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... | 114990.44 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | U'= \frac{1}{k\lambda}\left(\frac{1}{2}CV^2\right) |
[
{
"content": "Calculate the sum of the infinite series \\( \\sum_{n=1}^{\\infty} \\frac{2n+3}{n(n+1)(n+2)} \\).",
"role": "user"
}
] | \dfrac{7}{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... | 20/3 ft/s |
[
{
"content": "For which integer value of \\( a \\) does the ratio \\( \\frac{I_1}{I_0} = \\frac{3\\sqrt{2}}{4} \\), where \\( I_0 = \\iiint_G \\,dx\\,dy\\,dz \\) and \\( I_1 = \\iiint_G z\\,dx\\,dy\\,dz \\), and the region \\( G \\) is defined as \\( G = \\{ (x,y,z) : x^2 + y^2 + z^2 \\leq a^2, 0 \\leq z \\leq ... | 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... | 50.625e9*C^2 |
[
{
"content": "Find the number of automorphisms in the Galois group of the field extension \\( \\mathbb{Q}[\\sqrt{2}, \\sqrt{5}, \\sqrt{11}] \\).",
"role": "user"
}
] | 8 |
[
{
"content": "Determine the number of integer solutions to the Diophantine equation \\(x^4 - 4y^2 = 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... | 17.88% |
[
{
"content": "Solve for $x$ in the equation $3\\sin\\left(\\frac{x^2-x}{8}\\right)=3^x-3^{-x}$.",
"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... | The torque for the rod rotating about the center is =169.6Nm
The torque for the rod rotating about one end is =678.6Nm |
[
{
"content": "Determine the minimum number of tokens that must be placed in the cells of a $99 \\times 99$ grid such that every $4 \\times 4$ subgrid contains at least eight tokens.",
"role": "user"
}
] | 4901 |
[
{
"content": "Determine whether it is possible to draw $n$ circles ($n \\ge 3$, $n \\in \\mathbb{N}$) such that they all intersect at a single point, but no three circles intersect at any other point.",
"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... | sqrt5/3 m/s |
[
{
"content": "Let \\( f(x) = \\lim_{n\\rightarrow \\infty}\\frac{x^2+2(x+1)^{2n}}{(x+1)^{2n+1}+x^2+1} \\), where \\( n \\in \\mathbb{N} \\). Define \\( g(x) = \\tan \\left(\\frac{1}{2}\\arcsin\\left(\\frac{2f(x)}{1+(f(x))^2}\\right)\\right) \\). Determine the number of points where \\( g(x) \\) is not different... | 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... | 4.20 years |
[
{
"content": "A flag is made of three horizontal strips of fabric, each of a solid color, either red, white, blue or green. If no two adjacent strips can be the same color, how many distinct flags are possible? These two flags are different.\n\n[asy]draw((0,0)--(0,6),linewidth(1));\nfilldraw((0,3)--(4,3)--(4,4)... | 36 |
[
{
"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... | 2.34 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... | $30.89 |
[
{
"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... | 1.60 meters |
[
{
"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... | 35.0 ml |
[
{
"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... | 8379 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... | 80 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... | 2,430,300 |
[
{
"content": "Evaluate the limit as \\( k \\to \\infty \\):\n\\[ \\lim_{k\\rightarrow \\infty} 2k(m+2k)\\frac{e^{\\frac{-m}{2k}}(m+2k)^{\\frac{m}{2k}}}{(k!)^{\\frac{1}{k}}\\Gamma\\left(k+\\frac{p}{2}\\right)^{\\frac{1}{k}}c^{\\frac{m}{2k}}} \\]\nwhere \\(m, p \\in \\mathbb{N}\\) and \\(c > 0\\).",
"role": "... | 4e^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... | 1.4 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... | 3.46e-6 H |
[
{
"content": "A treasure chest contains $m$ coins. By morning, the number of remaining coins $N$ is given by the equation:\n\n$$\nN = \\frac{2^3}{3^3}(m-1) - \\frac{10}{9} = 3n\n$$\n\nThis leads to the system of equations:\n\n$$\n\\begin{aligned}\n8(m-1) - 30 &= 81k, \\\\\nm-1 &= 3t, \\\\\n8t - 27k &= 10, \\\\\... | 187 |
[
{
"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... | 1.6e-7 |
[
{
"content": "Let $T$ be a linear operator on $\\mathbb{R}^5$ such that $T^4 \\neq 0$ and $T^5 = 0$. Determine the rank of $T^2$. \\( \\text{Provide your answer as a single integer.} \\)",
