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... | 15.5 m/s, -15.625 m/s, 135 degrees to the +x axis |
[
{
"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... | cultural anthropology |
[
{
"content": "Determine whether the sequence \\( \\sum^n_{k=0}\\frac{x^k}{k^2} \\) is a Cauchy sequence in \\( n \\) for \\( x \\in [0,1] \\).",
"role": "user"
}
] | Yes |
[
{
"content": "Determine the value of \\( \\lambda \\) for which the integral \\[ I = \\int_{0}^{\\infty} \\left(\\frac{x+1}{3x^2 + \\lambda} - \\frac{\\lambda}{2x+1}\\right)dx \\] converges.",
"role": "user"
}
] | \dfrac{2}{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... | $130 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... | 3.2e-6 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... | 10042.83 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... | -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... | The masses of product are: m(CaCl_2) = 554.5g, m(CO_2) = 219.8g, m(H_2O) = 89.9g |
[
{
"content": "Determine the number of unique permutations of a list containing 5 A's and 6 B's, considering permutations equivalent under a cyclic shift. For example, the permutation [A, A, B, A, B, B, B, A, B, B, A] is considered equivalent to [A, B, A, B, B, B, A, B, B, A, A].",
"role": "user"
}
] | 42 |
[
{
"content": "Calculate the limit: $$ \\lim_{x \\to \\frac{\\pi}{2}^-} \\ln\\left(\\frac{2x}{\\pi}\\right)\\cdot e^{\\frac{1}{\\cos x}}$$",
"role": "user"
}
] | -\infty |
[
{
"content": "Determine whether a simple algebraic extension $\\mathbb{F}(a)$ of degree $n$ over a field $\\mathbb{F}$ can be embedded in the matrix ring $M_n(\\mathbb{F})$. 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... | 11E3 m/s |
[
{
"content": "Determine whether there exists a Galois extension $K/\\mathbb{Q}(i)$ such that $\\mathrm{Gal}(K/\\mathbb{Q}(i)) \\cong D_8$, where $D_8$ is the dihedral group of order 8.",
"role": "user"
}
] | Yes |
[
{
"content": "Let \\( \\rho:\\mathbb{R} \\to \\mathbb{R} \\) be a continuous function such that \\( \\rho(x) \\ge 0 ~\\forall ~x \\in \\mathbb{R} \\), and \\( \\rho(x)=0 \\) if \\( |x|\\ge 1 \\) and \\( \\int_{-\\infty}^{\\infty}\\rho(t)dt=1 \\). Let \\( f:\\mathbb{R}\\to\\mathbb{R} \\) be a continuous function... | f(0) |
[
{
"content": "Evaluate the integral of \\( \\log(z) \\) over the unit circle in the complex plane.",
"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... | -10 m/s^2 |
[
{
"content": "Three vertices of a cube are $P=(7,12,10)$, $Q=(8,8,1)$, and $R=(11,3,9)$. What is the side length of the cube?",
"role": "user"
}
] | 7 |
[
{
"content": "Determine the radius of convergence for the power series of a function $g$ that is analytic on the disk $D(Q,s)$ and has a singularity on the boundary of this disk.",
"role": "user"
}
] | 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... | 1.57e-3 |
[
{
"content": "Given the prior density function \\((B+1)Bq(1-q)^{B-1}\\) where \\(q\\) is the probability of death and is between 0 and 1, calculate the parameter \\(B\\) if the prior mean is 0.1.",
"role": "user"
}
] | 18 |
[
{
"content": "Determine the number of subsets of the set \\( A = \\{1, 2, 3, \\ldots, 35\\} \\) that contain exactly 26 elements and have a sum of elements divisible by 5.",
"role": "user"
}
] | 14121492 |
[
{
