prompt list | ground_truth string |
|---|---|
[
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to 0}\\left(\\frac{\\sin(x)}{x}\\right)^{1/x^2} \\] and show that it equals \\( e^{-1/6} \\).",
"role": "user"
}
] | e^{-1/6} |
[
{
"content": "Determine all values of \\( a \\) such that the system of equations \\[ \\begin{cases} 2^{b x}+(a+1) b y^{2}=a^{2} \\\\ (a-1) x^{3}+y^{3}=1 \\end{cases} \\] has at least one solution for any real number \\( b \\).",
"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... | 1.145 |
[
{
"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... | $10.83 |
[
{
"content": "A system consists of 7 components with independent, identically distributed lifetimes. The system will operate until any one of the components fails. If the lifetime \\( X \\) of each component has a probability density function given by:\n\n\\[f(x) = \\begin{cases} \\frac{3}{x^4}, & \\text{for } ... | \dfrac{21}{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... | -201 ft/s |
[
{
"content": "Determine the size of the set of all Turing machines.",
"role": "user"
}
] | \aleph_0 |
[
{
"content": "Find all quadratic polynomials \\( f(x, y) \\) such that:\n1. \\( f(1,2)=2 \\)\n2. \\( y f(x, f(x, y))=x \\cdot f(f(x, y), y)=(f(x, y))^{2} \\)",
"role": "user"
}
] | xy |
[
{
"content": "Evaluate $\\left\\lceil\\left(\\frac{7}{4}\\right)^2\\right\\rceil^2$",
"role": "user"
}
] | 16 |
[
{
"content": "Determine all values of $a$ such that the range of the function \\( y=\\frac{\\sqrt{a}-2\\cos x+1}{\\sin^2x+a+2\\sqrt{a}+1} \\) includes the interval \\([2, 3]\\).",
"role": "user"
}
] | 0 |
[
{
"content": "Let $A$ be a non-zero $4 \\times 4$ complex matrix such that $A^2 = 0$. What is the largest possible rank of $A$?",
"role": "user"
}
] | 2 |
[
{
"content": "Let $g: \\mathbb{R^{+}} \\rightarrow \\mathbb{R}$ be a differentiable function such that \\(2g(x) = g(xy) + g\\left(\\frac{x}{y}\\right)\\) for all \\(x, y \\in \\mathbb{R^+}\\), with \\(g(1) = 0\\) and \\(g'(1) = 1\\). Find the function \\(g(x)\\).",
"role": "user"
}
] | \ln x |
[
{
"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... | 2309.07 kN |
[
{
"content": "Let \\( f \\) be a differentiable function on \\( \\mathbb{R} \\) satisfying the functional equation \\( f(x+y) = f(x)f(y) \\). Given that \\( f(1) = 2 \\), find the function \\( f(x) \\).",
"role": "user"
}
] | 2^x |
[
{
"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... | Zn(s) + 2HCl(aq) rarr ZnCl_2(aq) + H_2(g) |
[
{
"content": "Evaluate the integral \\( \\int _{-2}^2\\left(3x^2-2\\right)\\cos \\left(x^3-2x\\right)\\sin \\left(x^3\\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... | 1850 s |
[
{
"content": "Find a polynomial \\( f(x) \\in \\mathbb{Q}[x] \\) of degree less than 4 such that \\( f(x) \\equiv 1 \\pmod{(x-1)^2} \\) and \\( f(x) \\equiv x \\pmod{x^2} \\).",
"role": "user"
}
] | -x^3 + x^2 + x |
[
{
"content": "For what value of $x$ will $\\frac{2x-1}{2x+2}$ and $\\frac{x-3}{x-1}$ be equal?",
"role": "user"
}
] | -7 |
[
{
"content": "Given a natural number $n$, determine whether there exists a natural number $x$ such that for each $i$ in the set $\\{1, 2, \\ldots, n\\}$, there exist distinct natural numbers $y_i$ and $z_i$ satisfying the equation $x^2 + y_i^2 = z_i^2$. Provide a justification for your answer.",
"role": "us... | Yes |
[
{
"content": "Find the condition on the parameter \\(p\\) such that the inequality \\(-9 < \\frac{3x^2+px-6}{x^2-x+1} < 6\\) holds for all real values of \\(x\\).",
"role": "user"
}
] | (-3, 6) |
[
{
"content": "Determine the number of solutions to the equation \\( x^{10} = \\begin{bmatrix}1 & 0 \\\\ 0 & 1 \\end{bmatrix} \\) in the space of 2x2 real matrices, \\( M_{2}(\\mathbb{R}) \\).",
