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... | 4.63 kN |
[
{
"content": "Calculate the limit: $$\\lim_{n\\to\\infty}\\dfrac{\\ln(n!)}{\\ln^{n+1}(n+1)}$$ using L'Hopital's rule.",
"role": "user"
}
] | 0 |
[
{
"content": "A sequence $(a_n)$ is defined as follows: $a_1 = 1,$ $a_2 = \\frac{1}{2},$ and\n\\[a_n = \\frac{1 - a_{n - 1}}{2a_{n - 2}}\\]for all $n \\ge 3.$ Find $a_{120}.$",
"role": "user"
}
] | \frac{20}{41} |
[
{
"content": "Determine the number of ways to select 13 cards from a standard 52-card deck such that each rank from 2 to A is represented exactly once, and no two cards form a 'straight flush' (i.e., no two cards of the same suit have adjacent ranks, considering A and 2 as adjacent).",
"role": "user"
}
] | 1594320 |
[
{
"content": "Suppose that a sequence of sets $\\{A_n : n \\in \\mathbb{N}\\}$ is increasing, and let $A = \\bigcup_{n=1}^\\infty A_n$. If $A$ is measurable, $\\mu(A) > 0$, and $\\mu$ is an atomless measure, does there exist an $n \\in \\mathbb{N}$ and a measurable set $B$ such that $B \\subset A_n$ and $\\mu(B... | Yes |
[
{
"content": "Determine the number of roots of the polynomial \\( p(x) = x^{12} + x^8 + x^4 + 1 \\) in the finite field \\( \\mathbb{F}_{11^2} \\).",
"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... | [PCl_3] = 0.0659 M
[Cl_2] = 0.0659 M
[PCl_5] = 2.47e-3 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... | 88750.39 |
[
{
"content": "Determine the number of homomorphisms from the group \\( \\mathbb{Z}_m \\oplus \\mathbb{Z}_n \\) to the group \\( \\mathbb{Z}_{mn} \\), where \\( m \\) and \\( n \\) are positive integers.",
"role": "user"
}
] | mn |
[
{
"content": "Determine if there exists an integer \\( m \\) such that the numbers \\( 100 + m \\cdot 101 \\) and \\( 101 - m \\cdot 100 \\) are not relatively prime.",
"role": "user"
}
] | Yes |
[
{
"content": "Let a sequence $b_1, b_2, \\ldots$ be defined as $b_1 = 1$, $b_2 = 2$, and $b_{n+1} = b_n b_{n-1}$. Calculate $b_{20}$",
"role": "user"
}
] | 2^{4181} |
[
{
"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... | 1114*10^(-5) kg/s |
[
{
"content": "Determine the minimum value of the function \\( f(a) = \\arccos(a) + \\arcsin(a) + \\arctan(a) \\) for \\( a \\in [-1, 1] \\).",
"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... | Net Income: $247 |
[
{
"content": "Fiona has an open chain with 2018 links. She can cut out links one by one, each time creating 2 or 3 new chains. What is the minimum number of links she needs to remove to be able to pay for any amount from 1 to 2018 links using some combination of her chains?",
"role": "user"
}
] | 10 |
[
{
"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... | 4.0756 mL |
[
{
"content": "Find all positive integers \\( n \\) such that the number \\( n^5 + 79 \\) consists of identical digits in its decimal representation.",
"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... | \frac{37t + \frac{33 + 12\sqrt{3}}{8}}{\sqrt{ \left( 0.75 + \frac{11}{2} t\right)^2 + \left(1 + \frac{3 \sqrt{3}}{2} t\right)^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... | 7.52e-6 |
[
{
"content": "Determine whether the following statement is true or false: If a function $g$ is discontinuous, then the sequence $(g_n)_n$, where $g_n(x) = g(x + 1/n)$, does not converge pointwise to $g$ on $\\mathbb{R}$.",
"role": "user"
}
] | B |
[
{
"content": "Given that the radius of convergence of the power series \\( \\sum_{n=1}^{\\infty}c_{n}z^{n} \\) is finite and nonzero, find the radius of convergence of the series \\( \\sum_{n=1}^{\\infty}c_{n}z^{n^{2}} \\).",
"role": "user"
}
] | 1 |
[
{
"content": "Find the maximum value of \\( P(X=1) \\) for a random variable \\( X \\) that takes non-negative natural number values, given that \\( E[X]=1 \\), \\( E[X^2]=2 \\), and \\( E[X^3]=5 \\). Use the Taylor expansion of the probability generating function at \\( z=1 \\).",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "Find the curvature of the ellipse given by the equation \\(\\frac{x^2}{a^2}+\\frac{y^2}{b^2}=1\\) at the point \\(P=(a,0)\\).",
