prompt list | ground_truth string |
|---|---|
[
{
"content": "Determine the Laurent series expansion of \\( f(z)=\\frac{1}{1+z^2} \\) about \\( z_0=i \\), valid in the region \\( 0<|z-i|<2 \\). Then, evaluate the integral \\( \\int_\\gamma \\frac{1}{1+z^2}dz \\), where \\( \\gamma(t)=i+e^{it}, 0\\leq t\\leq2\\pi \\).",
"role": "user"
}
] | \pi |
[
{
"content": "The function \\( u(x,t) = \\frac{2}{\\sqrt{\\pi}}\\int_{0}^\\frac{x}{\\sqrt{t}} e^{-s^2}ds \\) satisfies the partial differential equation \\( \\frac{\\partial u}{\\partial t} = K\\frac{\\partial^2u}{\\partial x^2} \\) where \\( K \\) is a positive constant. Determine the value of \\( K \\).",
... | \dfrac{1}{4} |
[
{
"content": "Find a function \\( f: \\mathbf{N}^{+} \\rightarrow \\mathbf{N}^{+} \\) such that \\( f(1) = 1 \\) and \\( f(n+1) \\geqslant f(n) + 2^n \\) for all \\( n \\in \\mathbf{N}^{+} \\).",
"role": "user"
}
] | 2^n - 1 |
[
{
"content": "Find the potential function \\( f(x, y, z) \\) such that its gradient \\( \\nabla f(x, y, z) = \\langle 2xy, 2yz + x^2, y^2 \\rangle \\).",
"role": "user"
}
] | x^2 y + y^2 z |
[
{
"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... | 545.50 |
[
{
"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... | $6,600.00 |
[
{
"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... | 1.6 kg |
[
{
"content": "An ellipse given by the equation \\(x^2 + 4y^2 = 4\\) is rotated anticlockwise through a right angle in its own plane about its center. Determine the value of \\(\\lambda + \\mu\\) if the locus of the point of intersection of a tangent to the ellipse in its original position with the tangent at th... | 11 |
[
{
"content": "Determine the conditions under which $2^x + 3^y$ is a perfect square, where $x$ and $y$ are positive integers.",
"role": "user"
}
] | (4, 2) |
[
{
"content": "Determine the dimension of the solution space for the differential equation \\( f^{(2n)}(x) + f(x) = 0 \\), where \\( x \\in \\mathbb{R} \\) and \\( n \\in \\mathbb{N} \\).",
"role": "user"
}
] | 2n |
[
{
"content": "Calculate the sum of the series: $$\\sum_{n=1}^\\infty \\frac{\\ln^{5} (2n^{7}+13) + 10 \\sin n}{n \\ln^6 (n^{7/8} + 2 \\sqrt n - 1) \\ln (\\ln (n + (-1)^{n}))}$$",
"role": "user"
}
] | \infty |
[
{
"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... | 45.05 degrees South of West |
[
{
"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... | 750.4km |
[
{
"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... | -1.43e10 J |
[
{
"content": "Determine whether a countable set can have uncountably many infinite subsets such that the intersection of any two distinct subsets is finite. Answer 'yes' or 'no'.",
"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... | 5.30e-33 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... | 9.813 m/s^2 |
[
{
"content": "Prove that a non-empty set in \\( \\mathbb{R} \\) that is both open and closed must be equal to \\( \\mathbb{R} \\).",
"role": "user"
}
] | \mathbb{R} |
[
{
"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... | speed: 23.391m/s; direction: -14.863 degree below x-axis |
[
{
"content": "Find the maximum value of the function \\( f(x) = \\frac{\\sin x}{x} \\) for \\( x \\neq 0 \\) without using L'Hopital's rule.",
"role": "user"
}
] | 1 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 0.148 |
[
{
