prompt list | ground_truth string |
|---|---|
[
{
"content": "Evaluate the integral \\( I = \\int_{C(i,2)} \\frac{1}{(z^2+4)^2} \\, dz \\), where \\( C(i,2) \\) is the circle centered at \\( i \\) with radius 2.",
"role": "user"
}
] | \dfrac{\pi}{16} |
[
{
"content": "Find the coefficient of \\(x^{17}\\) in the expansion of \\((3x^7 + 2x^5 - 1)^{20}\\).",
"role": "user"
}
] | -41040 |
[
{
"content": "Let $f:\\Bbb R\\to \\Bbb R$ be a bounded Lebesgue measurable function such that for all $a,b\\in \\Bbb R$ with $-\\infty<a<b<\\infty$, we have $\\int _a^b f = 0$. Prove that $f = 0$ almost everywhere.",
"role": "user"
}
] | 0 |
[
{
"content": "Consider \\(\\mathbb{R}^{2n}\\) with coordinates \\(x^{1},\\cdots,x^{2n}\\) and the differential form \\(\\omega^{2}=dx^{1}\\wedge dx^{n+1}+dx^{2}\\wedge dx^{n+2}+\\cdots+dx^{n}\\wedge dx^{2n}\\). Find the number of terms in the expansion of \\((\\omega^{2})^{n}\\) that correspond to \\(dx^{1}\\we... | n! |
[
{
"content": "Determine if the function \\( F(x,y) = \\sin(x)\\arctan(y) \\) is a Lipschitz function with respect to the infinity norm \\( \\|\\cdot\\|_\\infty \\). If it is, find an appropriate Lipschitz constant \\( L \\).",
"role": "user"
}
] | \dfrac{\pi}{2} |
[
{
"content": "Determine the number of roots of the equation \\(4\\cos(e^x) = 2^x + 2^{-x}\\).",
"role": "user"
}
] | 4 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $1,612 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 2.36 g |
[
{
"content": "Given that $X = h(Z, H)$, where $h$ is a measurable function, and $Z$ is a random variable independent of $H$, and knowing that $\\mathbb{E}[X | H = s] = f(s)$, find the value of $\\mathbb{E}[h(Z, s)]$. Express your answer in terms of $f(s)$. ",
"role": "user"
}
] | f(s) |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 2 mL |
[
{
"content": "Let $X$ be a smooth projective variety with an exact sequence:\n$$0\\mapsto Aut^{o}(X)\\rightarrow Aut(X)\\rightarrow H\\mapsto 0$$\nwhere $Aut^{o}(X)$ is the connected component of the identity and $H$ is the group of connected components of $Aut(X)$. Assume there is a GIT quotient $Y:=X//Aut^{o}... | Yes |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 350 m |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 3413 g HCl |
[
{
"content": "Find the real number \\( a \\) such that the conditional expectation \\( E(X | Y = aX) = \\frac{2}{9} \\), given the joint probability density function for random variables \\( X \\) and \\( Y \\) is:\n\\[\nf(x, y) = \\begin{cases} 2(x+y) & \\text{if } 0 < x < y < 1 \\\\ 0 & \\text{otherwise} \\en... | 3 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 735.75 N |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | [N2]=[O2]=0.0990 M
[NO]=2.30e-3 M |
[
{
"content": "Evaluate the limit: $$\\lim_{n\\rightarrow\\infty}\\sqrt[n]{3^n\\sum_{k=0}^{n}\\left( \\frac{4}{3}\\right)^k}$$",
"role": "user"
}
] | 4 |
[
{
"content": "Does there exist a continuous function $f$ defined on the compact interval $[0,1]$ that is not of bounded variation?",
"role": "user"
}
] | Yes |
[
{
"content": "Let $f: \\mathbb{C}^5 \\rightarrow \\mathbb{C}^7$ be a linear function such that $f(2i e_1 + e_3) = f(e_2)$ and $\\mathbb{C}^7 = X \\oplus \\text{Im}(f)$. Determine the dimension of $X$. \\( \\text{Im}(f) \\) denotes the image of $f$. \\( e_1, e_2, e_3 \\) are standard basis vectors in \\( \\mathb... | 3 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 1.62 T |
