prompt list | ground_truth string |
|---|---|
[
{
"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... | 4.07 x 10^{-4} s^{-1} |
[
{
"content": "Determine whether a convex closed curve in the plane can intersect itself at any point other than the starting and ending points. Answer with '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... | 411 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... | 1222.2 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... | h = 1/2*g*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... | 24 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... | -$126,647 |
[
{
"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... | 17.07 |
[
{
"content": "Consider the parabola given by the equation \\(y^2 = 4ax\\). A focal chord \\(PQ\\) is drawn, and the tangents to the parabola at points \\(P\\) and \\(Q\\) intersect at point \\(R\\), which lies on the line \\(y = 2x + a\\). Find the length of the chord \\(PQ\\).",
"role": "user"
}
] | 5a |
[
{
"content": "Let $\\mu$ be a strictly positive measure, and let $f:X\\rightarrow[0,\\infty]$ be an integrable function such that $\\int_Xfd\\mu=1$. Calculate the limit: \\[ \\lim_{n\\rightarrow \\infty}\\int_Xn\\log\\biggl(1+\\frac{f(x)}{n}\\biggl)d\\mu. \\]",
"role": "user"
}
] | 1 |
[
{
"content": "Let $M$ and $N$ be $d$-dimensional oriented Riemannian manifolds, and let $f: M \\to N$ be a smooth map. Assume $M$ is compact with boundary. Does there exist a vector field $X \\in \\Gamma(TM)$ such that $\\text{div} \\, X = \\det df$?",
"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... | 2.85e8 g C3H8 |
[
{
"content": "Determine the number of 4x4 matrices \\( A = [a_{ij}] \\) where \\( 1 \\le i,j \\le 4 \\), each element \\( a_{ij} = \\pm 1 \\), and the sum of the elements in each row and each column is zero.",
"role": "user"
}
] | 90 |
[
{
"content": "Suppose \\( f \\) is a differentiable function for which \\( \\lim_{x \\to \\infty} f(x) \\) and \\( \\lim_{x \\to \\infty} \\frac{df(x)}{dx} \\) both exist and are finite. Which of the following must be true?\n\n(A) \\( \\lim_{x \\to \\infty} \\frac{df(x)}{dx} = 0 \\)\n(B) \\( \\lim_{x \\to \\inf... | A |
[
{
"content": "Evaluate the limit: $$\\lim_{\\rho \\to 0} (\\rho^2)^{\\rho^4\\cos^2(\\phi)\\sin^2(\\phi)}.$$",
"role": "user"
}
] | 1 |
[
{
"content": "Evaluate the contour integral \\( \\oint_{|z|=2} \\tan{z}\\,dz \\). Here, \\( |z|=2 \\) represents a circle of radius 2 centered at the origin in the complex plane.",
"role": "user"
}
] | -4\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... | $1,323,784.17 |
[
{
"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... | 0.60 |
[
{
"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... | $32,175 |
[
{
"content": "Given a continuous function $f$ such that $f(f(x)) = 1 + x$, find the value of $f'(1)$. Provide a formal approach to solving this problem.",
"role": "user"
}
] | 1 |
[
{
"content": "Evaluate the integral \\( \\int_{0}^{\\infty} \\left( \\int_{0}^{\\infty} x e^{-x^2(1+y^2)} \\, d\\lambda(y) \\right) \\, d\\lambda(x) \\) using Tonelli's theorem.",
"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... | 1.07e4 N/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... | 13.79% |
[
{
"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... | 123,000 |
[
{
"content": "Determine the number of Q-field homomorphisms from the algebraic closure of the rational numbers to the complex numbers that keep the rational numbers fixed.",
"role": "user"
}
] | 2^{\aleph_0} |
[
{
"content": "Find the limit of the sequence \\(a_n = \\sqrt[n]{4^{2n} - 5^{n+2} - \\sin(\\sqrt{n})}\\) as \\(n\\) approaches infinity.",
"role": "user"
}
] | 16 |
[
{
"content": "Evaluate the improper integral: $$\\int_{0}^{1}\\ln(e^{1/x}+1)\\ dx$$",
"role": "user"
}
] | \infty |
[
{
"content": "Consider an operator $T$ on a finite-dimensional vector space that is upper-triangular with respect to some basis. Is $T$ also upper-triangular with respect to any orthonormal basis obtained from that basis using Gram-Schmidt orthogonalization?",
"role": "user"
}
] | Yes |
