prompt list | ground_truth string |
|---|---|
[
{
"content": "Determine all points where the function \\( f(z) = \\mathbb{Re}(z) \\cdot |z| \\) is complex differentiable.",
"role": "user"
}
] | 0 |
[
{
"content": "Evaluate the limit \\( \\lim_{s \\to \\infty} s^n P(|Y| > s) \\), where \\( Y \\) is a random variable such that \\( E[|Y|^n] < \\infty \\) for \\( n > 0 \\).",
"role": "user"
}
] | 0 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | horizontal |
[
{
"content": "Evaluate the limit: $$ \\lim_{n\\to \\infty}\\prod_{k=1}^{2n} \\left(1+ \\frac{ k}{n} \\right)^{1/n}. $$",
"role": "user"
}
] | \dfrac{27}{e^2} |
[
{
"content": "Let $M$ be a square matrix such that $M_{i,j}=c$ if $i\\not=j$ and $d$ if $i=j$, where $0<c<d$. If $A$ is a square matrix such that $AA^{t}=M$, is it necessarily the case that $A^{t}A=M$? Justify your answer.",
"role": "user"
}
] | No |
[
{
"content": "Given the sequence \\(\\{a_n\\}\\) defined by:\n\\[\na_1 = 0, \\quad a_2 = 1, \\quad a_n = \\frac{1}{2} n a_{n-1} + \\frac{1}{2} n(n-1) a_{n-2} + (-1)^n \\left(1 - \\frac{n}{2}\\right) \\quad (n \\geq 3),\n\\]\nfind the simplest expression for \\(f_n = a_n + 2 \\binom{n}{1} a_{n-1} + 3 \\binom{n}{... | 2n! - n - 1 |
[
{
"content": "Determine the number of complex solutions for the function \\( f(x) = (|x| + x^2)e^{-x} \\).",
"role": "user"
}
] | 3 |
[
{
"content": "Let $f(z)$ be an entire function such that $|f(z)| < K|z|$ for all $z \\in \\mathbb{C}$, where $K > 0$. Given that $f(1) = i$, determine the value of $f(i)$. Choose from the following options:\n\n1. $1$\n2. $-1$\n3. $i$\n4. $-i$",
"role": "user"
}
] | 2 |
[
{
"content": "Let $A$ be a $7\\times 7$ matrix with rank $3$, and let $a$ be a $7\\times 1$ column vector. Determine the least possible rank of the matrix $A + (a a^T)$. ($a^T$ denotes the transpose of the vector $a$.)",
"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... | 28.79% |
[
{
"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... | 1996.4 m/s |
[
{
"content": "Determine whether a symmetric matrix $A$ that commutes with every other symmetric matrix must be of the form $A = \\lambda I$ for some $\\lambda \\in \\mathbb{R}$. Provide a proof or counterexample.",
"role": "user"
}
] | A = \lambda I |
[
{
"content": "A regular triangular prism \\( A B C A_{1} B_{1} C_{1} \\) is inscribed in a sphere. The base of the prism is \\( A B C \\), and the lateral edges are \\( A A_{1}, B B_{1}, C C_{1} \\). Segment \\( C D \\) is the diameter of the sphere. Points \\( K \\) and \\( L \\) are the midpoints of edges \\(... | 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... | Year Hours Used Depreciation Book Value
1 2,900 47,676 183,324
2 1,800 29,592 153,732
3 1,900 31,236 122,496
4 2,100 34,524 87,972
5 1,900 31,236 56,736
6 1,900 31,236 25,500 |
[
{
"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... | 74,219.71 |
[
{
"content": "Let $L$ be a first order language, and let $M$ and $N$ be $L$-structures such that $N$ is finite. Suppose $h: N \\to M$ is an elementary embedding. Is $h$ an isomorphism of $L$-structures?",
"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... | $10,280 |
[
{
"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... | 129682.29 |
[
{
"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... | 5.4e2 nm |
[
{
"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... | 170.80 nm |
[
{
"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... | -16.66J |
[
{
"content": "Prove that the number of odd Catalan numbers $C_{i+j}$ in the $n \\times n$ matrix $M_n = \\left(C_{i+j}\\right)_{i,j=0}^{n-1}$ is $2n-1$. Here, $C_k = \\frac{1}{k+1}\\binom{2k}{k}$ denotes the Catalan numbers.",
"role": "user"
}
] | 2n - 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... | The magnitude of F2 and F3 is 9.6 kN and 13.6 kN, respectively. |
[
{
"content": "Given a linear function \\( f(x) \\), the distance between the points of intersection of the graphs \\( y = x^2 - 1 \\) and \\( y = f(x) \\) is \\( \\sqrt{30} \\), and the distance between the points of intersection of the graphs \\( y = x^2 \\) and \\( y = f(x) + 3 \\) is \\( \\sqrt{46} \\). Dete... | \sqrt{38} |
