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... | 30.6 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... | Debit Credit
DR Direct Materials Inventory $54,280
CR Direct Materials Price Variance $2,714
CR Accounts Payable $51,566 |
[
{
"content": "Given a function $\\mathbf{f}: \\mathbb{R}^n \\rightarrow \\mathbb{R}^n$ with a real, symmetric Jacobian matrix $\\mathbf{J} = \\frac{\\partial \\mathbf{f}}{\\partial \\mathbf{x}} \\in \\mathbb{R}^{n \\times n}$, determine if there exists a scalar function $y: \\mathbb{R}^n \\rightarrow \\mathbb{R... | Yes |
[
{
"content": "Let $A, B, C \\in k[x]^{n \\times n}$ be polynomial square matrices with $\\deg(A), \\deg(C) \\leq d$, where the degree of a matrix is defined as the maximum of the degrees of its entries. Assume all matrices have full rank $n$ over $k(x)$ and that $B$ is unimodular over $k[x]$, meaning $\\deg \\d... | Yes |
[
{
"content": "Let $f(x)$ be a polynomial function with non-negative coefficients such that $f(1)=f'(1)=f''(1)=f'''(1)=1$. Find the minimum value of $f(0)$. Use the Taylor series expansion to express $f(x)$ and determine $f(0)$. What is the minimum value of the constant term $f(0)$?",
"role": "user"
}
] | \dfrac{1}{3} |
[
{
"content": "Consider the infinite series:\n\n$$K=\\sum_{n=1}^{\\infty}\\frac{6^n}{(3^n-2^n)(3^{n+1}-2^{n+1})}$$\n\nCompute the last digit of $(K+6)^{(K+6)!}$.",
"role": "user"
}
] | 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... | $720,000 |
[
{
"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... | $79,705.11 |
[
{
"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... | 486.08 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... | 30.77% |
[
{
"content": "Determine whether the function \\( f(z) = \\frac{1}{e^{1/z} + 1} \\) has a removable singularity at \\( z = \\infty \\).",
"role": "user"
}
] | Yes |
[
{
"content": "Find the Radon-Nikodym derivative \\( \\frac{d\\mu}{dm} \\) where \\( m \\) is the Lebesgue measure on \\([0,1]\\), \\( f(x)=x^2 \\), and \\( \\mu(E)=m(f(E)) \\).",
"role": "user"
}
] | 2x |
[
{
"content": "In the infinite-dimensional Hilbert space $\\mathcal{L}_2(-\\pi, \\pi)$, can a complete orthonormal sequence exist with a countable number of vectors, given that each vector in the space has an uncountably infinite number of components?",
"role": "user"
}
] | Yes |
[
{
"content": "Determine the number of ordered pairs of integers \\((x, y)\\) with \\(1 \\le x < y \\le 100\\) such that \\(\\sin (x^\\circ) + \\sin (y^\\circ)\\) is a rational number.",
"role": "user"
}
] | 1 |
[
{
"content": "Evaluate the limit: $$\\lim_{n \\to \\infty} \\left( n - \\sum_{k = 1}^n e^{\\frac{k}{n^2}} \\right)$$",
"role": "user"
}
] | -\dfrac{1}{2} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 124.68 kJ |
[
{
"content": "Determine the limit of the sequence \\((b_n)_{n \\ge 1}\\) defined by the initial condition \\(b_1 > 0\\) and the recursive relation \\(b_{n+1} = \\frac{(n+1)b_n}{n+2+b_n^2}\\) for all \\(n \\ge 1\\).",
"role": "user"
}
] | 0 |
[
{
"content": "Let $\\mathcal{X}$ be a finite set. For any $k\\leq \\vert \\mathcal{X}\\vert$, define the function class $\\mathcal{G}_k=\\{g: \\vert \\{x\\in\\mathcal{X}: g(x)=0\\}\\vert \\leq k\\}$. Determine the VC dimension of the function class $\\mathcal{G}_k$. Provide your answer as a single integer.",
... | k |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 2 m/s^2 |
[
{
"content": "Suppose $A_{1}, A_{2}, \\ldots, A_{30}$ are thirty sets each containing five elements, and $B_{1}, B_{2}, \\ldots, B_n$ are $n$ sets each containing three elements. Let $\\bigcup_{i=1}^{30} A_{i} = \\bigcup_{j=1}^{n} B_{j} = S$. Assume that each element of $S$ belongs to exactly ten of the $A_{i}$... | 45 |
[
{
"content": "Determine the function $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ given that $f(a + b\\sqrt{2}) = (a + b\\sqrt{2})^{2k+1}$ for all integers $a$ and $b$, where $k$ is a non-zero natural number and $f$ is monotonic.",