[
{
"content": "Let \\( l \\) and \\( m \\) be two skew lines. On \\( l \\), there are three points \\( A, B, \\) and \\( C \\), such that \\( AB = BC \\). From points \\( A, B, \\) and \\( C \\), perpendiculars \\( AD, BE, \\) and \\( CF \\) are respectively drawn to \\( m \\), with feet of the perpendiculars at... | \sqrt{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... | 6.02% |
[
{
"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... | Trend Extrapolation |
[
{
"content": "Compute the expected value of the normal distribution given by the integral:\n\\[ \\frac{1}{\\sqrt{2\\pi\\sigma^{2}}}\\int_{-\\infty}^{\\infty} e^{-{(x - \\mu)^2/(2\\sigma^{2})}} x\\,dx. \\]\nCan this be done without using the error function (erf)?",
"role": "user"
}
] | \mu |
[
{
"content": "Determine whether the logical statement \\( \\forall x\\,P(x) \\to \\exists x\\,P(x) \\) is true.",
"role": "user"
}
] | true |
[
{
"content": "Let $r_1,$ $r_2,$ $\\dots,$ $r_7$ be the distinct complex roots of the polynomial $P(x) = x^7 - 7.$ Let\n\\[K = \\prod_{1 \\le i < j \\le 7} (r_i + r_j).\\]In other words, $K$ is the product of all numbers of the of the form $r_i + r_j,$ where $i$ and $j$ are integers for which $1 \\le i < j \\le... | 117649 |
[
{
"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... | 2.914 ms^-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... | 98.979 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... | 0.67 |
[
{
"content": "Given 95 numbers \\( a_{1}, a_{2}, \\cdots, a_{95} \\), each of which can only take on the values +1 or -1, find the smallest positive value of the sum of the products of every pair of these numbers:\n\\[\na_{1} a_{2} + a_{1} a_{3} + \\cdots + a_{94} a_{95}\n\\]",
"role": "user"
}
] | 13 |
[
{
"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... | 13.92 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... | monopolistic competition |
[
{
"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/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... | 0.95 |
[
{
"content": "In triangle \\( ABC \\), \\( \\angle BAC = 75^\\circ \\), \\( AB = 1 \\), and \\( AC = \\sqrt{6} \\). A point \\( M \\) is chosen on side \\( BC \\) such that \\( \\angle BAM = 30^\\circ \\). The line \\( AM \\) intersects the circumcircle of triangle \\( ABC \\) at a point \\( N \\) different fro... | 2 |
[
{
"content": "Determine the range of values for \\( a + b \\) such that the inequality \\( a \\cos x + b \\cos 2x \\geq -1 \\) holds for all \\( x \\in \\mathbb{R} \\).",
"role": "user"
}
] | [-1, 2] |
[
{
"content": "Consider a continuous and non-decreasing distribution function $F$ with an associated density function $f$. Let $S_1$ be the set where $F(s+\\varepsilon)> F(s-\\varepsilon)$ for all $\\varepsilon>0$, and $S_2$ be the closure of the set where $f(s)>0$. Determine if these two sets are equal, i.e., i... | S_1 = 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... | 2.52 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... | $183,900 |
[
{
"content": "Calculate the line integral \\( \\oint_L p \\, dx + q \\, dy \\) for the vector field \\( F = [p(x, y), q(x, y)] = [y^3 + e^{x^2}, x^3 + \\tan^2 y] \\), where \\( L \\) is the positively oriented circle defined by \\( x^2 + y^2 + 4y = 0 \\). Use Green's theorem to perform the calculation.",
"r... | -48\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... | 9.59 |
[
{
"content": "Evaluate the expression:\n\n$$\\sum_{k_1=0}^n\\sum_{k_2=0}^n\\cdots\\sum_{k_m=0}^n\\binom{n}{k_1}\\binom{n-k_1}{k_2}\\cdots\\binom{n-k_1-\\cdots-k_{m-1}}{k_m}$$\n\nwhere $m$ is a positive integer.",
"role": "user"
}
] | (m+1)^n |
[
{
"content": "If $2i$ is an eigenvalue of a real $2 \\times 2$ matrix $A$, find $A^2$. Express your answer in terms of the identity matrix $I$.",
"role": "user"
}
] | -4I |
[
{
"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... | 4.07 |
[
{
"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... | 11.39 m/s |
[
{
"content": "Solve the equation \\(\\sqrt{x}^{\\sqrt{x}^{\\sqrt{x}}} = 2^{512}\\).",
"role": "user"
}
] | 16 |
[
{