"role": "user"
}
] | \dfrac{a}{b^2} |
[
{
"content": "Determine whether the following statement is true or false: If a number written in the decimal system is read in the ternary system and the resulting number is divisible by 7, then the number written in the decimal system is also divisible by 7.",
"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... | -655 |
[
{
"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... | 1.0494e-10 N |
[
{
"content": "Evaluate the integral \\( \\int (10x^2 + 3y^3 - 4z^2 + 8) \\delta(2\\vec{r}) \\, d^3\\vec{r} \\), where \\( \\delta(2\\vec{r}) \\) is a three-dimensional delta function.",
"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... | t=2 |
[
{
"content": "Let $f : \\mathbb{R} \\to \\mathbb{R}$ be a function such that\n\\[f(f(x) + y) = f(x^2 - y) + 4f(x) y\\]for all real numbers $x$ and $y.$\n\nLet $n$ be the number of possible values of $f(3),$ and let $s$ be the sum of all possible values of $f(3).$ Find $n \\times s.$",
"role": "user"
}
] | 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... | $2,140.26 |
[
{
"content": "Let $a$ and $b$ be the roots of $k(x^2 - x) + x + 5 = 0.$ Let $k_1$ and $k_2$ be the values of $k$ for which $a$ and $b$ satisfy\n\\[\\frac{a}{b} + \\frac{b}{a} = \\frac{4}{5}.\\]Find\n\\[\\frac{k_1}{k_2} + \\frac{k_2}{k_1}.\\]",
"role": "user"
}
] | 254 |
[
{
"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": "Determine the minimum number of distinct slopes formed by connecting each point from a set of $x_1$ points on the positive x-axis to each point from a set of $x_2$ points on the negative y-axis.",
"role": "user"
}
] | x_1 + x_2 - 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... | $\Psi(x) = \exp(-\frac{m\omega}{2\hbar}x^2)$ |
[
{
"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... | 132 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... | 2.51 x 10^-3 T |
[
{
"content": "Evaluate the limit: $$\\lim_{x\\rightarrow 0} \\frac{\\sqrt{1+ x\\sin(x)} - \\sqrt{\\cos(2x)}}{ \\tan^2(x/2)}$$",
"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... | 3e9 meters |
[
{
"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... | 1055.55J |
[
{
"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... | $3,320.87 |
[
{
"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... | 289 K |
[
{
"content": "Define the function \\( f(x,y,z) = \\frac{x\\sin x + y\\sin y + z\\sin z}{x^2 + y^2 + z^2} \\) at \\((0,0,0)\\) such that it is continuous at that point.",
"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... | 0.673 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... | 17.18% |
[
{
"content": "Evaluate the double integral \\( \\int\\int_{R} xy \\, dx \\, dy \\) over the region \\( R \\) bounded by the lines \\( y = x \\), \\( y = 2x \\), and \\( x + y = 2 \\) using the change of variables \\( u = \\frac{y}{x} \\) and \\( v = \\frac{1}{x} \\).",
"role": "user"
}
] | \dfrac{13}{81} |
[
{
"content": "Identify a Banach space that is isometric to the quotient space $C[0,1]/M$, where $M=\\{f\\in C[0,1]:f(0)=0\\}$ and $C[0,1]$ is equipped with the supremum norm.",
"role": "user"
}
] | \mathbb{R} |
[
{
"content": "Find an equation of the plane that passes through the line of intersection of the planes \\(x-y=1\\) and \\(y+2z=3\\) and is perpendicular to the plane \\(x+y-2z=1\\).",
"role": "user"
}
] | x - y = 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... | 41000 |
[
{
"content": "Evaluate the limit using L'Hôpital's rule: $$\\lim_{x \\to 0^+} x^{x^x - 1}$$",
"role": "user"
}
] | 1 |
[
{
"content": "Find the minimum value of the function \\( f(t) = (t^2 - \\frac{1}{2})^2 + (t - 16)^2 \\).",
"role": "user"
}
] | \dfrac{833}{4} |
[
{
"content": "Let $f: X \\to Y$ be a linear map between normed spaces. If $f$ is an isometric isomorphism, does it follow that $f^{-1}$ is also an isometry?",
