prompt list | ground_truth string |
|---|---|
[
{
"content": "Given $D \\geq 2$ and $n \\leq \\frac{D}{2}$, consider the maximization problem:\n\n$$\\max_{x_1, \\ldots, x_n} \\sum_{i=1}^n x_i^2$$\nSubject to:\n$$\\sum_{i=1}^n x_i = D$$\n$$x_i \\geq 2, \\quad \\forall i$$\nwhere $x_i$ are real numbers. Let $x_1^*, \\ldots, x_n^*$ be an optimal solution, and d... | D - 2(n - 1) |
[
{
"content": "Determine the number of non-conjugate $6 \\times 6$ complex matrices with the characteristic polynomial $(x-5)^6=0$. Consider matrices that are not diagonalizable to the diagonal matrix $D$ with all diagonal entries equal to 5, but still have the same characteristic polynomial.",
"role": "user... | 10 |
[
{
"content": "Determine the ratio of the following two definite integrals involving the Gamma function: \\[ \\frac{\\int_{0}^{\\infty} e^{-x^2} x^3 \\, dx}{\\int_{0}^{\\infty} e^{-x^2} x^5 \\, dx} \\]",
"role": "user"
}
] | \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... | 50.03 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... | 53.75 |
[
{
"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... | 14.388 J/m^3 |
[
{
"content": "Given matrices \\(A, B \\in \\mathcal{M}_2(\\mathbb{R})\\) such that \\(A^2 + B^2 = AB\\), determine whether it must be true that \\(AB = BA\\).",
"role": "user"
}
] | AB = BA |
[
{
"content": "Determine the infimum of the set \\(\\{|f(2)| \\colon f\\colon \\Bbb C\\setminus\\{1\\}\\to \\Bbb C \\text{ is analytic and satisfies } |f(z)|\\le 3|z|^{\\sqrt3} \\}\\).",
"role": "user"
}
] | 0 |
[
{
"content": "Determine all natural numbers \\( k \\) for which the expression \\( (p_1 \\times p_2 \\times \\cdots \\times p_k) - 1 \\) is an exact power of a natural number greater than one, where \\( p_1, p_2, \\ldots, p_k \\) are the first \\( k \\) odd prime numbers.",
"role": "user"
}
] | \emptyset |
[
{
"content": "Evaluate the double integral: \\[ J = \\int\\limits_0^1 \\int\\limits_0^1 \\dfrac{x-y}{(x+y)^3} \\, dy \\, dx. \\]",
"role": "user"
}
] | \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... | 7.987% |
[
{
"content": "Let \\( p_{1}, p_{2}, \\cdots, p_{k} \\) be distinct prime numbers. Determine the number of expressions of the form \\( p_{1}^{\\alpha_{1}} p_{2}^{\\alpha_{2}} \\cdots p_{k}^{\\alpha_{k}} \\) where \\( \\alpha_{i} \\in \\mathbf{N} \\) such that \\( p_{1} p_{2} \\cdots p_{k} = \\alpha_{1} \\alpha_{... | k^k |
[
{
"content": "Let $R$ be a commutative ring with unity that has property P, meaning there exists a unique field among all proper subrings of $R$ (with the same unity as that of $R$). Assume $R$ is not a field. Does the polynomial ring $R[X]$ also have property P?",
"role": "user"
}
] | Yes |
[
{
"content": "Given that the logarithm is base 2, verify that \\((\\log(\\log n))^{(\\log_{\\log n}n^{-1})} = 1\\).",
"role": "user"
}
] | 1 |
[
{
"content": "What is the smallest positive integer $n$ such that, out of the $n$ unit fractions $\\frac{1}{k}$ where $1 \\le k \\le n$, exactly half of the fractions give a terminating decimal?",
"role": "user"
}
] | 12 |
[
{
"content": "Find the limit of the sequence \\(a_n = \\frac{1^k+2^k+\\ldots+n^k}{n^{k+1}}\\) as \\(n\\) approaches infinity, where \\(k\\) is a natural number.",
"role": "user"
}
] | \dfrac{1}{k+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... | 1.05e-5 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... | $9.85 mil |
[
{
"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... | $405,682 |
[
{
"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... | 0.297 m |
[
{