[
{
"content": "Calculate the limit of the following integral as \\( n \\to \\infty \\):\n\n\\[ \\lim_{n \\to \\infty } \\frac{1}{n^2} \\int_0^n \\frac{ \\sqrt{n^2-x^2} }{2+x^{-x}} \\, dx \\]",
"role": "user"
}
] | \dfrac{\pi}{8} |
[
{
"content": "Evaluate the definite integral: $$ \\int_{\\pi / 2}^{\\pi} 2^{8} \\cdot \\sin ^{8} x \\, dx $$",
"role": "user"
}
] | 35\pi |
[
{
"content": "If $7 - 4x = 15$, what is the value of $8x + 2$?",
"role": "user"
}
] | -14 |
[
{
"content": "Calculate \\( \\lim_{n\\to\\infty}\\int_{0}^{n}\\left(1+\\frac{x}{n}\\right)^n e^{-2x} \\, dx \\).",
"role": "user"
}
] | 1 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 8 A |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 1910.9456 acre.ft |
[
{
"content": "Is the set $C_f = \\{(x,y) \\in \\mathbb{R}^2 \\mid x > 0, y > 0, x^2 + y^2 = 1, (x,y) = f(a,b) \\text{ for some } (a,b) \\in P\\}$ dense in the segment of the unit circle $C = \\{(x,y) \\in \\mathbb{R}^2 \\mid x > 0, y > 0, x^2 + y^2 = 1\\}$?",
"role": "user"
}
] | Yes |
[
{
"content": "Let $(B_t)_t$ be a standard Brownian motion with $B_0=0$. Define $T_{a,b} = \\inf\\{t>0 : B_t \\not\\in(a,b)\\}$ where $a<0<b$. Find the expected first-passage time $\\mathbf{E}[T_{a,b}]$. ",
"role": "user"
}
] | -ab |
[
{
"content": "Compute the limit: \\[ L= \\lim\\limits_{ x\\to + \\infty} \\left[ (x+4)^{1+\\frac{1}{x}} - x^{1+\\frac{1}{x+\\cos x}} \\right] \\]",
"role": "user"
}
] | 4 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 61.2 m^3 |
[
{
"content": "Consider a graded set $A = (A_0, A_1, \\dots)$ where each $A_k$ is finite and consists of elements of size $k$. Is it possible to define an associative operation $*$ that is graded, meaning \\( *: A_k \\times A_\\ell \\to A_{k+\\ell} \\), such that there exists an element of size $0$ which is not ... | Yes |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 0.087 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 0.06 T |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 0.49 A |
[
{
"content": "Evaluate the integral \\( \\int_{|z|=3} \\left(\\frac{z-2}{z+2}\\right)^m dz \\) using Cauchy's Integral Formula.",
"role": "user"
}
] | -8\pi i m |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | -$7,866 |
[
{
"content": "Calculate the volume of the solid enclosed by the surface \\((x^2+y^2+z^2)^2 = x\\) using spherical coordinates \\((r, \\theta, \\phi)\\).",
"role": "user"
}
] | \dfrac{\pi}{3} |
[
{
"content": "Evaluate the integral \\( \\int_0^\\pi \\frac{d\\theta}{5+3\\cos\\theta} \\) using the method of residues.",
"role": "user"
}
] | \dfrac{\pi}{4} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $\bigtriangledown ^2f=\frac{1}{v^2}\frac{\partial^2 f}{\partial t^2}$ |
[
{
"content": "Given the matrix \\( A = \\begin{bmatrix} 101 & 2 & 3 & 4 & 5 \\\\ 1 & 102 & 3 & 4 & 5 \\\\ 1 & 2 & 103 & 4 & 5 \\\\ 1 & 2 & 3 & 104 & 5 \\\\ 1 & 2 & 3 & 4 & 105 \\end{bmatrix} \\), find all the eigenvalues of \\( A \\). It is known that one of the eigenvalues is 115. Determine the remaining eigen... | 100 |
[
{
"content": "Consider the polynomial \\( f(x) = x^7 - 4x^3 + x + 1 \\). Determine the number of roots of this polynomial that have a modulus less than 1.",