"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... | $1,380.13 |
[
{
"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... | Kinetic energy of particle A = 22.75 joules
Kinetic energy of particle B = 45.25 joules |
[
{
"content": "Given that $x^2 y + y^2 z + z^2 x = 2186$ and $xy^2 + yz^2 + zx^2 = 2188$, where $x, y, z$ are integers, find the value of $x^2 + y^2 + z^2$. \n",
"role": "user"
}
] | 245 |
[
{
"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... | 4.91 |
[
{
"content": "In statistical hypothesis testing, determine whether every unbiased test is also a Uniformly Most Powerful (UMP) test, assuming that the UMP test exists. Answer with \"True\" or \"False.\"",
"role": "user"
}
] | False |
[
{
"content": "Find the limit of the function \\( v(x,t) \\) as \\( t \\to \\infty \\), where \\( v(x,t) \\) is a bounded solution of the heat equation \\( v_t = v_{xx} \\) with the initial condition \\( v(x,0) = \\frac{e^{-x^2}-1}{e^{-x^2}+1} \\).",
"role": "user"
}
] | -1 |
[
{
"content": "Evaluate the limit \\( \\lim_{Z \\to Zi} (Z^2 - \\bar{Z}) \\).",
"role": "user"
}
] | -1 + i |
[
{
"content": "Two different integers from 1 through 20 inclusive are chosen at random. What is the probability that both numbers are prime? Express your answer as a common fraction.",
"role": "user"
}
] | \dfrac{14}{95} |
[
{
"content": "Determine the value of \\( \\sup \\{ c \\mid c > 0, n! \\in O((\\frac{n}{c})^n) \\} \\).",
"role": "user"
}
] | 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... | 3 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... | The Lewis structure of silicon tetrachloride:
<img src="/cimages/multimages/16/silicon_tetrachloride5410401761481837813.png"> |
[
{
"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... | 5.40% |
[
{
"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... | 1.84*10^9J |
[
{
"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... | 1.70e-4 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... | 7.18% |
[
{
"content": "Does there exist a separable metric compactification \\( \\overline{X} \\) of a separable metric space \\( X \\) such that \\( X \\) is open in \\( \\overline{X} \\)?",
"role": "user"
}
] | Yes |
[
{
"content": "Let $G$ be an infinite, at most countable discrete group acting freely on a standard probability measure space $X$ in a measure-preserving way. Given a natural number $N_0$ and a positive real number $\\varepsilon_0$, does there exist an $(\\varepsilon, N)$-fundamentalish domain with $\\varepsilon... | Yes |
[
{
"content": "Given positive integers $a$ and $b$, find the sum $a + b$ when $a, x, y, z, b$ are in both arithmetic and harmonic progression. It is known that $xyz = 55$ and $\\frac{1}{x} + \\frac{1}{y} + \\frac{1}{z} = \\frac{343}{55}$. The relations for arithmetic progression are $x = \\frac{3a + b}{4}$, $y =... | 8 |
[
{
"content": "Determine whether there exists a function that is non-zero only at a countable number of points, yet has a graph that is dense in \\( \\mathbb{R}^2 \\).",
"role": "user"
}
] | Yes |
[
{
"content": "Simplify $\\frac{10a^3}{55a^2}$ when $a=3$.",
"role": "user"
}
] | \frac{6}{11} |
[
{
"content": "How many degrees are in the acute angle formed by the hands of a clock at 3:30?",
"role": "user"
}
] | 75^\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... | 8.23 m/s |
[
{
"content": "Determine whether the series \\(\\sum _{n=1}^{\\infty }\\frac{(-1)^n}{(3n)!}\\) is convergent or divergent. If it is convergent, find the number of terms needed to approximate the sum to within \\(10^{-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... | 0.18 g |
[
{
"content": "Find the value of the infinite power tower \\(3^{3^{3^{\\dots}}}\\) modulo 100, where the power tower continues until the value stabilizes.",
"role": "user"
}
] | 87 |
[
{
"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... | 0.85 eV |
[
{