"role": "user"
}
] | \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... | P(CO2) = 2.78 atm, P(CO) = 2.06 atm |
[
{
"content": "Determine whether the series \\( \\sum_{k=0}^\\infty e^{-\\sqrt{k}} (-1)^k \\) converges faster than \\( e^{-\\sqrt{k}} \\) by evaluating the limit:\n\\[ \\lim_{N\\to\\infty} e^{\\sqrt{N}} \\sum_{k=N}^\\infty e^{-\\sqrt{k}} (-1)^k = 0. \\]",
"role": "user"
}
] | 0 |
[
{
"content": "Determine whether the function \\( \\mu(x,y) = e^{\\int h(xy) d(xy) } \\) can be expressed as a function of the product \\( xy \\). Specifically, can we write \\( \\mu(x,y) = \\mu(xy) \\)?",
"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... | 10.10% |
[
{
"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... | 8464kg |
[
{
"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... | 2.64 cm |
[
{
"content": "Find the value of \\(a\\) such that \\(\\lim_{x\\to\\infty}\\left(\\frac{x+a}{x-a}\\right)^x = e\\).",
"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... | 7453.03 m/s |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to 0} \\int_{x}^{\\sin x} \\frac{1}{t^3(1+t^{1/2})} \\, dt. \\]",
"role": "user"
}
] | -\dfrac{1}{6} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $40,352.68 |
[
{
"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... | 5.2 m/s |
[
{
"content": "Determine whether the number $2014! + 1$ is prime or composite.",
"role": "user"
}
] | Composite |
[
{
"content": "Given that $y(x)$ is a solution of the differential equation \\[ x^2 y^{\\prime \\prime}+x y^{\\prime}-4 y=x^2 \\] on the interval $(0, \\infty)$ such that $\\lim _{x \\rightarrow 0^{+}} y(x)$ exists and $y(1)=1$, find the value of $y^{\\prime}(1)$. \\[ y(x) = c_1 x^2 + c_2 \\frac{1}{x^2} + \\frac... | \dfrac{9}{4} |
[
{
"content": "Let $\\mathbf{a},$ $\\mathbf{b},$ $\\mathbf{c},$ $\\mathbf{d}$ be four distinct unit vectors in space such that\n\\[\\mathbf{a} \\cdot \\mathbf{b} = \\mathbf{a} \\cdot \\mathbf{c} = \\mathbf{b} \\cdot \\mathbf{c} =\\mathbf{b} \\cdot \\mathbf{d} = \\mathbf{c} \\cdot \\mathbf{d} = -\\frac{1}{11}.\\... | -\frac{53}{55} |
[
{
"content": "Calculate the area inside the loop defined by the equation \\((x+y)^3 = axy\\) using a double integral, where the parameter \\(a\\) is positive.",
"role": "user"
}
] | \dfrac{a^2}{60} |
[
{
"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... | 1207.68 |
[
{
"content": "Let $X$ be a compact metric space and let $f:X \\to X$ be a continuous map. If $f(X)$ is homeomorphic to $X$, does it imply that $f$ is injective? 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... | 4.47 |
[
{
"content": "Find all continuous functions \\( f \\) on the interval \\([0,1]\\) such that for every \\( x \\in (0,1) \\), the following equation holds:\n\\[ \\int_0^x f(t) \\, dt = \\int_x^1 f(t) \\, dt. \\]",
"role": "user"
}
] | 0 |
[
{
"content": "Given a \\(1 \\times n\\) rectangle divided into \\(n\\) unit squares, each square is colored either Red, Green, or Blue. Let \\(f(n)\\) denote the number of colorings where the Red color appears an even number of times. Calculate the value of \\(\\frac{f(9)}{f(3)}\\).",
"role": "user"
}
] | 703 |
[
{
"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... | 5.496% |
[
{
"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... | 110.48 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... | 4.801 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... | 38.0% |
[
{
"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... | 1.47% |
[
{
"content": "The function $f$ is differentiable and satisfies the equation \\( \\int_0^x (6f(t) + 4t) \\, dt = \\sin(x) \\). Find the value of \\( f'\\left(\\frac{\\pi}{6}\\right) \\).",
"role": "user"
}
] | -\dfrac{3}{4} |
