prompt list | ground_truth string |
|---|---|
[
{
"content": "Given two groups $G$ and $H$ with the same order, $|G| = |H| = p_1^{\\alpha_1} \\cdots p_n^{\\alpha_n}$, where each $p_i$ is a prime number, if the number of Sylow $p_i$-subgroups in $G$ equals the number in $H$ for all $i = 1, \\ldots, n$, does it necessarily imply that $G$ is isomorphic to $H$? ... | No |
[
{
"content": "Determine the number of natural numbers \\( n \\) such that \\( 5^n \\equiv 1 \\pmod{p} \\) does not hold for any prime \\( p > 3 \\).",
"role": "user"
}
] | 2 |
[
{
"content": "Evaluate the integral: \\[ \\int_{0}^{1}{\\sqrt[3]{2x^3-3x^2-x+1}}\\,\\mathrm{d}x. \\]",
"role": "user"
}
] | 0 |
[
{
"content": "Suppose $F:(0,\\infty) \\to [0,\\infty)$ is a continuous function satisfying $F(1)=0$. Assume that $F$ is strictly increasing on $[1,\\infty)$, strictly decreasing on $(0,1]$, and convex on $[1-\\epsilon,1]$ for some $\\epsilon>0$. Let $y_n \\in [1-\\epsilon,1)$ be a sequence converging to 1, and ... | Yes |
[
{
"content": "Calculate the line integral \\(\\int(10x^4 - 2xy^3) \\, dx - 3x^2y^2 \\, dy\\) along the curve defined by \\(x^4 - 6xy^3 = 4y^2\\) from the origin \\(O(0,0)\\) to the point \\(A(2,1)\\).",
"role": "user"
}
] | 60 |
[
{
"content": "Let \\( f:[2,4]\\to[3,5] \\) be a bijective decreasing function. Find the value of \\( \\int_{2}^{4}f(t) \\, dt - \\int_{3}^{5}f^{-1}(t) \\, dt. \\)",
"role": "user"
}
] | 2 |
[
{
"content": "Let $G$ be a connected self-centered graph with radius $r(G)$ and diameter $d(G)$ both equal to $m < \\infty$. Does $G$ always contain a cycle of length $C_{2m}$ or $C_{2m+1}$ as a subgraph?",
"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... | 33% |
[
{
"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... | 5.32e-8 A |
[
{
"content": "Determine the type of triangle formed by the intersection points of the hyperbola \\(xy = a\\) and the parabola \\(y = x^2 + ax - 1\\), given \\(a \\neq \\pm 1\\). Choose from the following options:\n\nA. Right triangle\n\nB. Equilateral triangle\n\nC. Obtuse triangle\n\nD. Scalene acute triangle"... | A |
[
{
"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... | 2110 g |
[
{
"content": "In triangle $\\Delta ABC$, the centroid $G$ is such that $GA = 2\\sqrt{3}$, $GB = 2\\sqrt{2}$, and $GC = 2$. Calculate the area of $\\Delta ABC$. \\(\\text{Express your answer in square units.}\\)",
"role": "user"
}
] | 6\sqrt{2} |
[
{
"content": "Given a semi-simple Lie Algebra $L$ with dimension $d$ over $\\mathbb{R}$, determine the dimension of its complexification $L_{\\mathbb{C}} = L \\otimes_{\\mathbb{R}} \\mathbb{C}$.",
"role": "user"
}
] | d |
[
{
"content": "Calculate the expected value $E[S(t)]$ for the expression $S(t) = \\sum_{i=1}^{N(t)} (X_i + T_i)^2$, where $X_i$ are independent and identically distributed random variables with density $f(x) = \\frac{1}{2}$ for $0 \\leq x \\leq 2$, and $T_i$ are the arrival times of a homogeneous Poisson process... | t^3 + 3t^2 + 4t |
[
{
"content": "Suppose $0 \\leq f(x)$ on $[a,b]$. If $f^2(x)$ is Riemann integrable on $[a,b]$, can we deduce that $f(x)$ is also Riemann integrable on $[a,b]$?",
"role": "user"
}
] | Yes |
[
{
"content": "Does a sequence of continuous functions \\(\\{u_n(x)\\}\\) defined on the closed interval \\([0,1]\\) that converges uniformly on the half-open interval \\([0,1)\\) necessarily converge uniformly on the entire closed interval \\([0,1]\\)?",
"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... | 11.85 |
[
{
"content": "Given the recurrence relation \\(x_{n+1} = 4x_n - x_{n-1}\\) for \\(n \\geq 1\\), with initial conditions \\(x_0 = 1\\) and \\(x_1 = 2\\), find an odd prime factor of \\(x_{2015}\\).",