[
{
"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... | 142.5 V |
[
{
"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... | 0.24 |
[
{
"content": "Consider the recurrent sequence $(x_n)_{n \\ge 0}$ defined by $x_0 \\in \\mathbb{R}$ and $x_{n+1} = 2^{\\frac{x_n}{2}}$. Determine the interval of initial values $x_0$ for which $\\lim\\limits_{n \\to \\infty} x_n = 2$. Choose from the following options: A. $x_0 \\in \\{2\\}$ B. $x_0 \\in [-2, 2]$... | E |
[
{
"content": "Evaluate the limit \\( \\lim_{x\\to a}\\frac{a^x-x^a}{x-a} \\) without using L'Hopital's rule or Taylor series, where \\( a > 0 \\).",
"role": "user"
}
] | a^a (\ln a - 1) |
[
{
"content": "Evaluate the integral \\( \\int_{C} (1 + z + z^2)(e^{\\frac{1}{z}}+e^{\\frac{1}{z-1}}+e^{\\frac{1}{z-2}}) \\, dz \\), where \\( C \\) is the circle \\(|z|=3\\) in the complex plane.",
"role": "user"
}
] | 32\pi i |
[
{
"content": "Diagonalize the quadratic form \\( q(x_1, x_2, x_3, x_4) = x_1x_2 + x_2x_3 + x_3x_4 \\) and determine its signature.",
"role": "user"
}
] | (2, 2) |
[
{
"content": "Find the value of \\(\\lambda\\) such that the expression \\(3x^2 + 5xy - 2y^2 - 3x + 8y + \\lambda\\) can be factored into two linear factors.",
"role": "user"
}
] | -6 |
[
{
"content": "Determine whether there exists a group \\((G, \\cdot)\\) such that the converse of the closure property holds, i.e., if \\(g_1 \\cdot g_2 \\in G\\), then \\(g_1, g_2 \\in G\\).",
"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... | exempt review |
[
{
"content": "Suppose you have a transitive countable model $M$ of $ZFC + \\mathbb{V} = \\mathbb{L}$ and you add a single Cohen generic $G$. If $A \\in M[G]$ is also Cohen generic over $\\mathbb{L}$ and $M[A] \\neq M[G]$, can you deduce that there exists a Cohen generic $G'$ over $M[A]$ such that $M[A][G'] = M[... | 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... | 10115.154 cal |
[
{
"content": "Determine the number of positive values among the numbers of the form \\( \\operatorname{ctg}\\left(\\left(15^{n}\\right)^{\\circ}\\right) \\), where \\( n \\) is a natural number from 1 to 2019.",
"role": "user"
}
] | 1010 |
[
{
"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... | 3.17e+02 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... | (3,-24.7) |
[
{
"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... | 4.06% |
[
{
"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... | $1,448,000 |
[
{
"content": "Determine whether the inequality \\( \\sum_{\\genfrac{}{}{0pt}{1}{p>N}{p\\text{ is prime}}}\\frac{1}{p^2} < \\frac{1}{N} \\) holds for every positive integer \\( N \\).",
"role": "user"
}
] | Yes |
[
{
"content": "Is an epimorphic endomorphism of a Noetherian commutative ring necessarily an isomorphism? Justify 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... | 3 m |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x\\to 0}\\frac{1}{x^4}\\int_{-x}^{0}\\sin(t^3)\\,dt. \\]",
"role": "user"
}
] | -\dfrac{1}{4} |
[
{
"content": "Let \\( f : \\mathbb{R} \\to \\mathbb{R} \\) be a function that satisfies the intermediate value property. For every integer \\( n \\ge 2 \\), does the function \\( f_n : \\mathbb{R} \\to \\mathbb{R} \\) defined by \\( f_n(x) = (f(x))^n \\) for all \\( x \\in \\mathbb{R} \\) also satisfy the inter... | 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... | 0.03 m |
[
{
"content": "Assume there is a totally ordered local basis at a point \\(x\\) in a topological space \\(X\\). Does this imply that there is a countable local basis at \\(x\\)?",
"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... | 0.1800 moles |
[
{
"content": "Evaluate the limit \\( \\lim _{x\\to \\infty} \\frac{1}{x} \\int_x^{2x} \\left|\\cos \\frac{1}{t} \\right| dt \\).",
"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... | 92,864.96 m/s |
[
{
