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... | $29,616 |
[
{
"content": "Given a fluid with a velocity field and density, calculate the rate of change of density in the Lagrangian frame, where the initial density \\( \\rho_0 = 2 \\) and \\( A = 1 \\).",
"role": "user"
}
] | -2 |
[
{
"content": "In a normed vector space, let \\( S \\) and \\( T \\) be disjoint compact sets. Define the distance between the sets as \\( d(S, T) = \\inf\\{||s - t|| : s \\in S, t \\in T\\} \\). Does there always exist a pair of elements \\( s \\in S \\) and \\( t \\in T \\) such that \\( d(S, T) = ||s - t|| \\... | Yes |
[
{
"content": "Evaluate the limit of the integral \\( \\lim_{n \\to \\infty} \\int_0^n \\frac{(1 - \\frac{x}{n})^n}{ne^{-x}} \\, dx \\) for \\( x > 0 \\) using the Bounded Convergence Theorem or Dominated Convergence Theorem.",
"role": "user"
}
] | 0 |
[
{
"content": "Determine whether the following statement is true or false: If $f : \\mathbb{C} \\to \\mathbb{C}$ is a holomorphic function such that $f(z) = 1$ for all $z \\in \\mathbb{C}$ with $|z| = 1$, then $f$ is a constant function with $f(z) = 1$ for all $z \\in \\mathbb{C}$. Provide a justification for yo... | 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... | 3.48 |
[
{
"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... | 360 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... | $3,165 |
[
{
"content": "What is the conditional expectation $E[Y|X,Z]$ if $Y$ is measurable with respect to the sigma-algebra generated by $X$ and $Z$?",
"role": "user"
}
] | Y |
[
{
"content": "Given an $n \\times p$ matrix $A$ and a $p \\times m$ matrix $B$, if the kernel of $A$ (ker$(A)$) is equal to the image of $B$ (im$(B)$), determine what can be concluded about the product $AB$.",
"role": "user"
}
] | AB = 0 |
[
{
"content": "Given a square matrix \\( A \\) of order \\( n \\), with the characteristic polynomial \\( p(\\lambda) = \\lambda^n + a_{n-1}\\lambda^{n-1} + \\cdots + a_0 \\), determine the trace of the adjoint of \\( A \\) using the coefficients of the characteristic polynomial.",
"role": "user"
}
] | (-1)^{n-1} a_1 |
[
{
"content": "Determine the radius of convergence of the complex power series \\( \\sum_{n=0}^{\\infty} z^{a^n} \\), where \\( a > 1 \\) is an integer.",
"role": "user"
}
] | 1 |
[
{
"content": "Given a set $E \\subset \\mathbb{R}$ with positive Lebesgue measure, determine whether there necessarily exists an $x \\in E$ such that $x = \\frac{1}{2}(y + z)$ for distinct $y, z \\in E$. 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... | 42.57 L |
[
{
"content": "Determine whether the statement is true: If $f:\\mathbb{R}_+ \\rightarrow \\mathbb{R}_+$ is a strictly increasing and concave function, then it satisfies the property $$f(ax+(1-a)y) \\le f(ax) + f((1-a)y)$$ for all $a \\in [0,1]$. Is this property correctly associated with concave functions?",
... | True |
[
{
"content": "Consider a continuous and surjective map \\( \\gamma: [0,T] \\rightarrow [0,1] \\times [0,1] \\). Compute the winding number of \\( \\gamma \\) around a point \\( a \\) where \\( a \\notin \\text{Im}(\\gamma) \\).",
"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... | 2485048.876 pound-feet |
[
{
"content": "In a network with a maximum flow of $|f|$, consider a minimum cut $(S,T)$ where an edge $e$ crosses the cut. If the edge $e$ is removed, what is the new maximum flow in the network? Express your answer in terms of $|f|$ and the capacity $c(e)$ of the edge $e$.",
"role": "user"
}
] | |f| - c(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... | 0.68 m/s |
[
{
"content": "Suppose we have a probability space \\((\\Omega, \\mathcal{F}, \\mu)\\), where \\(\\Omega\\) is a set, \\(\\mathcal{F}\\) is a \\(\\sigma\\)-algebra, and \\(\\mu\\) is a probability measure. Let \\(f: \\Omega \\rightarrow \\mathbb{R}\\) be a non-negative bounded function. Consider a sequence of se... | a |
[
{
"content": "Determine the number of different ways to completely cover a regular hexagonal chessboard, consisting of 24 small equilateral triangles, using 12 rhombus-shaped paper pieces, each composed of 2 small equilateral triangles.",
"role": "user"
}
] | 20 |
[
{
