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... | 43.8 m/s |
[
{
"content": "Given that \\( x, y, z > 0 \\) and \\( x + y + z = 1 \\), determine the maximum value of the function:\n\\[\nf(x, y, z) = \\sum \\frac{x(2y - z)}{1 + x + 3y}\n\\]",
"role": "user"
}
] | \dfrac{1}{7} |
[
{
"content": "Find the value of $a^3 + b^3$ for the solution $x = a + bi$ of the complex equation $x^3 = 9 + 46i$, where $a$ and $b$ are integers.",
"role": "user"
}
] | -19 |
[
{
"content": "Is every rational function $R(x)$ a real analytic function within its domain of definition $D \\subseteq \\mathbb{R}^n$?",
"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... | 106 g |
[
{
"content": "Solve the initial value problem: $$\\dfrac{d^2y}{dx^2}=\\left( \\dfrac{dy}{dx} \\right)^2 + 1$$ with initial conditions \\( y(0)=0 \\) and \\( \\dfrac{dy}{dx}(0)=0 \\) on the interval \\( \\left( \\dfrac{\\pi}{2},-\\dfrac{\\pi}{2}\\right) \\).",
"role": "user"
}
] | -\ln(\cos x) |
[
{
"content": "Find the largest value of the function \\( f(x) = \\sqrt{x^4-3x^2-6x+13} - \\sqrt{x^4-x^2+1} \\) for \\( x \\in \\mathbb{R} \\).",
"role": "user"
}
] | \sqrt{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... | 916.76 euros |
[
{
"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... | 931 |
[
{
"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... | 7.312% |
[
{
"content": "Calculate the number of non-isomorphic, simple, connected, labeled graphs with 4 vertices and 3 edges.",
"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... | 1.13 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... | 6300 |
[
{
"content": "Evaluate the limit: $$ \\lim_{(x,y) \\to (1,0)} \\frac{(x-1)^2\\ln(x)}{(x-1)^2 + y^2} $$",
"role": "user"
}
] | 0 |
[
{
"content": "Let $f \\in C_0^\\infty(\\mathbb{R}^n)$. Consider the Hilbert transform defined by\n\\[ Hf(x) = \\operatorname{p.v.}\\int_{\\mathbb{R}}\\frac{f(x-y)}{y} \\, dy. \\]\nDetermine if it is possible for $Hf = f$ almost everywhere, possibly after extending $H$ to an $L^p$ space.",
"role": "user"
}... | No |
[
{
"content": "Evaluate the limit: $$\\lim_{n \\to \\infty}\\frac{2^{-n^2}}{\\sum_{k=n+1}^\\infty 2^{-k^2}}$$",
"role": "user"
}
] | \infty |
[
{
"content": "Find the positive integer $p$ such that the function \\( f(x,y) = \\begin{cases} \\frac{(x-y)^{p}}{x^2+y^2} & \\text{if $(x,y) \\neq (0,0)$} \\\\ 0 & \\text{if $(x,y)= (0,0)$} \\end{cases} \\) is differentiable at the point $(0,0)$. \\( \\boxed{} \\)",
"role": "user"
}
] | 4 |
[
{
"content": "Find the radius of convergence of the series \\(\\sum_{n=1}^\\infty{\\frac{(n!)^3 \\cdot x^n}{(3n)!}}\\). Use the ratio test to determine the correct inequality for \\(|x|\\).",
"role": "user"
}
] | 27 |
[
{
"content": "Calculate the limit: $$\\lim_{x\\to 0} \\frac{1}{x} \\int_0^x \\cos(t^2)\\,dt.$$",
"role": "user"
}
] | 1 |
[
{
"content": "Solve the inequality $2x - 2 - (\\ln(x-1))(x-1) > 0$ and determine the solution interval.",
"role": "user"
}
] | (1, e^2 + 1) |
[
{
"content": "Determine the number of elements of order 5 in the alternating group $A_6$. Provide your answer as a single integer.",
"role": "user"
}
] | 144 |
[
{
"content": "Evaluate the limit:\n\n$$\\lim_{x \\to 0} \\frac{\\int_0^x(x-t)f(t)dt}{\\int_0^x tf(t)dt}$$\n\nwhere $f(x)$ is continuous and $\\lim_{x \\to 0} \\frac{f(x)}{x} = 1$. ",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "Calculate \\(2^{31} \\mod 2925\\) using the Chinese Remainder Theorem. First, factor 2925 into its prime components and find \\(2^{31}\\) modulo each of these prime factors. Then, use the Chinese Remainder Theorem to find the solution modulo 2925.",
