{2\pi})}{\Gamma(\frac{1}{4} + \frac{a}{2 \pi})} \Big]$ for $a=0$ to evaluate the integral $\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \ln ( \ln \tan x) \ dx$. | [
"$\\frac{\\pi}{2}\\ln \\left( \\frac{\\Gamma \\left( \\frac{3}{4} \\right)}{\\Gamma \\left( \\frac{1}{4} \\right)}\\sqrt{2\\pi} \\right)$"
] | [
"Use the result $\\int_{0}^{\\infty} \\frac{\\ln (a^{2}+x^{2})}{\\cosh x} \\ dx = 2 \\pi \\ln \\Big[\\sqrt{2 \\pi} \\ \\frac{\\Gamma (\\frac{3}{4} + \\frac{a}{2\\pi})}{\\Gamma(\\frac{1}{4} + \\frac{a}{2 \\pi})} \\Big]$ for $a=0$ to evaluate the integral $\\int_{\\frac{\\pi}{4}}^{\\frac{\\pi}{2}} \\ln ( \\ln \\tan x... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-b27179b0613f06536323b3c03a07166ca151f1cce5f6d24c63d0bb620f45ed71 | Use the result $x \ln x \to 0$ as $x \to 0$ to find $\lim_{x \to 0} f(x)$ where $f(x) = x^x$, and write down $\lim_{x \to 0} g(x)$ where $g(x) = x^{f(x)}$. | [
"\\(\\lim_{x \\to 0} f(x) = 1\\) and \\(\\lim_{x \\to 0} g(x) = 0\\)."
] | [
"Use the result $x \\ln x \\to 0$ as $x \\to 0$ to find $\\lim_{x \\to 0} f(x)$ where $f(x) = x^x$, and write down $\\lim_{x \\to 0} g(x)$ where $g(x) = x^{f(x)}$."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 29
} | 29 | 0 | 0 |
omr-ws-06b4c3268767d730022222f0c08a6a27dbb5d19918b0ec994c3d3ec34ecf95a1 | Use the result \[ \int_{0}^{a} f(x)\ dx = \int_{0}^{a} f(a-x)\ dx \] to evaluate \[ \int_{0}^{\pi} \frac{x}{2+\tan^{2}x}\ dx. \] | [
"\\(\\frac{\\pi^2}{4}(2 - \\sqrt{2})\\)"
] | [
"Use the result\n\n\\[ \\int_{0}^{a} f(x)\\ dx = \\int_{0}^{a} f(a-x)\\ dx \\]\n\nto evaluate\n\n\\[ \\int_{0}^{\\pi} \\frac{x}{2+\\tan^{2}x}\\ dx. \\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-977f6d12a1c6b7b14a2434bd90308521e19fd9810e68d5fcb9426e63dec4233c | Use the result from Problem 1 to estimate the largest number of terms of the $\tan^{-1} x$ expansion that you might need to use to compute $\tan^{-1} 0.2$ with an error of less than $10^{-10}$. | [
"6"
] | [
"Use the result from Problem 1 to estimate the largest number of terms of the $\\tan^{-1} x$ expansion that you might need to use to compute $\\tan^{-1} 0.2$ with an error of less than $10^{-10}$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 13
} | 45 | 0 | 0 |
omr-ws-8cb48e43b2bfbca1cc77daff3c223a64e5b7af7cad31b2a3054f751e53b647e2 | Use the result from Problem 1, together with a suitable value of $x$, to obtain a decimal estimate of the value of $\frac{100}{729}$. | [
"0.136"
] | [
"Use the result from Problem 1, together with a suitable value of $x$, to obtain a decimal estimate of the value of $\\frac{100}{729}$."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-97564f77dcf9f8f0c6ec50943f044de5aaa4364039bbf3b1ba984e57e68d33dc | Use the result from part b to determine the expected number of students who get their lucky pencil back, $E(X)$, by expressing $X$ in terms of the variables $X_{i}$. | [
"1"
] | [
"Use the result from part b to determine the expected number of students who get their lucky pencil back, $E(X)$, by expressing $X$ in terms of the variables $X_{i}$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-dd39d3790f36d307d34753fed57c6675c642b89d70995de760925b6dade44511 | Use the results from Problem 1 to find a sequence which has quadratic convergence to \( \sqrt[3]{N} \), and apply this to calculate \( \sqrt[3]{10} \) as accurately as possible. | [
"2.15443469"
] | [
"Use the results from Problem 1 to find a sequence which has quadratic convergence to \\( \\sqrt[3]{N} \\), and apply this to calculate \\( \\sqrt[3]{10} \\) as accurately as possible."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-2f675ae471a67fd0ddfaafbdd563bb5e7e304eadea5d17d8c2a9a35a32768bad | Use the roots of unity to find the sum of the coefficients of the terms whose power is divisible by 17 in the polynomial \( P(x) = (x+1)(x^2+1)(x^3+1)\cdots(x^{2007}+1) \). Specifically, find the sum of the coefficients of the terms \( x^0, x^{17}, x^{34}, x^{51}, \ldots, x^{2006} \). | [
"\\(\\frac{2^{2007}}{17}\\)"
] | [
"Use the roots of unity to find the sum of the coefficients of the terms whose power is divisible by 17 in the polynomial \\( P(x) = (x+1)(x^2+1)(x^3+1)\\cdots(x^{2007}+1) \\). Specifically, find the sum of the coefficients of the terms \\( x^0, x^{17}, x^{34}, x^{51}, \\ldots, x^{2006} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0927b69f87e1be992d41b64f0e1ddaf6647bbcef334ed0307422efc3c72825b8 | Use the series for \( f(x) = \arctan{x} \) to approximate the value of the integral \(\int_{0}^{\frac{1}{2}} \frac{\arctan{x^2}}{x} \, dx\) with an error bound \(\text{R}_\text{N} \leq 0.001\). | [
"-\\frac{1}{1152}"
] | [
"Use the series for \\( f(x) = \\arctan{x} \\) to approximate the value of the integral \\(\\int_{0}^{\\frac{1}{2}} \\frac{\\arctan{x^2}}{x} \\, dx\\) with an error bound \\(\\text{R}_\\text{N} \\leq 0.001\\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-933a4ec0f48dd0e2e786523caa4eddf2d282ff2aa09f17ccc7435678a804cdbe | Use the series method to solve the differential equation \( y'' + ay = 0 \) where \( a > 0 \). The expected solution is \( y = A \sin(\sqrt{a} x) + B \cos(\sqrt{a} x) \). In the recursion relation, you get \( -\frac{a C_n}{(n+2)(n+1)} = C_{n+2} \). How can you adjust the recursion relation to ensure that the coefficien... | [