"content": "Let $X$ be an infinite set with the co-finite topology, and let $\\varphi: G \\times X \\to X$ be a continuous group action. Consider a finite open cover $\\mathcal{U} = \\{U_1, U_2, \\ldots, U_n\\}$ of $X$. Define $A = \\cup_{i=1}^n (X - U_i)$, which is a finite set. Determine whether there exist... | 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... | 3.56 m/s |
[
{
"content": "Calculate the area bounded by the equation \\((3x + 4y + 7)^2 + (2x + y + 3)^2 = 1\\) using a double integral.",
"role": "user"
}
] | \dfrac{\pi}{5} |
[
{
"content": "Compute the integral \\( \\int_{|z|=2} \\frac{f'_n(z)}{f_n(z)} \\, dz \\), where \\( f_n(z) = 1 + \\frac{1}{z} + \\frac{1}{2!z^2} + \\cdots + \\frac{1}{n!z^n} \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Find the angle between the vector \\((2,2,1,1)\\) and the linear subspace spanned by the vectors \\((3,4,-4,-1)\\) and \\((0,1,-1,2)\\).",
"role": "user"
}
] | \dfrac{\pi}{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... | 0.028 |
[
{
"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... | 0.121 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... | 353.76 K |
[
{
"content": "Find the radius of convergence for the power series $$\\sum_{n=0}^{\\infty}{n^{1/n}x^{n}}.$$",
"role": "user"
}
] | 1 |
[
{
"content": "Consider the probability space \\((\\Omega,\\mathcal{F},P)\\) and the filtration \\(\\mathcal{F}_t\\). The Ito integral \\(I(f)(\\omega)=\\int_S^T f(t,\\omega)dB_t(\\omega)\\) is defined on a space \\(\\mathcal{V}(S,T)\\) of functions satisfying the following conditions:\n\n1. \\((t,\\omega)\\to f... | 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... | 7.20% |
[
{
"content": "Determine the values of $c$ for which the function $g(z) = f(z) - z^2 = 2 (c-i) xy$ is analytic in the complex plane $\\mathbb{C}$.",
"role": "user"
}
] | i |
[
{
"content": "Consider the discrete-time state-space system:\n\\[x(t+1) = Ax(t)+Bu(t), \\quad y(t) = Cx(t)\\]\nwith\n\\[\nA =\\begin{bmatrix}\n0 & 1 & 0 \\\\\n1 & 0 & 1 \\\\\n1 & 1 & 0 \\\\\n\\end{bmatrix}, \n\\quad B =\\begin{bmatrix}\n0 \\\\\n0 \\\\\n1 \\\\\n\\end{bmatrix}, \n\\quad C =\\begin{bmatrix}\n1 &... | 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... | //Plane.javapublic class Plane{private Airline airlineName;private String modelNumber;public Plane (Airline a, String m){airlineName = a;modelNumber = m;}public Airline getAirline(){return airlineName;}public String getModel(){return modelNumber;}public String toString(){String planedescription;planedescription = airli... |
[
{
"content": "Consider 7 points on a circle. Compute the number of ways to draw chords between pairs of points such that no two chords intersect and each point belongs to at most one chord. It is acceptable to draw no chords.",
"role": "user"
}
] | 127 |
[
{
"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... | .As: |
[
{
"content": "Evaluate the improper integral: $$\\int_0^{+\\infty}\\frac{x\\sin x}{x^2+1}\\,dx$$",
"role": "user"
}
] | \dfrac{\pi}{2e} |
[
{
"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... | 83.99 atm |
[
{
"content": "In a convex pentagon $ABCDE$, the sides have lengths $1, 2, 3, 4,$ and $5$, though not necessarily in that order. Let $F, G, H,$ and $I$ be the midpoints of the sides $AB$, $BC$, $CD$, and $DE$, respectively. Let $X$ be the midpoint of segment $FH$, and $Y$ be the midpoint of segment $GI$. The len... | 4 |
[
{
"content": "What is the greatest integer value of $x$ for which $6x^2 + x - 2 < 0$?",
"role": "user"
}
] | 0 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | -5i |
[
{
"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... | $11,000 |
[
{
"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... | $82,270 |
[
{
"content": "I have 5 red plates and 4 blue plates. If I randomly select two plates to serve dinner on, what is the probability that they're both the same color?",
"role": "user"
}
] | \frac{4}{9} |
[
{
"content": "Find the values of $k$ for which the function \\( g(x)=\\sqrt{ke^{-k^2(x-2)^2+4k(x-2)+4}} \\) is a probability density function.",
"role": "user"
}
] | 2\pi e^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... | 40000 |
[
{