"content": "Determine whether there exist two nonempty topological spaces $X$ and $Y$ such that their product space $X \\times Y$ is homeomorphic to $X$. 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... | 142.81 |
[
{
"content": "Calculate the algorithmic complexity of the most efficient algorithm for median smoothing applied to a time series with T observations and a window size of n, in terms of T and n.",
"role": "user"
}
] | O(T \log n) |
[
{
"content": "Determine the limit of the function \\( f(x,y) = xye^{-x^2y^2} \\) as \\( x^2 + y^2 \\to \\infty \\) when restricted to a straight line through the origin. Also, evaluate the limit \\( \\lim_{x^2+y^2 \\to \\infty} f(x,y) \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Determine the number of solutions in the third quadrant of the complex plane for the equation \\(z^{100} = 1+i\\).",
"role": "user"
}
] | 25 |
[
{
"content": "Given $\\tan \\theta = \\frac{1}{7},$ find\n\\[\\frac{1}{1 + \\cos \\theta} + \\frac{1}{1 - \\cos \\theta}.\\]",
"role": "user"
}
] | 100 |
[
{
"content": "Determine the number of solutions of the equation \\(x \\tan x = 1\\) in the interval \\([-2\\pi, 2\\pi]\\).",
"role": "user"
}
] | 4 |
[
{
"content": "Determine the total number of homomorphisms from a free group of rank 2 to the group of integers modulo 4.",
"role": "user"
}
] | 16 |
[
{
"content": "Let \\( P \\) be any point on the ellipse given by \\( \\frac{x^2}{4} + \\frac{y^2}{3} = 1 \\). The points \\( A \\) and \\( B \\) are fixed at \\( \\left( \\frac{1}{2}, 0 \\right) \\) and \\( (1, 1) \\) respectively. Determine the minimum value of the expression \\( \\frac{|PA|^2}{|PB|} \\).",
... | 1 |
[
{
"content": "Consider an operation $O$ on a field of sets. If $O$ satisfies the properties of being commutative, associative, idempotent, and $O(\\emptyset, X) = X$, does it follow that $O$ must be the set union operation? 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... | 10.5 A |
[
{
"content": "Find the limit of the expression as \\( n \\) approaches infinity: \\[ \\lim_{n\\to\\infty} \\left(\\frac{40}{40-41n^2}\\right)^{41n^2-n^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... | Project 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... | 2.9 rad/s^2 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 5e-25 |
[
{
"content": "Determine whether the function \\( f(z) = \\mathfrak{Re}(z) \\mathfrak{Im}(z) \\) is pointwise differentiable at \\( z=0 \\) using the limit definition: \n\\[ \\lim_{z\\to 0} \\frac{f(z) - f(0)}{z}. \\]",
"role": "user"
}
] | 0 |
[
{
"content": "Let $\\mathscr{C}$ be the class of compact subsets of the Euclidean plane $\\mathbb{R}^2$ with connected complement. If $K \\in \\mathscr{C}$ and $M \\subseteq \\mathbb{R}^2$ is homeomorphic to $K$, does it follow that $M$ belongs to $\\mathscr{C}$? Provide a justification for your answer.",
"... | Yes |
[
{
"content": "Calculate the expected value of the absolute difference between two independent uniform random variables on the interval [0, 1].",
"role": "user"
}
] | \dfrac{1}{3} |
[
{
"content": "Determine the number of real solutions to the equation \\( f(f(f(f(x)))) = 0 \\), where \\( f(x) = x^2 + 10x + 20 \\).",
"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... | 53790 |
[
{
"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... | 598 kg |
[
{
"content": "In a plane rectangular coordinate system, consider the points $A(0,\\frac43)$, $B(-1,0)$, and $C(1,0)$. The distance from a point $P$ to line $BC$ is the geometric mean of the distances from $P$ to lines $AB$ and $AC$. If line $L$ passes through the incenter $D$ of triangle $ABC$ and intersects th... | 7 |
