prompt list | ground_truth string |
|---|---|
[
{
"content": "Determine which of the following sets is recursively enumerable: \\( L_1 = \\{ M \\mid \\text{Turing Machine } M \\text{ terminates for at least 637 inputs} \\} \\) and \\( L_2 = \\{ M \\mid \\text{Turing Machine } M \\text{ terminates for at most 636 inputs} \\} \\).",
"role": "user"
}
] | L_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... | 2CO(g) + O2(g) -> 2CO2(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... | 1Cal//gm^@C |
[
{
"content": "Let $g(x)$ be a continuous function from $[-1,1]$ to $[0,1]$ with a fixed point $p$. Suppose $f(x)$ is a continuous function from $[-1,1]$ to $\\mathbb{R}$ satisfying the following properties:\n\n1. $f(x) = g(x) \\cdot f(g(x))$\n2. $\\lim\\limits_{x\\to1^{-}}\\dfrac{f(x)}{\\sqrt{1-x}}$ exists and ... | 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... | 2470614281.27 |
[
{
"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... | 1.8364e+3 A |
[
{
"content": "Given that vectors \\( a \\) and \\( b \\) are non-zero, and \\( a + 3b \\) is perpendicular to \\( 7a - 5b \\), and \\( a - 4b \\) is perpendicular to \\( 7a - 2b \\), find the angle between vectors \\( a \\) and \\( b \\).",
"role": "user"
}
] | \dfrac{\pi}{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... | 3150.66 pounds |
[
{
"content": "Evaluate the limit using Taylor series expansion: $$\\lim_{x \\to \\frac{\\pi}{4}}{\\left(\\frac{\\sin x + \\cos x}{\\sqrt{2}}\\right)}^{\\cot\\left(\\frac{\\pi}{4}-x\\right)}$$",
"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... | 2.5e-3 atm |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 11.1 |
[
{
"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... | 1.40e14 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... | 223.8 V |
[
{
"content": "Let \\( f: \\overline{\\mathbb{D}} \\to \\mathbb{C} \\) be a continuous function that is holomorphic on \\( \\mathbb{D} \\). If \\( f(\\mathbb{T}) \\subset \\mathbb{R} \\), is \\( f \\) constant? Justify your answer.",
"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... | 270.4MW |
[
{
"content": "Find all pairs of prime numbers \\(p\\) and \\(q\\) such that \\(p^3 + 19q^3 + 2018\\) is a cube of a prime number.",
"role": "user"
}
] | (3, 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... | $43.70 |
[
{
"content": "Express $\\sqrt[3]{7+5\\sqrt{2}}$ in the form $x + y\\sqrt{2}$, where $x$ and $y$ are rational numbers.",
"role": "user"
}
] | 1 + \sqrt{2} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | Neutral |
[
{
"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... | $475.25 |
[
{
"content": "Find the $n$-th derivative of the function $f(x) = x^{n-1} \\log x$. Choose the correct option: \\(\\text{A) } \\frac{(n-1)!}{x}\\), \\(\\text{B) } \\frac{n}{x}\\), \\(\\text{C) } (-1)^{n-1}\\frac{(n-1)!}{x}\\), \\(\\text{D) } \\frac{1}{x}\\).",
"role": "user"
}
] | A |
[
{
"content": "Find the range of values for $c$ such that the ellipses given by \\( \\frac{u^2}{(c+\\frac{1}{c})^2}+\\frac{v^2}{(c-\\frac{1}{c})^2}=1 \\) cover the entire plane, except for a specific segment on the $u$-axis.",
"role": "user"
}
] | (0, 1) \cup (1, \infty) |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | positive |
[
{
"content": "Provide an example of a differential form on \\(\\mathbb{R}^2\\) that is neither exact nor closed.",
"role": "user"
}
] | x \, dy |
[
{
"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... | 1.39 x 10-34 meter |
[
{
"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... | 1.8400 g/cm^3 |
[
{
"content": "Find the sum of the series: $$ \\sum_{n=1}^\\infty \\frac{\\sqrt{n-\\sqrt{n}}}{n^{1/n + 3/2}} $$",
"role": "user"
}
] | \infty |
[
{
"content": "Find the remainder when the expression \\( S = \\sum_{i=0}^{99} (n+i)^6 + 2^{2^{2558}} + 1 \\) is divided by 100, given that \\( 2^{2^{2558}} \\equiv 16 \\pmod{100} \\).",
"role": "user"
}
] | 67 |
[
{
