prompt list | ground_truth string |
|---|---|
[
{
"content": "Evaluate the improper integral: $$\\int_{-\\infty}^{\\infty} \\frac{\\cos^2 x}{(x^2-\\pi ^2)} dx$$",
"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... | 3.43 rad/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... | -0.3390 |
[
{
"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... | 1.57e13 W |
[
{
"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": "Let $c$ be a complex number. Suppose there exist distinct complex numbers $r$, $s$, and $t$ such that for every complex number $z$, we have\n\\[\n (z - r)(z - s)(z - t) = (z - cr)(z - cs)(z - ct).\n\\]Compute the number of distinct possible values of $c$.",
"role": "user"
}
] | 4 |
[
{
"content": "Determine whether an infinite product of copies of $\\mathbb{Z}$ is isomorphic to a direct sum of copies of $\\mathbb{Z}$. Provide a true or false answer.",
"role": "user"
}
] | 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... | Project A |
[
{
"content": "Given that $ab + bc + ca = 0$, find the value of $\\frac{1}{a^2-bc} + \\frac{1}{b^2-ca} + \\frac{1}{c^2-ab}$.",
"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... | 7.05% |
[
{
"content": "Let $K$ be a finite extension of $\\mathbb{Q}_p$, and consider the disk $D = \\{ x \\in \\overline{K} \\mid |x| < c < 1 \\}$, where $c \\in |K^\\times|$. Is the set of power series $f(T) \\in K[[T]]$ that converge on $D$ and have a finite supremum norm $\\|f\\|_\\infty = \\sup_{x \\in \\overline{K... | Yes |
[
{
"content": "Evaluate the limit: $$\\lim_{n\\rightarrow\\infty}n^{3/2}\\left(\\arctan\\left(\\sqrt{n+1}\\right)-\\arctan\\left(\\sqrt{n}\\right)\\right)$$",
"role": "user"
}
] | \dfrac{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... | 0.204 |
[
{
"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... | 1.0e-6 |
[
{
"content": "Suppose that $p(x)$ is a fifth-degree polynomial where $p'(x)$ is divisible by both $(x-2)^2$ and $(x+2)^2$. Determine the value of the definite integral \\(\\int_{-10}^{10}p(x)\\,dx\\).",
"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... | 18,536.90 |
[
{
"content": "Let $f: G \\to \\mathbb{C}$ be a function defined on a domain $G \\subset \\mathbb{C}$. If $f$ is continuously differentiable on $G$ and complex differentiable on a dense subset $H$ of $G$, determine whether the set of points in $G$ where $f$ is complex differentiable is open in $G$. Provide a jus... | Yes |
[
{
"content": "Given the function $f$ where $f(1) = 2005$ and $f(1) + f(2) + \\ldots + f(n) = n^2f(n)$ for all $n > 1$, determine the value of $f(2004)$. \\( \\boxed{} \\)",
"role": "user"
}
] | \dfrac{1}{1002} |
[
{
"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... | $1,600 U |
[
{
"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... | 399,244.05 |
[
{
"content": "Determine whether there exist finitely generated modules over a commutative ring that have constant finite rank but are not free. 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... | 1.637e18 kg |
[
{
"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... | $20,170 |
[
{
"content": "Evaluate the limit using the Lagrange Remainder Theorem: $$\\lim_{n\\to\\infty} n^2\\left(1-\\cos\\frac{x}{n} \\right)$$",
"role": "user"
}
] | \dfrac{x^2}{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... | $1,000 disadvantage |
[
{
"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... | 185 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... | 170 pm |
[
{
"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... | 463 K |
[
{
"content": "Let $f : \\mathbb{C} \\to \\mathbb{C}$ be a continuous function with $f(0)=0$. Let $\\{a_i\\}_{i\\in \\mathbb{N}}$ be a set of scalars in $\\mathbb{C}$ such that there exists a constant $C > 0$ with $|a_i| \\leq C$ for all $i \\in \\mathbb{N}$. Determine whether there always exists a holomorphic f... | No |
[
{
"content": "Given a circle $\\Omega$ with radius $r$, let $Z$ be the random variable representing the distance from the circle's center to a randomly chosen point, and $X$ be the random variable for the $x$-coordinate of the point. Determine the covariance $\\text{cov}(X,Z)$. The joint probability density fun... | 0 |