"role": "user"
}
] | 3 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 1.2e+04 J |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 9.8e5 N/C vertically downwards |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 674.762 degrees Celsius |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 0.30 N |
[
{
"content": "Find all functions \\( f: \\mathbb{Z}^+ \\to \\mathbb{Z}^+ \\) such that for all positive integers \\( x \\) and \\( y \\), the following equation holds:\n\\[ 2 y f\\left(f\\left(x^{2}\\right)+x\\right) = f(x+1) f(2xy). \\]",
"role": "user"
}
] | f(x) = x |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 0.044 |
[
{
"content": "Determine the values of \\( k \\) for which the matrix \\( \\begin{bmatrix}0&1&0\\\\0&0&1\\\\0&-k-7&k+8\\end{bmatrix} \\) is not diagonalizable over the complex numbers \\( \\Bbb C \\).",
"role": "user"
}
] | -7, -6 |
[
{
"content": "Determine whether the following statement is true or false: Every closed subspace of a Banach space is also a Banach space.",
"role": "user"
}
] | A |
[
{
"content": "Calculate the integral \\( I = \\frac{1}{2\\pi i } \\oint_C \\frac{f'(z)}{f(z)} dz \\) where \\( f(z) = \\frac{1+z^2}{1-\\cos{2\\pi z}} \\) and \\( C \\) is the circle around \\(|z| = \\pi\\). Use the argument theorem to determine the value of \\( I \\).",
"role": "user"
}
] | -12 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 14 years old |
[
{
"content": "Does there exist a real sequence \\((a_n)_{n \\in \\mathbb{N}} \\subset [0,1)\\) such that for each \\(n \\in \\mathbb{N}\\), there is a bijection \\(f: [n] \\to [n]\\) satisfying \\(a_{f(k)} \\in \\left[ \\frac{k-1}{n}, \\frac{k}{n} \\right)\\) for all \\(k \\in [n]\\)? In other words, can we con... | Yes |
[
{
"content": "Evaluate the integral \\( \\int_{\\frac{1}{3\\sqrt3}}^{3\\sqrt3}\\cfrac{1}{x^{4/3}+x^{2/3}}dx \\).",
"role": "user"
}
] | \dfrac{\pi}{2} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 0.0667 M |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $3,091.37 |
[
{
"content": "Determine the number of 3-Sylow subgroups in a group $G$ of order 126. Let $|G| = 126 = 2 \\cdot 3^2 \\cdot 7$. Use the Sylow theorems to find the possible values for the number of 3-Sylow subgroups, $s_3$. Additionally, consider a surjective group homomorphism $\\varphi: G \\rightarrow H$ where $... | 1 |
[
{
"content": "Evaluate the summation: \\[ \\sum_{n=0}^{\\infty} 10,000(n-1)\\frac{(3/2)^n e^{-\\frac{3}{2}}}{n!} \\]",
"role": "user"
}
] | 5000 |
[
{
"content": "Evaluate the limit: $$\\lim_{n\\to\\infty} \\frac{\\int_0^{\\pi/4}(\\tan x)^{2n}dx}{\\int_0^{\\pi/4}(\\tan x)^{2n+1}dx}$$",
"role": "user"
}
] | 1 |
[
{
"content": "Evaluate the definite integral \\( I = \\int_{0}^{\\infty} (x^2 - 3x + 1) e^{-x} \\ln^3(x) \\, dx \\) using the series expansion \\( e^{-x} = \\sum_{n=0}^{\\infty} \\frac{(-x)^n}{n!} \\).",
"role": "user"
}
] | -6\gamma |
[
{
"content": "Evaluate the limit: \\[ \\lim_{t \\to 0^+} \\frac{e^{-1/t}}{t^2} \\]",
"role": "user"
}
] | 0 |
[
{
"content": "A fair six-sided die is rolled repeatedly until a 6 appears. The sum of all numbers rolled, including the 6, is denoted as $S$. Calculate the standard deviation of $S$. Express your answer as a decimal rounded to two decimal places.",
"role": "user"
}
] | 16.73 |