"\\( C_1 = \\sqrt{a} \\) and \\( C_0 = 1 \\)"
] | [
"Use the series method to solve the differential equation \\( y'' + ay = 0 \\) where \\( a > 0 \\). The expected solution is \\( y = A \\sin(\\sqrt{a} x) + B \\cos(\\sqrt{a} x) \\). In the recursion relation, you get \\( -\\frac{a C_n}{(n+2)(n+1)} = C_{n+2} \\). How can you adjust the recursion relation to ensure t... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 3,
"tir": 4
} | 13 | 0 | 0 |
omr-ws-35ae566b80455332f23ef7ac78d65052267fa774c15b06839517649c0d3826a1 | Use the sign test at the 0.05 level of significance to test the doctor’s claim that the median waiting time for her patients is not more than 20 minutes, given the following waiting times (in minutes): 17, 15, 20, 20, 32, 28, 12, 26, 25, 25, 35, 24. | [
"0.1719"
] | [
"Use the sign test at the 0.05 level of significance to test the doctor’s claim that the median waiting time for her patients is not more than 20 minutes, given the following waiting times (in minutes): 17, 15, 20, 20, 32, 28, 12, 26, 25, 25, 35, 24."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 3
} | 3 | 0 | 0 |
omr-ws-7f4636ad2c93c808893bb9982f4d0fe39288d55607c14c76a3fd980578cd2087 | Use the simplex method to determine the optimal quantity of soursop and cashew juices to be produced per work shift, given the constraints on processing times and costs. Interpret the last tableau. | [
"0"
] | [
"Use the simplex method to determine the optimal quantity of soursop and cashew juices to be produced per work shift, given the constraints on processing times and costs. Interpret the last tableau."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-76b1c4773cf7350a351b77c1273a450acd9206fa522fa02f8ea807ebf273722e | Use the substitution $u = \arccos x$ to evaluate the integral $\int_{-1}^{1} T_{n}(x)\ dx$. | [
"\\frac{1 + (-1)^n}{1 - n^2}"
] | [
"Use the substitution $u = \\arccos x$ to evaluate the integral $\\int_{-1}^{1} T_{n}(x)\\ dx$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-b4d9adf973368b9655a5734edb224b85501ff17320703fe37a41ae0a57577dc9 | Use the substitution $x=\cosh^{2} u$ to find $\int\sqrt{\frac{x}{x-1}}dx$, giving your answer in the form $f(x) + \ln (g(x))$. | [
"\\(\\frac{\\sinh(2 \\ln(\\sqrt{x} + \\sqrt{x-1}))}{2} + \\ln(\\sqrt{x} + \\sqrt{x-1}) + C\\)"
] | [
"Use the substitution $x=\\cosh^{2} u$ to find $\\int\\sqrt{\\frac{x}{x-1}}dx$, giving your answer in the form $f(x) + \\ln (g(x))$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8b070dcf642d8e83f2235b70ca5f7146092c7429c5a20247ebe7b00364bdf1c3 | Use the substitution $y=xz$ to find the general solution of the differential equation \[ x\frac{dy}{dx} - y = x \cos \left(\frac{y}{x}\right), \] giving your answer in a form without logarithms. | [
"\\( x = 4(\\sqrt{2} - 1) \\left( \\sec \\left( \\frac{y}{x} \\right) + \\tan \\left( \\frac{y}{x} \\right) \\right) \\)"
] | [
"Use the substitution $y=xz$ to find the general solution of the differential equation\n\n\\[ x\\frac{dy}{dx} - y = x \\cos \\left(\\frac{y}{x}\\right), \\]\n\ngiving your answer in a form without logarithms."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8feb895f1c4e631fb25dc4cd3a547fb9b2ad8990a40508095e2559d5aefb0303 | Use the substitution $y=xz$, where $z$ is a function of $x$, to transform the differential equation $\frac{dy}{dx} = \frac{x^{2}-y^{2}}{xy}$ into the form $x\frac{dz}{dx} = \frac{1-2z^{2}}{z}$. | [
"\\( x \\frac{dz}{dx} = \\frac{1 - 2z^2}{z} \\)"
] | [
"Use the substitution $y=xz$, where $z$ is a function of $x$, to transform the differential equation $\\frac{dy}{dx} = \\frac{x^{2}-y^{2}}{xy}$ into the form $x\\frac{dz}{dx} = \\frac{1-2z^{2}}{z}$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-5298f4913f4d353686e6a3e16fc809a09b508649ea63409b154a10b4ab027060 | Use the substitution \( t = \tan \frac{x}{2} \) to find \( \int_0^{\frac{\pi}{2}} \frac{1}{5 + 3\cos x - 4\sin x} \, dx \). | [
"\\(\\frac{1}{2}\\)"
] | [
"Use the substitution \\( t = \\tan \\frac{x}{2} \\) to find \\( \\int_0^{\\frac{\\pi}{2}} \\frac{1}{5 + 3\\cos x - 4\\sin x} \\, dx \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 30
} | 53 | 0 | 0 |
omr-ws-bba3446746a3a4eb03716c93873292e097d72cd9d68817285b4413de196c307e | Use the substitution \( w = \sinh x \) to find the real root of the equation \( 4w^3 + 3w - 3 = 0 \). Give your answer in the form \( a \ln (b + \sqrt{c}) \) where \( a, b, \) and \( c \) are real numbers to be determined. | [
"\\(\\frac{1}{3} \\ln (3 + \\sqrt{10})\\)"
] | [
"Use the substitution \\( w = \\sinh x \\) to find the real root of the equation \\( 4w^3 + 3w - 3 = 0 \\). Give your answer in the form \\( a \\ln (b + \\sqrt{c}) \\) where \\( a, b, \\) and \\( c \\) are real numbers to be determined."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 12
} | 27 | 0 | 0 |
omr-ws-313f0f098357a0278d97d95efc4a56129cd676b1eff98fef7bcd05f920bbbe55 | Use the substitution \( x = \sin v \) to find the exact value of the integral \(\int \sqrt{1 - x^2} \, dx\). | [