[
{
"content": "Find the limit of the sequence \\((a_n)_{n \\in \\mathbb{N}}\\) defined by \\[a_n = \\sum_{k=1}^n\\frac{k^2}{n^3+k}.\\]",
"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... | CO2 + H2O -> H2CO3 -> HCO3- + H+ |
[
{
"content": "Determine the correct Jordan normal form of the matrix \\(\\begin{bmatrix}1&1&0\\\\-1&3&0\\\\-1&1&2\\end{bmatrix}\\) given that its characteristic polynomial is \\((X-2)^3\\) and its minimal polynomial is \\((X-2)^2\\). Choose from the following options:\n\n1) \\(\\begin{bmatrix}2&1&0\\\\0&2&0\\\\... | 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... | <img>/cimages/multimages/16/capture16937187321644569244.png</img> |
[
{
"content": "If a topological space is homeomorphic to a metric space, and another topological space is homeomorphic to the first space, is the second space also metrizable?",
"role": "user"
}
] | Yes |
[
{
"content": "Consider the function \\(f(m, n)\\) defined as follows: \\(f(m,1) = f(1,n) = 1\\) for \\(m, n \\geq 1\\), and \\(f(m,n) = f(m-1,n) + f(m,n-1) + f(m-1,n-1)\\) for \\(m > 1\\) and \\(n > 1\\). Let \\(S(n) = \\sum_{a+b=n} f(a,b)\\), where the sum is taken over all pairs of positive integers \\(a\\) a... | 2 |
[
{
"content": "Let $\\alpha_n \\in \\mathbb{R}$ be a fixed sequence of real numbers such that for all $t \\in \\mathbb{R}$, \\( \\lim_{n \\rightarrow \\infty}e^{i\\alpha_nt} = 1 \\). Determine whether it is necessarily true that \\( \\lim_{n \\rightarrow \\infty} \\alpha_n = 0 \\).",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate the limit as \\(n\\rightarrow\\infty\\) of the expression:\n\\[\nn\\left(1-\\frac{a\\log n}{n}\\right)^{n/2}\\left(1-\\frac{b\\log n}{n}\\right)^{n/2-1}\\]\nwhere \\(a,b>0\\) and \\(\\frac{a+b}{2}<1\\).",
"role": "user"
}
] | \infty |
[
{
"content": "Find the function \\( y(x) \\) that extremizes the boundary value problem given by the functional:\n\\[ I[y] = \\int_0^1 y'^2(x) \\, dx + y^2(0) - 2y^2(1) \\]",
"role": "user"
}
] | 0 |
[
{
"content": "Determine the Lebesgue measure of the set \\( E = \\{(a,b) \\in \\mathbb{R}^2 \\mid a-b \\in \\mathbb{Q}\\} \\) in \\( \\mathbb{R}^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... | 184 |
[
{
"content": "Given a Hamiltonian circuit, if the maximum weighted edge is removed, does this always result in a minimum weight Hamiltonian path? Answer yes or no.",
"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... | Date Account titles and Description Debit ($) Credit ($)
12/31 Postage 43.50
Office supplies 51.85
Equipment repair 49.00
Cash short or over (balancing figure) 1.40
Cash 145.75 |
[
{
"content": "Find the last three digits of \\(7^{7^{7^7}}\\).",
"role": "user"
}
] | 343 |
[
{
"content": "Find the area of the region bounded by the polar curve \\( r^2 = a^2 \\cos(2\\theta) \\).",
"role": "user"
}
] | a^2 |
[
{
"content": "Let $a, b, c$ be positive real numbers such that $\\frac{1}{a} + \\frac{1}{b} + \\frac{1}{c} = a + b + c$. Find the maximum value of the expression:\n\\[ \\frac{1}{(2a+b+c)^2} + \\frac{1}{(2b+c+a)^2} + \\frac{1}{(2c+a+b)^2}. \\]",
"role": "user"
}
] | \dfrac{3}{16} |
[
{
"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... | 66 units |
[
{
"content": "Consider an infinitely differentiable function $f: \\mathbb{R} \\to \\mathbb{R}$ that has zero values at $10$ distinct points in $\\mathbb{R}$. Let $f^{(n)}$ denote the $n$-th derivative of $f$ for $n \\geq 1$. Determine which of the following statements is true:\n\nA. $f^{(n)}$ has at least $10$ ... | B |
[
{
"content": "Let $M$ be a Von Neumann algebra and $\\phi$ be a completely bounded (c.b.) map on $M$. Consider the Hilbert space $H$ and the map $Id\\otimes \\phi$ on $B(H)\\overline{\\otimes} M$, where $B(H)$ is the algebra of bounded linear operators on $H$ and $\\overline{\\otimes}$ denotes the minimal tenso... | \|\phi\|_{cb} |
[
{
