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... | 0.153 |
[
{
"content": "Consider a matrix $A$ of size $100 \\times 4$. Let $B = A^T A$, where $B$ is a $4 \\times 4$ symmetric matrix. Is it possible for $B$ to be non-invertible? Justify your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate the series \\( \\sum^{\\infty}_{n=0}\\frac{1}{(n+3)\\cdot n!} \\).",
"role": "user"
}
] | e - 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... | 1.4e-15 m |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to \\infty} \\frac{1}{x} \\int_0^x \\frac{dt}{1+x^2 \\cos^2 t} \\]",
"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... | 17.41 g |
[
{
"content": "There are 30 men and 40 women in the Town Library Club. They wish to form a 7-person steering committee with 3 men and 4 women. In how many ways can they form the committee?",
"role": "user"
}
] | 371,\!043,\!400 |
[
{
"content": "Let $f:X\\rightarrow [0,1]$ be a continuous function. Determine whether the statement $a<b\\implies \\overline{f^{-1}[0,a)}\\subset f^{-1}[0,b)$ is true.",
"role": "user"
}
] | True |
[
{
"content": "Determine the condition under which the fraction \\( \\frac{t^a - 1}{t^b - 1} \\) is an integer, given three positive integers \\( t \\), \\( a \\), and \\( b \\).",
"role": "user"
}
] | b \mid a |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to 0} \\left( \\frac{1}{x} - \\frac{2}{e^{2x} - 1} \\right) \\]",
"role": "user"
}
] | 1 |
[
{
"content": "The function $f$ satisfies \\[\nf(x) + f(2x+y) + 5xy = f(3x - y) + 2x^2 + 1\n\\]for all real numbers $x,y$. Determine the value of $f(10)$.",
"role": "user"
}
] | -49 |
[
{
"content": "Compute the asymptotic relative efficiency (ARE) for estimating $e^{-\nabla}$ in a Poisson process, given by the expression:\n$$\\text{ARE} = \\left[\\frac{e^{-\\lambda}}{\\left(\\frac{n-1}{n}\\right)^{n\\lambda}\\log\\left( \\frac{n-1}{n}\\right)^n}\\right]^2$$",
"role": "user"
}
] | 1 |
[
{
"content": "Calculate the surface integral of the function \\( f(x,y,z) = x^2 + y^2 + z^2 \\) over the upper hemisphere of the sphere defined by \\( x^2 + y^2 + (z-1)^2 = 1 \\).",
"role": "user"
}
] | 6\pi |
[
{
"content": "Suppose $\\Omega$ is an open and connected subset of $\\mathbb{R}^n$ that is bounded. If $f$ is a continuous function on the closure of $\\Omega$ that vanishes on the boundary of $\\Omega$, determine whether $f$ can be approximated uniformly by a sequence of smooth functions with compact support i... | Yes |
[
{
"content": "Does the derivative of a differentiable function $f:(a,b) \\rightarrow \\mathbb{R}$, where $(a,b)$ is an open interval, correspond to the limit of a sequence of continuous 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... | 18.2 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... | 0.969 mol O2 |
[
{
"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... | 296.65ms^-1 |
[
{
"content": "Determine if there exists a sequence defined by $A_0=0$, $A_1=1$, and $A_{n+2}=pA_{n+1}-qA_{n}$, where $p$ and $q$ are fixed integers, that contains an infinite number of perfect cubes.",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate the triple integral \\( \\iiint\\limits_R{4y^2ze^{xy}dxdydz} \\) where \\( R \\) is the region bounded by \\( x=0,\\; y=1,\\; y=x,\\; z=0,\\; z=1. \\)",
"role": "user"
}
] | e - 2 |
[
{
"content": "Determine whether the limit \\( \\lim_{x\\to\\infty}\\frac{e^{x}+x-\\cos(2x)}{x^2} \\) converges or diverges.",
"role": "user"
}
] | \infty |
[
{
"content": "Find the smallest positive integer \\( b \\) such that there exists an integer \\( a \\) for which \\( \\frac{43}{197} < \\frac{a}{b} < \\frac{17}{77} \\).",
"role": "user"
}
] | 32 |
[
{
"content": "Evaluate the limit: $$\\lim_{x\\to 0} \\frac{1}{x}\\int_{x}^{2x}e^{-t^2}dt$$",
"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... | 19.88m |
[
{
"content": "Evaluate the integral \\( \\int_0^1 \\ln^n(x) \\, dx \\) for a non-negative integer \\( n \\).",
"role": "user"
}
] | (-1)^n n! |
[
{