[
{
"content": "Let \\( f: \\mathbb{R} \\to \\mathbb{R} \\) be a function such that \\( f(\\sin(1/n)) = \\cos(1/n) \\) for \\( n \\in \\mathbb{N} \\). Given that the derivative \\( f' \\) exists at \\( 0 \\), determine the value of \\( f'(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... | Dept. M Dept. O Dept. P Total
Sales $41,000 $30,000 $35,500 $106,500
Expenses
Avoidable 4,100 12,000 7,000 $23,100
Unavoidable 16,000 2,000 15,000 33,000
Total expenses 20,100 14,000 22,000 56,100
Net income (loss) $20,900 $16,000 $13,500 $50,400 |
[
{
"content": "Calculate the surface integral of the vector field \\( \\mathbf{F}(x,y,z) = (z,x,y) \\) over the surface \\( S \\), which is delimited by the elliptical cylinder \\( \\frac{x^2}{4} + \\frac{y^2}{9} = 1 \\) and the plane \\( z + y = 9 \\).",
"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... | 1s^(2)2s^(2)2p^(6)3s^(2)3p^(6)4s^(1)3d^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... | 9 mph |
[
{
"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... | $\frac{E}{E + M1 + M2}\times 13.6\% + \frac{M1 + M2}{E +M1 + M2}\times\:[ \frac{M1}{M1 + M2}\times d1 + \frac{M2}{M1 + M2}\times d2\:] (1 - 35\%)$ |
[
{
"content": "What is the distance between the center of the circle with equation $x^2+y^2=-4x+6y-12$ and the point $(1,7)$?",
"role": "user"
}
] | 5 |
[
{
"content": "Evaluate the sum of the series: \\[ \\sum_{n=0}^{\\infty} (n\\bmod 3)\\cdot 2^{-n} \\]",
"role": "user"
}
] | \dfrac{8}{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... | -0.802 N |
[
{
"content": "If $f$ is an odd meromorphic function, determine the residue of $f$ at the origin, denoted as $\\operatorname{res}(f,0)$. \\( \\boxed{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... | $17,930 |
[
{
"content": "A zoo has a menagerie containing four pairs of different animals, one male and one female for each. The zookeeper wishes to feed the animals in a specific pattern: each time he feeds a single animal, the next one he feeds must be a different gender. If he starts by feeding the male giraffe, how ma... | 144 |
[
{
"content": "Evaluate the limit: $$\\lim_{n\\to\\infty}\\left(1+\\frac{1}{2}\\right)^{\\tfrac{2}{n}}\\cdot\\left(1+\\frac{1}{3}\\right)^{\\tfrac{3}{n}}\\cdot\\left(1+\\frac{1}{4}\\right)^{\\tfrac{4}{n}}\\cdots\\left(1+\\frac{1}{n+1}\\right)^{\\tfrac{n+1}{n}}$$",
"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... | 4.213 |
[
{
"content": "Given the probability density function \\( f(x,y) = 3x \\) for \\( 0 \\le y \\le x \\le 1 \\) and \\( f(x,y) = 0 \\) elsewhere, find the probability \\( P(Y \\le X/2) \\).",
"role": "user"
}
] | \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... | staff model |
[
{
"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... | Date Account Titles and Explanation Debit Credit
01/01/14 Lease receivable $150,000.00
Asset $150,000.00
(To record inception of lease)
01/01/14 Cash $30,044.00
Lease receivable $30,044.00
(To record first lease payment) |
[
{
"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... | 15e12 J |
[
{
"content": "If \\( k \\) is an integer greater than 1 and \\( \\alpha \\) is a root of the equation \\( x^{2} - kx + 1 = 0 \\), and for any natural number \\( n \\) greater than 10, the units digit of \\( \\alpha^{2^{n}} + \\alpha^{-2^{n}} \\) is always 7, find the units digit of \\( k \\).",
"role": "use... | 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... | 12.94% |
[
{
"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... | 2.1e-7 N/C upwards |
[
{