"content": "Find the maximum value of the real number \\( m \\) such that for the quadratic equation \\( ax^2 + bx + c = 0 \\) with real coefficients and real roots, the inequality \\((a-b)^2 + (b-c)^2 + (c-a)^2 \\geq m a^2\\) holds.",
"role": "user"
}
] | \dfrac{9}{8} |
[
{
"content": "If $6a^2 + 5a + 4 = 3,$ then what is the smallest possible value of $2a + 1$?",
"role": "user"
}
] | 0 |
[
{
"content": "Determine whether, for any nonnegative number $h$, there exists a measure-preserving transformation with entropy equal to $h$. 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... | 17.7 g |
[
{
"content": "Evaluate the Lebesgue integral of the function \\( f(x) \\) over the interval \\([0,1]\\), where\n\\[ f(x) = \\begin{cases} 1 & \\text{if } x \\text{ is irrational}, \\\\ 0 & \\text{if } x \\text{ is rational}. \\end{cases} \\]",
"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... | 254 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... | 72 mm |
[
{
"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... | 0.82kgm^2s^-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... | 2,400 Units |
[
{
"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... | 4.90 |
[
{
"content": "Compute the integral \\( \\int_R (x^2-y^2)\\,dx\\,dy \\) over the region \\( R \\) in the first quadrant defined by the inequalities \\( 3 \\geq y-x \\geq 0 \\) and \\( 5 \\geq xy \\geq 2 \\). Use the change of variables \\( u = y-x \\) and \\( v = xy \\) to evaluate the integral.",
"role": "u... | -\dfrac{27}{2} |
[
{
"content": "Establish an isomorphism between the quotient space $V / \\ker(\\varphi)$ and the field $\\mathbf{F}$, where $\\varphi \\in \\mathcal{L}(V, \\mathbf{F})$, $V$ is an infinite-dimensional vector space over the field $\\mathbf{F}$, and $\\ker(\\varphi)$ is the kernel of $\\varphi$. Provide a detailed... | V / \ker(\varphi) \cong \mathbf{F} |
[
{
"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... | civil right |
[
{
"content": "By joining alternate vertices of a regular hexagon with edges $4$ inches long, two equilateral triangles are formed, as shown. What is the area, in square inches, of the region that is common to the two triangles? Express your answer in simplest radical form. [asy]\ndraw((0,3)--(0,8)--(4,11)--(8,8... | 8\sqrt{3} \text { square inches} |
[
{
"content": "Determine the norm of the linear functional $L: F \\rightarrow \\mathbb{R}$, where $F = \\{g \\in C([c,d]): g(c) = 0\\}$ and $L(g) = \\int_c^d g(y) \\, dy$. Use the norm $\\|g\\| = \\int_c^d |g(y)| \\, dy$.",
"role": "user"
}
] | 1 |
[
{
"content": "Find a function that is not bounded below and whose second derivative is bounded above.",
"role": "user"
}
] | -x^2 |
[
{
"content": "Evaluate the limit \\( \\lim_{x\\to +\\infty}\\frac {\\zeta(1+\\frac 1x)}{\\Gamma(x)} \\), where \\(\\zeta\\) is the Riemann zeta function and \\(\\Gamma\\) is the gamma function.",
"role": "user"
}
] | 0 |
[
{
"content": "Determine the values of \\( c \\) such that Alice wins for all sufficiently large \\( n \\) in the following game: Bob writes \\( n \\) integers on the board, not all equal. On each turn, a player erases two numbers and writes their arithmetic mean. Alice starts and can make at most \\( cn \\) mov... | \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... | 17.8 cm^3 |
[
{
"content": "Let $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ be a continuous function such that $\\lim _{n \\rightarrow \\infty} f^{n}(x)$ exists for every $x \\in \\mathbb{R}$, where $f^{n}(x)=f \\circ f^{n-1}(x)$ for $n \\geq 2$. Define $S = \\left\\{\\lim _{n \\rightarrow \\infty} f^{n}(x): x \\in \\mathbb{R}... | C |
[
{
"content": "Evaluate the limit \\( I = \\lim_{n \\to \\infty} \\int_{0}^{1} x^2 \\left(1+\\frac{x}{n}\\right)^n \\, dx \\).",
"role": "user"
}
] | e - 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... | 50.5 % |
[
{
"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... | 3 min |
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