"role": "user"
}
] | 1298 |
[
{
"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... | RM 44,587.16 |
[
{
"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... | 14920.78 |
[
{
"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... | 10778.3 rad/s |
[
{
"content": "Evaluate the integral: \\[ \\int_{0}^{1} \\left( 2x\\sin\\left(\\frac{1}{x}\\right) - \\cos\\left(\\frac{1}{x}\\right) \\right)\\, dx \\]",
"role": "user"
}
] | \sin 1 |
[
{
"content": "Find the floor of the limit \\( L = \\displaystyle\\lim_{m\\to \\infty}\\sum_{p=1}^m \\frac{p}{2p+m+m^2} \\).",
"role": "user"
}
] | 0 |
[
{
"content": "A tank is full of oil weighing 20 lb/ft³. The tank is an inverted right rectangular pyramid with a base width of 2 feet, a depth of 2 feet, and a height of 5 feet. Calculate the work required to pump the oil to a height of 3 feet above the top of the tank.",
"role": "user"
}
] | \dfrac{1700}{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... | 10.23% |
[
{
"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... | 3421 |
[
{
"content": "Let $A$ be an algebra over a commutative ring $R$, and suppose $A$ is projective as a module over $R$. Let $M$ be a right $A$-module. Is $M$ automatically projective as a right module over $R$?",
"role": "user"
}
] | No |
[
{
"content": "Evaluate the limit: $$\\lim_{n \\to \\infty}\\frac{(\\sum_{k=1}^{n} k^2 )*(\\sum_{k=1}^{n} k^3 )}{(\\sum_{k=1}^{n} k^6)}$$ by converting the sums into integrals.",
"role": "user"
}
] | \dfrac{7}{12} |
[
{
"content": "Let \\( p \\) be a prime number. Determine the largest integer \\( m \\) such that the set \\( \\{1, \\zeta_p, \\zeta_p^2, \\ldots, \\zeta_p^m\\} \\) is linearly independent over \\( \\mathbb{Q} \\), where \\( \\zeta_p \\) is a primitive \\( p \\)th root of unity.",
"role": "user"
}
] | p-2 |
[
{
"content": "Determine whether the inequality \\[ \\frac{\\displaystyle\\sum_{i=1}^{N-1}i\\cdot(N-i-1)!\\sum_{k=1}^{N-i}\\prod_{j=k}^{k+i}j}{N(N-1)N!} < 1 \\] holds for all integers \\( N \\geq 2 \\).",
"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... | 0.115 mL |
[
{
"content": "What is the sum of all the positive two-digit integers where one of the digits is four times the other?",
"role": "user"
}
] | 165 |
[
{
"content": "Find all positive integer triples \\((a, b, c)\\) such that:\n\\[ ab + c = \\gcd(a^2, b^2) + \\gcd(a, bc) + \\gcd(b, ac) + \\gcd(c, ab) = 239^2 \\]",
"role": "user"
}
] | (238, 238, 477) |
[
{
"content": "Evaluate the complex line integral \\( \\int_{\\gamma} |z|^2(\\bar{z} + i) \\, dz \\), where \\( \\gamma \\) is the circle given by \\( |z - i| = 1 \\) with counterclockwise orientation.",
"role": "user"
}
] | 4\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... | 19 mL water |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $72,152.56 |
[
{
"content": "Let $a$ and $b$ be nonzero complex numbers such that $a^2 + ab + b^2 = 0.$ Evaluate\n\\[\\frac{a^9 + b^9}{(a + b)^9}.\\]",
"role": "user"
}
] | -2 |
[
{
"content": "Determine whether there exists a polynomial-time algorithm for finding the chromatic sum of a tree.",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate the integral \\( \\int_{|z+2i|=1} \\frac{\\sin(z)}{z^3+16z} \\, dz \\).",
"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... | 2977 N |
[
{