"content": "Find the radius of convergence of the Taylor series for the function \\( f(x) = \\frac{1}{\\sqrt{x}} \\) centered at \\( x = 1 \\), given by:\n\n$$ f(x) = 1 + \\sum_{n=1}^\\infty (-1)^n \\cdot \\frac{(2n)!}{2^{2n} (n!)^2} \\cdot (x-1)^n $$",
"role": "user"
}
] | 1 |
[
{
"content": "Suppose \\( A \\) is a 3x3 non-singular matrix such that \\((A - 3I)(A - 5I) = O\\), where \\(I\\) is the 3x3 identity matrix and \\(O\\) is the 3x3 zero matrix. If \\(\\alpha A + \\beta A^{-1} = 4I\\), find the value of \\(\\alpha + \\beta\\).",
"role": "user"
}
] | 8 |
[
{
"content": "The point $A$ $(3,4)$ is reflected over the $x$-axis to $B$. Then $B$ is reflected over the line $y=x$ to $C$. What is the area of triangle $ABC$?",
"role": "user"
}
] | 28 |
[
{
"content": "Which of the following statements are true?\n\nA. 3 is a factor of 18.\n\nB. 17 is a divisor of 187 but not of 52.\n\nC. 24 is neither a divisor of 72 nor 67.\n\nD. 13 is a divisor of 26 but not of 52.\n\nE. 8 is a factor of 160.\n\nWrite your answer with the letters in alphabetical order, se... | \text{A,B,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... | 704,697.23 |
[
{
"content": "One hundred musicians are organizing a festival with several concerts. During each concert, some musicians perform on stage while the others are in the audience. What is the minimum number of concerts required so that each musician has the opportunity to listen to every other musician perform on s... | 9 |
[
{
"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... | 3e-14 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... | laissez-faire value transmission |
[
{
"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... | 10.31 |
[
{
"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... | \Delta m = \frac{V}{RT} (P_2 - P_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... | 900.0112 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... | 21.84% |
[
{
"content": "Find the number of positive integers \\( n \\) such that \\( n \\leq 2016 \\) and \\( \\left\\{\\frac{n}{2}\\right\\}+\\left\\{\\frac{n}{4}\\right\\}+\\left\\{\\frac{n}{6}\\right\\}+\\left\\{\\frac{n}{12}\\right\\}=3 \\), where \\( \\{x\\} = x - \\lfloor x \\rfloor \\) and \\( \\lfloor x \\rfloor ... | 168 |
[
{
"content": "Find the value of the real number \\( a \\) that minimizes the area bounded between the curves \\( y = \\frac{x^2}{4} \\) and \\( y = ax + 9 \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Find the residue of the function \\( f(z) = \\frac{z^2 e^{iz}}{(z^2+1)^2} \\) at the double pole \\( z = i \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Given the equation \\(x^3 = (3+4i)\\overline{x}\\), where \\(x\\) is a complex number, find the sum of the moduli of its roots.",
"role": "user"
}
] | 4\sqrt{5} |
[
{
"content": "In triangle ABC, an ex-inscribed circle is tangent to sides AB and BC at points F and G, respectively. Let G be on the extension of AB. The line GF is extended to intersect line AO at point H. Determine the measure of angle AHC.",
"role": "user"
}
] | 90^\circ |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 5.17% |
[
{
"content": "Determine the necessary and sufficient condition for the matrix \\( A = I - 2xx^T \\) to be orthogonal, where \\( x \\) is a non-zero column vector in \\( \\mathbb{R}^n \\).",
"role": "user"
}
] | x^T x = 1 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n\\to\\infty}\\sum_{k=1}^n \\frac{\\log(k)}{nk} \\]",
"role": "user"
}
] | 0 |
[
{