"content": "Let \\( S = \\sum_{k=1}^{100} X_k \\) where each \\( X_k \\sim \\text{Poisson}(1) \\). Use the Central Limit Theorem to approximate \\( P(S \\ge 200) \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Find the solution to the recurrence relation \\(T(n)=1+2\\sum_{i=1}^{n-1}T(i)\\) for \\(n > 1\\), with the initial condition \\(T(1)=1\\).",
"role": "user"
}
] | 3^{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... | 1.627e7 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... | $7,600 |
[
{
"content": "Let $R$ be a commutative ring with a unit element such that any ideal can be decomposed as a sum of an idempotent ideal and a small ideal (i.e., an ideal in the Jacobson radical $J(R)$). Determine whether the idempotents of $R/J(R)$ lift modulo $J(R)$. Specifically, if $x^2 - x \\in J(R)$ for $x \... | Yes |
[
{
"content": "Verify that \\( \\sqrt{12\\sqrt[3]{2} - 15} + \\sqrt{12\\sqrt[3]{4} - 12} = 3 \\).",
"role": "user"
}
] | 3 |
[
{
"content": "Evaluate the integral \\( \\iint_R \\sqrt{\\frac{x+y}{x-2y}} \\, dA \\) where \\( R \\) is the region bounded by the lines \\( y - \\frac{x}{2} = 0 \\), \\( y = 0 \\), and \\( x+y = 1 \\).",
"role": "user"
}
] | \dfrac{1}{3} |
[
{
"content": "Evaluate the sum:\n\\[ \\sum_{x=0}^\\infty x(x-1) {2+x \\choose x}(0.008)(0.8)^x \\]",
"role": "user"
}
] | 192 |
[
{
"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... | $0.41 |
[
{
"content": "Evaluate the limit: $$\\lim_{n \\to \\infty} \\sum_{k=0}^n \\frac{n!}{k!(n-k)!(k+2)}$$",
"role": "user"
}
] | \infty |
[
{
"content": "Given points $A(0,0,2)$ and $B(3,4,1)$ in the coordinate system $Oxyz$, find the minimum value of $AX + BY$ where $X$ and $Y$ are points in the $Oxy$ plane such that the distance $XY = 1$. Provide your answer as a single number.",
"role": "user"
}
] | 5 |
[
{
"content": "A section of a parallelepiped is a pentagon, and the ratio of any two sides of this pentagon is 1, 2, or 0.5. Determine the characteristics of the angles in this pentagon:\n\nA. Four angles are $120^{\\circ}$ and one angle is $60^{\\circ}$\nB. Three angles are $120^{\\circ}$ and two angles are $60... | A |
[
{
"content": "Let \\( f(z) \\) be a function with a continuous second derivative, and define \\( u(x,t) = f(x+ct) \\). It is known that \\( \\frac{\\partial^2 u}{\\partial t^2} - 12 \\frac{\\partial^2 u}{\\partial x \\partial t} + 36 \\frac{\\partial^2 u}{\\partial x^2} = 0 \\). Find the value of \\( c \\).",
... | 6 |
[
{
"content": "Is there a field extension $L/K$ such that it is an infinite algebraic extension of fields but the separable degree of $L$ over $K$ is finite?",
"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... | 112.50 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | Marginal revenue = (Change in total revenue) / (Change in total output) |
[
{
"content": "Let $G$ be a divisible group and $H$ be a subgroup of $G$. If $H$ has finite index in $G$, does it follow that $H=G$? Provide a proof or counterexample.",
"role": "user"
}
] | H = 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... | $53,849.76 |
[
{
"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... | 13.6mH |
[
{
"content": "In a two-player game played on a $9 \\times 9$ board, each square initially has one mark. Players take turns, with the first player picking a mark adjacent to at least one other mark and the second player removing one mark from an adjacent square. Determine the maximum number of marks the first pl... | 40 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n\\to \\infty}\\left[(n+1)\\int_{0}^{1}x^n\\ln(x+1)dx\\right] \\]",
"role": "user"
}
] | \ln 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... | $33,874 |
[
{
"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... | Discount on Note Receivable for $1,697.08 |
[
{
"content": "Determine the values of \\(a \\in \\mathbb{R}_+\\) for which the series \\(\\sum_{n>0} \\frac{a(a+1)(a+2)\\cdots(a+n-1)}{n!\\cdot n^2}\\) is convergent.",
"role": "user"
}
] | (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... | $2,000,000 |
[
{
"content": "Determine whether there exist any positive uniformly integrable martingales that converge almost surely to 0. Justify your answer.",
