prompt list | ground_truth string |
|---|---|
[
{
"content": "Let $n \\geq 3$ be an integer, and let $a_1, a_2, \\ldots, a_n$ be $n$ different real numbers whose sum is positive. A permutation $b_1, b_2, \\ldots, b_n$ of these numbers is called a \"good permutation\" if for each $k = 1, 2, \\ldots, n$, the sum $\\sum_{i=1}^{k} b_i > 0$. Determine the minimum... | (n-1)! |
[
{
"content": "Solve for \\( x \\) in degrees in the equation: \\[ \\sin 84^\\circ \\sin(54^\\circ-x) = \\sin 126^\\circ \\sin x. \\]",
"role": "user"
}
] | 30 |
[
{
"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... | 15 nT |
[
{
"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... | 20144.28 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... | Year: 2015
Debit Construction in Progress $1,200,000
Credit Accounts Payable $1,200,000
Description: To record costs incurred during the year
Year: 2015
Debit Accounts Receivable $1,000,000
Credit Billings $1,000,000
Description: To record the amount billed to client for long-term construction contract
Year: 2015
D... |
[
{
"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... | 0.247 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... | H2SO3, HNO2, H2CO3, HCN |
[
{
"content": "Evaluate the limit: $$ \\lim_{(x,y)\\to(1,1)} \\frac{(x-1)^{4/3}-(y-1)^{4/3}}{(x-1)^{2/3}+(y-1)^{2/3}} $$",
"role": "user"
}
] | 0 |
[
{
"content": "Let $a$ and $b$ be the real roots of\n\\[x^4 - 4x - 1 = 0.\\]Find $ab + a + b.$",
"role": "user"
}
] | 1 |
[
{
"content": "Calculate the limit \\( \\lim_{x\\to\\infty} \\frac{1}{x}\\int_0^xg(t)dt \\) where the function \\( g \\) alternates between \\( 2 \\) and \\( -2 \\) on intervals of the form \\([0^2, 1^2), [1^2, 2^2), \\ldots, [(n-1)^2, n^2)\\).",
"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... | 5.63e-4 V |
[
{
"content": "Find the minimum value of the function \\( g(x, y, z) = \\frac{1}{x} + \\frac{1}{y} + \\frac{1}{z} + 3x + 3y + 3z \\) for \\( x, y, z > 0 \\). Use the theorem that states: If \\( f:A\\rightarrow\\mathbb{R} \\) is convex and \\( \\xi \\) has a directional derivative in every direction with \\( \\na... | 6\sqrt{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... | 9 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... | 0.667 kW |
[
{
"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... | 56.174 N |
[
{
"content": "Suppose $H_t$ is a bounded and predictable process, and $(H \\cdot X)$ is a martingale. Is $X$ necessarily a martingale?",
"role": "user"
}
] | No |
[
{
"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... | 924.79 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... | 6.3 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... | Heterotrophic |
[
{
"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... | rate = -delta[S2O8^-2]/delta_t = 1/2 delta[S2O3^-2]/delta_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... | 11.64 sec |
[
{
"content": "Evaluate the integral \\( \\int_{0}^{1} x^2e^{\\lfloor x^2\\rfloor +\\lfloor x^4\\rfloor }\\,dx \\) using the concept of the greatest integer function.",
"role": "user"
}
] | \dfrac{1}{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... | 900.0m |
[
{
"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... | 42.234 g/mol |
[
{
"content": "Given the function \\( f(x) = \\frac{1+x}{2-x} \\), and for \\( n \\in \\mathbf{N} \\), define \\( f_{1}(x)=f(x) \\) and \\( f_{n+1}(x)=f\\left[f_{n}(x)\\right] \\). If \\( f_{13}(x)=f_{31}(x) \\), determine the expression for \\( f_{16}(x) \\). Choose from the following options:\n\nA. \\( \\frac{... | C |
[
{
"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... | 48.24 |
[
{
"content": "Evaluate the limit: $$\\lim_{(x, y) \\to (1, 2)} \\frac{(x-1)(y-2)-xy+3}{x^2-2x+y^2-4}$$",
"role": "user"
}
] | -1 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x\\to 0}\\frac{\\sin^2x-x\\tan x}{x^4} \\]",
"role": "user"
}
] | -\dfrac{2}{3} |
[
{
"content": "Consider an analytic function $f(z)$ defined for $\\text{Im}(z) \\leq 0$. Its conjugate $g(z) = \\overline{f(\\bar{z})}$ is analytic for $\\text{Im}(z) \\geq 0$. Define $h(z)$ as $f(z)$ for $\\text{Im}(z) \\leq 0$ and $g(z)$ for $\\text{Im}(z) > 0$. Is the contour integral of $h(z)$ around any clo... | No |
[
{
"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... | 6.22 mH |
[
{
"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... | 8.49e-3 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... | 17.33% |
[
{
"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... | May 1: Merchandise inventory: Debit $8,100, Accounts payable-Hauke: Credit $8,100.
