prompt list | ground_truth string |
|---|---|
[
{
"content": "A committee is to be made up from a group of 10 candidates. The committee must consist of a chairman and some (not necessarily nonzero) number of members-at-large. How many ways can the committee be chosen? (For example, suppose that Alex is chair while Betty, Carl, and D'Angelo make up the commit... | 5120 |
[
{
"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... | 1.3e+2 m |
[
{
"content": "Find the limit of the function \\( g(x,y) = \\frac{e^{\\frac{1}{\\sqrt{x^2+y^2}}}}{e^{x^2+y^2}+1} \\) as \\((x,y)\\) approaches \\((0,0)\\).",
"role": "user"
}
] | \infty |
[
{
"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... | 266 rpm |
[
{
"content": "Find the expected squared distance to the origin of a $d$-dimensional spherical Gaussian centered at the origin with variance $\\sigma^2$. Provide your answer in terms of $d$ and $\\sigma^2$. ",
"role": "user"
}
] | d \sigma^2 |
[
{
"content": "Find a number in the ring \\(\\mathbb{Z}[\\sqrt{14}, \\frac{1}{2}]\\) that is not in \\(\\mathbb{Z}[\\sqrt{14}]\\). Provide an example of such a number.",
"role": "user"
}
] | \dfrac{1 + \sqrt{14}}{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... | 4e-3 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... | 3.4 moles of Carbon, 4.5 moles of Hydrogen, and 3.4 moles of Oxygen |
[
{
"content": "Calculate the value of \\(\\sum_{(x,y,z) \\in S} g(x,y,z)\\), where \\(S\\) is the set of integer triplets \\((x, y, z)\\) satisfying \\(|x| + |y| + |z| = 2021\\) and \\(g(x, y, z) = x + y + z\\).",
"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... | -0.00621A/m^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... | 86.7 m/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... | 2.95 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... | 441 x 10^6 W |
[
{
"content": "In triangle \\( ABC \\), the incircle with center \\( O \\) touches the sides \\( AB \\), \\( BC \\), and \\( AC \\) at points \\( M \\), \\( N \\), and \\( K \\) respectively. Given that angle \\( AOC \\) is four times the size of angle \\( MKN \\), determine the measure of angle \\( B \\).",
... | 108^\circ |
[
{
"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... | 371 min |
[
{
"content": "Compute the limit: $$\\lim_{n \\rightarrow \\infty} \\left[n \\; \\arccos \\left( \\left(\\frac{n^2-1}{n^2+1}\\right)^{\\cos \\frac{1}{n}} \\right)\\right]$$",
"role": "user"
}
] | 2 |
[
{
"content": "Determine the order of the element \\(4U_5(105)\\) in the quotient group \\(U(105)/U_5(105)\\), where \\(U(105)\\) is the group of units modulo 105 and \\(U_5(105)\\) is the subgroup of elements congruent to 1 modulo 5.",
"role": "user"
}
] | 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... | $T=\frac{2\pi}{\sqrt{2GM}}r_0^{3/2}$ |
[
{
"content": "Consider a set of 10 labeled vertices \\( \\{1, 2, \\ldots, 10\\} \\). How many labeled trees can be formed such that exactly two of these vertices have a degree of 5?",
"role": "user"
}
] | 3150 |
[
{
"content": "Determine whether the limit \\( \\lim_{(x,y) \\to (0,0)} \\arctan\\left(\\frac{x^2+y^2}{x^4+y^4}\\right) \\) exists.",
"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... | 411.7V/m |
[
{
"content": "Find the number of complex numbers $z$ satisfying $|z| = 1$ and\n\\[\\left| \\frac{z}{\\overline{z}} + \\frac{\\overline{z}}{z} \\right| = 1.\\]",
"role": "user"
}
] | 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... | 11.34 years |
[
{
"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... | 1.18 |
[
{
"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... | F_0a/mk^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... | $77,710 |
[
{
"content": "Let $X_1, X_2, X_3,\\ldots$ be independent random variables with $P(X_n=1)=P(X_n=3)=\\frac{1}{4}$ and $P(X_n=0)=\\frac{1}{2}$. Calculate the limit of the following expression as $n$ approaches infinity:\n\n$$\\frac{3\\sqrt{n}(X_1+X_2+\\cdots+X_n)-5n}{n+(X_1+X_2+\\cdots+X_n)^2}$$",
