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... | 1.189 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... | 1.2 ft |
[
{
"content": "Compute the second moment of the chi-squared distribution with $n$ degrees of freedom.",
"role": "user"
}
] | n(n + 2) |
[
{
"content": "Determine the number of points where the functions \\( f(x) = x^2 \\) and \\( g(x) = x \\sin x + \\cos x \\) intersect.",
"role": "user"
}
] | 2 |
[
{
"content": "If two groups \\(G\\) and \\(H\\) have the same order \\(|G| = |H| = p^a m\\), where \\(p\\) is a prime and \\(\\gcd(p, m) = 1\\), do we necessarily have that the number of Sylow \\(p\\)-subgroups \\(n_p(G)\\) is equal to \\(n_p(H)\\)? Justify your answer.",
"role": "user"
}
] | No |
[
{
"content": "Let $A$ be an $E_{\\infty}$-ring. Consider a right $A$-module $M$ and a left $A$-module $N$. Define the relative tensor product $M \\otimes N = Bar_A(M, N)$, where $Bar_A$ denotes the bar construction. Since $A$ is an $E_{\\infty}$-ring, right and left $A$-modules are equivalent, allowing us to co... | Yes |
[
{
"content": "Determine the total number of 7-digit natural numbers whose digits sum to 20.",
"role": "user"
}
] | 139068 |
[
{
"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... | Debentures |
[
{
"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... | $25,865.64 |
[
{
"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... | 7.03 |
[
{
"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... | 9.78% |
[
{
"content": "Let $(X, \\mathcal{A}, \\mu)$ be a measure space and $(A_n)_{n \\in \\mathbb{N}}$ be a sequence of measurable sets. Show that if $\\sum_{n \\in \\mathbb{N}} \\mu(A_n) < \\infty$, then \\[ \\int_X \\limsup_{n \\to \\infty} \\mathbb{1}_{A_n} \\, d\\mu = 0. \\]",
"role": "user"
}
] | 0 |
[
{
"content": "Evaluate the limit of the following integral:\n\\[ \\lim_{n \\rightarrow \\infty} \\int_{[0, \\infty)} \\frac{n \\sin(\\frac{x}{n})}{x(1+x^2)}\\,\\mathrm dm(x) \\]",
"role": "user"
}
] | \dfrac{\pi}{2} |
[
{
"content": "Find the exact value of the root of the equation \\(x^3 + 5x - 18 = 0\\) using the method of expressing \\(x\\) as \\(a + b\\) where \\(3ab + 5 = 0\\). Show that the root is exactly 2.",
"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... | +7.5e-10 C |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 4.7 m |
[
{
"content": "Find all functions \\( f: \\mathbb{R} \\to \\mathbb{R} \\) that satisfy the equation \\( f(a+x) - f(a-x) = 4ax \\) for all \\( a, x \\in \\mathbb{R} \\).",
"role": "user"
}
] | f(x) = x^2 + C |
[
{
"content": "Determine the number of ordered pairs of integers \\((m, n)\\) that satisfy the equation \\(5m^2 + 9n^2 = 1234567\\).",
"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... | 439.68 feet |
[
{
"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... | $58.73 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 11.07% |
[
{
"content": "Given that \\((L^{-1}M)(L^{-1}M)^T=I\\) and both \\(L\\) and \\(M\\) are invertible lower triangular matrices with positive real diagonal elements, does this imply that \\(L = M\\)? If so, state the relevant property.",
"role": "user"
}
] | L = M |
[
{
"content": "Find the value of \\( c \\) that makes the function \\( f(x) = \\begin{cases}\\left(\\sin\\frac{\\pi}{2+x}\\right)^{1/x^2},& x \\neq 0\\\\c,& x = 0\\end{cases} \\) continuous at \\( x = 0 \\).",
"role": "user"
}
] | e^{-\pi^2/32} |
[
{