"role": "user"
}
] | 181 |
[
{
"content": "Find the limit of the expression \\( x = \\pm \\sqrt{\\frac{1 \\pm \\sqrt{1 - \\tan^2(\\theta)}}{2\\tan^2(\\theta)}} \\) as \\( \\theta \\) approaches 0, and show that it approaches \\( x = \\pm \\frac{1}{2} \\).",
"role": "user"
}
] | \pm \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... | 8.3 m/s |
[
{
"content": "Calculate the derivative of the conserved quantity \\( Q = u + \\frac{x^2}{2} \\) along the characteristics of the quasi-linear partial differential equation \\( \\frac{\\partial u}{\\partial t} + u\\frac{\\partial u}{\\partial x} = -xu \\).",
"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... | 49.23 months |
[
{
"content": "Consider the $n \\times n$ matrix $A = (a_{ij})$ where $a_{ij} = a_i a_j$ for real numbers $a_1, \\dots, a_n$. Determine which of the following statements is true:\n\n1. It is possible to choose $a_1, \\dots, a_n$ such that $A$ is non-singular.\n2. Matrix $A$ is positive definite if $(a_1, \\dots,... | 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... | 9.80 |
[
{
"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... | 0.247 N/m |
[
{
"content": "Evaluate the limit: $$\\lim_{n\\to\\infty} \\sum_{m=1}^n \\left(\\left(1+\\frac{m^3}{n^4}\\right)^{1/8}-1\\right).$$ Use the binomial theorem and the Squeeze theorem to determine the value of the limit.",
"role": "user"
}
] | \dfrac{1}{32} |
[
{
"content": "Let $f:(a, \\infty)\\to \\Bbb R$ be a differentiable function such that $\\lim_{x\\to\\infty}f(x)=l<\\infty$ and $\\lim_{x\\to\\infty}f'(x)$ exists (in the sense it can also be infinity). Determine whether we can conclude that $\\lim_{x\\to\\infty}f'(x)=0$. Provide a justification for your answer.... | 0 |
[
{
"content": "Given a Poisson process $N(t)$ with parameter $\\lambda > 0$, and independent and identically distributed random variables $X_1, X_2, \\ldots$ with a common mean $\\mu$ and positive variance, let \\( L(t) = \\sum_{i=1}^{N(t)} X_i \\). Find the expected value $E[L(t) | N(t) = n]$. Express your answ... | n\mu |
[
{
"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... | Late deceleration |
[
{
"content": "Given the polynomial function \\( f(x) = x^5 + ax^4 + bx^3 + cx^2 + dx + e \\) and the conditions:\n\\[ f(1) = 2, \\quad f(2) = 3, \\quad f(3) = 4, \\quad f(4) = 5, \\quad f(5) = 6, \\]\nfind the value of \\( b \\).",
"role": "user"
}
] | 85 |
[
{
"content": "Given the vector $y = \\begin{bmatrix} R(t) \\\\ F(t) \\end{bmatrix}$ representing the populations of rabbits $R(t)$ and foxes $F(t)$, which evolve according to the system of differential equations $\\dot y = Ay$, where $A = \\begin{bmatrix} 98 & -1170 \\\\ 7 & -83 \\end{bmatrix}$. The initial rab... | 4100 |
[
{
"content": "Determine the radius of convergence of the power series: $$\\sum_{n=1}^\\infty \\frac{(-1)^n z^{n(n+1)}}{n}$$",
"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... | 1.5 rad/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... | 35.23 degrees Celsius |
[
{
"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... | 13.21% |
[
{
"content": "Find the number of non-negative integer solutions to the equation \\(x_1 + x_2 + x_3 + x_4 = 12\\) given that \\(x_1 = 2x_2 + 2\\) and \\(x_3 \\le x_4\\).",
"role": "user"
}
] | 14 |
[
{
"content": "Let $A=\\{\\frac{(-1)^n+1}{n+1}+(-1)^n, n\\in\\mathbb{N}\\}$. It is given that $\\forall x\\in A, x\\in [-1,3]$. Find $x_{\\varepsilon}\\in A$ such that $\\forall\\varepsilon>0, 3-\\varepsilon< x_{\\varepsilon}\\leq 3$. Determine the value of $x_{\\varepsilon}$.",
"role": "user"
}
] | 3 |
[
{
"content": "Evaluate the infinite sum \\( \\sum_{n=0}^{\\infty} \\frac{2n}{8^n}{{2n}\\choose{n}} \\).",