"\\frac{1}{2} \\arcsin x + \\frac{1}{2} x \\sqrt{1 - x^2} + C"
] | [
"Use the substitution \\( x = \\sin v \\) to find the exact value of the integral \\(\\int \\sqrt{1 - x^2} \\, dx\\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-62cc83a23078f4b6cf5da52294d518664fc6359c07774e51795aae60757ec049 | Use the substitution \( x = e^t \) to transform the given Cauchy-Euler equation \( x^2y'' + 9xy' - 20y = 0 \) into a differential equation with constant coefficients. | [
"\\(\\frac{d^2y}{dt^2} + 8\\frac{dy}{dt} - 20y = 0\\)"
] | [
"Use the substitution \\( x = e^t \\) to transform the given Cauchy-Euler equation \\( x^2y'' + 9xy' - 20y = 0 \\) into a differential equation with constant coefficients."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 14
} | 29 | 0 | 0 |
omr-ws-08799e1e852b4d861775f1950328200f63c0d423ccee247f2414f74c18e8103f | Use the substitution \( x = u^2 \) (\( u > 0 \)) to find \( \int \frac{1}{x(1 + x\sqrt{x})} \, dx \). | [
"\\(\\frac{2}{3} \\left( \\frac{3}{2} \\ln x - \\ln (\\sqrt{x} + 1) - \\ln (x - \\sqrt{x} + 1) \\right) + C\\)"
] | [
"Use the substitution \\( x = u^2 \\) (\\( u > 0 \\)) to find \\( \\int \\frac{1}{x(1 + x\\sqrt{x})} \\, dx \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 6,
"tir": 14
} | 34 | 0 | 0 |
omr-ws-53214db2c13e6e256ab20ad053bab2385543603c80054a7cbff456d6f6fa1173 | Use the substitution \( x=\tan u \) to evaluate \( \int_{0}^{1} (1+x^{2})^{\frac{-3}{2}} \). Show that this is consistent with the value given by the reduction formula with \( n=\frac{-1}{2} \). | [
"\\(\\frac{\\sqrt{2}}{2}\\)"
] | [
"Use the substitution \\( x=\\tan u \\) to evaluate \\( \\int_{0}^{1} (1+x^{2})^{\\frac{-3}{2}} \\). Show that this is consistent with the value given by the reduction formula with \\( n=\\frac{-1}{2} \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-d5e5a09e4f8cc4d8101896e4dd48e6106b7c0fb14968c236ff62235a03f0eb93 | Use the substitution \( y = xu \) to solve the differential equation \[ \frac{dy}{dx} = \frac{2y + x}{y - 2x} \] given that the solution curve passes through the point \((1, 1)\). Give your answer in the form of a quadratic in \( x \) and \( y \). | [
"\\(-y^2 + 4yx + x^2 - 4x^3 = 0\\)"
] | [
"Use the substitution \\( y = xu \\) to solve the differential equation\n\n\\[\n\\frac{dy}{dx} = \\frac{2y + x}{y - 2x}\n\\]\n\ngiven that the solution curve passes through the point \\((1, 1)\\). Give your answer in the form of a quadratic in \\( x \\) and \\( y \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9927f1dd774ade7534108a1f5ee27336c77426b7e5800d0b4a6cb3ec99adbbe9 | Use the substitution \(v = \frac{y}{x}\) to solve the differential equation \(\frac{dy}{dx} = \frac{y^4 + 2xy^3 - 3x^2y^2 - 2x^3y}{2x^2y^2 - 2x^3y - 2x^4}\). | [
"\\(\\ln\\left|\\frac{y^2}{x^2 - y^2}\\right| - \\frac{2x}{y} = \\ln|Cx|\\)"
] | [
"Use the substitution \\(v = \\frac{y}{x}\\) to solve the differential equation \\(\\frac{dy}{dx} = \\frac{y^4 + 2xy^3 - 3x^2y^2 - 2x^3y}{2x^2y^2 - 2x^3y - 2x^4}\\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-41e1b6b507bef27f1d902bad618f786c27f4eb227fbbeb6d3ed793f3cb18a52a | Use the substitution \(v = \frac{y}{x}\) to solve the differential equation \(x^2 + 3xy + y^2 - x^2 \frac{dy}{dx} = 0\). | [
"\\(\\ln|x| + \\frac{x}{y + x} + c = 0\\)"
] | [
"Use the substitution \\(v = \\frac{y}{x}\\) to solve the differential equation \\(x^2 + 3xy + y^2 - x^2 \\frac{dy}{dx} = 0\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 5,
"tir": 2
} | 19 | 0 | 0 |
omr-ws-422c93f94eec325352e971b6587ecb65b9192901cfc93a6234a0acd634b73411 | Use the substitution \(x = \tan u\) to evaluate \(\int_{0}^{1} (1+x^{2})^{-\frac{3}{2}}\ dx\). Then, using the reduction formula for \(\int (1+x^2)^n dx\) with \(n = -\frac{3}{2}\), calculate the value of the integral. Find the sum of the numerator and denominator of the resulting fraction when expressed in lowest term... | [
"3"
] | [
"Use the substitution \\(x = \\tan u\\) to evaluate \\(\\int_{0}^{1} (1+x^{2})^{-\\frac{3}{2}}\\ dx\\). Then, using the reduction formula for \\(\\int (1+x^2)^n dx\\) with \\(n = -\\frac{3}{2}\\), calculate the value of the integral. Find the sum of the numerator and denominator of the resulting fraction when expre... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-fe7da62c546c931f700609ab67202b1828d255b164b51749c5f6c4f3046c27bb | Use the sum of the geometric series $1 - u + u^2 - \ldots + (-1)^{n-1}u^{n-1}$ to find an expression for the error in using the Maclaurin polynomial of degree $n$ as an approximation to $\ln(1+x)$. | [
"(-1)^{n} \\int_{0}^{x} \\frac{t^{n}}{1 + t} \\, dt"
] | [
"Use the sum of the geometric series $1 - u + u^2 - \\ldots + (-1)^{n-1}u^{n-1}$ to find an expression for the error in using the Maclaurin polynomial of degree $n$ as an approximation to $\\ln(1+x)$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c29511df93cdf70ca11fd0c2dea187a700fb476d92bf986a377941cfb5ddd5ad | Use the sum of the geometric series $1 - u + u^2 - u^3 + \cdots + (-1)^{n-1} u^{n-1}$ to find an expression for the error in using the Maclaurin polynomial of degree $n$ as an approximation to $\ln(1+x)$. | [