"content": "Find the smallest positive integer $M$ such that $M^{77} \\equiv 14 \\pmod{31}$. \\(\\text{Express your answer as a single integer.}\\)",
"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... | 0.15s |
[
{
"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... | 0.15 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... | 6.04e9 years |
[
{
"content": "Determine the values of \\( \\lambda > 0 \\) for which the integral \\( \\int_0^\\infty e^{\\sin x} \\frac{\\sin 2x}{x^\\lambda} \\, dx \\) converges.",
"role": "user"
}
] | (0, 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... | AB_2X_2 |
[
{
"content": "Evaluate the integral \\( \\int_{0}^{\\infty}\\frac{dx}{1+x^2+x^4} \\) and determine the value of \\( n \\) such that \\( \\int_{0}^{\\infty}\\frac{dx}{1+x^2+x^4} = \\frac{\\pi \\sqrt{n}}{2n} \\). Choose the correct option from the following: \\( A) \\ 1, \\ B) \\ 2, \\ C) \\ 3, \\ D) \\ 4, \\ E) ... | 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... | 0.11 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... | b, c, d |
[
{
"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... | 236 atm |
[
{
"content": "Given a function $f \\in L^p(\\mathbb{R})$ with $1 \\leq p < \\infty$ and a positive tolerance $\\epsilon$, determine if there exists a radius $R > 1$ such that the $L^p$ norm of $f$ over the set $B_R = \\{x: |x| \\geq R\\}$ is less than $\\epsilon$. Provide a justification for your answer.",
... | Yes |
[
{
"content": "Solve for $n$: $\\frac{2-n}{n+1} + \\frac{2n-4}{2-n} = 1$.",
"role": "user"
}
] | -\frac{1}{4} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 0.02678 |
[
{
"content": "In the parabola defined by the equation \\( y^2 = 2px \\) with focus \\( F \\) and directrix \\( l \\), a line through \\( F \\) intersects the parabola at points \\( A \\) and \\( B \\) such that \\( |AB| = 3p \\). Let \\( A' \\) and \\( B' \\) be the projections of \\( A \\) and \\( B \\) on the... | \dfrac{1}{3} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 1.3 kg |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 23% |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n\\to\\infty}\\frac{1+\\sqrt[n]{2}+\\sqrt[n]{3}+\\cdots+\\sqrt[n]{n}}{n} \\]",
"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... | 202.987 MJ |
[
{
"content": "Determine whether the limit of the function \\( f(x,y) = \\frac{(5\\cos y)(\\sin y-x)}{4\\left | x-\\sin y \\right |^{3/4}} \\) exists as \\((x,y)\\) approaches \\((0,0)\\).",
"role": "user"
}
] | 0 |
[
{
"content": "Determine the set of complex numbers $c$ for which there exists an entire function $f$ such that $f(1/k) = c^k$ for all positive integers $k$. Use the identity theorem in your analysis.",
"role": "user"
}
] | \{0, 1\} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 16.4 m/s |
[
{
"content": "The base of the pyramid \\( SABC \\) is a triangle \\( ABC \\) with \\( AB = AC = 10 \\) cm and \\( BC = 12 \\) cm. The face \\( SBC \\) is perpendicular to the base, and \\( SB = SC \\). Given that the height of the pyramid is 1.4 cm, calculate the radius of the sphere inscribed in the pyramid.",... | \dfrac{12}{19} |
[
{
"content": "Evaluate the limit \\( \\lim_{n \\to \\infty} \\left(\\cos\\left(\\frac{3\\pi n}{4n+2}\\right)\\right)^n \\) rigorously.",
"role": "user"
}
] | 0 |
[
{
"content": "Evaluate the improper integral \\( \\int_{-\\infty}^{\\infty} \\frac{1}{(1+x^2)^3} \\, dx \\) using the residue theorem.",
"role": "user"
}
] | \dfrac{3\pi}{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... | 5460 K |
[
{
"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... | 1.85 eV |
[
{
"content": "Determine the number of pairwise non-isomorphic 2-connected graphs with 20 vertices that do not contain a $K_4^-$ minor.",
"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... | 7 in |
[
{
"content": "Consider a finite set of uniformly distributed, independent random variables $\\mathbf{X} = \\{X_1, X_2, \\dots, X_n\\}$ on the unit interval. Calculate the expected value of the pairwise absolute differences between these variables, forming a new set $\\mathbf{Y}$. What is the expected value of t... | \dfrac{1}{3} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 4.42e6 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... | 6.67 |
[
{