"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... | Hg = +1, Cl = -1 |
[
{
"content": "Let \\( \\{X_n, n=1,2,\\ldots\\} \\) and \\( \\{Y_n, n=1,2,\\ldots\\} \\) be two sequences of random variables, where elements from \\( X_n \\) are independent of elements from \\( Y_n \\). Suppose \\( X_n \\) converges in probability to \\( X \\), and \\( Y_n \\) converges in probability to \\( Y... | Yes |
[
{
"content": "There are 4 representatives from each of 4 companies at a convention. At the start of the convention, every person shakes hands once with every person except the other representatives from their company. How many handshakes are there?",
"role": "user"
}
] | 96 |
[
{
"content": "In \"Dragon Poker,\" a deck has four suits. An ace is worth 1 point, a jack is worth 2 points, a 2 is worth \\(-2^{2}\\) points, a 3 is worth \\(-2^{3}\\) points, and so on, up to a 10, which is worth \\(-2^{10} = 1024\\) points. Kings and queens are absent. You can choose any number of cards from... | 0 |
[
{
"content": "Calculate the area of the region defined by the inequality \\( \\sin^2(\\pi x) + \\sin^2(\\pi y) \\le 1 \\) for \\( x, y \\in [-1, 1] \\).",
"role": "user"
}
] | 2 |
[
{
"content": "Identify the topological space that is homeomorphic to the partition of \\(\\mathbb{R}^2 - \\{0\\}\\) into lines of the form \\(y = mx\\) for all real \\(m\\), including vertical lines.",
"role": "user"
}
] | 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... | 240199 |
[
{
"content": "Evaluate the limit: $$\\lim_{n\\rightarrow\\infty}\\frac{\\prod_{k=1}^{n}\\left(1-\\frac{1}{2k}\\right)^2}{\\prod_{k=1}^{2n}\\left(1-\\frac{1}{2k}\\right)}$$",
"role": "user"
}
] | 0 |
[
{
"content": "Let \\((X_n)_{n \\geq 1}\\) be a sequence of i.i.d. real random variables with \\(\\text{Var}(X_1) = 1\\) and \\(E(X_1) = 0\\). Denote by \\(S_n\\) the partial sum \\(S_n = X_1 + X_2 + \\cdots + X_n\\). Prove that \\(\\lim_{n \\to \\infty} E(|S_n|) = \\infty\\).",
"role": "user"
}
] | \infty |
[
{
"content": "Find the maximum value of $k$ for which the sum \\( \\sum_{i=0}^k \\binom {10}{i} \\binom{15}{k-i} +\\sum_{i=0}^{k+1} \\binom {12}{i} \\binom {13}{k+1-i} \\) exists.",
"role": "user"
}
] | 25 |
[
{
"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... | 33.0 g |
[
{
"content": "In a math exam with $n$ participants, where $n>2$ and scores follow a normal distribution with mean $\\mu_X$ and variance $\\sigma^2$, the scores are denoted by $X_1, X_2, \\ldots, X_n$. The sample mean is given by $\\overline{X} = \\frac{1}{n}\\sum_{i=1}^n X_i$. Calculate the covariance between $... | 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... | 1.764 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... | 32.5KW |
[
{
"content": "Evaluate the integral \\( \\int_{-\\frac{\\pi}{2}}^{\\frac{\\pi}{2}} \\arctan(\\sin x)\\,\\mathrm dx \\) using symmetry properties of the integrand.",
"role": "user"
}
] | 0 |
[
{
"content": "Determine the cardinality of the set \\( B = \\{f : \\mathbb{R} \\to \\mathbb{R} \\mid \\forall x \\in \\mathbb{R}, f(\\lfloor x \\rfloor) = f(x)\\} \\), where \\(\\lfloor x \\rfloor\\) denotes the floor of \\(x\\). Is the cardinality of \\( B \\) equal to \\( \\aleph_0 \\)? If not, find an inject... | 2^{\aleph_0} |
[
{
"content": "Calculate the line integral \\( I = \\int \\limits_\\Gamma x^2 \\, dy - 2xy \\, dx \\), where \\( \\Gamma \\) is the part of the curve \\( x^4 - y^4 = 6x^2y \\) from point \\( A(-4\\sqrt{2}, 4) \\) to point \\( B(0,0) \\).",
"role": "user"
}
] | \dfrac{44}{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... | 314.1 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... | 47.16 atm |
[
{
"content": "The circle \\(x^2 + y^2 +2gx +2fy +c=0\\) bisects the circumference of the circle \\(x^2 + y^2 +2ax +2by +d=0\\). Which of the following conditions is true?\n\nA) \\(2a(g-a) +2b(f-b)=c-d\\)\n\nB) \\(2g(g-a) +2f(f-b)=d-c\\)\n\nC) \\(2a(g+a) +2b(f+b)=c+d\\)\n\nD) \\(2g(g+a) +2f(f+b)=c+d\\)",