"content": "Determine the correct representation of the real hyperbolic plane \\(\\mathbb{RH}^2\\) as a homogeneous space. Is it \\(SO(2,1)/SO(2)\\) or \\(SL(2)/SO(2)\\)?",
"role": "user"
}
] | SO(2,1)/SO(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... | 0.599 |
[
{
"content": "Determine the dimension of the real vector space $V$ consisting of all $3\\times3$ real matrices \\(\\begin{bmatrix}a_{11}&a_{12}&a_{13}\\\\a_{21}&a_{22}&a_{23}\\\\a_{31}&a_{32}&a_{33}\\end{bmatrix}\\) such that the sum of the elements in each row and each column is zero.",
"role": "user"
}
... | 4 |
[
{
"content": "Evaluate the limit: $$ \\lim_{x\\to +\\infty}\\frac{e^{-2x}(\\cos(x)+2\\sin(x))}{e^{-x}(\\cos(x)+\\sin(x))} $$ without using de l'Hopital's rule.",
"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... | {eq}F = \dfrac {Gm_1m_2} {d ^ 2}{/eq} |
[
{
"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... | 134.7 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... | 104.4 degree Celsius |
[
{
"content": "What is the least number of colors needed to shade the tessellation shown, such that no two tiles sharing a side are the same color?\n\n[asy]\ndraw((-8,-2)--(8,-2));\ndraw((-8,2)--(8,2));\ndraw((-8,6)--(8,6));\ndraw((-8,10)--(8,10));\ndraw((-8,14)--(8,14));\npath a=(-8,14)--(-6.5,12)--(-10,8)--(-6... | 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... | $66,750 |
[
{
"content": "In a completely regular space $X$, for every closed set $F \\subseteq X$, does there exist a nonzero continuous function $f: X \\to [0,1]$ such that $f = 0$ outside $F$?",
"role": "user"
}
] | No |
[
{
"content": "Find the range of negative values for \\( a \\) such that the inequality \\( \\sin^2 x + a \\cos x + a^2 \\geq 1 + \\cos x \\) is satisfied for all \\( x \\in \\mathbb{R} \\).",
"role": "user"
}
] | (-\infty, -2] |
[
{
"content": "Evaluate the limit: $$\\lim_{s \\rightarrow 0} \\int_{0}^s \\frac{1}{g(v)}dv$$ given that $g(v)$ is a continuous function around $0$ and $\\lim_{v \\rightarrow 0} g(v) = \\infty$. ",
"role": "user"
}
] | 0 |
[
{
"content": "The teacher asks Bill to calculate $a-b-c$, but Bill mistakenly calculates $a-(b-c)$ and gets an answer of 11. If the correct answer was 3, what is the value of $a-b$?",
"role": "user"
}
] | 7 |
[
{
"content": "How many ways are there to place nine different rings on four fingers, assuming the order of rings on each finger does not matter?",
"role": "user"
}
] | 262144 |
[
{
"content": "Find the values of \\(x\\) for which two of the numbers \\(\\log _{\\sqrt{x+34}}(2x+23)\\), \\(\\log _{(x+4)^{2}}(x+34)\\), and \\(\\log _{\\sqrt{2x+23}}(-x-4)\\) are equal, and the third number is greater by 1.",
"role": "user"
}
] | -9 |
[
{
"content": "Suppose $A$ and $B$ are $m \\times n$ matrices with orthonormal columns, and their column spaces are not the same. Determine whether there exists an orthogonal matrix $O$ such that $A^T O B$ is orthogonal.",
"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... | 108.4 rad |
[
{
"content": "Determine whether the function \\( f: [0,1 ] \\rightarrow \\mathbb{R} \\), defined by \\( f(x)= \\frac {1}{n} \\) when \\( x = \\frac {1}{n} \\) for \\( n \\in \\mathbb{N} \\) and \\( f(x) = 1 \\) otherwise, is Riemann integrable on \\([0,1]\\). If it is, find the value of the integral.",