"content": "Determine whether there exist infinite sets \\( X \\) and \\( Y \\) such that \\( X \\subseteq Y \\) and both \\( X \\) and \\( Y \\setminus X \\) have the same cardinality.",
"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... | 59 grams/mole |
[
{
"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... | 0.001259 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... | iron |
[
{
"content": "Determine whether an element in a monoid can have more than one multiplicative left inverse.",
"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... | $1,750.57 |
[
{
"content": "A triangle has two sides of lengths 4 cm and 9 cm. The radius of the inscribed circle that touches all three sides of the triangle is 1 cm. Determine the length of the third side of the triangle.",
"role": "user"
}
] | 6 |
[
{
"content": "Determine if the following manipulation is valid: Given the expression \\( h(t)=\\dfrac{1+o(1)}{(2 \\pi)^{3/2}\\sqrt{t}} \\) as \\( t\\downarrow0 \\), is it correct to state that \\((2 \\pi)^{3/2}\\sqrt{t}\\cdot h(t) -1 =o(1)\\) as \\( t\\downarrow 0 \\)?",
"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... | The torque (for the rod rotating about the center) is =8.73Nm
The torque (for the rod rotating about one end) is =34.91Nm |
[
{
"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... | $63,935.24 |
[
{
"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... | 1.52 atm^-1 |
[
{
"content": "A plane intersects a rhombus through one of its sides, forming angles of \\( \\alpha \\) and \\( 2\\alpha \\) with the diagonals of the rhombus. Determine the acute angle of the rhombus.",
"role": "user"
}
] | 60^\circ |
[
{
"content": "A linear stationary dynamic system is described by the equation \\(3 Y^{\\prime}(t) + Y(t) = 4 X^{\\prime}(t) + X(t)\\). The system receives a stationary random function \\(X(t)\\) with a correlation function \\(\\mathbf{k}_{x}(\\tau) = 6 e^{-2|\\tau|}\\). Determine the variance of the random func... | 10 |
[
{
"content": "Determine whether the inequality \\(|\\int_a^b f(x)dx| \\ge |f(b) - f(a)|\\) holds for all continuous functions \\(f(x)\\) on the interval \\([a, b]\\).",
"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... | 70.1 nm |
[
{
"content": "Given a sequence of positive terms $\\{a_{n}\\}$ such that $a_{n}^{n} + a_{n} - 1 = 0$ for $n \\in \\mathbf{N}^{*}$. Determine which of the following combinations of statements is correct:\n\n1. $0 < a_{n} \\leq \\frac{n}{n+1}$\n2. $\\frac{n}{n+1} \\leq a_{n} < 1$\n3. $\\{a_{n}\\}$ is an increasin... | A |
[
{
"content": "Consider a continuous and Riemann integrable function \\(f:\\mathbb{R}\\to\\mathbb{R}\\) and fix \\(a\\in\\mathbb{R}\\). Define \\(F:\\mathbb{R}\\to\\mathbb{R}\\) by \\(F(x) = \\int_{a}^{x}f(y)dy\\). Is it true that \\(F\\) is differentiable and \\(F'(x)=f(x)\\) for \\(x<a\\)?",
"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... | 26.12% |
[
{
"content": "Calculate the line integral \\( \\int_{\\gamma} \\frac{2xy\\,\\mathrm{dx}-(x^2+z^2)\\,\\mathrm{dy}+2yz\\,\\mathrm{dz}}{(x^2+z^2)^2} \\) where the path \\( \\gamma: [0,1] \\to \\mathbb{R}^3 \\) is defined by \\( \\gamma(t) = (\\cos(2\\pi t), \\sin(2\\pi t), t^2-t) \\) for all \\( t \\in [0,1] \\)."... | 0 |
[
{
"content": "Solve the congruence equation: \\(2^{53x} \\equiv 3 \\pmod{23}\\). Given that \\(2^{11} \\equiv 1 \\pmod{23}\\), find the value of \\(53x \\equiv 8 \\pmod{11}\\).",
"role": "user"
}
] | 7 |
[
{
"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... | 6.61e-16 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... | 313.5 N |
[
{
"content": "Determine the order of the automorphism group of \\( \\mathbb{Z}_5 \\times \\mathbb{Z}_5 \\).",
"role": "user"
}
] | 480 |
[
{
"content": "Given complex numbers \\( z, z_{1}, z_{2} \\) with \\( z_{1} \\neq z_{2} \\) such that \\( z_{1}^{2}=z_{2}^{2}=-2-2 \\sqrt{3} \\mathrm{i} \\), and \\(\\left|z-z_{1}\\right|=\\left|z-z_{2}\\right|=4\\), find \\(|z|\\).",
"role": "user"
}
] | 2\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... | 8496 L |
[
{
