prompt list | ground_truth string |
|---|---|
[
{
"content": "Two congruent right-angled isosceles triangles, each with a base length of 1, slide along a line. Determine the maximal area of overlap between the two triangles.",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "Determine the value of \\( a \\) for which the series \\( \\sum_{n=1}^{\\infty} n^{\\frac{1}{3}} \\left| \\sin\\left(\\frac{1}{n}\\right) - \\frac{1}{n^a} \\right| \\) converges.",
"role": "user"
}
] | 1 |
[
{
"content": "What is the smallest positive integer with factors of 16, 15, and 12?",
"role": "user"
}
] | 240 |
[
{
"content": "Construct a subset $N \\subseteq \\mathbb{R}$ with Lebesgue measure zero such that every point in $\\mathbb{R}$ is a condensation point of $N$. Is this possible?",
"role": "user"
}
] | Yes |
[
{
"content": "Given a real-valued function $f(x)$ that satisfies the functional equation \\[ f(x-y) = f(x) \\cdot f(y) - f(a-x) \\cdot f(a+y) \\] where $a$ is a constant, and knowing that $f(0) = 1$ and $f(2a) = -1$, find the value of $f(2a+x)$. ",
"role": "user"
}
] | -f(x) |
[
{
"content": "If a stack of eight quarters is exactly half an inch, how many quarters will be needed to make a stack one foot high?",
"role": "user"
}
] | 192 |
[
{
"content": "Find the minimum distance from the origin to the curve \\( \\frac{a^2}{x^2}+\\frac{b^2}{y^2}=1 \\).",
"role": "user"
}
] | a + b |
[
{
"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... | 4023 kg |
[
{
"content": "Determine the values of $p$ for which the integral \\( \\int_0^{\\infty} \\frac{1-\\cos x}{x^p} \\, dx \\) converges.",
"role": "user"
}
] | (1, 3) |
[
{
"content": "In $\\triangle ABC$, point $D$ is the midpoint of side $BC$. Point $E$ is on $AC$ such that $AE:EC =1:2$. Point $F$ is on $AD$ such that $AF:FD=3:1$. If the area of $\\triangle DEF$ is 17, determine the area of $\\triangle ABC$. [asy]\nsize(6cm);defaultpen(fontsize(11));\npair b =(0,0);pair c = ... | 408 |
[
{
"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... | 0.054 |
[
{
"content": "Determine the number of distinct ways to insert $k$ vertices along the edges of a polygon with $n$ sides, considering that the order of insertion matters and vertices can be placed anywhere on the edges.",
"role": "user"
}
] | n^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... | 5.077 m |
[
{
"content": "Let $f\\colon \\mathbb{C}\\to \\mathbb{C}$ be a nonconstant entire function. Determine whether there exists a function $\\bar{f}\\colon S^2\\to S^2$ such that $\\bar{f}|_\\mathbb{C}=f$. Answer with 'Yes' or 'No'.",
"role": "user"
}
] | No |
[
{
"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... | 630.52 g |
[
{
"content": "Given that $a$ and $b$ are real numbers satisfying the inequality $4a^2 + 3b^2 - 4ab - 4b + 2 \\le 0$, calculate the determinant of the matrix:\n\n$$\\begin{pmatrix} 4 & 4 & 4 \\\\ a & b & 2 \\\\ a^2 & b^2 & 4\\end{pmatrix}$$",
"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... | 18.70 m/s |
[
{
"content": "Determine the number of unique integer-sided right triangles (up to congruence) such that their area is equal to their perimeter.",
"role": "user"
}
] | 2 |
[
{
"content": "For a positive integer $n,$ let\n\\[H_n = 1 + \\frac{1}{2} + \\frac{1}{3} + \\dots + \\frac{1}{n}.\\]Compute\n\\[\\sum_{n = 1}^\\infty \\frac{1}{(n + 1) H_n H_{n + 1}}.\\]",
"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... | 73,884,962.71 |
[
{
"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... | 40.61 cm^2 |
[
{
"content": "Evaluate the integral \\( \\int\\limits_{-1/2}^{1/2}(\\sin^{-1}(3x-4x^3)-\\cos^{-1}(4x^3-3x))dx \\). Choose the correct answer from the following options:\n\n\\(A)\\ 0\\hspace{1cm}(B)\\ \\frac{-\\pi}{2}\\hspace{1cm}(C)\\ \\frac{\\pi}{2}\\hspace{1cm}(D)\\ \\frac{7\\pi}{2}\\)",
"role": "user"
... | B |
[
{
"content": "In triangle \\(\\triangle ABC\\), \\(\\angle A = 120^{\\circ}\\). Define \\(\\alpha = \\frac{\\overrightarrow{BA}}{|\\overrightarrow{BA}| \\cos A} + \\frac{\\overrightarrow{BC}}{|\\overrightarrow{BC}| \\cos C}\\) and \\(\\beta = \\frac{\\overrightarrow{CA}}{|\\overrightarrow{CA}| \\cos A} + \\frac... | 60^\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... | 38.00 g F |
[
{
"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... | 177 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... | $611.62 |
[
{
"content": "Determine whether the sum of two uniform random variables is uniformly distributed. Specifically, if \\(X_1, X_2 \\sim U[0,t]\\), does it imply that \\(2X_1 + X_2 \\sim U[0,3t]\\)? Provide a justification for your answer.",
"role": "user"
}
] | No |
[
{
"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. All atoms in the molecule belong to the same element.