"content": "Let $A, B \\in M_n(\\mathbb{R})$ be matrices such that $x^2 - 1$ and $x^2 + 1$ are their respective minimal polynomials. If $AB = BA$, find the minimal polynomial of the matrix product $AB$. \\( \\boxed{} \\)",
"role": "user"
}
] | 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... | 1.9 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... | 36250 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... | 8.613 kg |
[
{
"content": "Evaluate the limit \\( \\lim_{n \\to \\infty} \\frac{1}{n}\\sqrt[n]{\\frac{(2n)!}{n!}} \\) using Riemann integrals.",
"role": "user"
}
] | \dfrac{4}{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... | 1303.45 |
[
{
"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... | 40.8 cm |
[
{
"content": "A curve is defined by the parametric equation \\((x, y, z) = (-2 + 3t, 1 + 3t^2, 2t-3t^3)\\). Find the coordinates of the unique point \\(P\\) on the curve such that the tangent line at \\(P\\) passes through the point \\((-8, 10, 8)\\).",
"role": "user"
}
] | (-5, 4, 1) |
[
{
"content": "Let $X$ be a topological space and $(A_{n})$ be a decreasing sequence of open sets, i.e., $A_{n+1} \\subset A_{n}$. If the intersection of all $A_{n}$ is contained in a closed set $B$, is the closure of the intersection of all $A_{n}$ also contained in $B$?",
"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... | -$30,470.50 |
[
{
"content": "Calculate the remainder when \\(2017^{2016^{2015}}\\) is divided by 1000.",
"role": "user"
}
] | 281 |
[
{
"content": "Determine whether the following statement is true or false: If $f$ is a differentiable function on $\\mathbb{R}$ such that $f(0) = 0$ and $f(1) = 1$, then there exists a $c \\in (0,1)$ such that $f'(c) = 2c$. Justify your answer.",
"role": "user"
}
] | True |
[
{
"content": "Find all pairs of positive integers \\((m, n)\\) such that \\(2^{m+1} + 3^{n+1}\\) is a perfect square.",
"role": "user"
}
] | (3, 1) |
[
{
"content": "Let $f(x) = x^2 + bx + 9$ and $g(x) = x^2 + dx + e.$ If $f(x) = 0$ has roots $r$ and $s,$ and $g(x) = 0$ has roots $-r$ and $-s,$ compute the two roots of $f(x) + g(x) = 0.$",
"role": "user"
}
] | 3i,-3i |
[
{
"content": "Consider the cubic curve given by the equation \\( y = x^3 + ax^2y + bxy^2 + cy^3 \\). If \\((t, r)\\) and \\((t, s)\\) are two distinct points on this curve, is it necessarily true that \\(t = 0\\)? Provide a justification for your answer.",
"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... | 15.986 g |
[
{
"content": "Determine whether the following statement is true or false: If a function is strictly increasing and continuous, then it is differentiable.",
"role": "user"
}
] | B |
[
{
"content": "Determine the dimension of the image of the function \\(\\xi: F_n \\rightarrow M_n(\\mathbb{R})\\), where \\(F_n\\) is the set of real integrable functions on the interval \\([1, n]\\) and \\(M_n(\\mathbb{R})\\) is the set of \\(n \\times n\\) square matrices. The function \\(\\xi\\) maps a functi... | n-1 |
[
{
"content": "Evaluate the contour integral \\( \\oint_{|z-3|=4} \\frac{z}{\\cos z - 1} \\, dz \\) using the residue theorem.",
"role": "user"
}
] | -8\pi i |
[
{
"content": "Assuming the Continuum Hypothesis and the Axiom of Choice, consider the smallest uncountable ordinal number, denoted by $S$. Let $C(S)$ be the collection of all functions from $S$ into the continuum ($2^{\\aleph_0}$), such that the subset of $S$ not mapped to zero is countable at most. Determine i... | 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... | 3.34% |
[
{
"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... | -g/9 |
[
{
"content": "Find a non-recursive equation for the sequence $a_n = 3a_{n-1} - 3a_{n-2} + a_{n-3}$ for $n \\geq 3$, with initial terms $a_0 = 2$, $a_1 = 2$, and $a_2 = 4$, using a generating function.",
"role": "user"
}
] | n^2 - n + 2 |
[
{
"content": "Determine whether the group $S_3 \\oplus S_3$ contains an element of order 4, 6, 9, or 18.",
"role": "user"
}
] | 6 |
[
{
"content": "Let $V$ be an $n$-dimensional Euclidean space with inner product $\\langle\\cdot,\\cdot\\rangle$, and basis $e_1,\\dots,e_n$. The Gram matrix is $A=(a_{ij})$ with $a_{ij}=\\langle e_i,e_j\\rangle$. Suppose $\\langle e_i,e_j\\rangle\\leq 0$ for $i\\neq j$. Determine whether the following statement ... | 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... | ```c++