"\\( \\frac{|x^{n+1}|}{n+1} \\)"
] | [
"Use the sum of the geometric series $1 - u + u^2 - u^3 + \\cdots + (-1)^{n-1} u^{n-1}$ to find an expression for the error in using the Maclaurin polynomial of degree $n$ as an approximation to $\\ln(1+x)$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c1adee9a1c5f98c295ac04bc2f0e4e02655b467e98321780961a93463c1849d9 | Use the table of values to estimate the definite integral of \( f \) from 0 to 6 using three equal subintervals and the (a) left endpoints, (b) right endpoints, and (c) midpoints. If \( f \) is an increasing function, how does each estimate compare with the actual value? Explain your reasoning. Table: \[ \begin{array}{... | [
"Left endpoints: 64, Right endpoints: 236, Midpoints: 136"
] | [
"Use the table of values to estimate the definite integral of \\( f \\) from 0 to 6 using three equal subintervals and the (a) left endpoints, (b) right endpoints, and (c) midpoints. If \\( f \\) is an increasing function, how does each estimate compare with the actual value? Explain your reasoning.\nTable:\n\\[\n\... | [
"aops_c7_college_math",
"c7_college_math"
] | [
"tir"
] | {
"tir": 28
} | 28 | 0 | 0 |
omr-ws-f24bc5527bdd4cb809ba5adb97c201df25d3db3db5a58e2e3a4d18e6193a8bb8 | Use the tangent line to the graph of \( f(x) = \sin x \) at \( x = \pi/3 \) to approximate \( \sin(\pi/6) \). | [
"\\(\\frac{\\sqrt{3}}{2} - \\frac{\\pi}{12}\\)"
] | [
"Use the tangent line to the graph of \\( f(x) = \\sin x \\) at \\( x = \\pi/3 \\) to approximate \\( \\sin(\\pi/6) \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 3,
"tir": 12
} | 30 | 0 | 0 |
omr-ws-579273f4ac8a01e48cea5142541ad54ef39eac775139bc3c3d9a90477c892bef | Use the trapezoidal approximation with 3 subintervals of equal length to approximate the area under the curve \(y = 6x - x^2\) on the interval \([2, 5]\). | [
"\\dfrac{47}{2}"
] | [
"Use the trapezoidal approximation with 3 subintervals of equal length to approximate the area under the curve \\(y = 6x - x^2\\) on the interval \\([2, 5]\\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3fb2dfc6e3595292d0c91c854f02610121b9588c57588b061246f8fe94be8c71 | Use the trapezoidal rule with \( n=5 \) to approximate the area of the region \( S \) bounded above by the graph of \( f \) and below by the x-axis on the interval \([1,2]\). | [
"0.6956"
] | [
"Use the trapezoidal rule with \\( n=5 \\) to approximate the area of the region \\( S \\) bounded above by the graph of \\( f \\) and below by the x-axis on the interval \\([1,2]\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 6
} | 18 | 0 | 0 |
omr-ws-04578367303330f98f2f8fb5b37c8a23cc7d07ddf27301e8c5a732e98d22904f | Use the trig identity \( \cos^2 x = \frac{1 + \cos{2x}}{2} \) and the power series for \( \cos x \) to find a power series for \( f(x) = \cos^2 x \). | [
"\\( 1 + \\frac{1}{2} \\sum_{n=1}^{\\infty} \\frac{(-1)^n (2x)^{2n}}{(2n)!} \\)"
] | [
"Use the trig identity \\( \\cos^2 x = \\frac{1 + \\cos{2x}}{2} \\) and the power series for \\( \\cos x \\) to find a power series for \\( f(x) = \\cos^2 x \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 8,
"tir": 18
} | 33 | 0 | 0 |
omr-ws-9b3ccb949b950d1ea7730c85104f6b14827c4bdc9281d8155002db320c57c316 | Use the trigonometric form of the complex numbers \( z = \frac{-i}{2} \) and \( w = 1 - i \) to compute \( z^n \) and \( w^n \). | [
"2^{\\frac{n}{2}} \\left( \\cos \\frac{7n\\pi}{4} + i \\sin \\frac{7n\\pi}{4} \\right)"
] | [
"Use the trigonometric form of the complex numbers \\( z = \\frac{-i}{2} \\) and \\( w = 1 - i \\) to compute \\( z^n \\) and \\( w^n \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-358c27d11ee580bba031aa8645bdb44fef2ec75301e61e9b2c07a752969d6547 | Use the trigonometric form of the complex numbers \( z = \frac{-i}{2} \) and \( w = 1 - i \) to compute \( z^{\frac{1}{n}} \) and \( w^{\frac{1}{n}} \), giving all \( n \) values for the \( n \)-th roots of the complex numbers. Plot the results for \( n = 3 \) in the complex plane. | [
" 2^{\\frac{1}{2n}} \\left( \\cos\\left( \\frac{7\\pi + 8\\pi k}{4n} \\right) + i \\sin\\left( \\frac{7\\pi + 8\\pi k}{4n} \\right) \\right) "
] | [
"Use the trigonometric form of the complex numbers \\( z = \\frac{-i}{2} \\) and \\( w = 1 - i \\) to compute \\( z^{\\frac{1}{n}} \\) and \\( w^{\\frac{1}{n}} \\), giving all \\( n \\) values for the \\( n \\)-th roots of the complex numbers. Plot the results for \\( n = 3 \\) in the complex plane."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-091a31995b15893f44c9bb88eb35444e1cd0fb1822f3774e5346aee1fca788b1 | Use the value $k=8$ to find an approximation to five decimal places for $3^{0.5}$. | [
"1.73205"
] | [
"Use the value $k=8$ to find an approximation to five decimal places for $3^{0.5}$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6937c6a97cdec71567e561d6f3200ef7f5fc839b4575c177345fd1bdd5d686af | Use the washer method to find the volume of revolution generated when the region in the third quadrant enclosed between \( y = x^2 \) and \( y = x^3 \) is rotated about the line \( y = 1 \). | [