"content": "Determine the number of ways to roll a sum of 8 using two dice: one 4-sided and one 6-sided.",
"role": "user"
}
] | 3 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to 0} \\frac{\\sqrt{1+x\\cos(x)} - \\sqrt{\\sin(x)}}{x^3} \\]",
"role": "user"
}
] | \infty |
[
{
"content": "Suppose $n$ has at least two distinct odd prime factors. If $x^2 \\equiv y^2 \\pmod{n}$ and $\\gcd(x,y) = 1$, find the probability that $\\gcd(n, x-y)$ is a non-trivial factor of $n$. Express your answer as a fraction.",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "Determine the maximum number of points that can be placed inside a circle of radius 2 such that one of the points is at the center of the circle and the distance between any two points is at least 1.",
"role": "user"
}
] | 19 |
[
{
"content": "Determine all positive integers \\(n\\) such that:\n\n\\[ \\left\\lfloor\\frac{n}{2}\\right\\rfloor \\cdot \\left\\lfloor \\frac{n}{3} \\right\\rfloor \\cdot \\left\\lfloor \\frac{n}{4} \\right\\rfloor = n^2 \\]",
"role": "user"
}
] | 24 |
[
{
"content": "The arithmetic mean of nine numbers is 54. If two numbers $u$ and $v$ are added to the list, the mean of the eleven-member list becomes 66. What is the mean of $u$ and $v$?",
"role": "user"
}
] | 120 |
[
{
"content": "Evaluate the limit: $$\\lim_{x\\to\\frac\\pi4}\\frac{32^{\\frac12} - (\\cos x + \\sin x)^5}{1-\\sin(2x)}$$ Express your answer in terms of \\(\\sqrt{2}\\). Assume radians and avoid using derivatives or L'Hôpital's Rule unless it's unavoidable.",
"role": "user"
}
] | 5\sqrt{2} |
[
{
"content": "Given a set $A \\subseteq X \\subseteq \\mathbb{R}$ such that $X \\setminus A$ is dense in $X$, and a function $g: A \\rightarrow \\mathbb{R} \\setminus \\{0\\}$ with $\\sum_{a \\in A} |g(a)| < \\infty$, determine if it is possible to construct a function $G: X \\rightarrow \\mathbb{R}$ that is co... | Yes |
[
{
"content": "Consider the first-order linear differential equation $$\\frac{dy}{dx} + 10y = f(x), \\quad x > 0,$$ where $f(x)$ is a continuous function with $\\displaystyle \\lim_{x \\to \\infty} f(x) = 1$. Find $\\displaystyle \\lim_{x \\to \\infty} y(x)$. \\(\\text{Express your answer as a single number or e... | \dfrac{1}{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... | 2.70 s |
[
{
"content": "Find the points in the complex plane where the function \\( f(z) = |z|^2 + i\\bar{z} + 1 \\) is differentiable.",
"role": "user"
}
] | -i |
[
{
"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... | 612.0 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... | 39.6 m/s |
[
{
"content": "Determine whether the function $f(z) = \\frac{1}{z}$ has a removable singularity at $z=\\infty$ in the complex space $\\mathbb{C}_\\infty$.",
"role": "user"
}
] | Yes |
[
{
"content": "Let $A_n$, $B_n$, and $C$ be closed subsets in Euclidean space such that for each $n=1,2,...$, we have $A_n \\cap B_n \\neq \\varnothing$ and $A_n \\cap B_n \\cap C \\neq \\varnothing$. Given that $\\lim_{n\\to \\infty}A_n=A$, $\\lim_{n\\to \\infty}B_n=B$, $A\\cap C\\neq \\varnothing$, $B\\cap C\\... | A \cap B \cap C \neq \varnothing |
[
{
"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... | 0.8026 T |
[
{
"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... | 4.07% |
[
{
"content": "Three lines from the family of lines given by \\( C: x \\cos t + (y + 1) \\sin t = 2 \\) enclose an equilateral triangle \\( D \\). What is the area of the region \\( D \\)?",