"role": "user"
}
] | f(x) = x^{2k+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... | 958.78 |
[
{
"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... | 9.782 keV |
[
{
"content": "Evaluate the limit: $$ \\lim_{n\\to\\infty}\\frac{1}{n}\\sum_{i=\\left\\lceil \\sqrt{n}\\right\\rceil }^{n}\\ln\\left(\\frac{n}{i}\\right) $$.",
"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... | 92.0 g |
[
{
"content": "Compute the sum of the infinite series: \\[ \\sum_{n=1}^{\\infty} n \\cdot \\frac{1}{2^n} \\]",
"role": "user"
}
] | 2 |
[
{
"content": "Find the ratio of the coefficients $a_3$ and $a_4$ in the Laurent series expansion of $f(z)=\\dfrac{1}{2z^2-13z+15}$ in the annulus $\\frac{3}{2}<|z|<5$. \\(\\text{Express your answer as a simplified fraction.}\\)",
"role": "user"
}
] | 5 |
[
{
"content": "Find the ordered pair $(a,b)$ of real numbers such that the cubic polynomials $x^3 + ax^2 + 11x + 6 = 0$ and $x^3 + bx^2 + 14x + 8 = 0$ have two distinct roots in common.",
"role": "user"
}
] | (6,7) |
[
{
"content": "Evaluate the integral \\( \\int_{C_2} \\frac{dz}{z} \\), where \\( C_2 \\) is the circle of radius \\( R \\) centered at the origin, traversed once in the counterclockwise direction, with a branch cut along the positive real axis.",
"role": "user"
}
] | 2\pi i |
[
{
"content": "Let $n \\geq 2$ and $1 \\leq r \\leq n$. Consider the set $S_r = \\{ A \\in M_n(\\mathbb{Z}_2) \\mid \\text{rank}(A) = r \\}$. Compute the sum $\\sum_{X \\in S_r} X$. Provide your answer as a matrix.",
"role": "user"
}
] | 0 |
[
{
"content": "Evaluate the limit: $$\\lim_{n\\to \\infty} \\left(1+\\frac{1}{n!}\\right)^{2n} $$",
"role": "user"
}
] | 1 |
[
{
"content": "In a right triangle $ABC$ with $\triangle CAB = 90^\\circ$, points $M$ and $N$ are located on the hypotenuse $BC$. Given that $BM^2 + CN^2 = MN^2$, $AN = \\frac{\\sqrt{90}}{12}$, and $AM = \\frac{\\sqrt{80}}{12}$, find the measure of $\\angle MAN$. \\(\\text{Express your answer in degrees.}\\)",
... | 45 |
[
{
"content": "Determine the cardinality of the set of all bijections from \\(\\mathbb{N}\\) to \\(\\mathbb{N}\\).",
"role": "user"
}
] | 2^{\aleph_0} |
[
{
"content": "Let $X=\\langle x_1,\\cdots,x_n \\rangle$ be a vector of positive reals and $P=\\langle p_1,\\cdots,p_n \\rangle$ be a permutation of the numbers $\\{1,\\cdots,n\\}$. Define the vector $Y=\\langle y_1,\\cdots,y_n \\rangle$ by the mapping: $$ y_i=\\log\\left(1+\\frac{x_i}{1+\\sum\\limits_{p_j<\\,p_... | Yes |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 2.42 A |
[
{
"content": "For which values of \\( p > 0 \\) does the improper integral \\( \\int_{1}^{\\infty} x^{-p} \\sin{x} \\, dx \\) converge?",
"role": "user"
}
] | p > 0 |
[
{
"content": "Let $\\mathbf{a}$ and $\\mathbf{b}$ be unit vectors such that $\\mathbf{a} + 2 \\mathbf{b}$ and $5 \\mathbf{a} - 4 \\mathbf{b}$ are orthogonal. Find the angle between $\\mathbf{a}$ and $\\mathbf{b},$ in degrees.\n\nNote: A unit vector is a vector of magnitude 1.",
"role": "user"
}
] | 60^\circ |
[
{
"content": "Find a function \\( g: \\mathbb{N} \\to \\mathbb{P}(\\mathbb{N}) \\) such that the set \\( A = \\{ n \\in \\mathbb{N} \\mid n \\notin g(n) \\} \\) is finite.",
"role": "user"
}
] | g(n) = \mathbb{N} |
[
{
"content": "Given the group \\( G = C_{25} \\times C_{45} \\times C_{48} \\times C_{150} \\), where \\( C_n \\) denotes a cyclic group of order \\( n \\), determine the number of subgroups of order 5 in \\( G \\).",
"role": "user"
}
] | 31 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to \\infty} \\left( \\frac {7x+10}{1+7x} \\right)^{x/3} \\]",
"role": "user"
}
] | e^{\frac{3}{7}} |
[
{
"content": "Evaluate the limit: $$ \\lim_{x \\to +\\infty} \\int_x^{2x} \\sin \\left( \\frac{1}{t} \\right) \\, dt. $$",
"role": "user"
}
] | \ln 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... | 0.1116 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 26.2 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... | 128g |