"\\(\\frac{23\\pi}{210}\\)"
] | [
"Use the washer method to find the volume of revolution generated when the region in the third quadrant enclosed between \\( y = x^2 \\) and \\( y = x^3 \\) is rotated about the line \\( y = 1 \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 5,
"tir": 2
} | 13 | 0 | 0 |
omr-ws-ccd7972e213e64496cd979d461e1095ba17e0defa3777a957ef7775d96309e8b | Use the washer method to find the volume of revolution generated when the region in the third quadrant enclosed between \( y = x^2 \) and \( y = x^3 \) is rotated about the x-axis. | [
"\\(\\frac{2\\pi}{35}\\)"
] | [
"Use the washer method to find the volume of revolution generated when the region in the third quadrant enclosed between \\( y = x^2 \\) and \\( y = x^3 \\) is rotated about the x-axis."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 28
} | 42 | 0 | 0 |
omr-ws-0c1e6942bd9da564216b521c50dd8b854a8398f581421df652729d6fa4223f70 | Use trigonometric identities to find the power series for the function \( \sin(x) \cos(x) \). | [
"\\(\\sum_{n=0}^{\\infty} \\frac{(-1)^n 4^n x^{2n+1}}{(2n+1)!}\\)"
] | [
"Use trigonometric identities to find the power series for the function \\( \\sin(x) \\cos(x) \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 28
} | 51 | 0 | 0 |
omr-ws-4c819a44167a2ce06f72c4daa855dfdbbbbf5d8e9411cc72b16678553caaa356 | Useless Eustace believes that \( \int \sin^{n} x \, dx = \frac{\sin^{n+1} x}{n+1} \) for \( n = 1, 2, 3, \ldots \). Show that Eustace would obtain the correct value of \( I(\beta) \), where \( \cos \beta = -\frac{1}{6} \). | [
"\\(\\frac{245}{162}\\)"
] | [
"Useless Eustace believes that \\( \\int \\sin^{n} x \\, dx = \\frac{\\sin^{n+1} x}{n+1} \\) for \\( n = 1, 2, 3, \\ldots \\). Show that Eustace would obtain the correct value of \\( I(\\beta) \\), where \\( \\cos \\beta = -\\frac{1}{6} \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4abb7ccb9994980b1ab1298585a0a759f077cc68ca8f4009c55c7e7aa7a4b7ed | Using $L$-shaped triominos, and dominos, where each square of a triomino and a domino covers one unit, what is the minimum number of tiles needed to cover a $3$-by-$2017$ rectangle without any gaps? | [
"2017"
] | [
"Using $L$-shaped triominos, and dominos, where each square of a triomino and a domino covers one unit, what is the minimum number of tiles needed to cover a $3$-by-$2017$ rectangle without any gaps?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 8
} | 19 | 0 | 0 |
omr-ws-c027bafeff56aba99c1ab3a3d6da06ba07e567af2c9ec57cab2aa18d8564c745 | Using $\sin(a)$, calculate the product $\cos(a) \cdot \cos(2a) \cdot \ldots \cdot \cos(2^n a)$ for all $a \in \left(0, \frac{\pi}{2}\right)$ and $n \in \mathbb{N}$. | [
"\\( \\frac{\\sin(2^{n+1} a)}{2^{n+1} \\sin(a)} \\)"
] | [
"Using $\\sin(a)$, calculate the product $\\cos(a) \\cdot \\cos(2a) \\cdot \\ldots \\cdot \\cos(2^n a)$ for all $a \\in \\left(0, \\frac{\\pi}{2}\\right)$ and $n \\in \\mathbb{N}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-56fd90ac33f4978279bd7955902b3cb4da564f7f0ee84a4a5d171c12a75684c4 | Using (1) and (2) and the formulas for the sums of $n$ terms of an arithmetic and a geometric progression, calculate the following limits: $\lim\limits_{n\to \infty } \frac{1+2+\ldots +2^n}{1+5+\ldots +5^n}$. | [
"0"
] | [
"Using (1) and (2) and the formulas for the sums of $n$ terms of an arithmetic and a geometric progression, calculate the following limits: $\\lim\\limits_{n\\to \\infty } \\frac{1+2+\\ldots +2^n}{1+5+\\ldots +5^n}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-179a62f8e885f6c8844fd1c18b52a66d67c03f65cbdc3f92a8138ced76242f69 | Using (1) and (2) and the formulas for the sums of $n$ terms of an arithmetic and a geometric progression, calculate the following limits: $\lim\limits_{n\to \infty } \left(\frac{1+3+5+\ldots +(2n+1)}{n+1} -\frac{2n-1}{2}\right)$. | [
"1.5"
] | [
"Using (1) and (2) and the formulas for the sums of $n$ terms of an arithmetic and a geometric progression, calculate the following limits: $\\lim\\limits_{n\\to \\infty } \\left(\\frac{1+3+5+\\ldots +(2n+1)}{n+1} -\\frac{2n-1}{2}\\right)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a67698cb957a259d1b227622d48087cb44e19bbfdcf4f169add9cfd65ef86b39 | Using (1) and (2) and the formulas for the sums of $n$ terms of an arithmetic and a geometric progression, calculate the following limits: $\lim\limits_{n\to \infty } \left(\frac{1}{a_1a_2}+\frac{1}{a_2a_3} +\ldots +\frac{1}{a_na_{n+1}}\right)$,\\ where $(a_n)$, $n\in \mathbb{N}$, is an arithmetic progression with the ... | [
"$\\frac{1}{da_1}$"
] | [
"Using (1) and (2) and the formulas for the sums of $n$ terms of an arithmetic and a geometric progression, calculate the following limits: $\\lim\\limits_{n\\to \\infty } \\left(\\frac{1}{a_1a_2}+\\frac{1}{a_2a_3} +\\ldots +\\frac{1}{a_na_{n+1}}\\right)$,\\\\ where $(a_n)$, $n\\in \\mathbb{N}$, is an arithmetic pr... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a4708b9d88b4fed5b228fe965386bd3e678ac8020650f0bc18b494da22754623 | Using (1) and (2) and the formulas for the sums of $n$ terms of an arithmetic and a geometric progression, calculate the following limits: $\lim\limits_{n\to \infty } \left(\frac{1}{n^4}+\frac{8}{n^4}+\ldots + \frac{n^3}{n^4}\right)$. | [