"content": "Calculate the limit: \\[ \\lim_{n\\to\\infty} \\int_{0}^{\\pi} \\sqrt[n]{\\sin(x)}\\,dx. \\]",
"role": "user"
}
] | \pi |
[
{
"content": "Determine the values of $m$ for which the function $f(x) = e^x(x^2 + x + m)$ has exactly one local extremum in the interval $(-1, 1)$. Provide your answer in terms of $m$.",
"role": "user"
}
] | (-5, 1) |
[
{
"content": "Calculate the surface integral \\( \\int_B \\operatorname{curl}(a) \\cdot d\\vec{S} \\), where \\( B = \\{(x,y,z) \\in \\mathbb{R}^3 \\mid x^2 + y^2 + z^2 = 1, z \\geq 0\\} \\) is the surface of the upper half of a unit sphere centered at the origin, and the vector field \\( a(x,y,z) = (-y, x, z) ... | 2\pi |
[
{
"content": "Determine whether two scalene triangles are congruent if their longest sides, largest angles, and shortest sides are congruent. Does this specific case of side-side-angle (SSA) imply triangle congruence?",
"role": "user"
}
] | Yes |
[
{
"content": "For the infinite series \\(\\sum_{n=0}^{+\\infty}x^n\\) which converges to \\( \\frac{1}{1-x}\\), determine if this convergence property holds true when \\(x\\) is replaced by an \\(N \\times N\\) matrix. Provide the conditions under which the series converges.",
"role": "user"
}
] | \rho(A) < 1 |
[
{
"content": "Consider two urns, A and B. Initially, urn A contains 2 red marbles, and urn B contains 2 white marbles. In each trial, one marble is drawn from each urn and switched. Let \\( X_n \\) represent the number of red marbles in urn A after \\( n \\) trials. Given that urn A contains at least one red ma... | \dfrac{3}{5} |
[
{
"content": "For a polynomial \\(p(x)=1+\\sum_{j=1}^n b_j x^j\\) with \\(b_j>0\\), determine if there exist constants \\(C>0\\) and \\(D\\ge 0\\) such that the inequality\n\n$$x^2(x-1)\\left(\\frac{p''(x)}{p(x)}-\\frac{p'(x)^2}{p(x)^2}\\right)\\le C\\ln p(x)+D$$\n\nholds for sufficiently large \\(x>1\\).",
... | Yes |
[
{
"content": "Compute the integral of the 2-form $\\omega=\\mathrm{d}x\\wedge\\mathrm{d}y$ over the surface of the ellipsoid given by $\\frac{x^2}{a^2}+\\frac{y^2}{b^2}+\\frac{z^2}{c^2}=1$.",
"role": "user"
}
] | 0 |
[
{
"content": "Evaluate the integral \\( \\frac{2}{\\pi i}\\int_C \\left(ze^{1/z}+\\tan\\left(\\frac{z}{2}\\right)+\\frac{1}{(z-1)(z-3)^2}\\right)dz \\), where \\( C \\) is the positively oriented circle of radius 2 centered at the origin in the complex plane.",
"role": "user"
}
] | 3 |
[
{
"content": "Determine which of the following Taylor expansions is correct for a function $f: \\mathbb{R}^n \\to \\mathbb{R}$, when expanding around $x$ for $y = x + \\delta v$ with $\\delta > 0$ small:\n\nExpansion 1:\n\\[\nf(y) = f(x) + \\delta v^\\top \\nabla f(x) + \\frac{\\delta^2}{2}v^\\top\\nabla^2 f(x)... | 1 |
[
{
"content": "Determine the radius of convergence of the power series \\( \\frac{\\sin^2(z)}{z} \\).",
"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... | $172.36 |
[
{
"content": "Determine the number of points on a line XY that are equidistant from two intersecting lines AB and CD, given that XY is not parallel to the angular bisectors of AB and CD.",
"role": "user"
}
] | 2 |
[
{
"content": "Determine the number of elements in the field $\\mathbb{F}_{2}(a^2+a)$, where $F_{16} = \\mathbb{F}_{2}(a)$ and the minimal polynomial of $a$ over $\\mathbb{F}_2$ is $m_{a,\\mathbb{F}_2} = a^4+a+1$. \\( \\boxed{\\text{Answer}} \\)",
"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... | $900,000 |
[
{
"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... | 39600 N |
[
{
"content": "Evaluate the integral \\( \\int_{1}^{100} \\frac{f(x)}{x} \\, dx \\) given that the function \\( f(x) \\) satisfies \\( f(x) = f\\left(\\frac{100}{x}\\right) \\) for all real and positive \\( x \\), and \\( \\int_{1}^{10} \\frac{f(x)}{x} \\, dx = 5 \\).",
"role": "user"
}
] | 10 |
[
{
"content": "Let $f_1,..., f_n: \\mathbb{R}\\rightarrow\\mathbb{R}$ be Lebesgue-measurable functions and $F:\\mathbb{R}^n\\rightarrow\\mathbb{R}$ be a continuous function. Is the composition $g(x):=F(f_1(x),f_2(x),...,f_n(x))$ always Lebesgue-measurable?",
"role": "user"
}
] | Yes |
[
{
