prompt list | ground_truth string |
|---|---|
[
{
"content": "Evaluate the improper integral \\( \\int_0^\\infty \\frac{\\ln(1/x)}{1-x^2}\\,dx \\).",
"role": "user"
}
] | \dfrac{\pi^2}{4} |
[
{
"content": "Express the function \\( f(x) = |x+\\frac{\\pi}{2}| - 2|x| + |x-\\frac{\\pi}{2}| \\) as a Fourier series using the given coefficients \\( a_0 = \\frac{\\pi}{2} \\) and \\( a_n = \\frac{8}{\\pi n^2}\\sin^2(\\frac{n\\pi}{4}) \\). Evaluate the sum of the Fourier series at \\( x = 0 \\).",
"role":... | \pi |
[
{
"content": "Find the sum of the shifted sinc function given by:\n\\[ g(y) = \\sum_{n=-\\infty}^\\infty \\frac{\\sin(\\pi(n - y))}{\\pi(n-y)} \\]",
"role": "user"
}
] | 1 |
[
{
"content": "Evaluate the contour integral \\( \\int_{l}(z^2+\\bar{z}z)dz \\) where \\( l \\) represents the upper half of the unit circle, \\( |z|=1 \\), with \\( 0 \\leq \\arg(z) \\leq \\pi \\).",
"role": "user"
}
] | -\dfrac{8}{3} |
[
{
"content": "Given that $P(\\mu - t\\sigma < X < \\mu + t\\sigma) \\ge 1 - \\frac{1}{t^2} = 0.98$, find the smallest integer $k$ such that $38k = \\mu + t\\sigma = (0.75)(48k) + \\sqrt{50} \\cdot \\sqrt{48k \\cdot 0.75 \\cdot 0.25}$. Round up if necessary to ensure the probability is at least 0.98.",
"role... | 113 |
[
{
"content": "Find the limit of the sequence defined by $a_1 = 7$ and $a_{n+1} = \\frac{2^n + n^{(10^{17})}}{(1 + \\frac{1}{n})^{(n^2)} - \\frac{1}{n}} \\cdot a_n$ as $n$ approaches infinity.",
"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... | 2066.04 W |
[
{
"content": "Determine the radius of convergence of the power series solution to the differential equation \\((1-x^2)y''-2xy'+\\lambda y = 0\\) using the ratio test. The solution is given by \\(y(x) = \\sum_{n=0}^{\\infty} C_n x^n\\) with the recurrence relation \\(C_{n+2} = \\frac{n(n+1)-\\lambda}{(n+2)(n+1)}... | 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... | one half wavelength |
[
{
"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... | 5.24% |
[
{
"content": "Evaluate the limit \\( \\lim_{n\\to \\infty}\\int_{0}^{1}nx^nf(x)\\,dx \\) for a continuous function \\( f \\) defined on the interval \\([0,1]\\).",
"role": "user"
}
] | f(1) |
[
{
"content": "Is the hypothesis that a group $G$ is abelian necessary for a group with a totally ordered subgroup lattice to be isomorphic to $\\mathbb{Z}(p^k)$ for some prime $p$ and $k \\in \\{1,2,...\\}\\cup\\{\\infty\\}$?",
"role": "user"
}
] | No |
[
{
"content": "Find the limit of the sequence \\( x_n= \\frac{[(a+1)(a+2)...(a+n)]^{1/n}}{n} \\), where \\( a \\) is a fixed positive real number.",
"role": "user"
}
] | \dfrac{1}{e} |
[
{
"content": "Evaluate the integral \\( \\int_C \\frac{\\cos(z)}{z^2 - 4z + 5} \\, dz \\), where \\( C \\) is the contour consisting of the real line segment from \\(-r\\) to \\(r\\) and the upper semi-circle of radius \\(r\\).",
"role": "user"
}
] | \pi \cos(2 + i) |
[
{
"content": "What is \\(\\lim_{n \\to \\infty} m((n, \\infty))\\)?",
"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... | 3.75 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... | Dependent Clause |
[
{
"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... | $29,915.53 |
[
{
"content": "Let $G = \\Bbb Z \\times \\Bbb Z$ with group law given by addition. Let $H$ be the subgroup generated by $(2,3)$, so $H$ consists of all elements of the form $(2a,3a)$ for some $a \\in \\Bbb Z$. To which group is $G/H$ isomorphic?",
"role": "user"
}
] | \mathbb{Z} |
[
{
"content": "Evaluate the limit \\( \\lim\\limits_{n \\rightarrow +\\infty} \\int_{0}^{1} nxe^{-\\sqrt{n}x}e^{-x^2}dx \\). Can the Dominated Convergence Theorem be used to interchange the limit and the integral?",
"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... | -$157,250 |
[
{
"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... | 8.72e-9 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... | TCF/k where TCF is the terminal cash flow and k is the cost of capital |
[
{