"1/4"
] | [
"Using (1) and (2) and the formulas for the sums of $n$ terms of an arithmetic and a geometric progression, calculate the following limits: $\\lim\\limits_{n\\to \\infty } \\left(\\frac{1}{n^4}+\\frac{8}{n^4}+\\ldots + \\frac{n^3}{n^4}\\right)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c5a1a074f2e1c563b05b388de079c2ae268940c9d10f057460db19307e412370 | Using (1) and (2) and the formulas for the sums of $n$ terms of an arithmetic and a geometric progression, calculate the following limits: $\lim\limits_{n\to \infty } \left(\frac{7}{10}+\frac{29}{10^2}+\ldots + \frac{5^n+2^n}{10^n}\right)$. | [
"1.25"
] | [
"Using (1) and (2) and the formulas for the sums of $n$ terms of an arithmetic and a geometric progression, calculate the following limits: $\\lim\\limits_{n\\to \\infty } \\left(\\frac{7}{10}+\\frac{29}{10^2}+\\ldots + \\frac{5^n+2^n}{10^n}\\right)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6b052b918b4a459a4562aed0367f2328610ce76cfb87a4a834060726e693f1d4 | Using 14 sets of "-1" (where each can be interpreted as "subtract one" or "negative one"), unlimited pairs of parentheses, and one "^" for exponentiation (or LaTeX for exponentiation), construct an expression that yields the largest positive value. Each "-1" pair must remain together and cannot be split or regrouped. | [
"\\( 5^8 \\)"
] | [
"Using 14 sets of \"-1\" (where each can be interpreted as \"subtract one\" or \"negative one\"), unlimited pairs of parentheses, and one \"^\" for exponentiation (or LaTeX for exponentiation), construct an expression that yields the largest positive value. Each \"-1\" pair must remain together and cannot be split ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 2
} | 3 | 0 | 0 |
omr-ws-5c127e4514c4ee6b83a2631506b559861efcbc977f003af711629d8242020d3c | Using 150 2x2 tiles, how many different rectangular shapes can be formed? | [
"6"
] | [
"Using 150 2x2 tiles, how many different rectangular shapes can be formed?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1604241061cc35d2c2846deabf409de4b05b2de1a1aa52ef611d21e872cf476f | Using 2 cent, 3 cent, and 4 cent stamps, how many ways are there to arrange them in a row to get a total of 8 cents? | [
"8"
] | [
"Using 2 cent, 3 cent, and 4 cent stamps, how many ways are there to arrange them in a row to get a total of 8 cents?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-552282dab60a2cd1e975adaa9b99e42d51fa73dbe0768f5ae1f4ec281a6b0d87 | Using 2 green sticks of 3 cm each, 2 blue sticks of 4 cm each, and 2 red sticks of 5 cm each, how many non-congruent triangles can be formed where all sticks are used as the perimeter without overlapping? | [
"2"
] | [
"Using 2 green sticks of 3 cm each, 2 blue sticks of 4 cm each, and 2 red sticks of 5 cm each, how many non-congruent triangles can be formed where all sticks are used as the perimeter without overlapping?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3089512c83771a8d1485a4f3233e35f4e4935eba3d7bf95c159e8c7626fd702b | Using 21 tiles, some white and some black, a $3 \times 7$ rectangle is formed. Let $N$ be the minimum number of tiles of the same color that are located at the vertices of a rectangle. Find $N$. | [
"4"
] | [
"Using 21 tiles, some white and some black, a $3 \\times 7$ rectangle is formed. Let $N$ be the minimum number of tiles of the same color that are located at the vertices of a rectangle. Find $N$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 7
} | 10 | 0 | 0 |
omr-ws-959f281e6f7e293ab2eab19aedb74e99fe1f86d53b1d3ca427d1a295a545fed8 | Using 4 non-coplanar points, how many boxes can be made if the faces of the box may be parallelograms? | [
"4"
] | [
"Using 4 non-coplanar points, how many boxes can be made if the faces of the box may be parallelograms?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 4
} | 13 | 0 | 0 |
omr-ws-97140d9a814344248980bc22eb76a1f32494c802b751c0df74378359c609635b | Using 6 given colors, you color each side of a cube with a different color and write the numbers 6 and 1, 2 and 5, 3 and 4 on opposite faces. How many different cubes can you make, considering two cubes are the same if one can be rotated to match the other? | [
"30"
] | [
"Using 6 given colors, you color each side of a cube with a different color and write the numbers 6 and 1, 2 and 5, 3 and 4 on opposite faces. How many different cubes can you make, considering two cubes are the same if one can be rotated to match the other?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 7
} | 22 | 0 | 0 |
omr-ws-2f8201286f7816d5c3b744f9ac2956050231f5c3f1d9f2774d3b071dbd818624 | Using 6 given colors, you first color each side of a cube to a different color. Then, you write the first 6 counting numbers on it, one on each face, so that the numbers 6 and 1, 5 and 2, 4 and 3 are opposite each other. How many different cubes can you make? Note that two cubes are considered the same if you can rotat... | [
"1440"
] | [