"content": "Determine the values of \\( p > 0 \\) for which the integral \\[ T = \\sqrt{\\frac{1}{2g}}\\int_{0}^{b}\\frac{\\sqrt{1+y'(x)^2}}{\\sqrt{-y(x)}} \\, dx \\] is convergent, given that \\( y(x) = -2x^p \\) and \\( g \\) is a constant.",
"role": "user"
}
] | (0, 2) |
[
{
"content": "Given the Lorenz curve for a country is defined by the function \\( f(x) = x^k \\), find the value of \\( k \\) such that the Gini index is 0.9.",
"role": "user"
}
] | 19 |
[
{
"content": "Given the system of equations:\n\\[\n\\left\\{\n\\begin{array}{l}\na x + b y = 3, \\\\\na x^2 + b y^2 = 7, \\\\\na x^3 + b y^3 = 16, \\\\\na x^4 + b y^4 = 42,\n\\end{array}\n\\right.\n\\]\nfind the value of \\(a x^5 + b y^5\\).",
"role": "user"
}
] | 20 |
[
{
"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... | $3,618 |
[
{
"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... | 578.1 |
[
{
"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... | -1.09% |
[
{
"content": "Evaluate the integral \\( \\int_{-\\pi/2}^{\\pi/2} \\frac{\\sin(kn)}{1+k^2} \\, dk \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Compute the sum of the series: \\[ \\sum_{n=1}^\\infty n^2 \\cdot \\left(\\frac{2}{3}\\right)^n \\]",
"role": "user"
}
] | 30 |
[
{
"content": "Determine the value of $p$ for which the sequence $x_n = \\frac{1}{n}$ belongs to the space $l^p$.",
"role": "user"
}
] | p > 1 |
[
{
"content": "Determine the radius of convergence for the power series \\( \\sum_{n\\ge1}a_nx^n \\), where \\( a_n \\) is the number of divisors of \\( n^{50} \\).",
"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... | 113137.1 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... | 9*kg |
[
{
"content": "Calculate the value of $S_{10} + I_{10}$, where $S_n = \\sum_{i=1}^n \\frac{1}{i}$ and $I_n = \\int_{1}^{n} \\frac{x - \\lfloor x \\rfloor}{x^2} \\, dx$. Given that $I_{10} = \\ln(10) - \\int_{1}^{10} \\frac{\\lfloor x \\rfloor}{x^2} \\, dx$, evaluate the integral $\\int_{1}^{10} \\frac{\\lfloor x... | 1 + \ln 10 |
[
{
"content": "By partial fractions,\n\\[\\frac{1}{x(x + 1)(x + 2)(x + 3)(x + 4)} = \\frac{A}{x} + \\frac{B}{x + 1} + \\frac{C}{x + 2} + \\frac{D}{x + 3} + \\frac{E}{x + 4}\\]for some constants $A,$ $B,$ $C,$ $D,$ and $E.$ Find $A + B + C + D + E.$",
"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... | 8.4 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... | 1.3 mL |
[
{
"content": "Find the limit of the expression \\( \\frac{\\ln(\\ln n)}{\\ln n} - 1 \\) as \\( n \\to \\infty \\).",
"role": "user"
}
] | -1 |
[
{
"content": "Determine the number of 6-tuples \\((a_1,a_2,a_3,a_4,a_5,a_6)\\) such that each element \\(a_i\\) is chosen from the set \\(\\{1,2,3,4\\}\\) and the expressions \\[ a_j^2 - a_j a_{j+1} + a_{j+1}^2 \\] for \\(j = 1, 2, 3, 4, 5, 6\\) (with \\(a_7 = a_1\\)) are all equal to each other.",
"role": ... | 40 |
[
{
"content": "Determine the interval of absolute convergence for the series \\(\\sum_{n=1}^\\infty \\frac{n}{2^n(2n-1)}(x-1)^n\\).",
"role": "user"
}
] | (-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... | left |
[
{
"content": "Evaluate the limit: $$\\lim_{x\\to 0}\\frac{\\sin^2(3x)-3x\\tan(3x)}{(3x)^4}$$",
"role": "user"
}
] | -\dfrac{2}{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... | 90 mi/h |
[
{
"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... | {eq}\frac{v_2^2}{v_1^2}{/eq} |
[
{
"content": "Given an $L^1(\\mathbb{R})$ function $f$ such that $f(x) = -f(-x)$, does it follow that $\\lim_{\\xi \\to \\infty} \\hat{f}(\\xi) = \\hat{f}(0)$, given that $\\hat{f}(\\xi) = -\\hat{f}(-\\xi)$ and $\\lim_{\\xi \\to \\infty} \\hat{f}(\\xi) = 0$?",
"role": "user"
}
] | 0 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n \\to \\infty} n \\sin(2\\pi n!e) \\]",
"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... | 1.43e30 photons/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... | 3774.96 cm^3 |
[
{
"content": "Determine whether the number given by the sum \\( M = \\sum_{n=0}^{\\infty} b^{-n!} \\) is a Liouville number.",
"role": "user"
}
] | Yes |
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