"content": "Find a real-valued function \\( f \\) that satisfies the equation \\( f(x) + f(y) + 2xyf(xy) = \\frac{f(xy)}{f(x+y)} \\) for all positive real numbers \\( x \\) and \\( y \\).",
"role": "user"
}
] | \dfrac{1}{x^2} |
[
{
"content": "In the figure, AD = BE = CF and triangle DEF is equilateral. Determine whether triangle ABC must be equilateral.",
"role": "user"
}
] | Yes |
[
{
"content": "Given that the focal distance of a point on the parabola $y = x^2 - 4$ is $\\frac{25}{4}$, and the points are of the form $(\\pm \\sqrt{a}, b)$, find $a + b$ if the equation $a + b^2 + \\frac{225}{16} + \\frac{15b}{2} = \\frac{625}{16}$ is incorrect.",
"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... | 2.50 m/s |
[
{
"content": "Determine the values of \\(p\\) and \\(q\\) for which the integral \\(\\int_0^\\infty \\frac{x^p}{1+x^q}\\,dx\\) converges, given that \\(q > 0\\).",
"role": "user"
}
] | -1 < p < q - 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... | $440,000 |
[
{
"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... | $119,480 |
[
{
"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... | 29000 |
[
{
"content": "Find the conditions on $m$ such that the sequence $\\{b_n\\}_{n=1}^\\infty$ defined by \\( b_n = \\frac{n^m + 1}{n^{m+1} + 1} \\) is monotonically decreasing.",
"role": "user"
}
] | m \geq -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... | Re |
[
{
"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... | 0.51156 mol kg^(-1) |
[
{
"content": "Evaluate the triple integral \\( \\int_{-1}^{1} \\int_{-\\sqrt{1-x^2}}^{\\sqrt{1-x^2}} \\int_{0}^{\\sqrt{1-x^2-y^2}} x^2+y^2+z^2 \\, dz \\, dy \\, dx \\) using a change of coordinate system.",
"role": "user"
}
] | \dfrac{2\pi}{5} |
[
{
"content": "In the given diagram, $ST$ is the perpendicular bisector of $PR$, and $SP$ is the angle bisector of $\\angle QPR$. Given that $QS = 9\\text{ cm}$ and $SR = 7\\text{ cm}$, find the value of $x+y$ if $PR = \\frac{x}{y}$, where $x$ and $y$ are co-prime integers.",
"role": "user"
}
] | 31 |
[
{
"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... | Variable cost = $6 per gallon
Fixed cost = $50,000 |
[
{
"content": "Does Cauchy's estimate imply that if $K$ is a compact subset of the unit disc $\\mathbb{D}$ and $\\{f_n\\}_{n=1}^{\\infty}$ is a sequence of functions converging uniformly to $0$ on $K$ as $n \\to \\infty$, then the sequence of derivatives $\\{f_n'\\}_{n=1}^{\\infty}$ also converges uniformly to $... | 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... | 0.42 or 42% |
[
{
"content": "Find the form of the continuous function \\( f: \\mathbb{R} \\rightarrow \\mathbb{R} \\) that satisfies the equation \\( f(x) + x = 2f(2x) \\) for all \\( x \\in \\mathbb{R} \\), given that \\( f(0) = 0 \\).",
"role": "user"
}
] | \dfrac{x}{3} |
[
{
"content": "Determine the number of possible sizes of a non-empty family \\(G\\) of subsets of \\([n] = \\{1,2,...,n\\}\\) such that for every \\(C \\in G\\), its complement \\([n]\\setminus C\\) is also in \\(G\\), and for every \\(A,B \\in G\\), both \\(A \\cap B\\) and \\(A \\cup B\\) are in \\(G\\).",
... | n |
[
{
"content": "Given integers $n$ and $r_0$ with $r_0 < n$, define:\n\\[ r_1 = n \\mod r_0 \\]\n\\[ r_2 = n \\mod r_1 \\]\nwhere $0 < r_1 < r_0$ and $0 < r_2 < r_1$. Determine if it is possible for $r_2$ to be greater than $\\lceil \\frac{1}{2}r_0 \\rceil$. Provide a justification for your answer.",
"role": ... | Yes |
[
{
"content": "Find the limit of the sequence \\( y_n=\\sum_{k=1}^n\\frac{k \\cos^2k}{n^2+k \\sin^2k} \\) as \\( n \\to \\infty \\).",
"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... | 2.40 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... | 144,996,609 |
[
{
"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... | (P_1V_1)/T_1=(P_2V_2)/T_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... | 4.85e-19 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... | 9.958e6 electrons |
[
{
"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... | 99.60 m |
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