prompt list | ground_truth string |
|---|---|
[
{
"content": "Jan is thinking of a positive integer. Her integer has exactly 16 positive divisors, two of which are 12 and 15. What is Jan's number?",
"role": "user"
}
] | 120 |
[
{
"content": "Let $p$ be an integer, and let the roots of\n\\[f(x) = x^4 - 6x^3 + 26x^2 + px + 65\\]be $a_k + ib_k$ for $k = 1,$ $2,$ $3,$ $4.$ Given that the $a_k,$ $b_k$ are all integers, and that none of the roots are real, find $p.$",
"role": "user"
}
] | -46 |
[
{
"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... | 8e-19 J |
[
{
"content": "Let $r_1,$ $r_2,$ $r_3,$ and $r_4$ be the roots of\n\\[x^4 - 2x^3 - 5x^2 + 4x - 1 = 0.\\]Find the monic polynomial, in $x,$ whose roots are $\\frac{1}{r_1},$ $\\frac{1}{r_2},$ $\\frac{1}{r_3},$ and $\\frac{1}{r_4}.$",
"role": "user"
}
] | x^4 - 4x^3 + 5x^2 + 2x - 1 |
[
{
"content": "In a general topological space, can a point be a limit point of a sequence without any subsequence converging to it?",
"role": "user"
}
] | Yes |
[
{
"content": "Find the limit of the expression \\( \\sqrt[n]{2^n-n^2} \\) as \\( n \\to \\infty \\).",
"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... | 20.8 s |
[
{
"content": "Find the value of \\( \\lim_{x \\to 0} f(x) \\) if the function \\( f(x)=\\lim_{n\\to\\infty} \\frac{(1-\\cos(1-\\tan(\\frac{\\pi}{4}-x)))(1+x)^n+\\lambda\\sin((n-\\sqrt{n^2-8n})x)}{x^2(1+x)^n+x} \\) is continuous at \\( x=0 \\).",
"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... | 2.45e-4 |
[
{
"content": "Find the solution set of the equation \\(3^x = 2^x + 1\\) using algebraic methods.",
"role": "user"
}
] | 1 |
[
{
"content": "Evaluate the limit \\( \\lim_{n\\rightarrow\\infty}\\frac{1}{n}\\sum^n_{k=2} (k+1)^{\\frac{1}{k+1}} \\).",
"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... | 19.8kg |
[
{
"content": "Given that \\( a + b + c = 1 \\) where \\( a, b, c \\in \\mathbf{R}^{+} \\), determine the maximum value of \\( m \\) such that \\( \\sqrt{4a+1} + \\sqrt{4b+1} + \\sqrt{4c+1} > m \\).",
"role": "user"
}
] | 2 + \sqrt{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... | 0.0185 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{r \\to \\infty} \\frac{\\int_0^{\\pi/2}\\left(\\frac{x}{\\pi/2}\\right)^{r-1}\\cos x\\,dx}{\\int_0^{\\pi/2}\\left(\\frac{x}{\\pi/2}\\right)^r\\cos{x}\\,dx} \\]",
"role": "user"
}
] | 1 |
[
{
"content": "Suppose we have two fields $F$ and $F'$ with $F \\subseteq F'$. If a matrix $A \\in M_{n}(F)$ is diagonalisable over $F$, does it follow that $A$ is also diagonalisable over $F'$? Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "What is the ordered pair of integers $(x,y)$ for which $12x + 21y = 15$ and $21x + 12y = 51$?",
"role": "user"
}
] | (3,-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... | 23.5 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... | 32L |
[
{
"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... | 7.128 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... | 1*(000001)* |
[
{
"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... | 88.95 |
[
{
"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... | 11.025 m |
[
{
"content": "What is a norm on \\( \\mathbb{Z} \\) that is similar to the norm defined for \\( \\mathbb{Z}[w] \\) in the context of algebraic integers, where \\( w \\) is an element of \\( \\mathscr{O} \\)? Is it simply \\( N(n) = |n| \\)?",
"role": "user"
}
] | N(n) = |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... | -2.22 m/s |
[
{
"content": "Is a finite-dimensional topological vector space over the real numbers locally compact?",
"role": "user"
}
] | Yes |
[
{
"content": "Calculate the integral \\(\\int_{\\Omega}{y \\, dx \\, dy}\\) over the region \\(\\Omega=\\{(x,y)\\in \\mathbb{R}^2:0<x<y<2x<2\\}\\). Verify the correct boundaries for the integral to ensure an accurate result.",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "Find the smallest positive integer \\( k \\) such that \\( k! \\cdot S_{2016} \\) is an integer, where \\( S_n \\) is the sum of the reciprocals of the non-zero digits of all positive integers up to and including \\( n \\).",