"content": "Determine whether there exist positive integers $a$ and $b$ such that $a \\bmod p > b \\bmod p$ for all primes $p$ where $(p,a)=1$. Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Find the limit of the expression \\(((x^4-x^2)^\\frac{1}{4}-\\sqrt{x^3+x})\\) as \\(x\\) approaches infinity without using L'Hôpital's rule or conjugation.",
"role": "user"
}
] | -\infty |
[
{
"content": "Let $H$ be a closed hypersurface in $\\mathbb{A}^{n}$, where $n$ is sufficiently large, over $\\mathbb{C}$. Let $U$ be the open subset that is the complement of $H$. Given a point $x \\in H$, determine if there exists a curve $C \\subset \\mathbb{A}^{n}$ such that $C \\cap H = \\{x\\}$. Provide a ... | Yes |
[
{
"content": "Given that the ABC stock pays no dividend and is currently priced at $S_0 = \\$10$, with the stock price at expiry $T$ going up to $uS_0$ with probability $0 < p < 1$ and down to $dS_0$ with probability $1 - p$, where $d < 1 < u$. Assume no arbitrage and a zero interest rate. A European put option... | \dfrac{29}{9} |
[
{
"content": "Calculate the radius of convergence of the function $\\operatorname{sech}z$ about the origin.",
"role": "user"
}
] | \dfrac{\pi}{2} |
[
{
"content": "Calculate the expected value $E(X^3)$ for a random variable $X$ that follows a normal distribution with mean $\\mu$ and variance $\\mu^2$, i.e., $X \\sim N(\\mu, \\mu^2)$. Use this to find the covariance $\\operatorname{Cov}(X, X^2)$. Express your answer in terms of $\\mu$. \\( \\operatorname{Cov}... | 2\mu^3 |
[
{
"content": "Determine whether the following statement is true or false: If $p$ is an odd prime and $p$ divides the sum of two integers raised to the power of $p$, then $p^2$ also divides that sum.",
"role": "user"
}
] | True |
[
{
"content": "Given a function $f \\in L^1(\\mathbb{T})$ and $\\epsilon > 0$, determine whether there exists a continuous function $g \\in L^1(\\mathbb{T})$ such that $\\|f - g\\|_1 < \\epsilon$. Here, $\\mathbb{T} = \\mathbb{R}/\\mathbb{Z}$ is the unit circle, and the continuity of $g$ refers to the continuity... | Yes |
[
{
"content": "Let \\( A_1, A_2, \\ldots, A_n \\) be convex sets in a vector space. Suppose \\( x \\in \\operatorname{co}(A_1 \\cup A_2 \\cup \\dotsb \\cup A_n) \\). Is it true that \\( x = t_1 a_1 + \\dotsb + t_n a_n \\) such that \\( \\sum_{i=1}^n t_i = 1 \\), \\( t_i \\ge 0 \\), and \\( a_i \\in A_i \\) for e... | Yes |
[
{
"content": "In a metric space $X$, consider a ball $B(x, r)$ with center $x$ and radius $r$. Does the diameter of this ball always divide the radius in half, such that $B(x, r) = B\\left(x, \\frac{\\text{diam}(B(x, r))}{2}\\right)$? Answer yes or no.",
"role": "user"
}
] | No |
[
{
"content": "Evaluate the limit using Stirling's approximation: $$\\lim_{x \\to \\infty} \\frac{5^{-x} x!}{m! (x-m)!}$$ where $m$ is a positive integer less than $x$. Provide your answer in terms of $m$.",
"role": "user"
}
] | 0 |
[
{
"content": "Calculate the integral of the function \\( \\operatorname{sgn}(\\sin(\\pi/x)) \\) over the interval \\((0, 1)\\).",
"role": "user"
}
] | 1 - 2 \ln 2 |
[
{
"content": "Evaluate the integral: $$\\int_{-5}^5 \\dfrac{x^3 \\sin^2x}{x^4+2x^2+1}~dx$$",
"role": "user"
}
] | 0 |
[
{
"content": "Suppose $X_1$ and $X_2$ are isometric metric spaces. If $X_2$ is complete, does it imply that $X_1$ is also complete? Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Is the identity matrix an example of a matrix that is both stochastic and unitary?",
"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... | 8.23 x 10^6 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... | 141 g |