"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... | 5.20e-6 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... | 11.1 N/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... | 7389.85 |
[
{
"content": "Evaluate the limit \\( \\lim_{(x,y)\\to(0,0)} g \\left(\\frac{x^4 + y^4}{x^2 + y^2}\\right) \\) given that \\( \\lim_{z\\to 0}\\frac{g(z)}{z}=2. \\)",
"role": "user"
}
] | 0 |
[
{
"content": "Find the relative maximum and minimum values of the function \\( f(x, y) = x^4 + y^4 - 2x^2 + 4xy - 2y^2 \\).",
"role": "user"
}
] | -8 |
[
{
"content": "Find the values of $a \\in \\mathbb{R}$ for which the system of differential equations \\( \\begin{cases} x'(t) = x(t) + ay(t) \\\\ y'(t) = ay(t) \\end{cases} \\) has exactly one non-trivial solution \\((x(t), y(t))\\) that approaches \\((0, 0)\\) as \\(t \\rightarrow \\infty\\).",
"role": "us... | a < 0 |
[
{
"content": "Use the Plancherel formula to compute the integral \\( \\int_{-\\infty}^\\infty \\frac{1}{c^2+y^2} \\, dy \\) where \\( c > 0 \\).",
"role": "user"
}
] | \dfrac{\pi}{c} |
[
{
"content": "Given a natural number \\( x = 7^n + 1 \\), where \\( n \\) is an odd natural number, and it is known that \\( x \\) has exactly three distinct prime factors, one of which is 11, find the value of \\( x \\).",
"role": "user"
}
] | 16808 |
[
{
"content": "Given the function \\( f(x, y, z) \\) that satisfies the following conditions for any real number \\( t \\):\n\n\\[ f(x+t, y+t, z+t) = t + f(x, y, z) \\]\n\\[ f(t x, t y, t z) = t f(x, y, z) \\]\n\\[ f(x, y, z) = f(y, x, z) = f(x, z, y) \\]\n\nDetermine \\( f(2000, 2001, 2002) \\).",
"role": "... | 2001 |
[
{
"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... | $4.05 |
[
{
"content": "Let $R$ be a commutative ring with unity, which is a subring of a commutative ring $A$ with unity, and $A$ is a subring of $R[x,y]$. Is there an ideal $J$ of $A$ such that $A/J$ is isomorphic to $R$?",
"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... | 8666 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... | Copying someone else's writing word for word
Trying to pass off another writer's work as your own |
[
{
"content": "Evaluate the integral \\(\\int_0^\\infty \\frac{\\mathrm{d} x}{(1+x^2)(1+x^d)}\\) for an arbitrary value of \\(d > 0\\).",
"role": "user"
}
] | \dfrac{\pi}{4} |
[
{
"content": "A sphere, lying in the octant where all the coordinates are nonnegative, is tangent to the $xy$-, $xz$-, and $yz$-plane. A point on the sphere has distances of 50, 29, and 41 from the $xy$-, $xz$-, and $yz$-planes, respectively. Enter all possible values for the radius of the sphere, separated b... | 27,93 |
[
{
"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. Disruption of proofreading activity during protein synthesis, leading to the production of proteins with inappropriate amino acids.
2. Inhibition of protein synthesis by blocking the formation of peptide bonds.
3. Inhibition of protein synthesis by preventing the formation of the initiation complex during translatio... |
[
{
"content": "Evaluate the limit: $$\\lim_{n\\rightarrow\\infty}{\\int\\int}_{(0,\\infty)\\times (0,\\infty)} \\frac{\\sin^2(nxy)xy}{e^{(x^2+y^2)}}\\,dx\\,dy.$$ Use convergence theorems such as the Dominated Convergence Theorem to justify your solution.",
"role": "user"
}
] | \dfrac{1}{8} |
[
{
"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... | 7000 |
[
{
"content": "Determine whether there exists a transcendental number whose simple continued fraction expansion is bounded.",
"role": "user"
}
] | Yes |
[
{
"content": "Determine the asymptotic behavior of the sum \\( \\sum_{i=1}^{n^2 - 1} \\frac{i^2}{\\left[\\frac{n^3}{3}\\right]^2} \\) as \\( n \\to \\infty \\).",
"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... | 492kHz |
[
{
"content": "Let $k \\ge 1$ be an integer. Determine the smallest value of $N$ as a function of $k$ such that for any set $S$ of positive integers with $|S| = N$, there exists a subset $T \\subset S$ with $|T| = k+1$ where for every $a,b \\in T$, the number $a^2-b^2$ is divisible by $10$. What is this smallest... | 6k + 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... | 6 eV |