[
{
"content": "Determine the values of $a$ for which the integral \\( \\int_0^{1/e}\\frac{x^a}{\\ln{x}} \\,dx \\) converges.",
"role": "user"
}
] | a > -1 |
[
{
"content": "A function \\( f: \\mathbb{N^+} \\rightarrow \\mathbb{N^+} \\), defined on the set of positive integers \\( \\mathbb{N^+} \\), satisfies the following properties:\n\\[f(n)=\\begin{cases} f(n/2) & \\text{if } n \\text{ is even}\\\\ f(n+5) & \\text{if } n \\text{ is odd} \\end{cases}\\]\nLet \\( R=\... | 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... | 13.3 kgm^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... | 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... | 1674.42 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... | 7.2e33 kg m^2 s^-1 |
[
{
"content": "A particular number written in base 3 requires three digits (${\\_ \\_ \\_}_3$). When the number is written in base 3 and 4, the digits are the reverse of each other. What is this number expressed in base 10?",
"role": "user"
}
] | 22 |
[
{
"content": "Calculate the limit: $$ \\lim_{h\\rightarrow0}\\frac{f(a+3h)-f(a-2h)}{f(a-5h)-f(a-h)} $$ where $f(x)$ is a differentiable function and $f'(a) \\ne 0$ for some $a$. Provide your answer in terms of $f'(a)$. ",
"role": "user"
}
] | -\dfrac{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... | 0.0114 rad/s |
[
{
"content": "What is the maximum number of zeros for the Laplace transform of a function that is the difference of two sums, each containing $k$ exponential terms with positive exponents?",
"role": "user"
}
] | 2k - 1 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to \\infty} \\left(\\sqrt[3]{(x-2)^2(x-1)} - x\\right) \\]",
"role": "user"
}
] | -\dfrac{5}{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... | 20.12% |
[
{
"content": "Let \\(g : [p,q] \\to \\mathbf{R}\\) be a differentiable function on \\([p,q]\\). Given that \\(g'(p) > n > g'(q)\\) for some \\(n \\in \\mathbf{R}\\), determine whether there exists a point \\(d \\in (p,q)\\) such that \\(g'(d) = n\\).",
"role": "user"
}
] | Yes |
[
{
"content": "Determine the least possible radius of a graph with 100 vertices and a maximum degree of 4.",
"role": "user"
}
] | 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... | 1.33 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... | 7400 |
[
{
"content": "Two perpendicular normals to a variable circle are tangent to a fixed circle \\( C_1 \\) with radius 2. The locus of the center of the variable circle is the curve \\( C_2 \\). Find the product of the maximum and minimum distances between the curves \\( C_1 \\) and \\( C_2 \\).",
"role": "user... | 4 |
[
{
"content": "Determine if the function \\(f(x,y) = \\begin{cases} \\frac{x^3y^3}{x^2+y^2} & x^2+y^2 \\neq 0 \\\\ 0 & x = y = 0 \\end{cases}\\) is differentiable at \\((0,0)\\).",
"role": "user"
}
] | Yes |
[
{
"content": "Let $f(x) = \\sqrt{x}$ and $g(x) = x^2.$ Find $f(g(f(g(f(8))))).$",
"role": "user"
}
] | 2\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... | 332.22 Newton |
[
{
"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... | $2,237.39 |
[
{
"content": "Find the smallest positive integer \\( n \\) such that the symmetric group \\( S_n \\) has a subgroup of order \\( p^2 \\), where \\( p \\) is a prime number.",
"role": "user"
}
] | 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... | 8.0 V |
[
{
"content": "Evaluate the integral \\( \\int_{1}^{\\infty}\\frac{e^{-x}-e^{x}}{1-e^{2x}}dx \\).",
"role": "user"
}
] | \dfrac{1}{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... | 11,666.67 kg |
[
{
"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... | 105 utils |
[
{