"rol... | 1 |
[
{
"content": "Evaluate the limit: $$\\lim_{n\\to \\infty}\\left(\\frac{\\ln x_n}{\\root n \\of{e}-1}-n\\right)$$ where $x_n=\\sum_{k=1}^n \\frac{1}{(k+1)!}$. Provide your answer in a simplified form.",
"role": "user"
}
] | -\infty |
[
{
"content": "Let $(X,\\mathcal{M},\\mu)$ be a measure space and $1\\leq p\\leq\\infty$. Suppose $f,g\\in L^p(X,\\mathcal{M},\\mu)$ with $f(x)g(x)=0$ for almost all $x\\in X$ and $\\Vert f\\Vert_p=\\Vert g \\Vert_p=1$. Verify by direct calculation that \\(\\Vert f+g\\Vert_p^2+\\Vert f-g\\Vert_p^2=2(\\Vert f\\Ve... | 2 |
[
{
"content": "What is the minimum value of $z$ if $z=x^2+2y^2+6x-4y+22?$",
"role": "user"
}
] | 11 |
[
{
"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... | i = 0.768, Q = 4.658 |
[
{
"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... | 4.62 m/s |
[
{
"content": "Solve the equation \\(3^{(x-3)^{(x+3)}} = (3x+2)^{(x-3)}\\) for \\(x\\).",
"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... | U = m * g * d * \theta |
[
{
"content": "Determine whether \\(c^c - a^a\\) is divisible by \\(2^a b^2\\) when \\(a = 2k\\), \\(b = 2k+1\\), and \\(c = 2k+2\\) for some integer \\(k\\).",
"role": "user"
}
] | Yes |
[
{
"content": "Find the limit of the sequence defined recursively by:\n\\[ x_n = x_{n-1} \\cdot \\left(2-\\frac{\\tan^{-1}(n) \\cdot n^2 - 3}{3 + n^2}\\right) \\]\nwith initial condition \\( x_1 = 2-\\frac{\\tan^{-1}(1) \\cdot 1^2 - 3}{3 + 1^2} \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Determine the order of the group $G$ defined by the relations $\\langle a, b; a^5, b^4, ab = ba^{-1}\\rangle$. Provide your answer as a single integer.",
"role": "user"
}
] | 20 |
[
{
"content": "Find the closed form of the infinite product \\( P = \\prod\\limits_{k=3}^{\\infty}\\frac{k}{2\\pi}\\sin{\\left(\\frac{2\\pi}{k}\\right)} \\), which represents the product of the areas of all regular polygons inscribed in a circle with area 1.",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "Determine whether $f(n) = O(g(n))$ or $g(n) = O(f(n))$, where $f(n) = (\\sqrt{n})^{\\sqrt{n}}$ and $g(n) = (\\lg n)^n$. Provide your answer with justification.",
"role": "user"
}
] | f(n) = O(g(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... | $47.58 per unit |
[
{
"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... | $53,490.30 |
[
{
"content": "Evaluate the limit \\(\\displaystyle\\lim_{n\\to\\infty}\\left(1+\\dfrac{1}{n}\\right)\\left(1+\\dfrac{1}{2n}\\right)\\left(1+\\dfrac{1}{4n}\\right)\\ldots\\left(1+\\dfrac{1}{2^{n-1}n}\\right)\\).",
"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... | 84.76 m |
[
{
"content": "Find the function \\( f(x) \\) that satisfies the functional equation \\( f(x+1) = x f(x) \\).",
"role": "user"
}
] | \Gamma(x) |
[
{
"content": "Find all solutions to the equation \\(x^5 = 3\\) in the field \\(\\mathbb{K}\\) with 25 elements, which is an extension of \\(\\mathbb{Z}_5\\) through the element \\(\\sqrt{2}\\), without using brute force.",
"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... | = (C# / D#) * 1000 |
[
{
"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... | 2410 rev/s |
[
{
"content": "Determine whether the condition \\(\\sum\\limits_{k=1}^{\\infty} (a_{k+1} - a_k)^2 <\\infty\\) implies that \\(\\frac{1}{n^2} \\sum\\limits_{k=1}^n a_k^2 \\to 0\\) 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... | 13188 |