"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... | $251,528 |
[
{
"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... | 609.4 N.m.s^2 |
[
{
"content": "Points \\(A_1, B_1, C_1,\\) and \\(D_1\\) are taken on the planes of the faces of the tetrahedron \\(ABCD\\) such that the lines \\(AA_1, BB_1, CC_1,\\) and \\(DD_1\\) are parallel. Determine the ratio of the volumes of tetrahedrons \\(ABCD\\) and \\(A_1B_1C_1D_1\\).",
"role": "user"
}
] | \dfrac{1}{3} |
[
{
"content": "Determine the smallest natural number \\( n \\) such that the group \\( \\mathbb{Z}_{6} \\times \\mathbb{Z}_{20} \\times \\mathbb{Z}_{45} \\) is isomorphic to the direct product of \\( n \\) cyclic groups.",
"role": "user"
}
] | 2 |
[
{
"content": "Given $n$ independent Gaussian random variables $x_i \\sim N(0,1)$ and constants $c_i \\geq 0$, let the set $K$ contain $k$ indices corresponding to the smallest $c_i$. What is the probability that $\\sum_{i \\in K} c_ix_i \\leq 0$?",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "Find the maximum value of the product $abc$, given that $a$, $b$, and $c$ are real numbers satisfying $a+b+c=5$ and $a^2+b^2+c^2=11$.",
"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... | 238 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... | 2.27 |
[
{
"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... | 891 m/s |
[
{
"content": "Determine whether there exists a non-empty set $X$ and a collection ${\\cal F} \\subseteq {\\cal P}(X) \\setminus \\{\\emptyset\\}$ of non-empty subsets of $X$ such that:\n\n1. For every $a \\in {\\cal F}$, $|a| \\geq 2$.\n2. For any distinct $a, b \\in {\\cal F}$, $|a \\cap b| \\leq 1$.\n3. There... | yes |
[
{
"content": "Given the equations \\(x^2 + y^2 + z^2 = 49\\), \\(x + y + z = x^3 + y^3 + z^3 = 7\\), find the value of \\(xyz\\).",
"role": "user"
}
] | -112 |
[
{
"content": "Let \\( a_{n} = 6^{n} + 8^{n} \\). Find the remainder when \\( a_{2018} \\) is divided by 49.",
"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... | Self Employment Tax liability: $19,916.39
Tax deductible portion: $9,958.20 |
[
{
"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... | 1.03e-22 |
[
{
"content": "Let $f$ and $g$ be two homomorphisms from $\\mathrm{SO}(3)$ to $PGL(2,\\mathbb{C})$. Determine whether there exists an element $S \\in \\mathrm{PGL}(2,\\mathbb{C})$ such that for all $X \\in \\mathrm{SO}(3)$, the equation $g(X) = Sf(X)S^{-1}$ holds.",
"role": "user"
}
] | Yes |
[
{
"content": "In a diagram involving two functions, the straight line $y = -mx + c$ is tangent to the graph of $y = -bx^2 + d$ at the point $P(2,4)$. Here, $c$ is a positive real number, and $m$ is a real number with $0 < m < 2$. The line intersects the $x$ and $y$ axes at points $A$ and $B$, respectively, and ... | 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... | 11.11% |
[
{
"content": "Evaluate the limit: $$\\lim_{x \\to 0}\\frac{\\cos x-\\sqrt{1+x^2}}{x^4}$$ using Maclaurin's expansions or L'Hospital's rule.",
"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... | 28424.46N |
[
{
"content": "Find the largest possible set \\( A \\) consisting of 2-digit natural numbers (from 01 to 99) such that every subset of \\( A \\) has a unique sum. What is the size of this set?",
"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... | $46,089.86 |
[
{