"role": "user"
}
] | No |
[
{
"content": "Compute\n\\[\\frac{(10^4+324)(22^4+324)(34^4+324)(46^4+324)(58^4+324)}{(4^4+324)(16^4+324)(28^4+324)(40^4+324)(52^4+324)}.\\]",
"role": "user"
}
] | 373 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to \\infty} \\frac{x^x}{(2x)!} = 0 \\]",
"role": "user"
}
] | 0 |
[
{
"content": "Calculate the value of \\(321^{654} \\mod 1013\\).",
"role": "user"
}
] | 863 |
[
{
"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... | 90,990.39 |
[
{
"content": "There are more than 1,000,000 ways to arrange \\( n \\) identical black socks and \\( 2n \\) identical gold socks in a row such that there are at least 2 gold socks between any 2 black socks. Find the sum of the digits of the smallest possible value of \\( n \\).",
"role": "user"
}
] | 9 |
[
{
"content": "Determine the values of \\( k \\) for which the inequality \\( (\\cos A - \\frac{1}{5})(\\cos B - \\frac{1}{5})(\\cos C - \\frac{1}{5}) \\leq k \\) holds for all triangles \\( ABC \\).",
"role": "user"
}
] | \dfrac{4}{125} |
[
{
"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... | 14 mL |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n\\to\\infty} \\sum_{k=1}^{n/2} \\left(\\frac{en}{2k}\\right)^{2k} \\frac{1}{\\sqrt{\\pi k}}^{\\frac{3n}{\\log_2{n}}} \\]",
"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... | 4NO_2(g) + O_2(g) + 2H_2O(l) -> 4HNO_3(aq) |
[
{
"content": "Find all real numbers $a$ such that the equation\n\\[x^3 - ax^2 - 2ax + a^2 - 1 = 0\\]has exactly one real solution in $x.$",
"role": "user"
}
] | \left( -\infty, \frac{3}{4} \right) |
[
{
"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... | -11N |
[
{
"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... | 2.0718 m |
[
{
"content": "A triangle is formed with edges along the line $y=\\frac{2}{3}x+5$, the $x$-axis, and the line $x=k$. If the area of the triangle is less than $20$, find the sum of all possible integral values of $k$.",
"role": "user"
}
] | -120 |
[
{
"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... | 0.00016 |
[
{
"content": "In \\( \\triangle ABC \\), the incenter is \\( I \\) and the side lengths are \\( AB = 5 \\), \\( BC = 12 \\), and \\( CA = 13 \\). A circle \\( \\omega \\) centered at \\( I \\) intersects \\( \\triangle ABC \\) at 6 points. Given that the sum of the green marked angles is \\( 180^\\circ \\), fin... | \dfrac{16}{3} |
[
{
"content": "Determine whether it is possible to arrange 17 lines on a plane such that no three lines intersect at a single point and there are exactly 101 intersection points.",
"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... | $7.97 |
[
{
"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. The economy must have only one Phillips curve.
2. There must exist an inverse permanent tradeoff between the inflation rate and the unemployment rate in the economy. |
[
{
"content": "Determine the number of 16-digit integers $t$ such that $12t+5$, $15t+8$, $20t+11$, and $30t+17$ are all prime numbers.",
"role": "user"
}
] | 0 |
[
{
"content": "Consider a function \\( f: \\Bbb R^k \\supset A \\rightarrow \\Bbb R^m \\) such that for all \\( x, y \\in A \\), the condition \\( |f(x) - f(y)| \\le |x - y| \\) holds. Is it true that such a function can always be extended to a function \\( f': \\Bbb R^k \\rightarrow \\Bbb R^m \\) that satisfies... | Yes |
[
{
"content": "If \\( f \\) is a strictly increasing function over the positive reals with \\( f(f(x)) = x^2 + 2 \\), find the value of \\( f(3) \\).",
"role": "user"
}
] | 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... | 1400 N |
[
{
"content": "Calculate the surface area of the portion of the cone \\(z^2 = x^2 + y^2\\) that is contained within the sphere \\(x^2 + y^2 + z^2 = 8\\).",
"role": "user"
}
] | 8\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... | [Cl2] = 1.3 M, [NO] = 2.6 M |
[
{
"content": "Determine the number of different ways to order the set of rational numbers, \\(\\mathbb{Q}\\).",
"role": "user"
}
] | 2^{\aleph_0} |
[
{
"content": "Evaluate the integral \\( \\int_{|z|=2} \\frac{dz}{z^2 + 1} \\) without using partial fraction decomposition.",