"role": "user"
}
] | \sqrt{2} |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n\\to \\infty}\\left(\\sum_{k=1}^{2n} \\frac{1}{\\binom{2n}{k}}\\right)^n \\]",
"role": "user"
}
] | e |
[
{
"content": "Find the order of the subgroup of \\( \\mathbb{C}^\\times \\) generated by \\( \\{i, e^{\\frac{2i \\pi}{5}}, -1\\} \\).",
"role": "user"
}
] | 20 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | <img>/cimages/multimages/16/screen_shot_2020-11-30_at_9.58.58_pm637364464423824527.png</img> |
[
{
"content": "The equation $x^2-kx-12=0$ has only integer solutions for certain positive integers $k$. What is the sum of all such values of $k$?",
"role": "user"
}
] | 16 |
[
{
"content": "How many base-10 integers are exactly 4 digits in their base-3 representation and exactly 2 digits in their base-6 representation?",
"role": "user"
}
] | 9 |
[
{
"content": "Determine the number of field extensions between the fields $\\mathbb{Q}(\\zeta^3)$ and $\\mathbb{Q}(\\zeta)$, where $\\zeta$ is a primitive 12th-root of unity.",
"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... | 122.7 J/K |
[
{
"content": "Let \\( f:[-1,1]\\to \\mathbb{R} \\) be a continuous function satisfying the functional equation \\( f(2x^2-1)=(x^3+x)f(x) \\). Determine the value of \\( \\lim_{x\\to 0}\\frac {f(\\cos x)}{\\sin 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... | 3.66% |
[
{
"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... | 0.005 (C)^-1 |
[
{
"content": "The exact amount of fencing that enclosed the four congruent equilateral triangular corrals shown here is reused to form one large equilateral triangular corral. What is the ratio of the total area of the four small corrals to the area of the new large corral? Express your answer as a common fract... | \frac{1}{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... | Rank the projects based on their IRR in descending order: B(15.24%), D(14.29%), C(13.00%), A(12.38%), E(11.65%). If all projects have IRR greater than the required rate of return, all project are acceptable. |
[
{
"content": "Let $a, b, c$, and $d$ be positive real numbers such that\n\\[\\begin{array}{c@{\\hspace{3pt}}c@{\\hspace{3pt}}c@{\\hspace{3pt}}c@{\\hspace{3pt}}c}a^2+b^2&=&c^2+d^2&=&2008,\\\\ ac&=&bd&=&1000.\\end{array}\\]\nIf $S=a+b+c+d$, compute the value of $\\lfloor S\\rfloor$.",
"role": "user"
}
] | 126 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 67.667 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... | 129.6ms^-1 |
[
{
"content": "Evaluate the limit: $$\\lim_{n\\to\\infty}\\int_{n}^{n+1} e^{-x^2}dx $$.",
"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... | lower |
[
{
"content": "Given the polynomial \\( p(x) = x^4 + ax^3 + bx^2 + cx + d \\) with the conditions \\( p(1) = 10 \\), \\( p(2) = 20 \\), and \\( p(3) = 30 \\), compute \\( \\frac{p(12) + p(-8)}{10} \\).",
"role": "user"
}
] | 1984 |
[
{
"content": "Is any $F$-stable maximal torus contained in some $F$-stable Borel subgroup of an affine algebraic group $G$ over the algebraic closure of a finite field $\\mathbb{F}_q$? Assume $F$ is a Frobenius morphism defining an $\\mathbb{F}_q$ structure on $G$. It is known that each $F$-stable Borel subgrou... | 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... | -17000 V |
[
{
"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... | Current Liabilities: $20,000,000
Notes payable: $10,000,000
Long-term debt: $27,000,000
Common stock: $56,000,000
Total Capital: $113,000,000 |
[
{
"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... | 2PO_4^{3-}(aq) + 3Pb^{2+}(aq) o Pb_3(PO_4)_2(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... | -14 |