"Using 6 given colors, you first color each side of a cube to a different color. Then, you write the first 6 counting numbers on it, one on each face, so that the numbers 6 and 1, 5 and 2, 4 and 3 are opposite each other. How many different cubes can you make? Note that two cubes are considered the same if you can ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 5
} | 16 | 0 | 0 |
omr-ws-3dac9ec0775a3bdd944b78b4833132a1b53355775c5c46d1fd9413d2717851ab | Using 600 cards, where 200 cards have the number 5, 200 cards have the number 2, and 200 cards have the number 1, what is the maximum number of groups that can be created such that the sum of the card numbers in each group is 9? | [
"150"
] | [
"Using 600 cards, where 200 cards have the number 5, 200 cards have the number 2, and 200 cards have the number 1, what is the maximum number of groups that can be created such that the sum of the card numbers in each group is 9?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 33,
"tir": 21
} | 54 | 0 | 0 |
omr-ws-471729b6b979ec1cee4f246c325d27b530f188ef9e32a093fbdaaf1b09ceee90 | Using 7 different kinds of jigsaws formed by 4 unit squares, what is the maximum number of different kinds of jigsaws that can be chosen to form a 7×4 rectangle? | [
"6"
] | [
"Using 7 different kinds of jigsaws formed by 4 unit squares, what is the maximum number of different kinds of jigsaws that can be chosen to form a 7×4 rectangle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 7
} | 16 | 0 | 0 |
omr-ws-05a5310fdf9c7f7dc2f40391d34906f177820ee4d4b1c4e5ab3d1abb8601b2b6 | Using Bayes' rule, simplify the expression for \(\text{Pr}(s' \mid a, z, s)\) and find the value of \(X\) in the following equation: \[ \text{Pr}(s' \mid a, z, s) = \frac{\text{Pr}(z \mid s', a, s) \text{Pr}(s' \mid a, s)}{X} \] | [
"\\text{Pr}(z \\mid a, s)"
] | [
"Using Bayes' rule, simplify the expression for \\(\\text{Pr}(s' \\mid a, z, s)\\) and find the value of \\(X\\) in the following equation:\n\\[ \\text{Pr}(s' \\mid a, z, s) = \\frac{\\text{Pr}(z \\mid s', a, s) \\text{Pr}(s' \\mid a, s)}{X} \\]"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-49f6e8b8fa49bc20002f380b57b617131b659fcaa570e0697fc4d5d1b2057fa9 | Using Burnside's lemma, calculate the number of all possible strings of length 6 that can be formed using the characters from the string "AABBBCCCC". | [
"410"
] | [
"Using Burnside's lemma, calculate the number of all possible strings of length 6 that can be formed using the characters from the string \"AABBBCCCC\"."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 2
} | 7 | 0 | 0 |
omr-ws-be9667df7d32b1cb172c816ff402c385b2724928361361ec93dd5df09ebb2c7a | Using Cauchy's integral formula or the integral formula for derivatives, calculate $\int_L \frac{e^z}{z} \, dz$, where $L$ is a closed curve that does not pass through the point $z=0$. | [
"\\(2\\pi i\\) if \\(z=0\\) is inside \\(L\\), and \\(0\\) if \\(z=0\\) is outside \\(L\\)."
] | [
"Using Cauchy's integral formula or the integral formula for derivatives, calculate $\\int_L \\frac{e^z}{z} \\, dz$, where $L$ is a closed curve that does not pass through the point $z=0$."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 19
} | 19 | 0 | 0 |
omr-ws-c13c842cf708c9626cc1c4ff385c45ff7721f266a18367309fbef99263551266 | Using Chebyshev's Inequality, find a lower bound for the probability that the dealer will sell more than 44 and less than 56 cars tomorrow, given that the number of used cars sold each day has a mean of 50 and a variance of 25. | [
"\\(\\frac{25}{49}\\)"
] | [
"Using Chebyshev's Inequality, find a lower bound for the probability that the dealer will sell more than 44 and less than 56 cars tomorrow, given that the number of used cars sold each day has a mean of 50 and a variance of 25."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ccc24531c527257a5d5928c30502085357d50015a0cc6918f71c5503c799a5ca | Using Cramer's rule, determine the set of values of $k$ such that the following system of linear equations has integer solutions: \[ 2x - y + 3z \equiv k \] \[ 2x + y - z \equiv 1 \] \[ 6x - 3y + z \equiv 3k \] | [
"k \\equiv 3 \\pmod{4}"
] | [
"Using Cramer's rule, determine the set of values of $k$ such that the following system of linear equations has integer solutions:\n\\[ 2x - y + 3z \\equiv k \\]\n\\[ 2x + y - z \\equiv 1 \\]\n\\[ 6x - 3y + z \\equiv 3k \\]"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-990bc8512b78b3ba30a579a08c94a16cb73bd5bd37b3f1a138cd5b2356fd0735 | Using De Moivre's formula, calculate the value of \( \cos\left(\frac{\pi}{7}\right) - \sin\left(\frac{2\pi}{7}\right) + \cos\left(\frac{3\pi}{7}\right) \). | [
"\\dfrac{1}{2}"
] | [
"Using De Moivre's formula, calculate the value of \\( \\cos\\left(\\frac{\\pi}{7}\\right) - \\sin\\left(\\frac{2\\pi}{7}\\right) + \\cos\\left(\\frac{3\\pi}{7}\\right) \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-d8971a9bb62abb3e614757a9cfe60d07eed0b0ee362ead06058647feec69c099 | Using De Morgan's Laws and other properties of set algebra, simplify the expression \((A' \cap B \cap C) \cup [A' \cup (B \cap C')]' \) and express the result in terms of \(A\), \(B\), and \(C\). What is the simplified form? | [