"role": "use... | 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... | 1113 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... | 0.5% |
[
{
"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... | F = \frac{XY}{X + Y} |
[
{
"content": "Evaluate the limit using polar coordinates: $$\\lim_{(x,y)\\rightarrow (0,0)} \\frac{x^4-2x^2y^2+y^4}{x^2+y^2}$$",
"role": "user"
}
] | 0 |
[
{
"content": "Determine whether the function \\( g(x) = \\int_0^1 f(t) \\, t^x \\, dt \\), where \\( f : [0,1] \\to [0,1] \\) is a smooth (class \\( C^\\infty \\)) but not necessarily real-analytic function, is necessarily real-analytic on the interval \\((-1, \\infty)\\).",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x\\to+\\infty} \\frac{\\sqrt{x}(\\sin{x}+\\sqrt{x}\\cos{x})}{x\\sqrt{x}-\\sin(x\\sqrt{x})} \\]",
"role": "user"
}
] | 0 |
[
{
"content": "Determine the correct differential equation to solve the following problem: A line of initial length 1 is reduced in a process where, in each step, two points $x_1$ and $x_2$ are selected on the remaining line, and the segment from the left end to the maximum of $x_1$ and $x_2$ is removed. What is... | 1 |
[
{
"content": "Calculate the expectation value of $X^{2n+1}$ for a standard normal distribution using the Gaussian integral technique.",
"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... | 0.083 mg |
[
{
"content": "Find the limit of the function \\( f(h) = \\frac{\\ln^2(1+h)\\sin\\left( \\frac{1}{h}\\right)}{h} \\) as \\( h \\) approaches 0.",
"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... | [Kr]5s^24d^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... | -4.10e-12 C |
[
{
"content": "Find all positive integer triples \\((x, y, z)\\) that satisfy the equation: \n\n\\[ 1 + 2^x + 3^y = z^3. \\]",
"role": "user"
}
] | (2, 1, 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... | -0.70% |
[
{
"content": "Let $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ be a continuous function. For every $n \\in \\mathbb{N}_0$, define the function $$ f_n : \\mathbb{R} \\rightarrow \\mathbb{R}: x \\mapsto f(x + 1/n). $$ Determine whether the sequence $(f_n)_n$ converges pointwise to $f$ on $\\mathbb{R}$. Answer \"true... | true |
[
{
"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... | -$9,731 |
[
{
"content": "Let \\( \\{X_n, n \\geq 1\\} \\) be a sequence of random variables with finite means satisfying the condition:\n\\[ E(X_{n+1} | X_0, X_1, \\ldots, X_n) = aX_n + bX_{n-1}, \\quad n \\geq 1 \\]\nwhere \\( a > 0, b < 1 \\), and \\( a + b = 1 \\). Determine the value of \\( \\alpha \\) such that \\( S... | \dfrac{1}{b} |
[
{
"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... | 10.95% |
[
{
"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... | 2.09e9 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... | -22.544 m |
[
{
"content": "Given the equations \\(\\log _{4} x-\\left(\\frac{1}{4}\\right)^{x}=0\\) and \\(\\log _{\\frac{1}{4}} x-\\left(\\frac{1}{4}\\right)^{x}=0\\) with roots \\(x_{1}\\) and \\(x_{2}\\) respectively, determine the relationship between \\(x_{1} x_{2}\\).\n\n(A) \\(0 < x_{1} x_{2} < 1\\)\n\n(B) \\(x_{1} x... | A |
[
{
"content": "Calculate the determinant of the matrix $nI_{n-1} - J_{n-1}$ using eigenvalues, where $I_{n-1}$ is the identity matrix of size $(n-1)$ and $J_{n-1}$ is the matrix of all ones of size $(n-1)$. Provide your answer as a single numerical value.",
"role": "user"
}
] | n^{n-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... | 2519.46 g |
[
{
"content": "A piece of string fits exactly once around the perimeter of a square whose area is 144. Rounded to the nearest whole number, what is the area of the largest circle that can be formed from the piece of string?",
"role": "user"
}
] | 183 |
[
{
"content": "Evaluate the integral \\( \\int _{0}^{2\\pi} e^{-it} e^{e^{it}} \\, dt \\).",
"role": "user"
}
] | 2\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... | 16837.63 |