"role": "user"
}
] | 12\sqrt{3} |
[
{
"content": "Given that the function \\( y \\) satisfies the differential equation \\( \\frac{dy}{dx} = 1 - y^2 \\) and is symmetric about the point \\((\\ln(4), 0)\\), find the value of \\( y(\\ln(2)) \\).",
"role": "user"
}
] | -\dfrac{3}{5} |
[
{
"content": "Find the angle between the tangents drawn to the curve \\(3x^2+4xy+5y^2-4=0\\) at the points of intersection with the lines \\(y=-\\frac{3}{2}x\\) and \\(y=-\\frac{2}{5}x\\).",
"role": "user"
}
] | \dfrac{\pi}{2} |
[
{
"content": "Calculate the limit \\(\\lim_{x\\to 0}{f(x)-f(\\ln(1+x))\\over x^3}\\) given that \\(f(x)\\) is \\(C^1\\) in some neighborhood about \\(x=0\\), \\(f'(0)=0\\), and \\(f''(0)=1\\). Determine if the limit exists and find its value if it does.",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "Determine the asymptotic decay rate for the sequence \\( \\{b_n\\}_{n=1}^{\\infty} \\) where \\( b_n \\) is the \\( n \\)-th largest number in the sequence \\( a_{i,j} = \\frac{1}{i^2 j^2} \\) arranged in decreasing order. Specifically, find if \\( b_n = O(n^{-k}) \\) for some \\( k \\).",
"ro... | 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... | (9.00^n + 0.407^e)m/s |
[
{
"content": "Calculate the operator norm of the operator $A$ on the Hilbert space $L^2([0, \\pi])$, where $A$ is defined by\n\\[ A(f)(x) = \\int_0^\\pi \\sin(x-y) f(y) \\, dy, \\quad 0 \\le x \\le \\pi. \\]\nDetermine \\( \\|A\\| \\).",
"role": "user"
}
] | \dfrac{\pi}{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... | Metal A |
[
{
"content": "Find the number of permutations \\(a_1, a_2, \\ldots, a_{100}\\) of the numbers \\(1, 2, \\ldots, 100\\) such that:\n\\[|a_1 - 1| = |a_2 - 2| = \\ldots = |a_{100} - 100|\\]",
"role": "user"
}
] | 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... | 1.14 |
[
{
"content": "Calculate the limit: $$\\lim_{x \\rightarrow 0}\\frac{x^{2}\\ln\\left(1+2x\\right)}{2\\sin\\left(x\\right)\\left[ \\cos\\left(3x\\right)-1\\right]} $$",
"role": "user"
}
] | -\dfrac{2}{9} |
[
{
"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... | 7.67 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... | 2,3-dimethylbut-2-ene |
[
{
"content": "Calculate the distance between the matrix \\( A = \\begin{bmatrix} 1&-1 \\\\ 1&4\\end{bmatrix} \\) and the subspace \\( W = \\text{Span} \\{ \\begin{bmatrix} 0&-1 \\\\ 1&2\\end{bmatrix}, \\begin{bmatrix} 1&0 \\\\ 0&2\\end{bmatrix} \\} \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Determine the rank of the equivalence class representing the integer \\(-4\\) in the context of ordered pairs of naturals, where \\((a, b)\\) represents the integer \\(a - b\\). Is it \\(\\omega\\) or \\(\\omega + 1\\)?",
"role": "user"
}
] | \omega |
[
{
"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 g |
[
{
"content": "Evaluate the limit: $$\\lim _{x \\to \\infty} \\frac {1} {x} \\int_1^x \\frac {\\ln t} {t} \\, \\mathrm d t$$ using integration by parts.",
"role": "user"
}
] | 0 |
[
{
"content": "Evaluate the limit using L'Hopital's Rule: $$\\lim_{x\\rightarrow 0}\\frac{\\sinh^2(2x)-4x^2}{x^2\\sinh^2(2x)}$$",
"role": "user"
}
] | \dfrac{4}{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... | 1.07e-25 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... | 2K_3PO_4 + 3Ca(NO_3)_2 -> Ca_3(PO_4)_2 + 6KNO_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... | Pentecostalism |