[
{
"content": "Let $R$ be a unique factorization domain (UFD) and $p(x) \\in R[x]$ be a polynomial of degree at least 1. Suppose $p(x)$ has a root $\\alpha$ in the fraction field $K$ of $R$. Determine if it is possible to express $p(x)$ as a product of a linear polynomial $(x-\\alpha)$ and another polynomial in ... | Yes |
[
{
"content": "Find all functions \\( f: \\mathbb{R} \\to \\mathbb{R} \\) that satisfy the functional equation \\((x-y)(f(x)+f(y))=(x+y)f(x-y)\\), given that \\(f(0)=0\\) and \\(f\\) is an odd function.",
"role": "user"
}
] | f(x) = kx |
[
{
"content": "Given the parametric equations \\( x = 2t + t^2 \\sin\\left(\\frac{1}{t}\\right) \\) for \\( t \\neq 0 \\) and \\( x = 0 \\) for \\( t = 0 \\), and \\( y = \\frac{\\sin(t^2)}{t} \\) for \\( t \\neq 0 \\) and \\( y = 0 \\) for \\( t = 0 \\), find \\( \\frac{dy}{dx} \\) at \\( t = 0 \\).",
"role... | \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... | 1. Look at the information given. If the equation has "energy" or "heat" on the left-hand side of the reaction arrow, the reaction is endothermic. 2. Look at the energy diagram for the reaction. If the products are at a higher energy level than the reactants, the reaction must have absorbed energy. 3. Do an experimen... |
[
{
"content": "Provide an example where \\(E\\left[\\lim_{n \\to \\infty}X_n\\right] \\neq \\lim_{n \\to \\infty}E\\left[X_n\\right]\\), with \\(E\\) representing expectation and \\(X_n\\) being a random variable. Explain why this discrepancy occurs.",
"role": "user"
}
] | E\left[\lim_{n \to \infty}X_n\right] \neq \lim_{n \to \infty}E\left[X_n\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... | 46 mph |
[
{
"content": "Let $G$ be a finite group with normal subgroups $H$ and $K$. Given that $[G:H]=2$ and $[G:K]=3$, find the index $[G:H \\cap K]$.",
"role": "user"
}
] | 6 |
[
{
"content": "Let $p, q, r$ be non-vanishing non-constant entire holomorphic functions such that $p + q + r = 0$. Determine whether there exists an entire function $h$ such that $p, q, r$ are constant multiples of $h$.",
"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... | PH2 = PI2 = 0.716 atm, PHI = 0.0155 atm |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n \\to \\infty} \\sum_{i = 1}^{n}{1 \\over n} \\,\\left\\lfloor\\,\\sqrt{\\,{4i \\over n}\\,}\\,\\right\\rfloor \\] where \\( \\left\\lfloor\\,x\\,\\right\\rfloor \\) is the greatest integer function.",
"role": "user"
}
] | \dfrac{3}{4} |
[
{
"content": "Solve the differential equation \\( y'' + y = t \\) using Green's functions, subject to the boundary conditions \\( y(0) = 0 \\) and \\( y(1) = 1 \\).",
"role": "user"
}
] | 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... | C5H10 |
[
{
"content": "Find the limit of the sequence \\( \\left(\\frac{n!}{n^n}\\right)^{\\frac{1}{n}} \\) as \\( n \\to \\infty \\) without using L'Hopital's rule.",
"role": "user"
}
] | \dfrac{1}{e} |
[
{
"content": "In triangle \\( ABC \\) with orthocenter \\( H \\), \\( AB = 13 \\), \\( BC = 21 \\), and \\( CA = 20 \\). The perpendicular bisector of \\( CH \\) meets \\( BC \\) at \\( P \\). Lines \\( PH \\) and \\( AB \\) meet at \\( Q \\). The line through \\( Q \\) perpendicular to \\( PQ \\) meets \\( AH ... | 173 |
[
{
"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... | -6.70 m/s^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... | 5.9479 s-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... | 12.8 g |
[
{
"content": "Determine whether there exists an uncountable subset of \\(\\mathbb{R}\\) with measure zero that can be constructed as the limit of an increasing nested sequence of uncountable sets with measure zero.",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate the limit: $$\\lim_{n\\to \\infty} \\frac{1}{n^{2020}} \\sum_{k=1}^n k^{2019}$$",