"role": "user"
}
] | 7 |
[
{
"content": "Calculate the volume of the solid bounded above by the plane \\( z = y \\) and below by the paraboloid \\( z = x^2 + y^2 \\).",
"role": "user"
}
] | \dfrac{\pi}{32} |
[
{
"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... | 4.00 atm |
[
{
"content": "Compute the limit of the following integral: \\[ \\lim_{N \\rightarrow \\infty} \\frac{1}{\\sqrt{N}} \\int_{-b}^b e^{-\\frac{Nx^{2}}{2}}g(x)\\,dx \\] where $b > 0$ and $g(x)$ is a continuous function defined on $[-b, b]$. Use a change of variables to simplify the computation.",
"role": "user"
... | 0 |
[
{
"content": "A number $x$ is 3 larger than its reciprocal. What is the value of $\\left(x-\\frac{1}{x}\\right)^4$?",
"role": "user"
}
] | 81 |
[
{
"content": "Suppose $656_7=3ab_{10}$, where $a$ and $b$ represent base-10 digits. Find $\\frac{a\\cdot b}{15}$.",
"role": "user"
}
] | 1 |
[
{
"content": "Determine whether there exist distinct natural numbers \\( a \\), \\( b \\), and \\( c \\) such that \\( \\frac{a + 1}{a} = \\frac{1}{2} \\left( \\frac{b + 1}{b} + \\frac{c + 1}{c} \\right) \\).",
"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... | $18.45, $19.65 and $61.03 respectively |
[
{
"content": "Solve the equation for x in terms of k: \\[ 0 = (x+k)e^{-(x+k)^2}+(x-k)e^{-(x-k)^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... | Technological Developments |
[
{
"content": "Given the Maurer-Cartan equation: \\(\\frac{\\partial f}{\\partial\\theta_{2}}-f=f\\frac{\\partial g}{\\partial\\theta_{1}}\\), where \\(f\\) is not identically zero, determine if \\(f\\) must be discontinuous on \\(S^{1}\\times S^{1}\\).",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate the double integral \\( \\iint_D 2x \\cos(2(x-y)x) \\, dx \\, dy \\), where \\( D \\) is the rectangle with corners at \\((\\pi/6, 2+\\pi/6), (\\pi/6, 4+\\pi/6), (\\pi/4, 4+\\pi/4), \\text{ and } (\\pi/4, 2+\\pi/4)\\).",
"role": "user"
}
] | -\dfrac{5}{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... | 1/2 * (PM-PC) * (QC-QM) |
[
{
"content": "Given the function \\( V(x(t)) = \\frac{x''}{x} \\), where the prime denotes differentiation with respect to time, find \\( \\frac{dV(x(t))}{dx} \\).",
"role": "user"
}
] | \dfrac{x''' x - x'' x'}{x^2 x'} |
[
{
"content": "Determine whether the Global Approximation Theorem holds for domains with piecewise $C^1$ boundaries that satisfy the segment condition.",
"role": "user"
}
] | Yes |
[
{
"content": "If $u_n: G \\rightarrow \\mathbb{R}$ is a sequence of harmonic functions that converges uniformly to $u: G \\rightarrow \\mathbb{R}$, must $u$ also be harmonic?",
"role": "user"
}
] | Yes |
[
{
"content": "In the Erdős–Rényi model $G(n, p)$, calculate the expected number of different spanning trees of the graph $G$. Provide your answer as a function of $n$ and $p$.",
"role": "user"
}
] | n^{n-2} p^{n-1} |
[
{
"content": "Solve the equation \\(\\frac{x(\\log_{2}(x))^2}{2} = x^{3/2}\\) for \\(x\\).",
"role": "user"
}
] | 4 |
[
{
"content": "Can every smooth curve (i.e., smooth real 1-dimensional manifold) be embedded in the real plane \\( \\mathbb{R}^2 \\)?",
"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... | Ag |
[
{
"content": "A mouse is sitting in a toy car on a negligibly small turntable. The car cannot turn on its own, but the mouse can control when the car is launched and when the car stops. When launched, the car travels in a straight line at 1 meter per second. The turntable spins at 30 rpm. Determine the area, in... | \dfrac{\pi}{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... | 32956 |
[
{
"content": "Let \\( G \\) be a finite group and \\( \\chi \\) its character. If the degree of \\( \\chi \\) is greater than 1, is it always true that \\( \\langle \\chi, \\chi \\rangle \\ge 1 \\)? Justify your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Given a bivariate normal distribution $(X, Y)$ where both $X$ and $Y$ have a mean of $0$ and variance of $1$, and a correlation coefficient of $\\rho$, find the correlation between $X^2$ and $Y^2$ using conditional expectations.",