"content": "Show that the function \\( f(x,y) = \\frac{x^3\\sin(x+y) - y^4\\ln(x^2+y^2)}{x^2+y^2} \\) is continuous at the point \\((0,0)\\), given that \\( f(0,0) = 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... | 4676.537 N/C |
[
{
"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... | 0.0217 rad/sec |
[
{
"content": "How many distinct, non-equilateral triangles with a perimeter of 60 units have integer side lengths $a$, $b$, and $c$ such that $a$, $b$, $c$ is an arithmetic sequence?",
"role": "user"
}
] | 9 |
[
{
"content": "In the context of knot theory, consider a free product with amalgamation $G \\ast_K H = \\mathbb{Z} \\oplus \\mathbb{Z}$, where $K = \\mathbb{Z} \\oplus \\mathbb{Z}$. Given that $G$ is a free group, determine if $H$ must also be a free group.",
"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... | 3896501.4 m |
[
{
"content": "For a positive integer $n$ and nonzero digits $a$, $b$, and $c$, let $A_n$ be the $n$-digit integer each of whose digits is equal to $a$; let $B_n$ be the $n$-digit integer each of whose digits is equal to $b$, and let $C_n$ be the $2n$-digit (not $n$-digit) integer each of whose digits is equal t... | 18 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n \\to \\infty} \\frac{\\lceil 4^{n+\\frac{\\log \\frac{8}{3}}{\\log 4}} \\rceil}{\\lceil 4^{n+0.707519} \\rceil} \\] Determine if this limit equals 1.",
"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... | 49.52 kN |
[
{
"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... | x = 30, y = 0 |
[
{
"content": "If $X$ is a topological space and $Y \\subseteq X$ is homeomorphic to $X$, with $Y$ being dense in $X$, must it be true that $X = Y$?",
"role": "user"
}
] | No |
[
{
"content": "Let $H$ and $H'$ be two Hopf algebras, and let $\\phi: H \\to H'$ be a bialgebra map. Is $\\phi$ automatically a Hopf algebra map?",
"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... | 11.57 |
[
{
"content": "Suppose $a, b, c > 0$ and $a^2 + b^2 + c^2 = 2abc + 1$. Determine the maximum value of $(a - 2bc)(b - 2ca)(c - 2ab)$. ",
"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... | priming |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 90.0% |
[
{
"content": "Suppose $E\\subset\\mathbb{R}^{n}$ with $m(E) = 0$. Determine if $m(\\{(x,y)\\in\\mathbb{R}^{n}: x-y\\in E \\})=0$, where $m$ denotes the Lebesgue measure.",
"role": "user"
}
] | 0 |
[
{
"content": "The sequences of positive integers $1,a_2, a_3, \\dots$ and $1,b_2, b_3, \\dots$ are an increasing arithmetic sequence and an increasing geometric sequence, respectively. Let $c_n=a_n+b_n$. There is an integer $k$ such that $c_{k-1}=100$ and $c_{k+1}=1000$. Find $c_k$.",
"role": "user"
}
] | 262 |
[
{
"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... | $2,484.51 |
[
{
"content": "Given a sequence \\( \\{x_n\\} \\) of real numbers defined by \\( x_1 = \\frac{2014}{3} \\) and \\( x_{n+1} = x_n \\left(1 + \\frac{1}{x_n - \\lfloor x_n \\rfloor}\\right) \\) for \\( n \\in \\mathbb{N} \\), find \\( \\lim_{n \\to \\infty} \\left( \\sum_{r=1}^n \\frac{1}{x_r} \\right) \\).",
"... | \dfrac{2}{1007} |
[
{
"content": "Determine the possible values of $n$ such that $\\phi(n)$ is even and $n > 6$.",
"role": "user"
}
] | n > 6 |
[
{
"content": "Evaluate the integral \\( \\int_{0}^{\\pi} \\frac{\\ln (1+a\\cos (x))}{\\cos (x)} \\, \\mathrm{d}x \\) using the rule of differentiation under the sign of integration.",
"role": "user"
}
] | \pi \arcsin 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... | $2,754 |
[
{
"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... | 21.312e-22 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... | 3861.62 |
[
{
"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... | 16.46% |
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