"content": "In an $n$-dimensional vector space $V$ with a basis $\\{v_1, v_2, \\ldots, v_n\\}$ and its dual basis $\\{v_1^*, v_2^*, \\ldots, v_n^*\\}$ for the dual space $V^*$, where $v_i^*(v_j) = \\delta_{ij}$, suppose $\\{f_1, f_2, \\ldots, f_n\\}$ is a basis for $V^*$. Is it possible to construct a basis $... | 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... | 67.47 miles/hour |
[
{
"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... | 18 years |
[
{
"content": "Evaluate the integral \\( \\int_{0}^{2\\pi}\\cos(m\\theta)\\,d\\theta \\) using path integration, where \\( m \\) is a nonzero integer.",
"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... | 6.09% |
[
{
"content": "In triangle \\( ABC \\), \\( AB \\) is longer than \\( AC \\). Point \\( N \\) is on \\( BC \\) such that \\( AN \\) bisects \\( \\angle BAC \\). The centroid \\( G \\) of \\( \\triangle ABC \\) satisfies \\( GN \\perp BC \\). Given \\( AC = 6 \\text{ cm} \\), \\( BC = 5 \\sqrt{3} \\text{ cm} \\),... | 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... | 6.5e+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... | Ricky's Piano Rebuilding Company
Unadjusted Trial Balance
For the year ending December 31
Account Debit Credit
Cash $6,025
Accounts Receivable 35,000
Supplies 2,575
Equipment 10,500
Land 6,650
Building 27,500
Accounts Payable $11,360
Unearned Revenue (deposits) 5,100
Notes Payable 40,7... |
[
{
"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... | 8.5 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... | 1.87 m |
[
{
"content": "Suppose that $x$ and $y$ are positive real numbers such that $x^2-xy+2y^2=8$. Then the maximum possible value of $x^2+xy+2y^2$ can be expressed in simplest form as $\\frac{a + b \\sqrt{c}}{d},$ where $a,$ $b,$ $c,$ $d$ are positive integers. Find $a + b + c + d.$",
"role": "user"
}
] | 113 |
[
{
"content": "Can there exist two distinct irrational numbers $x$ and $y$ such that $y$ is a final segment of $x$?",
"role": "user"
}
] | Yes |
[
{
"content": "Compute the integral of the 2-form $\\omega = \\mathrm{d}x \\wedge \\mathrm{d}y$ over the ellipsoid defined by $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} + \\frac{z^2}{c^2} = 1$. Use the parametrization:\n\n$$F:D=[0,\\pi]\\times[0,2\\pi]\\rightarrow\\mathbb{R^3}$$\n$$(\\varphi,\\theta)\\mapsto(a\\sin\\v... | 0 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $161,600 |
[
{
"content": "Consider a sequence of random variables $V_n = V(W_1,...,W_n)$, where $W_i$ are independent and identically distributed random variables, and $W$ is a random variable such that $W \\sim W_i$. Define a non-negative function $f(V_n,W)$. If for any fixed $\\epsilon > 0$, we have\n$$ P_W\\{f(V_n,W) > ... | 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... | 0.198 s |
[
{
"content": "Find all continuous functions \\( f(x) \\) that satisfy the equation:\n\\[ \\int_0^{1} f(x) \\, dx = \\frac{1}{3} + \\int_0^{1} f^2(x^2) \\, dx \\]",
"role": "user"
}
] | \sqrt{x} |
[
{
"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... | 38.0 g/mol |
[
{
"content": "Consider the sequence \\( a_n \\) defined by the recurrence relations:\n\\[ a_0 = 0, \\quad a_1 = 3, \\]\n\\[ a_n = 8a_{n-1} + 9a_{n-2} + 16 \\text{ for } n \\ge 2. \\]\nDetermine the smallest positive integer \\( h \\) such that \\( a_{n+h} - a_n \\) is divisible by 1999 for all \\( n \\ge 0 \\).... | 1998 |
[
{