[
{
"content": "Evaluate the integral \\( \\int_{0}^{\\pi}\\frac{1}{\\sin^4 2x+\\cos^4 2x}\\,\\text{d}x \\) using complex analysis and the unit circle.",
"role": "user"
}
] | \pi \sqrt{2} |
[
{
"content": "Given a trapezoid $ABCD$ with $AD \\parallel BC$ and $AB = CD$, the diagonals $AC$ and $BD$ intersect at point $P$, and the lines $AB$ and $CD$ intersect at point $Q$. Points $O_1$ and $O_2$ are the centers of the circumcircles of triangles $ABP$ and $CDP$, respectively, with radius $r$. Construct... | 2 |
[
{
"content": "Let $M$ be a compact manifold and $f : M \\rightarrow M$ be a homeomorphism isotopic to the identity. Can $f$ be fragmented, meaning for any open cover $\\mathcal{U}$ of $M$, there exist homeomorphisms $f_1,\\ldots,f_n$ isotopic to the identity and open sets $U_1,\\ldots,U_n \\in \\mathcal{U}$ suc... | 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... | $9,836.90 |
[
{
"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... | $\lambda = \dfrac { d } { \sqrt { 2 } }$ |
[
{
"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... | 1.56e8 V/m |
[
{
"content": "Let $p$ be a prime and $n$ a positive integer not divisible by $p$. If $p^k$ fully divides $x-y$, where $x$ and $y$ are distinct integers not divisible by $p$, does it follow that $p^k$ fully divides $x^n-y^n$? 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... | $8.28 |
[
{
"content": "Evaluate the limit \\( \\lim_{(x,y)\\rightarrow (0,0)}\\frac{e^{-\\frac{1}{x^2+y^2}}}{x^6+y^6} \\).",
"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... | 0.09652 mg |
[
{
"content": "Determine whether the limit \\(\\lim_{(x,y)\\to (0,0)} \\frac{x^4y^4}{(x^2+y^2)^2}\\) exists.",
"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... | 1e-14V |
[
{
"content": "Compute the probability $P(4X - 3Y > 5Z)$, where $X$, $Y$, and $Z$ are independent random variables, each following a normal distribution with means $2$, $1$, and $0$, respectively, and a common variance of $2$. Provide your answer as a probability value.",
"role": "user"
}
] | 0.6915 |
[
{
"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... | Silver |
[
{
"content": "Find all natural number solutions \\((x, y)\\) to the equation \\(3^x - 7^y = 2\\).",
"role": "user"
}
] | (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... | 5.1e6 N/C |
[
{
"content": "What is the remainder when \\(3!^{{{5!}^{...}}^{2013!}}\\) is divided by 11?",
"role": "user"
}
] | 1 |
[
{
"content": "Let $X$ be a compact metric space and $\\mu$ be a positive, finite, atomless, regular measure on the $\\sigma$-algebra of Borel subsets of $X$. Does there exist a set $C \\subset X$ such that $C$ is uncountable and has $\\mu$-measure zero?",
"role": "user"
}
] | Yes |
[
{
"content": "If a group $G$ has a central subgroup $H$ such that $G/H$ is cyclic, does this imply that $G$ is abelian? 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... | $113,648.76 |
[
{
"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... | 0.80 |
[
{
"content": "Find the value of \\( n \\) such that the area under the curve \\( y = x^n \\) from \\( x = 1 \\) to \\( x = a \\) is one third of the area under the curve from \\( y = 1 \\) to \\( y = a^n \\).",