"content": "Calculate the number $x$ that equals the sum of the prime positive divisors of the number of zeros at the end of the product of the first $784$ integers when expressed in base $15$. Then, find $x - 60$. Choose the correct answer from the following options: \\(\\begin{array}{ll} 1.&\\text{33}\\\\ 2... | B |
[
{
"content": "Find the expression for $w''(z)$ given the boundary conditions $w'(0) = 0$ and $w'(L) = \\frac{qm}{kc}$.",
"role": "user"
}
] | \dfrac{qm}{kcL} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 32.4 N |
[
{
"content": "Given $N$ indistinguishable balls, where only one ball is heavier, and a balance scale that can determine which of two groups of balls is heavier, find the minimal number of weighings required to identify the heaviest ball. Each ball can be weighed at most $K$ times, where $K \\geq 1$. For example... | 3 |
[
{
"content": "Determine the number of solutions to the equation \\(1+e^{\\cot^2x} = \\sqrt{2| \\sin x | -1} + \\frac{1-\\cos2x}{1 + \\sin^4x}\\) for \\(x\\in(0,5\\pi)\\).",
"role": "user"
}
] | 5 |
[
{
"content": "Is it consistent with ZFC that there exists a non-identity automorphism of $P(\\omega_1)/\\mathrm{NS}_{\\omega_1}$?",
"role": "user"
}
] | Yes |
[
{
"content": "Calculate the expectation \\( E\\left(\\int^1_0 W(s)\\,ds \\int^1_0 tW(t)\\,dt\\right) \\), where \\( W(t) \\) is a Wiener process. The integrals are evaluated from 0 to 1.",
"role": "user"
}
] | \dfrac{5}{24} |
[
{
"content": "Suppose you and I each have a box of 600 marbles. In my box, 4 of the marbles are black, while 3 of your marbles are black. We each draw 300 marbles with replacement from our own boxes. Using Poisson approximation, calculate the probability that you and I draw the same number of black marbles.",
... | 0.216 |
[
{
"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. Create a dematerialization account with a depository participant (DP).
2. Submit physical certificates along with a dematerialization request form to the DP.
3. Wait for the securities to be credited to the demat account. The DP will send documents to the registrar (Reliance Industries Limited) who will credit the securities to the holder's account through the National Securities Depository Limited. |
[
{
"content": "Identify the units in the polynomial ring $R[x]$, where $R$ is a domain.",
"role": "user"
}
] | R^\times |
[
{
"content": "Calculate the norm \\( \\lVert \\left[ \\sin\\frac{x}{2} \\right] \\rVert_Q \\) in the quotient space \\( Q = L^\\infty(\\mathbb{R})/S \\), where \\( S \\) is the subspace of \\( 2\\pi \\) periodic functions in \\( L^\\infty(\\mathbb{R}) \\). The norm is defined as \\( \\lVert [f] \\rVert_{Q} = \\... | 1 |
[
{
"content": "Determine whether a continuous and piecewise $C^{\\infty}$ function can have an infinite number of isolated zeroes. Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "In a cyclic quadrilateral $\\square$ ABDC, $\\triangle$ ABC is an equilateral triangle with side length $a$. Express $ \\overline{DA}^4 + \\overline{DB}^4 + \\overline{DC}^4 $ in terms of $a$. Given that $ \\overline{DA} = x, \\overline{DB} = y, \\overline{DC} = z$, and using the Cosine Law, the f... | 2a^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... | 1.34e5 revolutions/min |
[
{
"content": "Each day, two out of the three teams in a class are randomly selected to participate in a MATHCOUNTS trial competition. What is the probability that Team A is selected on at least two of the next three days? Express your answer as a common fraction.",
"role": "user"
}
] | \frac{20}{27} |
[
{
"content": "Evaluate \\( \\lim_{x\\to 0} \\frac {\\arcsin(x) \\sqrt{\\sin(x)}}{\\sqrt{2x-x^2}} \\).",