"content": "In the diagram, $ABCD$ is a parallelogram where $DC$ is tangent to the circumcircle of $\\triangle ABC$ at point $E$, which is also on $AD$. Given that $AB = 4$ and $BE = 5$, determine the length of $DE$. Express your answer as a single number.",
"role": "user"
}
] | 6 |
[
{
"content": "Evaluate the improper integral \\( I = \\int^{\\infty}_0 (y^{\\frac{1}{2}}e^{-\\frac{1}{2}y})dy \\) using the fact that \\( \\int^{\\infty}_0 (e^{-x^2})dx = \\frac{\\sqrt{\\pi}}{2} \\).",
"role": "user"
}
] | \sqrt{2\pi} |
[
{
"content": "Determine whether the following statement is true:\n\nGiven the equation:\n\\[ t = \\int \\frac{du}{u(C_{1}-\\gamma \\ln(u)+x_{0}\\beta u)} \\]\nwhere \\( u \\) is a function of time, is it true that differentiating both sides with respect to time results in:\n\\[ 1 = \\frac{1}{u(C_{1}-\\gamma \\l... | True |
[
{
"content": "Given that the graph of the function $f(x)$ is defined over a certain interval and each side of the quadrilateral in the figure is equal to $a$, suppose that the product of the integers satisfying the inequality $(f\\circ f)(x)\\times f'(x)<0$ is less than the largest element in the image set of t... | C |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $72,300 |
[
{
"content": "Suppose \\( S \\) is a subset of \\( \\mathbb{R} \\) and that \\( S \\) is not closed. Must there exist a convergent sequence in \\( S \\) that converges to some \\( l \\notin S \\)?",
"role": "user"
}
] | Yes |
[
{
"content": "Let \\( f \\) be a function that has derivatives of all orders for all real numbers, and let \\( P_3(x) \\) be the third-degree Taylor polynomial for \\( f \\) about \\( x=0 \\). The Taylor series for \\( f \\) about \\( x=0 \\) converges at \\( x=1 \\), and \\( |f^{(n)}(x)|\\leq\\frac{n}{n+1} \\)... | \dfrac{1}{30} |
[
{
"content": "Consider two finite-dimensional normed vector spaces $X$ and $Y$. If $D: X \\rightarrow Y$ is an isometric isomorphism and $X$ is a complete space, is $Y$ also complete? Justify 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... | $119.1141 |
[
{
"content": "Evaluate the limit \\(\\lim_{x\\rightarrow 0}\\frac{(1+2x+3x^2)^{\\frac{1}{x}}-(1+2x-3x^2)^{\\frac{1}{x}}}{x}\\).",
"role": "user"
}
] | 6e^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... | 535.05 |
[
{
"content": "Determine the values of \\(\\theta\\) for which the set \\(S = \\{x \\in \\mathbb{R}^n : |x-a| \\leq \\theta|x-b|\\}\\) forms a ball.",
"role": "user"
}
] | [0, 1) |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 2e^4 + 2e^2 - 4 |
[
{
"content": "The surface given by the equation \\( z = x^3 + xy^2 \\) intersects the plane \\( 2x - 2y = 1 \\) in a curve. Determine the slope of this curve at the point where \\( x = 1 \\) and \\( y = \\frac{1}{2} \\).",
"role": "user"
}
] | \dfrac{17}{4} |
[
{
"content": "For positive real numbers $p$ and $q$, determine the condition under which the series \\[ \\frac{2^p}{1^q} + \\frac{3^p}{2^q} + \\frac{4^p}{3^q} + \\ldots \\] is convergent. Choose the correct option:\n\n(a) $p < q - 1$\n(b) $p < q + 1$\n(c) $p \\ge q - 1$\n(d) $p \\ge q + 1$",
"role": "user"
... | A |
[
{
"content": "Determine the values of $u_0 \\geq 0$ for which the sequence defined by \\( u_{n} = \\frac{1}{n} + \\sqrt{u_{n-1}} \\) for all \\( n \\in \\mathbb{N} \\) converges.",
"role": "user"
}
] | [0, \infty) |
[
{
"content": "Evaluate the integral \\( \\int _{0}^{4} \\frac{1}{(2x+8)\\, \\sqrt{x(x+8)}}\\, dx. \\)",
"role": "user"
}
] | \dfrac{\pi}{24} |
[
{