[
{
"content": "Determine whether there exist continuous functions \\( P(x) \\) and \\( Q(x) \\) such that the differential equation \\( y'' + P(x)y' + Q(x)y = 0 \\) has a solution \\( y = \\sin(2x) \\).",
"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... | 25 s |
[
{
"content": "Evaluate the expression \\( \\frac{\\int_0^{\\pi}x^3\\log(\\sin x)\\,dx}{\\int_0^{\\pi} x^2 \\log(\\sqrt{2} \\sin x)\\,dx} \\).",
"role": "user"
}
] | \dfrac{3\pi}{2} |
[
{
"content": "Determine whether the following statement is true or false: If $k \\rightarrow \\infty$ and $\\sqrt{\\frac{1}{x_k}} = o(y_k)$, then $\\frac{1}{x_k} = o(y_k)$, given that $x_k \\rightarrow \\infty$ and $y_k \\rightarrow 0$. Provide a justification for your answer.",
"role": "user"
}
] | True |
[
{
"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... | 1.69 g/L |
[
{
"content": "Consider a net \\((x_\\alpha^*)\\) in \\((X\\times X)^*\\), where \\(x_\\alpha^* = (x_{\\alpha 1}^*, x_{\\alpha 2}^*)\\). Suppose that \\(\\sup_\\alpha |x_\\alpha^*(x)|<\\infty\\) for all \\(x = (x_1,x_2)\\in X\\times X\\). Does it follow that \\(\\sup_\\alpha (\\|x_{\\alpha 1}^*\\|_{op} + \\|x_{\... | Yes |
[
{
"content": "Consider the function \\( f = \\sum_{k=1}^\\infty a_k \\chi_{A_k} \\), where \\( a_k \\in \\mathbb{R} \\) and \\( \\chi_{A_k} \\) is the indicator function on the set \\( A_k \\). Can this function \\( f: \\mathbb{R} \\to \\mathbb{R} \\) produce uncountably many distinct values?",
"role": "use... | Yes |
[
{
"content": "In a circle with a diameter of 8 meters, points A and B are located on the diameter, each 1 meter from the center. Through point B, draw any chord PC. Calculate the sum of the squares of the medians of triangle APC.",
"role": "user"
}
] | \dfrac{147}{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... | 0.029 m |
[
{
"content": "Evaluate the integral \\( \\int_0^1 r(x) \\, dx \\), where \\( r(x) = \\begin{cases} 1 & \\text{if } x \\text{ is rational}, \\\\ 0 & \\text{if } x \\text{ is irrational}. \\end{cases} \\)",
"role": "user"
}
] | 0 |
[
{
"content": "Evaluate the line integral \\( \\oint_{\\mathcal C}\\begin{pmatrix}x_2^2 \\cos x_1 \\\\ 2x_2(1+\\sin x_1)\\end{pmatrix} \\cdot d\\vec{x} \\) where \\( \\mathcal C \\) is the unit circle in \\( \\mathbb{R}^2 \\) traced counterclockwise starting and ending at the point (1,0). Use the parameterizatio... | 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... | 724.88N |
[
{
"content": "Use Green's theorem to find the area enclosed by the ellipse \\(\\frac{x^2}{4} + \\frac{y^2}{9} = 1\\). Determine appropriate functions \\(P\\) and \\(Q\\) such that \\(\\frac{\\partial Q}{\\partial x} - \\frac{\\partial P}{\\partial y} = 1\\).",
"role": "user"
}
] | 6\pi |
[
{
"content": "Let $A$ be an $n \\times n$ matrix with real entries. Determine whether the statement \"If $A$ is similar to a symmetric matrix, then $A$ is symmetric\" is true or false.",
"role": "user"
}
] | B |
[
{
"content": "Palindromic primes are two-digit prime numbers such that the number formed when the digits are reversed is also prime. What is the sum of all palindromic primes less than 50?",
"role": "user"
}
] | 109 |
[
{
"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... | $3,240 |
[
{
"content": "Let $X\\subset\\mathbb{P}^N$ be a smooth projective variety. Fix a general point $q \\in X$, and let $C\\subseteq X$ be a smooth curve passing through $q$. Consider a flat family of linear spaces $\\Lambda_{\\xi, q}$, with $\\xi \\in C$, parametrized by $C$. Assume that for a general $\\xi\\in C$,... | 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... | The Griess test |
[
{
"content": "Given that \\( a = 3^p \\), \\( b = 3^q \\), \\( c = 3^r \\), and \\( d = 3^s \\) where \\( p \\), \\( q \\), \\( r \\), and \\( s \\) are positive integers, find the smallest value of \\( p + q + r + s \\) such that \\( a^2 + b^3 + c^5 = d^7 \\).",
"role": "user"
}
] | 106 |
[
{