"ro... | 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... | -4.55 kJ/mol |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $364.3 |
[
{
"content": "Evaluate the integral \\( \\int_0^{\\frac{\\pi}{2}}\\int_0^{\\frac{\\pi}{2}}\\left(\\frac{\\sin\\phi}{\\sin\\theta}\\right)^{\\frac{1}{2}}\\,d\\theta\\,d\\phi \\).",
"role": "user"
}
] | \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... | $2,350.93 |
[
{
"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... | $140,640 |
[
{
"content": "A criminal must pass 10 lie detector tests successfully to escape from jail. The probabilities of each lie detector detecting a lie are as follows: 0, \\( \\frac{1}{2} \\), \\( \\frac{2}{3} \\), \\( \\frac{3}{4} \\), \\( \\frac{4}{5} \\), \\( \\frac{5}{6} \\), \\( \\frac{6}{7} \\), \\( \\frac{7}{8... | 9864100 |
[
{
"content": "Find all positive integer solutions $(a, b, c)$ such that $a! + b! = c!$.",
"role": "user"
}
] | (1, 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... | $48,102,618.92 |
[
{
"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... | 570 cm/s |
[
{
"content": "Find all natural number solutions to the equation: \\[ \\frac{x-1}{1+(x-1) y}+\\frac{y-1}{2 y-1}=\\frac{x}{x+1}. \\]",
"role": "user"
}
] | (2, 2) |
[
{
"content": "Given a symmetric Gaussian multivariate random variable $(X,Y)$ with covariance matrix \\( \\begin{bmatrix} 4 & 2 \\\\ 2 & 2 \\end{bmatrix} \\), find the probability \\( P(X > 2Y) \\).",
"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... | 5.4% |
[
{
"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... | 44,145sqrt2 J |
[
{
"content": "Calculate \\( \\lim_{n\\to \\infty} \\int_0^1 \\frac{nx\\sin(x)}{n^3x^2 +1}\\,dx \\).",
"role": "user"
}
] | 0 |
[
{
"content": "In triangle $ABC$, where $AB=8$, $BC=7$, and $CA=6$, point $E$ is located on side $BC$ such that $\\angle BAE=3\\angle EAC$. Calculate the value of $\\frac{4(AE)^2}{5}$.",
"role": "user"
}
] | 27 |
[
{
"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... | 4.67 atm |
[
{
"content": "Let $f:[a,b] \\to \\mathbb{R}$ be a Riemann integrable function that is not constant on any open subinterval. Suppose that for any sequence of partitions $P_n$ of $[a,b]$, the lower Darboux sums satisfy $\\lim_{n \\to \\infty}L(f,P_n)=\\int_a^b f(x)dx$. Does it follow that $\\lambda(P_n) \\to 0$? ... | Yes |
[
{
"content": "Consider two circles $k_1$ and $k_2$ centered at points $A$ and $B$ with radii $r_1$ and $r_2$, respectively. A third circle $k$ is tangent to both $k_1$ and $k_2$ from the inside, with radius $r_3$. Determine the limit of the ratio $r_3 / r_1$ as the radius $r_1$ approaches 0, while keeping the c... | \dfrac{1}{2} |
[
{
"content": "Given that the numbers \\( x, y, z \\) form an arithmetic progression with a common difference \\( \\alpha = \\arccos \\frac{2}{3} \\), and the numbers \\( \\frac{1}{\\sin x}, \\frac{6}{\\sin y}, \\frac{1}{\\sin z} \\) also form an arithmetic progression in the given order, find the value of \\( \... | \dfrac{5}{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 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... | 0.015 mV |
[
{
"content": "Find $q(x)$ if the graph of $\\frac{4x-x^3}{q(x)}$ has a hole at $x=-2$, a vertical asymptote at $x=1$, no horizontal asymptote, and $q(3) = -30$.",
"role": "user"
}
] | -3(x+2)(x-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... | F-N=N-F |
[
{
"content": "Find the length of the curve defined by the function \\( f(t) = \\frac{e^t + e^{-t}}{2} \\) over the interval \\([0, 2\\pi]\\).",
"role": "user"
}
] | \sinh(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... | 1. Investment in shares of IBM Account: Debit $100,000
Bank Account: Credit $100,000
Being shares purchased
2. Investment in shares of Bell Labs Account: Debit $125,000
Bank Account: Credit $125,000
Being shares purchased
3. Bank Account: Debit $2,000
Dividend Income Account: Credit $2,000
Being dividend received |
[
{