"content": "Determine the largest exponent \\(d\\) such that the volume of the set \\(X(r)\\), consisting of matrices \\(A \\in M(n \\times m)\\) with \\(n \\leq m\\), where the largest singular value of \\(A\\) is at most 1 and the smallest singular value is less than \\(r\\), divided by \\(r^d\\) remains un... | m - n + 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... | 15.7 seconds |
[
{
"content": "Determine the number of 3-to-1 functions from the set \\( A \\) to the set \\( B \\), where \\( |A| = 15 \\) and \\( |B| = 5 \\).",
"role": "user"
}
] | \dfrac{15!}{(3!)^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... | v = \sqrt{\dfrac{2gL}{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... | $6,987.5 |
[
{
"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... | $63,437 |
[
{
"content": "Evaluate the integral \\( \\int_0^a \\arcsin\\sqrt{\\frac{a}{a+x}}\\,dx \\) using the substitution \\( x = a\\tan^2\\theta \\).",
"role": "user"
}
] | a |
[
{
"content": "Consider a circle with radius $R$. Two points, $A$ and $B$, are chosen at random within this circle. Determine the probability that the circle with center $A$ and radius equal to the distance $AB$ is entirely contained within the original circle. The circles may be tangent.",
"role": "user"
... | \dfrac{1}{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... | 67.14% |
[
{
"content": "Calculate the surface area of the portion of the cone \\(x^2 + y^2 = z^2\\) that lies inside the sphere \\(x^2 + y^2 + z^2 = 6z\\).",
"role": "user"
}
] | 9\sqrt{2}\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... | 25.6352 months |
[
{
"content": "Determine whether the function $f(n) = n^{\\log(n)}$ is $O(\\log(n)^n)$. Justify your answer by comparing the growth rates of the functions.",
"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... | \frac{3a}{2b\sqrt T }\ln \left[ {\frac{{{V_1}\left( {{V_2} + b} \right)}}{{{V_2}\left( {{V_1} + b} \right)}}} \right] |
[
{
"content": "Evaluate the sum \\( \\tan\\left(\\frac{\\pi}{16}\\right)+\\tan\\left(\\frac{5\\pi}{16}\\right)+\\tan\\left(\\frac{9\\pi}{16}\\right)+\\tan\\left(\\frac{13\\pi}{16}\\right) \\) using tangent addition formulas.",
"role": "user"
}
] | -4 |
[
{
"content": "Determine the dimension of $H^0(T, \\mathcal{L})$, where $T$ is a complex torus of dimension 2 and $\\mathcal{L}$ is a line bundle that gives $T$ an $(m,n)$-polarization.",
"role": "user"
}
] | mn |
[
{
"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... | HCl |
[
{
"content": "Evaluate the integral: \\[ \\int_{0}^{2\\pi} \\frac{\\cos(3\\theta)}{5-4\\cos(\\theta)}d\\theta \\] without using the residue theorem.",
"role": "user"
}
] | \dfrac{\pi}{12} |
[
{
"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... | 90.272 m^2/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... | 9.11x10^6 m |
[
{
"content": "Evaluate the integral \\( \\oint_{C} \\frac{3z^2+z}{z^2-1} \\, dz \\) using the residue theorem, where \\( C \\) is the circle \\( |z|=2 \\).",
"role": "user"
}
] | 2\pi 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... | nRln(V_2/V_1) |
[
{
"content": "Determine whether every group action can be represented as a conjugation by unitaries. Specifically, let $G$ be a countable discrete group, and define $l^\\infty(G) := \\{f \\colon G \\to \\mathbb{C} \\mid \\|f\\|_\\infty < \\infty\\}$. Consider the group action of $s \\in G$ on $f \\in l^\\infty(... | Yes |
[
{
"content": "Let $f(x_1, \\dots, x_n)$ be the joint probability density function of random variables $X_1, \\dots, X_n$, where $f(x_1, \\dots, x_n) > 0$ if $X_i \\in D$ for $i=1, \\dots, n$ for some set $D$, and $0$ elsewhere. Suppose $f$ is symmetric, meaning its value is the same for any permutation of the v... | \dfrac{1}{n!} |
[
{
"content": "Evaluate the limit of \\((x^2+y^2)^{x^2y^2}\\) as \\((x,y)\\) approaches \\((0,0)\\).",
"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... | Ni(NH3)6^2+ + 2OH- |
[
{
"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... | 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... | Mix 449.15 ml of ethylene glycol in 1 L of water. |
[
{
"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... | $91,500 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 4.138% |
[
{