"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... | 2.934 m/s^2 towards south |
[
{
"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... | 5000.00 |
[
{
"content": "Let \\( a, b, c \\in \\mathbf{C} \\). Determine which of the following statements is correct:\n\n1. If \\( a^{2} + b^{2} > c^{2} \\), then \\( a^{2} + b^{2} - c^{2} > 0 \\).\n2. If \\( a^{2} + b^{2} - c^{2} > 0 \\), then \\( a^{2} + b^{2} > c^{2} \\).\n\nChoose the correct option:\nA. Both (1) and... | D |
[
{
"content": "Given vectors \\(a\\) and \\(b\\) such that \\(a \\times b = 2i + j - k\\) and \\(a + b = i - j + k\\), find the least value of the magnitude of vector \\(a\\). Here, \\(i\\), \\(j\\), and \\(k\\) are unit vectors in the directions of the \\(x\\), \\(y\\), and \\(z\\) axes, respectively, and \\(\\... | \sqrt{2} |
[
{
"content": "Evaluate the integral \\( \\int_{C} \\frac{z}{z^2+1} \\, dz \\), where \\( C \\) is the path \\( |z-1| = 3 \\), using the Residue Theorem.",
"role": "user"
}
] | 2\pi i |
[
{
"content": "A $3491$ by $3491$ square has its length decreased by $60$ and its width increased by $60$. By how much does its area change?",
"role": "user"
}
] | 3600 |
[
{
"content": "Let $g(x)$ be a non-negative function integrable on $(0,1)$ such that \\( \\int_0^1 g(x) \\, dx = 1 \\). If \\( g(x) = \\frac{1}{x} \\int_0^x g(t) \\, dt \\) almost everywhere on \\((0,1)\\), determine whether $g(x)$ must be identically equal to 1.",
"role": "user"
}
] | g(x) \equiv 1 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 2*m*g*h |
[
{
"content": "Solve the Volterra Integral Equation of the First Kind:\n\\[ \\int_{0}^{x} J_{0}(x-t)y(t) \\, dt = \\sin(x) \\]\nwhere \\( J_{0}(x) = \\sum_{k=0}^{\\infty} \\frac{(-1)^k}{(k!)^2} \\left(\\frac{x}{2}\\right)^{2k} \\) is the Bessel Function of the First Kind.",
"role": "user"
}
] | J_0(x) |
[
{
"content": "Determine the dimension of the subspace $W$ of the vector space $V$ of all infinitely differentiable functions on $\\mathbb{R}$, where $W$ consists of functions that are solutions to the initial value problem (IVP): \\[ f'' + 4f = 0, \\quad f(\\pi) = 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... | 32.55m |
[
{
"content": "Determine the radius of convergence of the power series \\( \\sum_{n=1}^\\infty n^{n^{1/3}}z^n \\).",
"role": "user"
}
] | 1 |
[
{
"content": "Given that \\( f(f(x)) = x^2 + \\frac{1}{4} \\), compute the value of \\( f\\left(\\frac{1}{2}\\right) \\).",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 560.81 J |
[
{
"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... | 6264 |
[
{
"content": "Does there exist a non-unital $C^*$ algebra $A$ such that the center of the quotient $A/I$, denoted as $\\mathcal{Z}(A/I)$, is non-zero?",
"role": "user"
}
] | Yes |
[
{
"content": "A single fair die is rolled repeatedly. What is the probability that two sixes in a row will appear before a five followed by a six?",
"role": "user"
}
] | \dfrac{5}{12} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 37.8 degrees Celsius |
[
{
"content": "Evaluate the limit \\( \\lim_{z \\to 0} \\frac{\\bar{z}^2}{z} \\) as \\( z \\) approaches 0.",
"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... | $225,000 |
[
{
"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... | 0.214 M |
[
{
"content": "Determine whether the language \\(L = \\{0^{2^n} \\mid n \\geq 0\\}\\) can be decided in polynomial time. Specifically, is there a polynomial-time algorithm that can verify if a given string of zeros is of the form \\(0^{2^n}\\) for some non-negative integer \\(n\\)?",