"content": "Determine the cardinality of the set \\( \\mathbb{F} \\) of all entire functions \\( f \\) such that \\( f(x) = |x^k| \\) for all \\( x \\in (-1, 1) \\), where \\( k \\) is an odd integer with \\( k \\ge 1 \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Evaluate the limit: $$\\lim _{x\\to \\infty }\\left[\\left(\\frac{x+2}{x+3}\\right)^b-\\cos\\left(\\frac{2}{x}\\right)\\right]^x$$",
"role": "user"
}
] | 0 |
[
{
"content": "Find the coefficient $a_{-2}$ in the Laurent series expansion of the function $$f(z)=\\frac{1}{2-e^z}$$ around the singularity $z_0=\\log(2)$. ",
"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... | 6.24 mH |
[
{
"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... | 0.817 |
[
{
"content": "Find all prime numbers $p$ such that $8p^4 - 3003$ is also a prime number.",
"role": "user"
}
] | 5 |
[
{
"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... | (111) |
[
{
"content": "Determine the remainder when \\(15^{100!}\\) is divided by 5000, without using the fact that \\(15^r \\equiv 625 \\pmod{5000}\\) for even \\(r \\geq 4\\).",
"role": "user"
}
] | 625 |
[
{
"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... | 11024.08 |
[
{
"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... | 18000 |
[
{
"content": "Compute \\( \\lim\\limits_{n\\to\\infty}\\int\\limits_0^{\\infty} \\frac{1}{x^{3/2}}\\sin\\left(\\frac{x}{n}\\right) 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... | 9.6e-9 N |
[
{
"content": "Let $g$ be a continuous function on $[0, \\infty)$ such that $\\lim\\limits_{x \\to \\infty} g(x) = L$, where $L$ is a finite number. Suppose there exists a $d \\geq 0$ such that $g(d) < L$. Does $g$ necessarily have a global minimum on $[0, \\infty)$?",
"role": "user"
}
] | Yes |
[
{
"content": "In the diagram, square $ABCD$ has sides of length $4,$ and $\\triangle ABE$ is equilateral. Line segments $BE$ and $AC$ intersect at $P.$ Point $Q$ is on $BC$ so that $PQ$ is perpendicular to $BC$ and $PQ=x.$ [asy]\npair A, B, C, D, E, P, Q;\nA=(0,0);\nB=(4,0);\nC=(4,-4);\nD=(0,-4);\nE=(2,-3.464)... | 105^\circ |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 64 m/s |
[
{
"content": "Evaluate the limit of the function \\( f(x,y) = \\frac{(x-1)^2\\ln(x)}{(x-1)^2 + y^2} \\) as \\((x,y) \\to (1,0)\\).",
"role": "user"
}
] | 0 |
[
{
"content": "Let $X$ be a random variable uniformly distributed in the unit ball. Consider the random variable $Y = X + (0, \\rho, 0)$, where $\\rho \\neq 0$. Is the probability density function (pdf) of $Y$ still constant inside the translated region?",
"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... | $1,708.59 |
[
{
"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 g |
[
{
"content": "Determine the radius of convergence for the series \\( F(x)= \\sum_{n=0}^\\infty \\frac{n!x^n}{n^n} \\).",
"role": "user"
}
] | e |
[
{
"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... | 8.78% |
[
{
"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... | $\frac{4.9(a^2-b^2)+6(b-a)}{b-a}$ |
[
{
"content": "Compute the limit of the determinant of the following \\( n \\times n \\) matrix as \\( n \\to \\infty \\):\n\\[\n\\lim_{n \\rightarrow \\infty} \\begin{vmatrix} \n1+x & -x & 0 & 0 & \\cdots & 0 & 0 \\\\\n-\\frac{1}{2} & 1+\\frac{x}{2} & -x & 0 & \\cdots & 0 & 0 \\\\\n0 & -\\frac{1}{3} & 1+\\frac{... | e^x |
[
{
"content": "Find the maximum value of $x^2 + y^2 + z^2$ given that $x + y + z = -1$ and $xyz = 1$.",
"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... | _{80}^{198} Hg + _0^1 n\to _{79}^{198} Au + _1^1 H |
[
{
"content": "Given a 6th degree polynomial \\( f(x) \\) such that \\( f(0) = 0 \\), \\( f(1) = 1 \\), \\( f(2) = 2 \\), \\( f(3) = 3 \\), \\( f(4) = 4 \\), \\( f(5) = 5 \\), and \\( f(6) = 7 \\), find the value of \\( f(8) \\).",
"role": "user"
}
] | 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... | 0.00063096 |
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