"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... | $116,401.82 |
[
{
"content": "Find the maximum constant \\(M\\) such that for the side lengths \\(a \\leq b < c\\) of a right triangle, the inequality \\(\\frac{1}{a}+\\frac{1}{b}+\\frac{1}{c} \\geq \\frac{M}{a+b+c}\\) holds.",
"role": "user"
}
] | 5 + 3\sqrt{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... | Episco, INC
Balance Sheets
December 31, 2013 and 2012
Assets 2013 2012 Liabilities 2013 2012
Current assets Current liabilities
Cash $304 $240 Note payable $392 $320
Accounts receivable $1,008 $960 Accounts payable $984 $880
Inventory $1,928 $1,840 Total current liabilities $1,376 $1,200
Total current assets $3,24... |
[
{
"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... | 8.8 |
[
{
"content": "In linear regression, given a noisy target $y = (w^{*})^T \\textbf{x} + \\epsilon$, where $\\epsilon$ is a noise term with zero mean and $\\sigma^2$ variance, independently generated for every example $(\\textbf{x},y)$, determine the expected error of the best possible linear fit to this target.",... | \sigma^2 |
[
{
"content": "Consider smooth vector fields $X_i$, $1 \\le i \\le n$, on an open set $U \\subset \\mathbb{R}^n$, which are linearly independent at each point. Define $[X_i, X_j] = \\sum_{k=1}^n c_{ij}^kX_k$ with constant coefficients $c_{ij}^k$. Let $\\overline{X}_i$, $1 \\le i \\le n$, be another copy on $\\ov... | 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... | broad spectrum |
[
{
"content": "Evaluate the contour integral \\( \\oint_{C_r}\\frac{\\pi^2}{\\sin^2\\pi w}(w-N)dw \\), where \\( N \\) is a fixed integer and \\( C_r \\) is a circle centered at \\( N \\) with radius \\( 0 < r < 1 \\).",
"role": "user"
}
] | 2\pi 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... | optional |
[
{
"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... | 2.64 rpm |
[
{
"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... | 0.735 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... | 285 kJ |
[
{
"content": "Determine whether the following statement is true or false: For any function \\( f(x) \\) that is either strictly increasing or strictly decreasing on the interval \\((a,b)\\), where \\(a\\) and \\(b\\) are integers, the inequality \\( \\int _{a}^{b} f(x)dx < \\sum_{n=a}^{b}f(n) \\) holds.",
"... | False |
[
{
"content": "Given the joint density function \\( f_{X,Y}(x,y) = \\frac{2}{3} (x+2y) \\) for \\( 0 < x < 1, 0 < y < 1 \\), find the probability \\( P(X > Y) \\).",
"role": "user"
}
] | \dfrac{4}{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... | 10.93% |
[
{
"content": "Let \\( A_1 A_2 \\cdots A_n \\) be a regular \\( n \\)-gon inscribed in the unit circle. Calculate the value of \\( \\sum_{1 \\leq i < j \\leq n} |A_i A_j|^2 \\).",
"role": "user"
}
] | n^2 |
[
{
"content": "Given real numbers \\( x \\) and \\( y \\) satisfying the equation \\(\\frac{x^{2}}{3}+y^{2}=1\\), determine the range of values of the expression \\( P = |2x+y-4|+|4-x-2y| \\).",
"role": "user"
}
] | [2, 14] |
[
{
"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... | $334,000 |
[
{
"content": "On each of the first three days of January, there is a $\\frac{1}{3}$ chance that it will snow where Bob lives. On each of the next four days, there is a $\\frac{1}{4}$ chance that it will snow. What is the probability that it snows at least once during the first week of January?",
"role": "us... | \frac{29}{32} |
[
{
"content": "Determine whether non-commuting matrices can share a simultaneous eigenvector. 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... | 450 g |
[
{
"content": "Given the equation \\( \\frac{a_1}{k^2+2}+\\cdots+\\frac{a_5}{k^2+6}=\\frac{1}{k^2+1} \\) for all \\( k = \\{1,2,3,4,5\\} \\), find the value of \\( \\frac{a_1}{3}+\\cdots+\\frac{a_5}{7} \\).",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "Find the values of $z$ that satisfy the equation $e^{2z} = -e^{i\\pi}$. Express your answer in terms of $z$. ",
"role": "user"
}
] | z = i\pi 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... | 0.051 m |
[
{