2. Dipoles balance each other within the molecule, which requires a symmetric molecular structure. |
[
{
"content": "If the asymptotic densities of sets $A$ and $B$ satisfy $d(A) + d(B) = 1$, does this imply that $A \\cap B$ is non-empty? Answer 'yes' or 'no' and justify your answer.",
"role": "user"
}
] | no |
[
{
"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... | $18508.30 |
[
{
"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... | 0.00809 |
[
{
"content": "Consider a sequence \\( \\{a_0, a_1, a_2, \\ldots\\} \\) that has no limit point. Does this imply that \\( |a_n| \\to \\infty \\) as \\( n \\to \\infty \\)? Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Given the initial list of numbers: $48, 99, 120, 165, 273$, determine the maximum possible size of the list after playing the game optimally. In the game, a player can choose any two numbers $a$ and $b$ from the list, compute $c = |a - b|$, and add $c$ to the list if $c$ is non-zero and not alread... | 91 |
[
{
"content": "Find the derivative of the function \\( g(t) = \\int_t^{t^2} \\cos(tx) \\, dx \\) and evaluate it at \\( t = 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... | 23 V |
[
{
"content": "Determine $f'(0)$ for the function $f(x) = \\int_{0}^{x} \\cos\\left(\\frac{1}{t}\\right)dt$ using appropriate mathematical techniques.",
"role": "user"
}
] | 0 |
[
{
"content": "Let \\( \\mathbf{A} \\) be an \\( m \\times n \\) matrix and \\( \\mathbf{B} \\) be a symmetric \\( n \\times n \\) matrix, where \\( m < n \\). Given that \\( \\operatorname{Rank}(\\mathbf{A}) = \\operatorname{Rank}(\\mathbf{B}) = m \\), determine the rank of the matrix product \\( \\mathbf{ABA}^... | m |
[
{
"content": "Use the Divergence Theorem to evaluate the surface integral \\( \\iint_S F \\cdot dS \\), where \\( F = (x^3y + z)i + xze^yj + (xy − 3z^4)k \\) and \\( S \\) consists of the five lower faces of the cube \\([0, 1]\\times[0, 1]\\times[0, 1]\\), excluding the face with \\( z = 1 \\). Hint: Close the ... | \dfrac{e}{4} |
[
{
"content": "Determine the fundamental group of a 2-sphere with the identification $x = -x$. What is the fundamental group of this space?",
"role": "user"
}
] | \mathbb{Z}/2\mathbb{Z} |
[
{
"content": "Compute the last four digits of \\(2013^{2014^{2015}}\\).",
"role": "user"
}
] | 9361 |
[
{
"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... | $13,821.40 |
[
{
"content": "Determine whether a non-continuous function $f:[0,3] \\to \\mathbb{R}$ can be bijective. Answer with '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... | 0.075 mol |
[
{
"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... | 20.038% |
[
{
"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... | 10.867 dynes |
[
{
"content": "Determine the number of zeros of the function \\( f(z) = e^z - 2i \\).",
"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... | 3.776 ms^{-2} |
[
{
"content": "Through the focus \\( F \\) of the parabola \\( y = ax^2 \\) (where \\( a > 0 \\)), a straight line intersects the parabola at points \\( P \\) and \\( Q \\). If the lengths of the line segments \\( PF \\) and \\( FQ \\) are \\( p \\) and \\( q \\) respectively, what is \\( \\frac{1}{p} + \\frac{1... | C |
[
{
"content": "In triangle \\(ABC\\), a triangle is formed from the medians and another triangle is formed from the altitudes. Both of these triangles are similar to triangle \\(ABC\\). Determine the ratio of the lengths of the sides of triangle \\(ABC\\).",
"role": "user"
}
] | 1:1:1 |
[
{
"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... | 1.69 |
[
{
"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... | 0.44 m/s |
[
{
"content": "A biased coin with a probability of 0.51 for heads is tossed repeatedly. The tossing stops when there are more heads than tails. Determine if the probability of ultimately getting more heads than tails in this series of tosses is 1.",
"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... | 195.9 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... | 108/5 m/min |