"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... | 85.891% |
[
{
"content": "Evaluate the limit: $$\\lim _{n\\to \\infty }\\left( e^{\\frac{1}{\\sqrt{n+2}} -\\frac{1}{\\sqrt{n}}} -1\\right) \\times \\sqrt{n+1}$$",
"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... | -1,720 |
[
{
"content": "Determine the radius of convergence of the series \\( \\sum_{n=0}^{\\infty} \\frac{(n!)^3}{(3n)!}z^{3n} \\).",
"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... | 146.5 m |
[
{
"content": "Find the natural number \\( n \\) such that the equation \\( F_{n-1} \\cdot x^2 - F_n \\cdot y^2 = (-1)^n \\) has a solution in positive integers \\( x \\) and \\( y \\), where \\( \\{F_n\\} \\) is the Fibonacci sequence defined by \\( F_1 = F_2 = 1 \\) and \\( F_{n+1} = F_n + F_{n-1} \\) for \\( ... | 3 |
[
{
"content": "Calculate the value of the sum \\( f(1/2009) + f(2/2009) + \\ldots + f(2008/2009) \\) where \\( f(x) = \\frac{e^{2x-1}}{1+e^{2x-1}} \\).",
"role": "user"
}
] | 1004 |
[
{
"content": "Determine the interval of convergence of the power series \\( \\sum_{n=0}^\\infty (\\frac{\\pi}{2} - \\arctan(n))(x)^{n} \\).",
"role": "user"
}
] | [-1, 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... | 1751.64 L |
[
{
"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... | 0.243 g |
[
{
"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... | 0.017 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... | 1.4 s |
[
{
"content": "The image of the point with coordinates $(1,1)$ under the reflection across the line $y=mx+b$ is the point with coordinates $(9,5)$. Find $m+b$.",
"role": "user"
}
] | 11 |
[
{
"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... | 9049.47 |
[
{
"content": "Let $M$ be an $n \\times n$ matrix with real entries such that $M^3 = I$. Suppose $Mv \\neq v$ for any non-zero vector $v$. Determine which of the following statements is true:\n\nA. $M$ has real eigenvalues.\nB. $M + M^{-1}$ has real eigenvalues.",
"role": "user"
}
] | B |
[
{
"content": "Evaluate the limit \\( \\displaystyle \\lim_{x \\to 0} \\frac{ \\log ((k+x)^{b} - (k-x)^{b} )}{\\log x} \\), where \\(k \\in (0,1)\\) and \\(b \\in (0,1)\\) are constants.",
"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... | 37 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... | 222.5 GPa |
[
{
"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... | 12.24 degrees |
[
{
"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... | 0.841 R_0 |
[
{
"content": "Calculate the derivative of \\( g(x) \\) at \\( x=0 \\), where \\( g(x) = f(f(f(x))) \\) and \\( f(x) = x(x+1)(x+2)\\cdots(x+100) \\).",
"role": "user"
}
] | (100!)^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... | 151 pJ |
[
{
"content": "Given a convex figure in the Cartesian plane that is symmetric with respect to both axes, a rectangle $A$ is constructed inside it with maximum area. The rectangle is then enlarged with its center at the center of the rectangle and a ratio of lambda such that it covers the convex figure. Determine... | 2 |
[
{
"content": "Calculate \\( \\lim_{n \\to \\infty} \\int_0^1 nx^n (f(x)-f(0)) \\, dx \\), where \\( f(x) \\) is continuous on \\([0,1]\\).",
"role": "user"
}
] | f(1) - f(0) |
[
{
"content": "Evaluate the limit: $$\\lim_{x \\rightarrow 0} e^{-\\frac{1}{x^2}}\\cdot \\frac{2}{x^3}$$",
"role": "user"
}
] | 0 |
[
{
"content": "Estimate the expected value of \\(|v_1 + v_2 + \\cdots + v_n|^2\\), where \\(v_1, v_2, \\ldots, v_n\\) are randomly selected unit vectors in \\(\\mathbb{R}^n\\).",
"role": "user"
}
] | n |
[
{
"content": "Can the columns of a complex $(m \\times n)$-matrix $A$ with linearly independent columns form an orthogonal basis for $\\operatorname{Im}(A)$?",
"role": "user"
}
] | Yes |
[
{
"content": "Determine the limit superior of the sequence \\(\\left\\{\\frac{1}{n}\\sum\\limits_{k=1}^{n}(-1)^{k-1}(k+1)\\right\\}\\).",
"role": "user"
}
] | \dfrac{1}{2} |
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