[
{
"content": "Evaluate \\( \\lim_{x\\to\\frac{\\pi}{2}} \\left(\\frac{\\sin x}{1-\\sin x}\\right)^{\\frac{1}{1-\\sin x}} \\).",
"role": "user"
}
] | \infty |
[
{
"content": "Is there a nonstandard model $M$ of Peano arithmetic and a nonstandard $c \\in N$ such that the operations $\\oplus_{<c}$ and $\\otimes_{<c}$ are computable?",
"role": "user"
}
] | No |
[
{
"content": "Determine whether there exists a measurable set $B$ with positive Lebesgue measure $\\mu(B)>0$ and a constant $M>0$ such that for all $x\\in B$, a given Lebesgue measurable function $f: [0,1] \\to \\mathbb{R}$ satisfies $f(x) < M$. Provide a justification for your answer.",
"role": "user"
}
... | Yes |
[
{
"content": "Let \\( G = (\\mathbb{Z} \\bigoplus \\mathbb{Z}) \\star (\\mathbb{Z} \\bigoplus \\mathbb{Z}) \\), where \\( \\star \\) denotes the free product, and let \\( F \\) be the commutator subgroup of \\( G \\). By Kurosh's theorem, \\( F \\) is a free group. Find a proper normal subgroup of \\( F \\) (ot... | [F, F] |
[
{
"content": "A polynomial of 6th degree, \\( f(x) \\), satisfies \\( f(x) = f(2-x) \\) for all \\( x \\in \\mathbb{R} \\). If \\( f(x) = 0 \\) has 4 distinct roots and two equal roots, find the sum of the roots of \\( f(x) = 0 \\).",
"role": "user"
}
] | 6 |
[
{
"content": "What is the last digit of \\(\\operatorname{lcm}(3^{2003}-1,3^{2003}+1)\\)?",
"role": "user"
}
] | 4 |
[
{
"content": "In a Cartesian coordinate system, points \\( A(1,2) \\), \\( B(3,0) \\), and \\( P \\) lie on the circle defined by \\( (x-3)^{2}+(y-2)^{2}=1 \\). Given that \\( \\overrightarrow{O P}=\\lambda \\overrightarrow{O A}+\\mu \\overrightarrow{O B} \\) where \\( \\lambda, \\mu \\in \\mathbb{R} \\), deter... | 12 |
[
{
"content": "Find the limit inferior of the sequence \\((-1-\\frac{2}{n})^n\\) as \\(n\\) approaches infinity.",
"role": "user"
}
] | -e^2 |
[
{
"content": "Determine the automorphism group ${\\rm Aut}(Q_{8})$ for the quaternion group $Q_{8}$.",
"role": "user"
}
] | S_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... | $4200 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 47.3 ^° C |
[
{
"content": "Evaluate the integral \\( \\int_{0}^{1}\\int_{0}^{1-x_1}...\\int_{0}^{1-\\sum_{i=1}^{n-1}{x_i}}1 \\,dx_n\\,dx_{n-1}...\\,dx_1 \\) over the region defined by \\( \\sum_{i=1}^{n}{x_i}=1 \\). Determine the value of this integral in terms of \\( n \\).",
"role": "user"
}
] | \dfrac{1}{n!} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 2.97kgm^2s^-1 |
[
{
"content": "If the sequence $\\{\\ln(a_n)\\}$ of positive terms converges to $a$, does it imply that the sequence $\\{a_n\\}$ converges to $e^a$? Justify your answer with a formal $\\epsilon-\\delta$ proof.",
"role": "user"
}
] | e^a |
[
{
"content": "Find the marginal probability mass function of $x$ given the joint mass function:\n\\[ f_{x,y} (x,y) = \\begin{cases} \\frac{1}{x} & 0< y<x < 1 \\\\ 0 & \\text{else} \\end{cases} \\]",
"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... | $48,500 |
[
{
"content": "Given three independent and identically distributed random variables $X, Y, Z$ following a Uniform $(0,1)$ distribution, find the probability $\\Pr(X > YZ)$. Provide your answer as a single probability value.",
"role": "user"
}
] | \dfrac{3}{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... | 77.448 h |
[
{
"content": "Let $a$ and $b$ be non-square elements in a finite field. If a polynomial over the finite field has a root $\\sqrt{a} + \\sqrt{b}$, does it also have $\\sqrt{a} - \\sqrt{b}$ as a root? Answer with 'Yes' or 'No'.",
"role": "user"
}
] | No |
[
{
"content": "Find the spectral radius of the following $n \\times n$ matrix:\n\n$$ \\begin{matrix} 1 & 0 & . & . & . &-1 \\\\ 0 & 1 & . & . & . &0 \\\\ . & . & . & & &. \\\\ . & . & & . & &. \\\\ . & . & & & . &. \\\\ -1 & 0 & . & . & . &1 \\\\ \\end{matrix} $$",
"role": "user"
}
] | 2 |
[
{
"content": "Evaluate the double integral \\( \\iint_{x^2+y^2<\\sqrt{x}}\\sqrt{x^2+y^2}\\,dx\\,dy \\).",
"role": "user"
}
] | \dfrac{2}{3} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | Assets:
Non-current assets:
Equipment: $46,000
Current assets:
Cash: $12,000