[
{
"content": "Evaluate the limit \\( \\lim_{(x,y)\\rightarrow (0,0)}\\dfrac{xy^3}{x^2+y^4} \\).",
"role": "user"
}
] | 0 |
[
{
"content": "In a first-order language L with only the predicate = and no constants or function symbols, determine the smallest number of variables required to write a sentence A that is valid in a structure M if and only if M has at least n elements in its domain.",
"role": "user"
}
] | n |
[
{
"content": "Evaluate the sum \\(\\sum_{n=1}^{\\infty} \\frac{1}{(2n-1)(2n+1)(2n+3)}\\).",
"role": "user"
}
] | \dfrac{1}{12} |
[
{
"content": "Determine the points in the complex plane where the function \\( f(x + yi) = x^2 - y^2 \\) is complex differentiable.",
"role": "user"
}
] | 0 |
[
{
"content": "Let \\( a \\) and \\( b \\) be positive integers such that \\( 15a + 16b \\) and \\( 16a - 15b \\) are both perfect squares. Determine the smallest possible value among these squares.",
"role": "user"
}
] | 231361 |
[
{
"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... | 75 |
[
{
"content": "Let $a, b, c \\ge 0$ such that $ab + bc + ca = 1$. Determine the maximum value of \\[ P = \\sqrt{\\frac{1}{a+1} + \\frac{1}{b+1}} + \\sqrt{\\frac{1}{a+1} + \\frac{1}{c+1}} + \\sqrt{\\frac{1}{c+1} + \\frac{1}{b+1}}. \\]",
"role": "user"
}
] | 1 + \sqrt{6} |
[
{
"content": "What is the greatest common divisor of $1729$ and $1768$?",
"role": "user"
}
] | 13 |
[
{
"content": "Consider the functions \\( \\varphi, \\psi : \\mathbb{R} \\rightarrow \\mathbb{R} \\) defined by \\( \\varphi(x) = x^3 \\) and \\( \\psi(x) = x \\) for \\( x < 0 \\) and \\( \\psi(x) = 2x \\) for \\( x \\geq 0 \\). Do the atlases \\( A_1 = \\{ \\text{id}_\\mathbb{R} \\} \\), \\( A_2 = \\{ \\varphi... | 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... | 1.41 s |
[
{
"content": "Given the equations \\(x^4 + y^4 = 5\\) and \\(x^2 + xy + y^2 = 10\\), find the value of \\((x+y)^4\\).",
"role": "user"
}
] | 195 |
[
{
"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": "Let \\( f(x) = \\begin{cases} 0 & \\text{if } -1 \\leq x \\leq 0 \\\\ x^4 & \\text{if } 0 < x \\leq 1 \\end{cases} \\). If \\( f(x) = \\sum_{k=0}^{n} \\frac{f^{(k)}(0)}{k!}x^{k} + \\frac{f^{(n+1)}(\\xi)}{(n+1)!}x^{n+1} \\) is the Taylor's formula for \\( f \\) about \\( x=0 \\) with the maximum po... | \xi \in (0, x) |
[
{
"content": "Determine the Krull dimension of the ring \\( B = A[x,y,z]/\\langle xy, xz \\rangle \\), where \\( A \\) is a Noetherian commutative ring.",
"role": "user"
}
] | \dim A + 2 |
[
{
"content": "Determine whether the following statement is true or false: If $f(n) = O(g(n))$ and $f(n) \\neq \\Omega(g(n))$, then $f(n) = o(g(n))$. Justify your answer.",
"role": "user"
}
] | False |
[
{
"content": "Determine the radius of convergence for the series \\( \\sum _{{k=0}}^{\\infty }{\\frac {x^{k}}{p_{k+1}\\#}} \\), where \\( p_k\\# \\) is the product of the first \\( k \\) primes.",
"role": "user"
}
] | \infty |
[
{
"content": "Let \\( a, b, c, d, e \\) be positive integers such that their sum is 2018. Define \\( M = \\max (a+b, b+c, c+d, d+e) \\). Determine the smallest possible value of \\( M \\).",
"role": "user"
}
] | 673 |
[
{
"content": "Determine whether the expression \\( a = \\frac{36^{36} + 41^{41}}{77} \\) is an integer.",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate the limit: $$\\lim_{n\\to\\infty} \\int_{0}^{\\pi/2} \\sqrt[n]{\\sin^nx+\\cos^nx} \\,dx$$",
"role": "user"
}
] | \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... | 9.944mg |
[
{
"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... | $21,290.20 |
[
{
"content": "Find the coefficient $a_{-3}$ in the Laurent expansion of the function \\( f(z) = \\frac{1}{2z^3} - \\frac{1}{z^3+i} \\) in the region \\( 0 < |z| < 1 \\).",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "Compute the average value of the function \\( f(x,y,z) = xyz \\) over the region \\( W \\) defined by \\( 0 \\le z \\le y \\le x \\le 1 \\).",