"(A \\cap B') \\cup (B \\cap C)"
] | [
"Using De Morgan's Laws and other properties of set algebra, simplify the expression \\((A' \\cap B \\cap C) \\cup [A' \\cup (B \\cap C')]' \\) and express the result in terms of \\(A\\), \\(B\\), and \\(C\\). What is the simplified form?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 8
} | 11 | 0 | 0 |
omr-ws-0859b50a462124f9bdbaa80961abc24391b06555b140ee59f2eaff0ab9290c4e | Using Descartes' Rule of Signs, determine the maximum number of positive and negative real roots for the equation \( x|x| + px + q = 0 \) and reconcile this with the graphical analysis of the equation. | [
"3"
] | [
"Using Descartes' Rule of Signs, determine the maximum number of positive and negative real roots for the equation \\( x|x| + px + q = 0 \\) and reconcile this with the graphical analysis of the equation."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-464604a7a5e2d0008b67e772c927d5e37af7b6e0c9f1b64be6f6918067af9770 | Using Descartes' Rule of Signs, determine the possible numbers of positive, negative, and imaginary roots of the polynomial $x^3 - x + 2$. | [
"Positive roots: 0 or 2, Negative roots: 1, Imaginary roots: 2 or 0"
] | [
"Using Descartes' Rule of Signs, determine the possible numbers of positive, negative, and imaginary roots of the polynomial $x^3 - x + 2$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-c6d22f1d924df51b140b3e70864cb2135740007032a109a96f49d5c4fa9775ec | Using Euclid's algorithm, find gcd(111...111, 11...11), where there are one hundred 1's in the decimal representation of the first number and sixty 1's in the decimal representation of the second number. | [
"111...11 (twenty 1's)"
] | [
"Using Euclid's algorithm, find gcd(111...111, 11...11), where there are one hundred 1's in the decimal representation of the first number and sixty 1's in the decimal representation of the second number."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 15
} | 38 | 0 | 0 |
omr-ws-20e257417bf935923070155459700467073380c6cad56dbd1730542f3cf81a86 | Using Euclidean tools, inscribe a square in a given triangle such that one side of the square lies on a given side of the triangle. | [
"\\dfrac{b h}{b + h}"
] | [
"Using Euclidean tools, inscribe a square in a given triangle such that one side of the square lies on a given side of the triangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 5
} | 8 | 0 | 0 |
omr-ws-27093444017a502c77e2784fa5aeed77a70dbeed4887523b11295476526d5fa3 | Using Euler's Theorem, simplify \(2^{2013} \mod 100\). | [
"92"
] | [
"Using Euler's Theorem, simplify \\(2^{2013} \\mod 100\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-71af905b1bd21263b463fd23fb06f0cffd29227b22afac78e4fcda3e56772f05 | Using Euler's method, find the value of the series \[ S = \sum_{n=0}^{\infty} \frac{1}{(2n+1)^2} \] by: a. Determining the zeros of \(1 - \sin x\). b. Expressing the Taylor series of \(1 - \sin x\) about \(x = 0\). c. Let \( p(x) = a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n \) and \(\alpha_i\) be the zeros of \(p\) repea... | [
"\\dfrac{\\pi^2}{8}"
] | [
"Using Euler's method, find the value of the series\n\\[ S = \\sum_{n=0}^{\\infty} \\frac{1}{(2n+1)^2} \\]\nby:\na. Determining the zeros of \\(1 - \\sin x\\).\nb. Expressing the Taylor series of \\(1 - \\sin x\\) about \\(x = 0\\).\nc. Let \\( p(x) = a_0 + a_1 x + a_2 x^2 + \\cdots + a_n x^n \\) and \\(\\alpha_i\\... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-e70a585020673b68db63ba0e0c6fef337f4d1b9f479c033aecc4b319f21800d3 | Using Gaussian elimination, determine the value of \( a \) so that the system of linear equations \[ ax + y + z = 1 \] \[ x + ay + z = 1 \] \[ x + y + az = 1 \] has: (i) no solution (ii) a unique solution (iii) infinitely many solutions | [
"(i) \\( a = -2 \\)\n(ii) \\( a \\neq -2, a \\neq 1, a \\neq -1 \\)\n(iii) \\( a = 1 \\)"
] | [
"Using Gaussian elimination, determine the value of \\( a \\) so that the system of linear equations \n\\[ ax + y + z = 1 \\]\n\\[ x + ay + z = 1 \\]\n\\[ x + y + az = 1 \\]\nhas:\n(i) no solution\n(ii) a unique solution\n(iii) infinitely many solutions"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-9cbb4368631a9515c62b0bec316e8cae584f31c03a46ecd3731147a2ae3f3c2e | Using Green's Theorem, calculate the line integral of the vector field \( \mathbf{F}(x,y) = (4xy, 2x^2 - 3xy) \) over the curve \( C \), where \( C \) is formed by the circumference \( x^2 + y^2 = 1 \) from the point \( (1,0) \) to the point \( (-1,0) \), by the line segment connecting the point \( (-1,0) \) to the poi... | [
"-2"
] | [
"Using Green's Theorem, calculate the line integral of the vector field \\( \\mathbf{F}(x,y) = (4xy, 2x^2 - 3xy) \\) over the curve \\( C \\), where \\( C \\) is formed by the circumference \\( x^2 + y^2 = 1 \\) from the point \\( (1,0) \\) to the point \\( (-1,0) \\), by the line segment connecting the point \\( (... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 4
} | 15 | 0 | 0 |
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