"role": "user"
}
] | \rho^2 |
[
{
"content": "Consider a family of functions \\( f_n : \\mathbb{N} \\cup \\{0\\} \\to \\mathbb{N} \\cup \\{0\\} \\) that follows the relations: \\( f_n(a+b) = f_n(f_n(a)) + f_n(b) \\) and \\( f_n(kn) = 0 \\) where \\( n \\) is any positive integer greater than 1. Find the minimum number of elements in the range... | 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... | 1.42 Hz |
[
{
"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... | 5.01 m/sec |
[
{
"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... | belts of volcanoes |
[
{
"content": "Given that \\( \\lim _{ x\\rightarrow 0 }{ \\frac { f(x) }{ { x }^{ 2 } } =a } \\) and \\( \\lim _{ x\\rightarrow 0 }{ \\frac { f(1-\\cos x) }{ g(x)\\sin^2x } = b }\\), where \\(b \\neq 0\\), find \\( \\lim _{ x\\rightarrow 0 }{ \\frac { g(1-\\cos2x) }{ x^4 } }\\).",
"role": "user"
}
] | \dfrac{a}{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... | Cl2CHCH2OH < ClCH2CH2OH < CH3CH2OH |
[
{
"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... | $31,758.57 |
[
{
"content": "Find the curvature of the curve \\( c(t) = \\int^t_0 r(u) \\, du \\), where \\( r(t) \\) is a unit speed curve on a sphere of radius \\( R \\), in terms of \\( R \\). Do not use the formula for curvature of a non-unit speed curve.",
"role": "user"
}
] | \dfrac{1}{R^2} |
[
{
"content": "Does there exist a non-recursive sequence of natural numbers that has no recursive subsequence?",
"role": "user"
}
] | Yes |
[
{
"content": "Let $f(x)$ be an irreducible polynomial of degree 4 over a field $F$. If the splitting field of $f(x)$ over $F$ has a Galois group isomorphic to the dihedral group $D_4$, determine the possible number of distinct roots of $f(x)$. \\( \\boxed{4} \\)",
"role": "user"
}
] | 4 |
[
{
"content": "Evaluate the limit: $$\\lim_{n \\to \\infty} \\int_0^1 \\frac{dx}{x^n + 1}$$",
"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... | 10360 km |
[
{
"content": "Evaluate \\( \\lim _{n \\rightarrow \\infty} \\int_{0}^{1} n x^{n} e^{x^{2}} d x \\).",
"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... | 22.4 m |
[
{
"content": "Calculate the sum of the first 99 terms of the sequence defined by \\( T_n = \\frac{1}{5^{2n-100} + 1} \\).",
"role": "user"
}
] | \dfrac{99}{2} |
[
{
"content": "The solid shown has a square base of side length $s$. The upper edge is parallel to the base and has length $2s$. All other edges have length $s$. Given that $s=6\\sqrt{2}$, what is the volume of the solid?\n[asy] size(180); import three; pathpen = black+linewidth(0.65); pointpen = black; currentp... | 288 |
[
{
"content": "Let $J$ be a Jordan block of size $n$ with eigenvalue $\\alpha$ such that $|\\alpha| = 1$. If $\\lim_{k\\to\\infty} J^k v = 0$, does it imply that the $n$th entry of $v$, denoted as $v_n$, is zero?",
"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... | 1.73*10^-6 C |
[
{
"content": "Determine whether the Lindenbaum algebra generated by an infinite set $P$ of propositional variables is atomless.",
"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... | 21.74e3 W |
[
{
"content": "Given the polynomial equation \\( x^4 + 3x^3 - 13x^2 - 51x - 36 = 0 \\), determine the coefficient of \\( x^3 \\) in the new polynomial equation whose roots are each 2 more than the roots of the original equation.",
"role": "user"
}
] | -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... | 0.65 |
[
{
"content": "In the figure, triangles $ABC$ and $BCD$ are equilateral triangles. What is the value of $AD \\div BC$ when expressed in simplest radical form?\n\n[asy]\ndraw((0,0)--(5,8.7)--(10,0)--cycle);\ndraw((10,0)--(15,8.7)--(5,8.7));\nlabel(\"$A$\",(0,0),SW);\nlabel(\"$B$\",(5,8.7),N);\nlabel(\"$C$\",(10,0... | \sqrt{3} |
[
{