[
{
"content": "True or false: If a group $G$ acts transitively on a set $\\Omega$, then any subgroup of $G$ that fixes some element of $\\Omega$ must be normal in $G$.",
"role": "user"
}
] | False |
[
{
"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... | 20600 yr |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n \\to \\infty} \\int_{0}^{1} \\frac{2nx^{n-1}}{1+x} \\, dx. \\]",
"role": "user"
}
] | 1 |
[
{
"content": "Given integers $n$, $p$, and $q$ greater than 0, and $a, b \\in \\mathbb{Z}/(n)$, determine the number of $p \\times q$ matrices with elements in $\\mathbb{Z}/(n)$ such that the sum of entries in each row is equal to $a$ and the sum of entries in each column is equal to $b$. Assume that $pa \\equi... | n^{(p-1)(q-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... | 13.9999815 |
[
{
"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... | -14.79 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... | 1300 |
[
{
"content": "Does a Lipschitz continuous function guarantee differentiability at every point within an open interval? Specifically, if a function \\( f: [a,b] \\to [c,d] \\) is Lipschitz continuous on the interval \\([a,b]\\), does this imply that \\( f \\) is differentiable at every point within the open inte... | No |
[
{
"content": "Prove or disprove the following conjecture: For all integers \\( n \\geq 3 \\), the determinant of the \\( n \\times n \\) matrix \\( \\mathbf{M}_n \\), defined by \\( \\mathbf{M}_n = \\begin{bmatrix} F_1 & F_2 & \\dots & F_n \\\\ F_{n+1} & F_{n+2} & \\dots & F_{2n} \\\\ \\vdots & \\vdots & \\ddot... | 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... | NO_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... | 66.45 |
[
{
"content": "Calculate the limit \\( \\lim_{x\\to 0^+} f(x) \\) for a function \\( f(x) \\) that is bounded on the interval \\([0,1]\\) and satisfies the property \\( f(ax) = bf(x) \\) for \\(0 < x < \\frac{1}{a}\\), where \\(a, b > 1\\).",
"role": "user"
}
] | 0 |
[
{
"content": "A particle moves so that it is at $\\left( 3 \\sin \\frac{t}{4}, 3 \\cos \\frac{t}{4} \\right)$ at time $t.$ Find the speed of the particle, measured in unit of distance per unit of time.",
"role": "user"
}
] | \frac{3}{4} |
[
{
"content": "The chart below gives the air distance in miles between selected world cities. If two different cities from the chart are chosen at random, what is the probability that the distance between them is less than $7000$ miles? Express your answer as a common fraction.\n\n\\begin{tabular}{|c|c|c|c|c|}\n... | \frac{2}{3} |
[
{
"content": "Find the value of \\(\\displaystyle\\sum_{k=0}^{2n}(-1)^k\\cdot (k+2n)\\cdot \\binom{2n}{k}^2\\) in terms of \\(n\\) and \\(A\\), where \\(A=\\displaystyle \\sum_{k=0}^{2n}(-1)^k\\cdot \\binom{2n}{k}^2\\).",
"role": "user"
}
] | 3nA |
[
{
"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 g/mol |
[
{
"content": "Evaluate the limit: $$\\lim_{n \\to \\infty} \\left[\\left(\\frac{n+3}{n+1}\\right)\\log(n+1) - \\frac{n+4}{n+2}\\log(n+2)\\right]$$",
"role": "user"
}
] | 0 |
[
{
"content": "Evaluate the complex integral \\( \\int_{C}\\frac{z^2+1}{z^4+4z^2} \\, dz \\) using the Cauchy Integral Theorem, where \\( C \\) is a simple closed curve enclosing the origin.",
"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... | 19.91% or 0.1991 |
[
{
"content": "Determine the interval of convergence for the series: \\[ \\sum_{n=1}^{\\infty}\\frac{((n+1)x)^n}{n^{n+1}} \\]",
"role": "user"
}
] | [-1, 1) |
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