"content": "Is it necessary for a finite field with order \\(p^n\\), where \\(p\\) is a prime number, to be a field extension of a subfield isomorphic to \\(\\Bbb{Z_p}\\) and consequently have characteristic \\(p\\)?",
"role": "user"
}
] | Yes |
[
{
"content": "Given the equations $x + y = xy = x^2 + y^2$ where $x$ and $y$ are real numbers, determine the value of these expressions.",
"role": "user"
}
] | 0 |
[
{
"content": "Evaluate the limit: $$\\lim\\limits_{x\\to 0^{+}}\\lim\\limits_{n\\to \\infty}\\frac{d^n}{dx^n}\\left(\\frac{e^{\\sqrt{x}}+e^{-\\sqrt{x}}}{x^n}\\right)$$",
"role": "user"
}
] | 0 |
[
{
"content": "Determine whether there exists a surface $S$ of degree $d \\geq 4$ in the complex projective space $\\mathbb{P}^3_{\\mathbb{C}}$ that does not contain any lines.",
"role": "user"
}
] | Yes |
[
{
"content": "In a Hilbert space, determine whether the annihilation operator $c(f)$ is a bounded linear operator on the fermionic Fock space. If it is bounded, find its norm.",
"role": "user"
}
] | \|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... | Pb^{2+}(aq) + 2NH_3(aq) + 2H_2O(aq) o Pb(OH)_2(s) + 2NH_4^+(aq) |
[
{
"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... | 0.8898 |
[
{
"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... | 539 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... | $132,481.57 |
[
{
"content": "Let \\( f(x) \\) be a function satisfying \\( f(x) \\geq |x|^{\\alpha} \\) for all \\( x \\) in a deleted neighborhood of zero, where \\( \\frac{1}{2} < \\alpha < 1 \\). If \\( f(x)f(2x) \\leq |x| \\) holds, determine the limit of \\( f(x) \\) as \\( x \\to 0 \\). Is it \\( \\frac{1}{\\alpha} \\),... | 0 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 0.38 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... | 47.3 cm |
[
{
"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... | -2.1 ft/s |
[
{
"content": "Let $V$ be the vector space of polynomials in the variable $t$ of degree at most $2$ over $\\mathbb{R}$. An inner product on $V$ is defined by \\(\\langle f,g \\rangle = \\int_0^1 f(t)g(t)\\,dt .\\) Let \\(W=\\operatorname{span}\\{1+t^2,1-t^2\\}\\) and $W^{\\perp}$ be the orthogonal complement of ... | 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... | The Federal Reserve System (Fed) |
[
{
"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... | 0.06181 |
[
{
"content": "Evaluate the limit of the integral: \\[ \\lim \\limits_{x \\to \\infty} \\int_0^\\infty \\sin \\left(e^{xt}\\right)\\,dt \\]",
"role": "user"
}
] | 0 |
[
{
"content": "Consider a real and positive semi-definite matrix $A \\in \\mathbb{R}^{n \\times n}$ with rank $p < n$. The matrix $A$ can be decomposed as \\( A = L D L^\\top \\), where $L$ is a lower unit triangular matrix and $D$ is a diagonal matrix with the number of its non-zero diagonal entries equal to th... | Yes |
[
{
"content": "Suppose $G$ is a group with exactly $8$ elements of order $3$. Determine the number of subgroups of order $3$ in $G$. Express your answer as an integer.",
"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... | 346.5 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... | 12.142 m |
[
{
"content": "Let $X$ be an open, simply connected subset of $\\mathbb{R}^2$ containing the point $0$. For every pair of points $p, q \\in X \\setminus \\{0\\}$, is there an open, simply connected subset $A_{pq} \\subseteq X \\setminus \\{0\\}$ such that $p, q \\in A_{pq}$?",
"role": "user"
}
] | Yes |
[
{
"content": "Given a quadratic polynomial \\( f(x) = x^2 + \\cdots \\) such that \\( f(1) = 0 \\), find the value of \\( |13a + 7b + 3c + d| \\), where \\( Q(x) = x^4 + ax^3 + bx^2 + cx + d \\) is the quotient when \\( \\{f(x)\\}^3 \\) is divided by \\( x^2 + 1 \\). Assume there is no information about the rem... | 21 |
[
{
"content": "Find the positive integer \\( n \\) such that the following equation holds:\n\\[ \\lfloor \\sqrt{9n^2+1}\\rfloor + \\lfloor\\sqrt{9n^2+2}\\rfloor + \\cdots + \\lfloor\\sqrt{9n^2+24n}\\rfloor = 349. \\]",
"role": "user"
}
] | 2 |
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