[
{
"content": "Find the distance between the foci of an ellipse if the major axis is 17 units long, and the area of the triangle formed by any point of intersection 'P' of the ellipse and a circle (which shares the same center with the ellipse and passes through the foci $F_1$ and $F_2$) is 30 square units.",
... | 13 |
[
{
"content": "In the town of Discretion, duels are rarely fatal. Each dueler arrives at a random time between 5 a.m. and 6 a.m. and leaves exactly 5 minutes later, unless their opponent arrives during this interval, in which case they fight. What fraction of duels lead to violence?",
"role": "user"
}
] | \dfrac{23}{144} |
[
{
"content": "Determine if (n, m) = (18, 7) is the only positive integer solution to the equation \\( n^2 + n + 1 = m^3 \\).",
"role": "user"
}
] | (18, 7) |
[
{
"content": "Calculate the volume of the tetrahedron with vertices at the points \\( A_{1}(1, -1, 1) \\), \\( A_{2}(-2, 0, 3) \\), \\( A_{3}(2, 1, -1) \\), and \\( A_{4}(2, -2, -4) \\).",
"role": "user"
}
] | \dfrac{11}{2} |
[
{
"content": "Determine whether the sequence \\( v_n = \\int_0^\\pi x \\cdot \\cos(nx) \\,dx \\), where \\( n \\in \\mathbb{N} \\), converges.",
"role": "user"
}
] | 0 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 6.51e-10 m |
[
{
"content": "Compute the Legendre symbol \\( \\left(\\frac{3^{24671}}{105953}\\right) \\).",
"role": "user"
}
] | -1 |
[
{
"content": "Let $ f(x) = x^3 + x + 1$. Suppose $ g$ is a cubic polynomial such that $ g(0) = - 1$, and the roots of $ g$ are the squares of the roots of $ f$. Find $ g(9)$.",
"role": "user"
}
] | 899 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | W = (y - x^2) + (z - y^2)\frac{1}{3} + (x - z^2)\frac{1}{4} |
[
{
"content": "Given real numbers $a, b, c, d$ such that $a + b + c + d = 0$ and $abcd = 1$, find the minimum value of the expression $E = a^4 + b^4 + c^4 + d^4 + a^2 + b^2 + c^2 + d^2$. Use Vieta's formulas to express $E$ in terms of $s$, where $a, b, c, d$ are roots of the quartic equation $x^4 + sx^2 + tx + 1... | 8 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | End of Year 3 |
[
{
"content": "Given two independent lighthouses A and B on a one-dimensional sea, each emitting a noisy light signal, the received signal strength depends on the distance from the lighthouse. You have a symmetric probability density function (PDF) for the probable distance from each lighthouse based on the rece... | Yes |
[
{
"content": "Thirteen people wish to have dinner together at a round table for several nights, ensuring that each person has different neighbors every night. What is the maximum number of nights they can achieve this arrangement?",
"role": "user"
}
] | 6 |
[
{
"content": "Evaluate the limit: $$\\lim_{n \\to \\infty}n\\sin(2\\pi\\sqrt{1+n^2}),n\\in \\mathbb{N}$$",
"role": "user"
}
] | \pi |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 1.90 A |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 0.950 J |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 157.1Nm |
[
{
"content": "Evaluate the limit:\n\\[ \\lim_{n \\to \\infty} \\frac{\\sqrt{a(4^0n)}+\\sqrt{a(4^1n)}+\\cdots+\\sqrt{a(4^{10}n)}}{\\sqrt{a(2^0n)}+\\sqrt{a(2^1n)}+\\cdots+\\sqrt{a(2^{10}n)}} \\]\nwhere \\( a(k) = (2k+3)(k-1) \\) for every \\( k \\in \\mathbb{N} \\).",
"role": "user"
}