"role": "user"
}
] | \dfrac{1}{2020} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $25.91 |
[
{
"content": "Can a set consisting solely of isolated points be uncountable in a second countable space?",
"role": "user"
}
] | No |
[
{
"content": "Let \\(a_n\\) be a sequence of real numbers such that \\(\\lim_{n\\to +\\infty}a_n=A\\). Determine whether the following limit is true:\n\\[\\lim_{n\\to +\\infty}\\left(1+\\frac{a_n}{n}\\right)^n=e^A.\\]",
"role": "user"
}
] | e^A |
[
{
"content": "Determine the order of the pole of the function \\( f(z) = \\frac{1}{\\sin(z^2)} \\) at \\( z = 0 \\).",
"role": "user"
}
] | 2 |
[
{
"content": "Determine whether there exist two distinct prime numbers \\( p \\) and \\( q \\) such that the equation \\( 2^a p^b = q^c - 1 \\) has no solution in natural numbers \\( a, b, \\) and \\( c \\).",
"role": "user"
}
] | Yes |
[
{
"content": "Compute \\( \\mathbb{E}\\left(\\int_{0}^{T} |B_{t}|^{2} dt\\right) \\) where \\((B_t)_{t\\geq0}\\) is a standard Brownian motion and \\(T\\) is the first time the Brownian motion hits \\(1\\) or \\(-1\\).",
"role": "user"
}
] | \dfrac{1}{6} |
[
{
"content": "Evaluate the integral \\( \\int_{0}^{1}\\sum_{n=1}^{\\infty}\\frac{9}{\\pi n^{2}}\\cos^{3}\\left(\\frac{2 \\pi n x}{3}\\right) d x \\).",
"role": "user"
}
] | \dfrac{\pi}{4} |
[
{
"content": "Evaluate the limit: $$\\lim_{(x,y)\\to (0,0)} \\frac{x^2-2|x|-2|y|}{|x|+|y|}$$",
"role": "user"
}
] | -2 |
[
{
"content": "Given a field $k$, suppose every polynomial in $k[x]$ is a factor of the characteristic polynomial of some matrix over $k$. Does it follow that $k$ is algebraically closed? 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... | 6.6e7 m/s |
[
{
"content": "Evaluate the limit: \\[ \\lim_{h \\to 0} \\int_{x}^{x+h} \\ln(t) \\, dt. \\]",
"role": "user"
}
] | 0 |
[
{
"content": "Determine the range of the real number \\( m \\) such that for any real number \\( x \\), at least one of the functions \\( f(x) = 2mx^2 - 2mx - 8x + 9 \\) or \\( g(x) = mx - m \\) is positive.",
"role": "user"
}
] | (0, 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... | 1/r |
[
{
"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... | -18890 |
[
{
"content": "Calculate the arc length of the curve defined by the parametric equations \\( \\sigma(t) = \\cos(t) \\mathbf{i} + \\sin(t) \\mathbf{j} \\) and \\( r(t) = e^{t} \\) from \\( t=0 \\) to \\( t=\\infty \\).",
"role": "user"
}
] | \infty |
[
{
"content": "In decimal (base-10), a six-digit number is called a \"special square number\" if it meets the following conditions: (i) None of its digits are zero; (ii) It is a perfect square; (iii) The first two digits, the middle two digits, and the last two digits are all perfect squares (when considered as ... | B |
[
{
"content": "Evaluate the integral \\( \\int_0^\\pi \\sin x \\cdot P_n(\\cos x ) \\, dx \\) where \\( P_n \\) is the \\( n \\)-th Legendre polynomial, and determine if it equals zero.",
"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... | $2,231.25 |
[
{
"content": "Let $R$ be a principal ideal domain (PID) and $M$ a free $R$-module of finite rank $a$. If $N$ is a free $R$-submodule of $M$ with $\\operatorname{rank}(N) = a$, is it true that $N = M$? Justify your answer.",
"role": "user"
}
] | No |
[
{
"content": "Evaluate the limit \\( \\lim_{z\\to0} \\left(\\frac{1}{e^{z}-1}-\\frac{1}{z}\\right) \\) using the Taylor series expansion of \\( e^z \\).",
"role": "user"
}
] | -\dfrac{1}{2} |
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