"content": "Calculate the limit: \\[ \\lim\\limits_{(x,y) \\to (0,0)} \\dfrac{2 - \\cos(x) - \\cos(y)}{x^2+y^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... | 5.03 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... | 23.87% |
[
{
"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... | 56 g/mol |
[
{
"content": "The numbers \\( x \\) and \\( y \\) satisfy the equations \\(\\operatorname{tg} x + \\operatorname{tg} y = 4\\) and \\(3 \\sin (2x + 2y) = \\sin 2x \\sin 2y\\). Find the value of \\(\\operatorname{ctg} x \\operatorname{ctg} y\\).",
"role": "user"
}
] | \dfrac{7}{6} |
[
{
"content": "Evaluate the integral \\( \\int_c \\left( e^{z^2} \\left( \\frac{1}{z^2} - \\frac{1}{z^3} \\right) \\right) \\, dz \\) where \\( c \\) is the unit circle.",
"role": "user"
}
] | -2\pi i |
[
{
"content": "Let $X$ be a continuous uniform random variable with support $[-1,1]$, and let $Y = X^2$. Calculate the expected value of $X$ conditioned on $Y$, i.e., $E[X \\mid Y]$. Justify why the answer is $0$. Provide a clear explanation of the calculation.",
"role": "user"
}
] | 0 |
[
{
"content": "For what values of \\( \\gamma \\in \\mathbb{C} \\) do there exist non-singular matrices \\( A, B \\in \\mathbb{C}^{n \\times n} \\) such that \\( AB = \\gamma BA \\)?",
"role": "user"
}
] | \gamma^n = 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... | 2.89 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... | $8,592 |
[
{
"content": "Determine the cardinality of the set of all binary relations from the set of natural numbers \\( \\mathbb{N} \\) to the Cartesian product \\( \\mathbb{N} \\times \\mathbb{N} \\).",
"role": "user"
}
] | 2^{\aleph_0} |
[
{
"content": "Consider a function \\(g:[a,b]\\rightarrow [a,b]\\) that satisfies \\(|g(x)-g(y)|\\leq k|x-y|\\) for all \\(x,y \\in [a,b]\\), where \\(0<k<1\\). If we randomly choose an initial point \\(y_0\\in[a,b]\\) and define a sequence recursively as \\(y_{n+1}=g(y_n)\\), does the sequence \\(\\langle y_n\\... | Yes |
[
{
"content": "Determine the asymptotic density of positive integers \\( n \\) such that \\( \\gcd(n,\\sigma(n^2))=\\gcd(n^2,\\sigma(n^2)) \\).",
"role": "user"
}
] | 1 |
[
{
"content": "Find the constant \\( g \\) such that the size of the test is \\( \\beta \\) for the following scenario: You have a random sample \\( X_1, X_2, \\dots, X_n \\) drawn from a uniform distribution on \\([\\varphi, \\varphi + 1]\\). To test the null hypothesis \\( H_0 : \\varphi = 0 \\) against the al... | 1 - \beta^{1/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... | 5.12 g |
[
{
"content": "Calculate the fundamental group \\( \\pi_1 \\) of the space \\( S^2 - \\{N, S\\} \\), where \\( N \\) and \\( S \\) are the north and south poles of the 2-sphere \\( S^2 \\).",
"role": "user"
}
] | \mathbb{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... | 1. Initial velocity
2. Final velocity
3. Average velocity
4. Acceleration
5. Distance traveled |
[
{
"content": "Let the point $A$ lie on the exterior of the circle $k(R)$. From $A$, draw the tangents $AB$ and $AC$ to $k$. If the triangle $ABC$ is equilateral, find the side length of $\\triangle ABC$ in terms of $R$.",
"role": "user"
}
] | R\sqrt{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... | -1.014 rad |
[
{
"content": "Evaluate the limit: \\( \\lim_{x \\to 0^-}{\\frac{-e^{1/x}}{x}} \\)",
"role": "user"
}
] | 0 |
[
{
"content": "Find the smallest integer $n$ such that $10^{17} < n!$. Provide your answer as an integer.",
"role": "user"
}
] | 19 |
[
{
"content": "Two teams, each with seven players numbered 1 to 7, compete in a series of games. In the first game, the players numbered 1 from each team play against each other. The loser of each game is eliminated and replaced by the next player from the same team. This process continues until all players from... | 432 |
[
{
"content": "Given integers \\(a, b, c\\) with \\(a < c\\) and satisfying the equation \\(a^2 + c^2 = 2b^2\\), prove that \\(c^2 - a^2\\) is divisible by 48.",
"role": "user"
}
] | 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... | 0.16 m |
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