"role": "user"
}
] | 0 |
[
{
"content": "Determine whether the outer measure of the real numbers, \\( m^*(\\mathbb{R}) \\), is infinite. Recall the definition of outer measure: \\[ m^*(E) = \\inf\\left\\{\\sum_{n=1}^{\\infty}|I_n| : E \\subseteq \\bigcup_{n=1}^{\\infty} I_n\\right\\} \\] where \\( I_n \\) are intervals covering \\( E \\)... | \infty |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n \\to \\infty} \\sqrt[n]{\\frac{(2n - 1)!}{n!}} \\]",
"role": "user"
}
] | \infty |
[
{
"content": "Consider the graph of $$\\frac{x^3-x^2+x}{6x^2-9x}.$$Let $a$ be the number of holes in the graph, $b$ be the number of vertical asympotes, $c$ be the number of horizontal asymptotes, and $d$ be the number of oblique asymptotes. Find $a+2b+3c+4d$.",
"role": "user"
}
] | 7 |
[
{
"content": "Are all finite simple groups isomorphic to a subgroup of some symmetric group?",
"role": "user"
}
] | Yes |
[
{
"content": "Does there always exist a field \\( \\Omega \\) which contains both fields \\( L \\) and \\( E \\) as \\( K \\)-subfields?",
"role": "user"
}
] | Yes |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 5.424 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... | Five types of analysis can be done using financial ratios: Liquidity, Turnover, Earnings, Solvency, and Profitability. |
[
{
"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... | 114 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... | 1.13 J |
[
{
"content": "Find the area of the part of the plane $5x+2y+z=10$ that lies inside the cylinder $x^2 + y^2 = 25$. Compute the area of this surface.",
"role": "user"
}
] | 25\pi\sqrt{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... | 0.97 |
[
{
"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... | list |
[
{
"content": "Given the parabola \\( y^2 = 8(x + 2) \\) with focus \\( F \\), a line with an inclination angle of \\( 60^\\circ \\) is drawn through \\( F \\). If this line intersects the parabola at points \\( A \\) and \\( B \\), and the perpendicular bisector of chord \\( AB \\) intersects the axis at point ... | A |
[
{
"content": "Given \\( z = \\frac{\\sqrt{3} - i}{2} \\), find the smallest positive integer \\( n \\) such that \\((z^{95} + i^{67})^{94} = z^n\\), where \\( i = \\sqrt{-1} \\).",
"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... | 0.0132 |
[
{
"content": "Given the function \\( f(x) = a + x - b^x \\) has a zero point \\( x_0 \\in (n, n+1) \\) where \\( n \\in \\mathbf{Z} \\), and the constants \\( a \\) and \\( b \\) satisfy the conditions \\( 2019^a = 2020 \\) and \\( 2020^b = 2019 \\), find the value of \\( n \\).",
"role": "user"
}
] | -1 |
[
{
"content": "Determine the complex polynomial in the form \\( p(z) = (z-(a+bi))^n + q \\) that has roots at the vertices of a regular hexagon, with one root located at \\( 2 + 0i \\).",
"role": "user"
}
] | z^6 - 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... | 0.0377 kg |
[
{
"content": "Suppose $f \\in L^\\infty(\\mathbb{R})$ and $f_h(x) = f(x + h)$ satisfies $$\\lim_{h \\to 0} \\|f_h - f\\|_\\infty = 0.$$ Does there exist a uniformly continuous function $g$ on $\\mathbb{R}$ such that $f = g$ almost everywhere?",
"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... | Complementary sequence and annealing temperature |
[
{
"content": "Calculate the probability that the distance of a randomly chosen point A from the origin in the unit disk is greater than twice the distance of another randomly chosen point B from the origin in the same disk. Express this probability as \\( P(R_1 > 2R_2) \\), where \\( R_1 \\) and \\( R_2 \\) are... | \dfrac{1}{8} |
[
{
"content": "Find the derivative of the function $g(x)$ that satisfies the integral equation: $$g(x) = x^3 + \\int_{0}^{x} (x-t)g(t) \\, dt.$$",
"role": "user"
}
] | 3e^x + 3e^{-x} - 6 |
[
{
"content": "Determine the value of $f(0)$ such that the function \\( f(x) = \\frac{1 - \\cos(1 - \\cos x)}{x^4} \\) is continuous for all real numbers \\( x \\).",
"role": "user"
}
] | \dfrac{1}{8} |
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