Accounts receivable: $2,500
Inventory: $22,000
Total assets: $82,500
Liabilities:
Current liabilities:
Accounts payable: $6,800
Wages payable: $2,600
Long-term liabilities:
Mortgage: $22,000
Total liabilities: $31,400
Owner's equity: $51,100
T... |
[
{
"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... | $29.884 |
[
{
"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... | $7,055.51 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 5360 m |
[
{
"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": "What is the units digit of the smallest integer $n$ such that $7n$ has 1984 digits?",
"role": "user"
}
] | 3 |
[
{
"content": "How many integers satisfy the inequality $(x+3)^{2}\\leq1$?",
"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... | 92164 L |
[
{
"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... | 0.012 M |
[
{
"content": "In a profinite group $G$ with order $lp^n$, where $l$ and $p$ are distinct primes with $l < p$, determine if the $p$-Sylow subgroups are necessarily normal.",
"role": "user"
}
] | Yes |
[
{
"content": "Suppose we have $n$ independent normal random variables $X_1, X_2, \\ldots, X_n$ and another independent normal random variable $Y$. What is the probability that $Y$ is less than all of $X_1, X_2, \\ldots, X_n$?",
"role": "user"
}
] | \dfrac{1}{n+1} |
[
{
"content": "Determine whether the function $e(x,y) = d(f(x),f(y))$ defines a metric on the space $(X,d)$ for any function $f: X \\rightarrow X$. Justify your answer by checking the properties of a metric.",
"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... | 5.77% |
[
{
"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... | -$5,150 UF |
[
{
"content": "Determine if the function \\( g(x,y) = \\begin{cases} 1, & \\text{if } (x,y) = \\left(\\frac{p}{3^n},\\frac{q}{3^n}\\right), 0<p,q<3^n \\\\ 0, & \\text{otherwise} \\end{cases} \\) is integrable over the region \\( E = [0,1] \\times [0,1] \\). If it is integrable, compute the integral \\( \\int_E g... | 0 |
[
{
"content": "Find the minimum value of the function \\( f(a, b, c) = \\int_0^1 (a + bx + cx^2)^2 \\, dx \\) subject to the constraint \\( a + b + c = 1 \\). The integral expands to \\( a^2 + a(b + \\frac{2c}{3}) + \\frac{b^2}{3} + \\frac{bc}{2} + \\frac{c^2}{5} \\).",
"role": "user"
}
] | \dfrac{1}{9} |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to +\\infty} \\frac{\\ln(\\ln x)}{\\ln(x - \\ln x)} \\]",
"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... | 6.64% |
[
{
"content": "What is the ratio of the area of a triangle $ABC$ to the area of the triangle whose sides are equal in length to the medians of triangle $ABC$?",
"role": "user"
}
] | \dfrac{4}{3} |
[
{
"content": "Robin bought a four-scoop ice cream cone having a scoop each of vanilla, chocolate, strawberry and cherry. In how many orders can the four scoops be stacked on the cone if they are stacked one on top of the other and chocolate cannot be on the bottom?",
"role": "user"
}
] | 18 |
[
{
"content": "Determine the order of the quotient group $G/Z(G)$, where $G = \\{a^k, a^k b \\mid 0 \\le k < 9\\}$ with $o(a) = 9$, $o(b) = 2$, and $ba = a^{-1}b$. The center $Z(G)$ consists of elements that commute with all elements in $G$. What is the order of $G/Z(G)$?",
"role": "user"
}
] | 18 |
[
{
"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... | <img>/cimages/multimages/16/iupac8207331213046676996.png</img> |
[
{
"content": "Find the approximate number of pairs of integers $(a, b)$ such that $0 \\leq a \\leq b < 64$ and $ab \\geq 128$.",
"role": "user"
}
] | 1756 |
[
{
"content": "Determine whether a Riemann-integrable function can have uncountably many discontinuities in its domain of integration.",
"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... | Erythropoiesis |
[
{
"content": "Determine whether the following statement is correct:\n\nGiven a continuous function \\(f(x,y): \\mathbb{R}^+ \\times \\mathbb{R}^+ \\rightarrow \\mathbb{R}^+_0\\) and a function \\(g(x)\\) such that \\(f(x,g(x)) \\geq f(x,z)\\) for all \\(z \\geq 0\\), is \\(f(x,g(x))\\) continuous?",
"role":... | Yes |
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