"content": "Let \\( L: \\mathbb{R}^n \\to \\mathbb{R}^m \\) be a linear function. Prove that if \\( \\lim_{\\hat{h} \\to \\hat{0}} \\frac{\\|L(\\hat{h})\\|}{\\|\\hat{h}\\|} = 0 \\), then \\( L \\equiv 0 \\).",
"role": "user"
}
] | L \equiv 0 |
[
{
"content": "Calculate the outward flux of the vector field \\( F = (x^3, y^3, z^2) \\) across the surface of the region enclosed by the circular cylinder \\( x^2 + y^2 = 49 \\) and the planes \\( z = 0 \\) and \\( z = 2 \\).",
"role": "user"
}
] | 7399\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... | Long-term or non-current liabilities |
[
{
"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... | 72600 |
[
{
"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 net present value of system A and B are -$378,249.05 and -$516,277.65 respectively. |
[
{
"content": "In a tennis tournament, there are participants from $n$ different countries. Each team consists of a coach and a player who need to be accommodated in a hotel. The rooms for coaches are separate from those for players. Each player wants to share a room with only those from countries that have a de... | n + 1 |
[
{
"content": "Determine the number of inflection points of the function \\((x-2)^6(x-3)^9\\).",
"role": "user"
}
] | 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... | 13.5 |
[
{
"content": "Determine the number of abelian groups up to isomorphism with an order of \\(6^5 = 2^5 \\cdot 3^5\\).",
"role": "user"
}
] | 49 |
[
{
"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... | 29.7 degrees Celsius |
[
{
"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... | 7.741% |
[
{
"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... | 108 mL |
[
{
"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... | 7.13 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... | 4.66e-4 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... | 30.63 degrees Celcius |
[
{
"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... | 400L |
[
{
"content": "Determine whether the sequence \\( a_n = \\frac{1}{n} + \\frac{1}{n+1} + \\dots + \\frac{1}{2n} \\) is convergent. If it is, find its limit.",
"role": "user"
}
] | \ln 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... | 200J |
[
{
"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... | 17.05 x 10^6 m |
[
{
"content": "Let \\( A = (a_{ij})_{i,j=1,\\dots,n} \\) be a matrix such that \\( a_{ij} \\ge 0 \\) for all \\( i,j = 1, \\dots, n \\), and \\( A \\) is positive definite. Let \\( I \\) be the identity matrix, and \\( \\pmb{1} \\) the vector containing only ones. Suppose that the solution \\( x = (x_i)_{i=1,\\d... | Yes |
[
{
"content": "For which \\( \\alpha > 0 \\) does the equality \\( \\int_0^1 \\sum_{n=0}^\\infty x^\\alpha e^{-nx} \\, dx = \\sum_{n=0}^\\infty \\int_0^1 x^\\alpha e^{-nx} \\, dx \\) hold?",
"role": "user"
}
] | \alpha > 0 |
[
{
"content": "Find the value of $b$ that minimizes the function \\( z = h(u) = u^3\\left(\\frac{1}{A} - \\frac{B}{A^2}\\right) + u^2\\left(\\frac{\\epsilon B}{A^2} - 1 - \\frac{\\epsilon}{A} + \\frac{B}{A}\\right) - \\frac{u}{A} + \\frac{b}{B} \\) subject to the conditions \\( A > 0 \\), \\( B > 0 \\), \\( \\ep... | 0 |
[
{
"content": "Is it true that if \\((T_n)\\) is a sequence of consistent estimators for a parameter \\(\\theta\\), then \\(f(T_n)\\) is a sequence of consistent estimators for \\(f(\\theta)\\), given that \\(f\\) is a continuous function?",
"role": "user"
}
] | Yes |
[
{
"content": "Find an irreducible polynomial with integer coefficients that has \\(\\sqrt{2} + \\sqrt{3}\\) as a root.",
"role": "user"
}
] | x^4 - 10x^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... | 9.09 g |
[
{
"content": "Let $\\Gamma$ be a non-elementary hyperbolic group acting on the Gromov boundary $\\partial\\Gamma$. If $a \\in \\Gamma$ is a torsion element, does $a$ fix every element of $\\partial\\Gamma$? Answer yes or no.",
"role": "user"
}
] | No |
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