"role": "user"
}
] | 3 |
[
{
"content": "Evaluate the expected value \\( \\mathbf{E}[\\cos(B(t))] \\) where \\( B(t) \\) is a standard Brownian motion.",
"role": "user"
}
] | e^{-t/2} |
[
{
"content": "Suppose $G_1$ and $G_2$ are groups with orders $3p$ and $3q$, respectively, where $p$ and $q$ are distinct primes. If there exists a non-trivial homomorphism $\\phi: G_1 \\to G_2$, determine the order of the image of $\\phi$. \\( \\boxed{\\text{Order of the image}} \\)",
"role": "user"
}
] | 3 |
[
{
"content": "Determine all polynomial functions \\( p(x) \\) with complex coefficients such that \\( p(x) \\) maps real numbers to real numbers and non-real complex numbers to non-real complex numbers.",
"role": "user"
}
] | p(x) = ax + b |
[
{
"content": "Determine the cardinality of the set of epsilon numbers (\\(\\epsilon\\)-numbers) less than \\(\\omega_1\\), the first uncountable ordinal. Is this set countable or uncountable?",
"role": "user"
}
] | \aleph_1 |
[
{
"content": "Given $n$ i.i.d Gaussian random variables $x_i$ with mean $0$ and variance $1$, construct two Chi-squared random variables $Y=\\sum_{i \\in I_1} x_i^2$ and $Z=\\sum_{i \\in I_2} x_i^2$ with $|I_1|=|I_2|$. Compute $\\Pr(Y \\le Z)$ when $Y$ and $Z$ are not independent.",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "Evaluate the integral \\( \\int_{0}^{1} \\mathrm{~d} z \\int_{0}^{\\sqrt{1-z^{2}}} \\mathrm{~d} y \\int_{0}^{\\sqrt{1-z^{2}-y^{2}}}\\left(x^{2}+y^{2}+z^{2}\\right)^{-\\frac{3}{2}} \\mathrm{~d} x \\) using spherical coordinates.",
"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... | [Ar]3d^2 4s^2 |
[
{
"content": "Determine the number of distinct ways to fill a 3 × 3 grid with 0s and 1s, given that the center cell must contain a 0 and two grids are considered identical if one can be transformed into the other through a symmetry operation.",
"role": "user"
}
] | 51 |
[
{
"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... | 1.873 calories |
[
{
"content": "Solve the inequality \\(8\\cdot 3^{\\sqrt{x}+\\sqrt[4]{x}} + 9^{\\sqrt[4]{x}+1} \\geq 9^{\\sqrt{x}}\\) and express the solution set for \\(x\\).",
"role": "user"
}
] | [0, 16] |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 7.79e-8 s |
[
{
"content": "Evaluate the triple integral \\( \\iiint \\frac{1}{\\sqrt{x^2+y^2+(z-2)^2}} \\, dV \\) over the solid region \\( E \\) bounded by the sphere \\( x^2+y^2+z^2 = 1 \\).",
"role": "user"
}
] | \dfrac{2\pi}{3} |
[
{
"content": "Determine the least possible degree of a polynomial with real coefficients that has the roots $2\\omega, 2+3\\omega, 2+3\\omega^2$, and $2-\\omega-\\omega^2$, where $\\omega$ is a primitive cube root of unity.",
"role": "user"
}
] | 5 |
[
{
"content": "Determine whether every compact submanifold with corners in a Riemannian manifold is locally star-shaped with respect to some point in its boundary.",
"role": "user"
}
] | Yes |
[
{
"content": "Let \\( f(x) = ax^2 + bx + c \\). Given that \\( |f(x)| \\le 1 \\) for all \\( x \\) in \\([0,1]\\), and \\( f(0) = 1 \\), \\( f(1) = 0 \\), and \\( f\\left(\\frac{1}{2}\\right) = \\frac{1}{4} \\), determine the maximum possible value of \\(|a| + |b| + |c|\\).",
"role": "user"
}
] | 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... | 192 |
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