May 2: Accounts receivable: Debit $5,200, Sales: Credit $5,200; Cost of goods sold: Debit $3,900, Merchandise inventory: Credit $3,900.
May 5: Accounts payable-Hauke: Debit $600, Merchandise inventory: Credit $600.
May 9: Cash: Debit $5,... |
[
{
"content": "Determine whether the condition $A\\otimes\\mathbb{Q}=0$ implies that an abelian group $A$ is finite.",
"role": "user"
}
] | No |
[
{
"content": "Determine whether the existence of $f''(c) > 0$ at a point $c$ where $f'(c) = 0$ implies a local minimum at $c$, even if $f''$ is not continuous near $c$. Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate the integral \\( \\int_{-\\pi}^{\\pi}(\\sin x)^4dx \\) using Parseval's identity.",
"role": "user"
}
] | \dfrac{3\pi}{4} |
[
{
"content": "Let $k$ be a field and consider the formal power series ring $R = k[[t]]$. Suppose $A$ is a ring and there exist two ring homomorphisms $\\phi, \\psi: R \\to Z(A)$, where $Z(A)$ denotes the center of $A$, endowing $A$ with two distinct structures of an $R$-algebra. These structures induce $R$-modu... | No |
[
{
"content": "Evaluate the integral \\( \\int _C ze^{\\frac{3}{z}}dz \\), where \\( C \\) is the contour \\( |z|=2 \\) oriented in the anti-clockwise direction.",
"role": "user"
}
] | 9\pi i |
[
{
"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... | 37.059 m/s |
[
{
"content": "Evaluate the triple integral \\( \\iiint\\limits_D \\sin(xy) - \\sin(xz) + \\sin(yz) \\, dx \\, dy \\, dz \\), where \\( D = \\{ (x,y,z): 1 \\leq x^2 + y^2 + z^2 \\leq 4 \\} \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Determine the radius of convergence of the complex series \\( \\sum_{n=0}^{\\infty} \\sqrt{n}(4 + (-1)^n)z^n \\).",
"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... | 3.3 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... | 2.2s |
[
{
"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... | 1.11e7 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... | 69 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... | 28.7 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... | 629000 |
[
{
"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... | Magnitude is 20625 N/C and direction is -x axis |
[
{
"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... | 0.0025 |
[
{
"content": "Evaluate the integral \\( \\int_{-\\pi}^{\\pi} \\cos^n(x) \\sin(nx) \\, dx \\) using the binomial theorem and the parity of the cosine function.",
"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... | 1.42 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... | 248.85 s |
[
{
"content": "Let $U$ be a square with side length 1. Two points are randomly chosen on the sides of $U$. The probability that the distance between these two points is at least $\\frac{1}{2}$ is $\\frac{a - b \\pi}{c}\\left(a, b, c \\in \\mathbf{Z}_{+}, (a, b, c)=1\\right)$. Find the value of $a + b + c$.\\n\\n... | 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... | $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... | 30.6 g/mol |
[
{
"content": "Let $(x_n)$ be a sequence in a Banach space $X$. If $\\sum^{\\infty}_{n=1} \\|x_n − x_{n+1}\\|<\\infty$, does it necessarily follow that $(x_n)$ is a Cauchy sequence? Provide a justification for your answer.",
"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... | 1.66e-18 |
[
{
"content": "Calculate the integral \\( \\int_{[0,1]\\times [0,1]} xy \\, dx \\, dy \\) using the definition of the Riemann integral. Use a partition of \\([0,1]\\times[0,1]\\) and compute the Riemann sum to find the value of the integral.",
"role": "user"
}
] | \dfrac{1}{4} |
[
{
"content": "Given a function $f(x)$, if for any $k \\in (0,1)$ the following conditions hold:\n\n1. \\( \\lim_{x\\to \\infty} \\left( f(x) - k\\ln(x) \\right) \\leq 0 \\)\n2. \\( \\lim_{x\\to \\infty} \\left( f(x) - \\ln(x) \\right) \\geq 0 \\)\n\nProve that:\n\n\\[ \\lim_{x\\to \\infty} \\left( f(x) - \\ln(x... | 0 |