"role": "user"
}
] | \dfrac{1}{8} |
[
{
"content": "Evaluate the limit \\( \\lim_{n \\to \\infty} n t_n \\), where \\( t_n = \\sum_{i=1}^{2n} \\frac{(-1)^{i+1}}{2n+i} \\). This sequence is given by \\( \\frac{1}{2n+1} - \\frac{1}{2n+2} + \\cdots + \\frac{1}{4n-1} - \\frac{1}{4n} \\). Use the hint that this limit can be related to a suitable Riemann... | \dfrac{1}{8} |
[
{
"content": "Calculate the expected value of $X(X-1)$ for a Poisson distribution with parameter $\\lambda$ using the probability-generating function.",
"role": "user"
}
] | \lambda^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... | 1.12 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... | 665 g |
[
{
"content": "A bag contains sticks of integer lengths. If any 4 sticks are chosen and laid end-to-end, they cannot form a quadrilateral. Given that the longest stick has a length of 150, determine the maximum possible number of sticks in the bag.",
"role": "user"
}
] | 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... | 46,648,862 |
[
{
"content": "Find the function \\( f(x) \\) such that \\( f(x) = f'(x)\\int{f(x)}\\,dx \\).",
"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... | $31.13 |
[
{
"content": "What is the smallest field containing both the rational numbers \\( \\mathbb{Q} \\) and the transcendental number \\( e \\)? Is it the field of real numbers \\( \\mathbb{R} \\) or a proper subfield of \\( \\mathbb{R} \\)?",
"role": "user"
}
] | \mathbb{Q}(e) |
[
{
"content": "Let $g:[-1,1] \\to \\mathbb{R}$ be a function such that $g^{(n)}(x)$ exists and $|g^{(n)}(x)| \\leq 2$ for every $n \\geq 1$ and for every $x \\in [-1,1]$. Determine whether $g$ has a convergent power series expansion in a neighborhood of every $x \\in [-1,1]$. Provide a justification for your ans... | Yes |
[
{
"content": "Determine whether the following inequality holds:\n\\[ \\int_{l}^{w} \\frac{t^{a-1}dt}{(1+ut)^{p+1}} \\le \\int_{0}^{\\infty} \\frac{t^{a-1}dt}{(1+ut)^{p+1}} = \\frac{1}{u^a}B(a,p+1-a) \\]\nwhere \\(0 < l < w < \\infty\\), \\(0<a\\), and \\(1<p\\).",
"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... | $MnO_4^- + 4H^+ + 3e^- \rightarrow MnO_2 + 2H_2O$ |
[
{
"content": "Find the limit: $$ \\lim\\limits_{x\\rightarrow 0} \\frac{e^{-1/x^2}}{x}. $$",
"role": "user"
}
] | 0 |
[
{
"content": "Given the cubic equation \\(x^3 + 2x^2 + 3x + 1 = 0\\) with roots \\(a, b, c\\), it is known that \\(\\frac{1}{a^3} + \\frac{1}{b^3} - \\frac{1}{c^3}\\), \\(\\frac{1}{a^3} + \\frac{1}{c^3} - \\frac{1}{b^3}\\), and \\(\\frac{1}{c^3} + \\frac{1}{b^3} - \\frac{1}{a^3}\\) are the roots of another cubi... | 5 |
[
{
"content": "Determine the number of roots of the function \\( f(z) = 1 + 6z^3 + 3z^{10} + z^{11} \\) within the annulus defined by \\( 1 < |z| < 2 \\) using Rouche's theorem.",
"role": "user"
}
] | 7 |
[
{
"content": "In a graph with a total of $n$ vertices, determine the maximum number of edges that can connect a subset of $p$ vertices to the remaining $q$ vertices.",
"role": "user"
}
] | pq |
[
{
"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 Pa < (1 torr = 1 mm Hg) < 1 in Hg < 1 psi < 1 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... | Fe |
[
{
"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... | 5.42 m/s |
[
{
"content": "Does there exist a Galois extension $L/\\mathbb{Q}$ with Galois group $A_4$ such that all the decomposition groups are cyclic?",
"role": "user"
}
] | Yes |
[
{
"content": "Given that $x$ and $y$ are real numbers satisfying $x^2 + 2xy - y^2 = 6$, determine the minimum value of $(x^2 + y^2)^2$. Provide your answer as a single number.",
"role": "user"
}
] | 18 |
[
{
"content": "Find the center of mass of a thin sheet in the shape of the cone \\(z = \\sqrt{x^2 + y^2}, z \\le 4\\), with density function \\(\\rho(x, y, z) = z\\).",
"role": "user"
}
] | (0, 0, 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... | 0.19 |
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