] | 683 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 0.027 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | -$14,700 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $375,690 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 5.45e-4 N |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | -7.2 N |
[
{
"content": "What is the 125th digit beyond the decimal point in the decimal representation of $\\frac47$?",
"role": "user"
}
] | 2 |
[
{
"content": "Determine whether the following statement is true or false: For any finite group $G$, if the exponent of $G$ is even, then $G$ must be abelian.",
"role": "user"
}
] | False |
[
{
"content": "Given that \\(3^a = 5^b = m\\) and \\(\\frac{1}{a} + \\frac{1}{b} = 2\\), find the value of \\(m\\).",
"role": "user"
}
] | \sqrt{15} |
[
{
"content": "Calculate the work done by the force field \\( \\mathbf{F} = \\frac{2x}{y}\\mathbf{i} - \\frac{x^2}{y^2}\\mathbf{j} \\) in moving a particle from the point \\((-1, 1)\\) to the point \\((3, 2)\\).",
"role": "user"
}
] | \dfrac{7}{2} |
[
{
"content": "Determine the number of semi-direct products $H \\rtimes Q$ that can be constructed for $H = C_{42}$ and $Q = C_{3}$, where $C_{n}$ denotes the cyclic group of order $n$. Specifically, find the number of homomorphisms $\\theta: C_{3} \\rightarrow \\text{Aut}(C_{42})$. Given that $\\text{Aut}(C_{42... | 3 |
[
{
"content": "Given a sample of size \\( n \\) from a Poisson distribution with the probability mass function (PMF): \\[ p(k; \\lambda) = \\frac{e^{-\\lambda}\\lambda^k}{k!}, \\quad k = 0, 1, 2, \\ldots \\] find the maximum likelihood estimator (MLE) of \\( \\lambda \\).",
"role": "user"
}
] | \overline{K} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 0.0684 cm |
[
{
"content": "A regular 20-sided polygon is inscribed in the unit circle in the complex plane, with vertices corresponding to the complex numbers \\( z_{1}, z_{2}, \\cdots, z_{20} \\). Determine the number of distinct points represented by the complex numbers \\( z_{1}^{1995}, z_{2}^{1995}, \\cdots, z_{20}^{199... | 4 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 983 MJ |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | -$1,094,402 |
[
{
"content": "Determine the largest integer $n$ such that $2^n$ divides $63^{2018} - 1$. Provide your answer as a single integer.",
"role": "user"
}
] | 7 |
[
{
"content": "For any odd number $n$, determine if it is possible to orient the edges of a complete graph with $n$ vertices such that each vertex has an equal number of edges oriented towards it and away from it.",
"role": "user"
}
] | Yes |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | -2.67 m/s^2 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 232 kJ |
[
{
"content": "Determine whether the limit \\( \\lim\\limits_{x\\rightarrow\\infty} f(x) \\) exists given that \\( \\lim\\limits_{x\\rightarrow \\infty} (f'(x)+f(x)) = L < \\infty \\).",
"role": "user"
}
] | L |
[
{
"content": "A ball was floating in a lake when the lake froze. The ball was removed (without breaking the ice), leaving a hole $24$ cm across as the top and $8$ cm deep. What was the radius of the ball (in centimeters)?\n$\\textbf{(A)}\\ 8 \\qquad \\textbf{(B)}\\ 12 \\qquad \\textbf{(C)}\\ 13 \\qquad \\textbf... | 13 |
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