[
{
"content": "The polynomial $p(x) = x^2+ax+b$ has distinct roots $2a$ and $b$. Find $a+b$.",
"role": "user"
}
] | -1 |
[
{
"content": "For $a > 0$, consider the function \\( f(x) = \\frac{2}{a}\\sqrt{1-x}\\left(a^2+\\frac{1}{1-2x}\\right) \\) defined for \\( 0 \\leq x < \\frac{1}{2} \\). Determine the value of \\( a \\) for which \\( f(x) \\) attains its minimum value.",
"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... | $1,000,125.64 |
[
{
"content": "Find all functions $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ such that for all real numbers $x$ and $y$, the equation \\( f(x^2 + f(y)) = y + (f(x))^2 \\) holds.",
"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... | 3.4 m/s |
[
{
"content": "Circle $\\omega$ has radius 5 and is centered at $O$. Point $A$ lies outside $\\omega$ such that $OA=13$. The two tangents to $\\omega$ passing through $A$ are drawn, and points $B$ and $C$ are chosen on them (one on each tangent), such that line $BC$ is tangent to $\\omega$ and $\\omega$ lies out... | 17 |
[
{
"content": "True or False: Let $G$ and $H$ be finite groups. Is it true that any subgroup of $G \\times H$ is equal to $A \\times B$ for some subgroups $A < G$ and $B < H$? Provide a justification for your answer.",
"role": "user"
}
] | False |
[
{
"content": "Determine the greatest positive integer \\( x \\) such that \\( x \\) divides \\( 7^y + 12y - 1 \\) for every positive integer \\( y \\).",
"role": "user"
}
] | 18 |
[
{
"content": "Find the limit of the sequence \\(a_n = \\frac{\\sqrt[n]{(n+1)(n+2)...(2n)}}{n}\\).",
"role": "user"
}
] | \dfrac{4}{e} |
[
{
"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 |
[
{
"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 N |
[
{
"content": "Two people agree to meet between 12:00 and 13:00. Each person's arrival time is independently and identically distributed, following a uniform distribution over this interval. Given that one person has already arrived, calculate the probability that the waiting time for the other person is at leas... | \dfrac{1}{4} |
[
{
"content": "In triangle \\(ABC\\), \\(BR = RC\\), \\(CS = 3SA\\), and \\(\\frac{AT}{TB} = \\frac{p}{q}\\). If the area of \\(\\triangle RST\\) is twice the area of \\(\\triangle TBR\\), find the value of \\(\\frac{p}{q}\\). Choose from the following options:\n(A) \\(\\frac{2}{1}\\)\n(B) \\(\\frac{8}{3}\\)\n(C... | E |
[
{
"content": "Find the second directional derivative of the function \\( f(x, y) = x^3 + 5x^2y + y^3 \\) at the point \\((2, 1)\\) in the direction of the vector \\( u = \\left< \\frac{3}{5}, \\frac{4}{5} \\right> \\).",
"role": "user"
}
] | \dfrac{774}{25} |
[
{
"content": "Can a finite game be designed using an honest coin (with equal probabilities of heads and tails) such that the probability of one player winning is an irrational number \\( \\alpha \\), where \\( 0 < \\alpha < 1 \\)? A finite game is one that inevitably ends in a finite number of moves with probab... | 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... | 10.3538 kN |
[
{
"content": "Can the Maximum Modulus Principle be applied to an analytic function $f$ defined on a domain $D$ that includes a disk $D_R(z_0)$ and its boundary, if $|f(z_0)| \\leq |f(z)|$ for all $z$ on the boundary of the disk?",
"role": "user"
}
] | Yes |
[
{
"content": "Find the constant \\( a \\) such that the Euler-Lagrange equations for the functional \\( S[y_{1}, y_{2}]=\\int [y_{1}'^2+2y_{2}'^2+(2y_{1}+y_{2})^2]dx \\) can be transformed into decoupled equations using the linear transformation \\( z_{1}=y_{1}+ay_{2} \\) and \\( z_{2}=2y_{1}+y_{2} \\). After t... | -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... | 0.3 atm |
[
{
"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... | pV = \muRT |
[
{