#include <iostream>
#include <math.h>
#include <iomanip>
using namespace std;
int main(){
int n[5];
for(int i = 1; i <= 2; i++)
{
cout<<"Enter integer "<<i<<": "; //input integer
cin>>n[i];
if(n[i] < 1 || n[i] > 20) //check condition for input value
{
cout<... |
[
{
"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... | E=C/r^3, where C = 8.76*10^9 Vm |
[
{
"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... | 540 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... | 20.52 MeV |
[
{
"content": "Given that the function \\( f \\) satisfies the condition for all integers \\( x \\) and \\( y \\):\n\n\\[ \n f(x - y^2) = f(x) + (y^2 - 2x) \\cdot f(y) \n\\]\n\nFind the value of \\( f(1984) \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Find the limiting probability as \\( N \\to \\infty \\) that if two natural numbers \\( x \\) and \\( y \\) (from 0 to \\( N-1 \\)) are chosen at random, their sum of squares, \\( x^2 + y^2 \\), is divisible by 10.",
"role": "user"
}
] | \dfrac{9}{50} |
[
{
"content": "There are seven people, each with a uniquely colored hat. After entering a room, their hats are shuffled and redistributed. In how many ways can the hats be distributed such that none of the first three people receive their original hat?",
"role": "user"
}
] | 3216 |
[
{
"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... | 11.75% |
[
{
"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... | 31811.515 rad/s |
[
{
"content": "Determine whether there exists a polynomial $f(x)$, with no constant term, such that for all natural numbers $p$ and $q$, if $\\gcd(p, q) = 1$, then $\\gcd(f(p), f(q)) = 1$. Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Let \\( f:\\mathbb{N}\\mapsto\\mathbb{R} \\) be the function \\[f(n)=\\sum_{k=1}^\\infty\\dfrac{1}{\\operatorname{lcm}(k,n)^2}.\\] It is well-known that \\( f(1)=\\tfrac{\\pi^2}{6} \\). Determine the smallest positive integer \\( m \\) such that \\( m\\cdot f(10) \\) is the square of a rational mu... | 42 |
[
{
"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... | $17,240 |
[
{
"content": "Calculate the surface area of the region of intersection of the two cylinders given by the equations \\(y^2 + z^2 = 1\\) and \\(x^2 + y^2 = 1\\).",
"role": "user"
}
] | 16 |
[
{
"content": "Determine the limits of integration for the volume enclosed by the paraboloid $z = x^2 + y^2$ and the plane $z = 3 - 2y$. Evaluate the triple integral to show that the volume is $8\\pi$. Use the hint that $\\cos^4\\theta = \\frac{1}{8}\\cos4\\theta + \\frac{1}{2}\\cos2\\theta + \\frac{3}{8}$ if ne... | 8\pi |
[
{
"content": "In an equilateral triangle $\\triangle CDE$ inscribed in a circle, each side has a length of $16$. Point $F$ is the midpoint of side $DE$. Points $G$ and $H$ lie on the circle such that $\\triangle FGH$ is also equilateral. Find the side length of $\\triangle FGH$ and express it in the form $a\\sq... | 4\sqrt{5} - 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... | 3.17 km North |
[
{
"content": "Karlson has 29 ones written on a board. Each minute, he erases any two numbers and writes their sum on the board, then eats a number of candies equal to the product of the two erased numbers. What is the maximum number of candies Karlson could eat in 29 minutes?",
"role": "user"
}
] | 406 |
[
{
"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... | 2.43 |
[
{
"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... | 7.6 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... | $28,993.79 |
[
{
"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... | 222 grams |
[
{
"content": "Let $f$ be a function defined on $(a,b)$, integrable on any $[c,d]$ with $a<c<d<b$, but not integrable in the standard Riemann sense on the entire $[a,b]$. Define the function $F$ as an improper integral:\n\\[ F(u) = \\int_{a}^{u} f(x) \\, dx = \\lim_{c \\to a} \\int_{c}^{u} f(x) \\, dx. \\]\nAssu... | 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... | Net Income + Non-cash Expenses + Losses - Gains + Decrease in Current Assets - Increase in Current Assets + Increase in Current Liabilities - Decrease in Current Liabilities |
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