"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... | 0.049 |
[
{
"content": "Suppose \\(f\\in C^1([0,1])\\) and \\(f'(0)\\neq 0\\). For \\(x\\in(0,1]\\), let \\(\\theta(x)\\) be such that \\[\\int_0^x f(t)dt = f(\\theta(x))x\\] Find \\[\\lim_{x\\to 0^{+}} \\frac{\\theta(x)}{x}\\]",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "Determine whether the following statement is true or false: If $E \\subseteq \\mathbb{R}$ is such that $m^*(E) > 0$, where $$ m^*(E) = \\inf\\left\\{\\sum_{n=1}^\\infty|b_n - a_n|\\, :\\mid\\, E \\subseteq \\bigcup_{n = 1}^\\infty(a_n,b_n)\\right\\} $$ then there exists a Borel set $B \\subseteq E... | B |
[
{
"content": "Let G be a finite group of odd order n, and let g be an element in a real conjugacy class C, meaning $C = C^{-1}$ and there exists an h $\\in G$ such that $h^{-1}gh = g^{-1}$. Prove that $h^{-2}gh^2 = g$, implying $h^2 \\in C_G(g)$. Furthermore, given that the order of h is odd, show that h $\\in ... | h \in C_G(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... | 7.158 s |
[
{
"content": "A circle of radius 1 is tangent to a circle of radius 2. The sides of $\\triangle ABC$ are tangent to the circles as shown, and the sides $\\overline{AB}$ and $\\overline{AC}$ are congruent. What is the area of $\\triangle ABC$?\n\n[asy]\nunitsize(0.7cm);\npair A,B,C;\nA=(0,8);\nB=(-2.8,0);\nC=(2.... | 16\sqrt{2} |
[
{
"content": "Determine the order of growth of the sum \\(\\sum_{k=1}^n\\frac{1}{\\sqrt{k+2}}\\) as \\(n\\) approaches infinity.",
"role": "user"
}
] | \Theta(\sqrt{n}) |
[
{
"content": "Consider a homomorphism \\( \\varphi \\) from a cyclic group \\( G = \\mathbb{Z}/d\\mathbb{Z} \\) to an arbitrary group \\( H \\). If \\( \\varphi \\) is defined by choosing an element \\( h \\in H \\) such that \\( \\varphi(g_G) = h \\) for a generator \\( g_G \\) of \\( G \\), determine whether ... | 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... | The magnitude of the acceleration is approximately 9.43. The direction is along the unit vector #(1/9.43)(64hati - 4.99hatj)# or #(1/9.43)(8hati + (1-(8sqrt(3)/2 + (16pi)/6))hatj)# |
[
{
"content": "If $x, y, z$ are positive real numbers such that $x + y + z = 1$, determine whether the inequality \\( \\left(1+\\frac{1}{x}\\right)\\left(1+\\frac{1}{y}\\right)\\left(1+\\frac{1}{z}\\right) \\ge 64 \\) holds true.",
"role": "user"
}
] | 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... | $69.41 |
[
{
"content": "In triangle $XYZ$, angle $XZY$ is $45$ degrees, and angle $ZYX$ is $80$ degrees. Let $P$ be the foot of the perpendicular from $X$ to $YZ$, $Q$ the center of the circle circumscribed around triangle $XYZ$, and $R$ the other end of the diameter which goes through $X$. Determine the measure of angle... | 45 |
[
{
"content": "Determine whether a unital *-endomorphism of a unital finite-dimensional C*-algebra can have a non-zero eigenvalue with an absolute value greater than 1.",
"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... | 7.9 |
[
{
"content": "Determine whether the function \\( f : \\mathbb{R} \\rightarrow \\mathbb{R} \\) defined by \\( f(x) = 0 \\) if \\( x \\in \\mathbb{Q} \\) and \\( f(x) = x \\) otherwise, is Lebesgue measurable.",
"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... | 26.1 degrees |
[
{
"content": "For odd integers A and x greater than 1, determine if it is always true that \\(A^x - A\\) is divisible by both \\((A+1)/2\\) and \\((A-1)/2 + 1\\).",
"role": "user"
}
] | Yes |
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