"content": "Find the smallest value of the sum of squares \\( P = \\sum_{i=1}^{19} x_i^2 \\), where \\( x_1, x_2, \\ldots, x_{19} \\) are positive integers satisfying \\( \\sum_{i=1}^{19} x_i = 2020 \\) and \\( x_i \\geq 2 \\) for all \\( i \\).",
"role": "user"
}
] | 214762 |
[
{
"content": "Calculate the Frechet derivative of the function \\( f(x) = Ax \\) with respect to the vector \\( x \\), where \\( A \\) is a given matrix.",
"role": "user"
}
] | A |
[
{
"content": "Determine the limit of the sequence \\(a_n = \\left(1 + \\frac{1}{n^2}\\right)\\left(1 + \\frac{2}{n^2}\\right) \\dots \\left(1 + \\frac{n}{n^2}\\right)\\) as \\(n\\) approaches infinity.",
"role": "user"
}
] | \sqrt{e} |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x\\rightarrow 1}\\left(\\frac{\\sqrt[3]{7+x^3}-\\sqrt[2]{3+x^2}}{x-1}\\right) \\] without using L'Hôpital's rule.",
"role": "user"
}
] | -\dfrac{1}{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... | 3.05 s |
[
{
"content": "Evaluate the integral \\( \\int_C \\vec{F} \\cdot d\\vec{r} \\) using Stokes' theorem, where \\( \\vec{F}=[-z, 5x, -y] \\), \\( C \\) is the ellipse \\( x^2 + y^2 = 4 \\) with \\( z = x + 2 \\), and the direction of integration is clockwise as observed from the origin. Determine the surface integr... | -24\pi |
[
{
"content": "Given a closed set $C$ in $\\mathbb{R}^m \\times \\mathbb{R}^n$, determine if the set $\\{y \\in \\mathbb{R}^n: (x_0, y) \\in C\\}$ is closed for any $x_0 \\in \\mathbb{R}^m$. Justify your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Calculate the integral \\( \\int_{[0,\\frac{1}{4}]\\times[0,2\\pi]} x\\cos(xy)\\cos(2\\pi x)\\ \\text{d}x\\wedge\\text{d}y \\).",
"role": "user"
}
] | \dfrac{1}{4\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... | 1.39807 |
[
{
"content": "Find the limit: \\[ \\lim_{n \\to \\infty} \\frac{\\frac{\\sin(1)}{1} + \\frac{\\sin(2)}{2} + \\frac{\\sin(3)}{3} + \\cdots + \\frac{\\sin(n)}{n}}{n} \\]",
"role": "user"
}
] | 0 |
[
{
"content": "Find all limit points of the sequence \\(a_n = (-n)^{(-n)^n}\\) for \\(n \\in \\mathbb{N}\\).",
"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... | 7.62 |
[
{
"content": "Given the expressions:\n\n\\[ a = 1 + \\frac{w^2x^2}{2!} + \\frac{w^4x^4}{4!} + \\cdots \\]\n\n\\[ b = -x - \\frac{w^2x^3}{3!} - \\frac{w^4x^5}{5!} - \\cdots \\]\n\nwhere \\( w \\) is a non-real, complex cube root of unity, and \\(!\\) denotes factorial, find the value of \\( a^2 - b^2w^2 \\).\n\n... | 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... | ($279,512) |
[
{
"content": "In triangle ABC, a circle with radius r is inscribed. Tangent lines parallel to the sides of the triangle form three smaller triangles: $\\triangle A_cB_cC$, $\\triangle A_bBC_b$, and $\\triangle AB_aC_a$. The inradii of these smaller triangles are 1, 2, and 3, respectively. Find the radius r of t... | 6 |
[
{
"content": "Given an arithmetic sequence $\\{a_n\\}$ with the first term $a_1 > 0$, and the conditions:\n\\[ a_{2013} + a_{2014} > 0, \\quad a_{2013} a_{2014} < 0, \\]\nfind the largest natural number $n$ for which the sum of the first $n$ terms, $S_n > 0$, holds true.\n\n(A) 4025\n\n(B) 4026\n\n(C) 4027\n\n(... | B |
[
{
"content": "Let $F$ be a finite field of order $q$. Consider $U(3,F)$, the multiplicative group of $3 \\times 3$ upper-triangular matrices with entries from $F$ and all diagonal entries equal to 1. Determine whether the number of distinct conjugacy classes of $U(3,F)$ is a polynomial in $q$ with integer coeff... | q^2 + q - 1 |
[
{
"content": "What is the value of $b$ if $-x^2+bx-5<0$ only when $x\\in (-\\infty, 1)\\cup(5,\\infty)$?",
"role": "user"
}
] | 6 |
[
{
"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... | 1.14 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... | 2.53 km/hr |
[
{
"content": "Find the coefficient of \\(x^{25}\\) in the expansion of \\((1 + x^2 + x^4 + x^7)^{10}\\).",
"role": "user"
}
] | 34750 |
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