"content": "Determine the sign of the determinant of the matrix \\( B \\) given by:\n\\[\nB = \\begin{bmatrix} -b_{12}-b_{13} & b_{12} & b_{13} \\\\ b_{21} & -b_{21}-b_{23} & b_{23} \\\\ b_{31} & b_{32} & -b_{31} - b_{32} \\end{bmatrix},\n\\]\nwhere all \\( b_{ij} \\)'s are positive real numbers. Can this be ... | 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... | 123.44 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... | 1.570 m |
[
{
"content": "Evaluate the line integral \\( \\int z \\sin z \\, dz \\) over the path \\( \\gamma(t) = \\pi t + (t^2-t)i \\) for \\( 0 \\leq t \\leq 1 \\).",
"role": "user"
}
] | \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... | 3782 Joules, 903.92 Calories |
[
{
"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... | 3.55 moles |
[
{
"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... | 13.60% |
[
{
"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... | 3.20 |
[
{
"content": "Evaluate the line integral \\( \\int 3yz \\, dx + z^3 \\, dy + 3yz^2 \\, dz \\) along the curve parameterized by \\( r(t) = (\\sin^2 t - \\cos^2 t) \\cdot \\hat{i} + \\sin t \\cdot \\hat{j} + \\cos t \\cdot \\hat{k} \\) for \\( 0 \\leq t \\leq 2\\pi \\), which lies on the surface \\( x = y^2 - z^2... | 3\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... | 145.6 mi |
[
{
"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... | Hydrogen bonding and dispersion forces |
[
{
"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... | 1190 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... | 33.66 m |
[
{
"content": "Let \\( f : \\mathbb{R}^+ \\to \\mathbb{R} \\) satisfy the equation \\( f(xy) = e^{xy - x - y} (e^y f(x) + e^x f(y)) \\) for all \\( x, y \\in \\mathbb{R}^+ \\). Given that \\( f'(1) = e \\), find the function \\( f(x) \\).",
"role": "user"
}
] | e^x \ln x |
[
{
"content": "Evaluate the integral \\( \\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} \\exp \\left( -\\frac{A}{2}(x-y)^2 + B(x-y) \\right) \\, dx \\, dy \\), where \\(A\\) and \\(B\\) are real and positive constants.",
"role": "user"
}
] | \infty |
[
{
"content": "Evaluate the limit of the series as \\( n \\to \\infty \\): \\[ \\lim_{n\\to\\infty} \\left(\\frac{1}{n+1}+\\frac{1}{n+2}+\\frac{1}{n+3}+\\ldots+\\frac{1}{n^2}\\right) \\]",
"role": "user"
}
] | \infty |
[
{
"content": "Determine the number of integer solutions to the equation \\(x+y+z=120\\) where \\(x, y, z \\geq 5\\) and \\(x \\leq y \\leq z\\). Use the action of a group to find the solution.",
"role": "user"
}
] | 972 |
[
{
"content": "Determine whether the function \\( f: \\mathbb{R} \\rightarrow \\mathbb{R} \\) defined by \\[ f(x) = \\begin{cases} 0 & \\text{if } 0 \\leq x < 1 \\\\ 1 & \\text{if } x = 1 \\end{cases} \\] is a Baire class one function.",
"role": "user"
}
] | Yes |
[
{
"content": "Suppose that \\(\\int_{0}^{\\infty}f(x)dx<\\infty\\) and \\(f(x)\\geq0\\) for all \\(x\\geq0\\). Prove that \\(\\lim_{t \\to \\infty}\\int_{\\frac{t}{2}}^{t} f(x)\\,dx=0\\).",
"role": "user"
}
] | 0 |
[
{
"content": "Given the joint probability density function (PDF) of random variables $X$ and $Y$: $f_{X,Y}(x,y) = \\frac{4}{15}x^2$ for $1 \\leq x \\leq 2$ and $0 \\leq y \\leq x$, find the conditional expectation of $\\frac{1}{X^2 Y}$ given that $Y = \\frac{5}{4}$. The marginal PDF of $Y$ is $f_Y(y) = \\frac{2... | \dfrac{64}{215} |
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