[
{
"content": "Evaluate the limit: $$\\lim_{x \\to 0}\\left(\\frac{\\sin^{-1}(x)}{x}\\right)^{\\frac{1}{\\sin^2(x)}}$$",
"role": "user"
}
] | e^{\frac{1}{6}} |
[
{
"content": "If \\( \\alpha, \\beta, \\gamma \\) are the roots of the equation \\( x^3 - 3x^2 + 3x + 7 = 0 \\), and \\( \\omega \\) is a cube root of unity, find the value of the expression:\n\\[\n\\frac{\\alpha - 1}{\\beta - 1} + \\frac{\\beta - 1}{\\gamma - 1} + \\frac{\\gamma - 1}{\\alpha - 1}\n\\]\nChoose ... | D |
[
{
"content": "Determine the leading order term of the asymptotic expansion for \\( \\int_{1/x}^1 e^{-t} \\, dt \\) as \\( x \\to \\infty \\).",
"role": "user"
}
] | 1 - \frac{1}{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... | 0.155 atm, 0.493 atm, 0.820 atm, and 1.310 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... | 0.168kg |
[
{
"content": "Let $f:\\mathbb{R}\\rightarrow\\mathbb{R}$ be a function such that $f'(x)$ is continuous and $|f'(x)|\\le|f(x)|$ for all $x\\in\\mathbb{R}$. If $f(0)=0$, find the maximum value of $f(5)$. \\( \\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... | 2.618% |
[
{
"content": "Determine whether there exists a positive integer with exactly 2019 digits, all of which are positive, such that the sum of the squares of its digits is a perfect square.",
"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... | 20.8% |
[
{
"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... | 2 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... | 42,128.69 |
[
{
"content": "Consider the function $f(n)$ defined by:\n\n$$f(n) = 1 + (n!)^2 - \\sigma(n!)(n!)^2 + 2\\sum_{k=1}^{\\sigma(n!)-1} \\left\\lfloor \\frac{k(1 + (n!)^2)}{\\sigma(n!)} \\right\\rfloor$$\n\nwhere $\\sigma(n)$ is the divisor function that sums the divisors of the integer $n \\geq 1$. Determine whether ... | 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... | 68.071 Newtons |
[
{
"content": "Given a power series \\(\\sum_{n=1}^{\\infty}a_nx^n\\) with radius of convergence \\(R\\), determine if the power series \\(\\sum_{n=1}^{\\infty}\\frac{a_n}{n}x^n\\) always has the same radius of convergence. Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Given that $\\sum_{i=1}^{n}a_i = 1$ and $a_i > 0$ for all $i$, find the value of $a_i$ that minimizes the summation $\\sum_{i=1}^{n}a_i^2$. What is the optimal value for each $a_i$?",
"role": "user"
}
] | \dfrac{1}{n} |
[
{
"content": "Determine the dimension of the kernel of $X^3$, where $X$ is the nilpotent matrix $J_n$. Assume $J_n$ is the standard nilpotent Jordan block of size $n$. Provide your answer as a single integer.",
"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... | 1.7999e-11 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... | 417.21e-7 C/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... | 0.429 g |
[
{
"content": "Find the sum of the series \\( \\sum_{k=1}^{\\infty} \\frac{a_k}{k} \\), where \\( a_k = \\frac{k+1}{2k+4} - \\frac{k}{2k+2} \\) for \\( k \\geq 2 \\).",
"role": "user"
}
] | \dfrac{1}{8} |
[
{
"content": "The lateral edges of a triangular pyramid are mutually perpendicular, and the sides of the base are \\(\\sqrt{85}\\), \\(\\sqrt{58}\\), and \\(\\sqrt{45}\\). The center of the sphere, which touches all the lateral faces, lies on the base of the pyramid. Find the radius of this sphere.",
"role"... | \dfrac{14}{9} |
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