[
{
"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 billion light years |
[
{
"content": "Determine the values of $a$ for which the integral \\( \\int_{B(0,1)} \\frac{dx}{|x|^a} \\) is finite, where \\( B(0,1) \\) is the unit ball in \\( \\mathbb{R}^n \\) and \\( |x| \\) is the Euclidean norm of \\( x \\).",
"role": "user"
}
] | a < 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... | 4480 N |
[
{
"content": "Provide an example of a set with order type $\\omega^\\omega$. What is the structure of such a set?",
"role": "user"
}
] | \omega^\omega |
[
{
"content": "Solve the equation: \\(4\\cdot 25^x - 25\\cdot 4^{x+1} = 9\\cdot 10^x\\).",
"role": "user"
}
] | 2 |
[
{
"content": "Determine the radius of convergence, \\( R \\), for the power series:\n\\[ \\sum_{n=2}^{\\infty} \\frac{(-1)^n z^{2n}}{(2n-1)!} \\]",
"role": "user"
}
] | \infty |
[
{
"content": "How many positive perfect cube factors does $3^65^{10}$ have?",
"role": "user"
}
] | 12 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 2.90 |
[
{
"content": "Let $X_{1}, X_{2}, \\ldots, X_{n}$ be independent and identically distributed random variables with a finite expected value. Determine the value of $\\mathbb{E}[X_{1}|(X_{1}+X_{2}+ \\ldots + X_{n})=x]$. Provide your answer in terms of $x$ and $n$.",
"role": "user"
}
] | \dfrac{x}{n} |
[
{
"content": "Find the constant $a$ such that the function \\( f(x) = \\begin{cases} x\\left(1+2x\\sin\\left(\\frac{1}{x}\\right)\\right) & \\text{if } x \\neq 0 \\\\ a & \\text{if } x = 0 \\end{cases} \\) is differentiable everywhere.",
"role": "user"
}
] | 0 |
[
{
"content": "Determine the number of solutions to the polynomial congruence \\( x^4 - 5x - 6 \\equiv 0 \\pmod{100^{100}} \\).",
"role": "user"
}
] | 8 |
[
{
"content": "Determine the number of solutions to the equation \\(x^y - y^x = x + y\\) where both \\(x\\) and \\(y\\) are prime numbers.",
"role": "user"
}
] | 1 |
[
{
"content": "Find the minimum value of the expression \\(\\left(\\sqrt{x^2+1}-y\\right)^2+\\left(\\sqrt{y^2+1}-x\\right)^2\\) for real numbers \\(x\\) and \\(y\\).",
"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... | 42,342 |
[
{
"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... | 0.012466 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... | -176V |
[
{
"content": "Determine whether the following statement is true or false: An open subset $U$ of a compactly generated space $X$ is compactly generated if each point of $U$ has an open neighborhood in $X$ with closure contained in $U$. Provide your answer as either \"True\" or \"False.\"",
"role": "user"
}... | True |
[
{
"content": "Calculate the sum of the products of the digits for all five-digit numbers composed of the digits from 0 to 7.",
"role": "user"
}
] | 17210368 |
[
{
"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... | \rho g \frac{196}{3} Pa |
[
{
"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... | 10004 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... | 4.81% |
[
{
"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... | 375 sqrt(2) joules |
[
{
"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 workers |
[
{
"content": "Calculate the triple integral \\( I = \\int_{R} (x-y) \\,dV \\) over the region \\( R = \\{(x,y,z) \\in \\mathbb{R^3} \\mid 1 \\leq x^2 + y^2 \\leq 16, 0 \\leq z \\leq y + 4\\} \\). Express your answer to three decimal places.",
"role": "user"
}
] | -200.277 |
[
{
"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... | -$34,000 |
[
{
"content": "Determine the length of the longest simple circuit in a wheel graph $W_n$ for $n > 1$.",
"role": "user"
}
] | 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... | 2.45 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... | 211.8 g |
[
{
"content": "Determine whether the sequence \\( a_n = \\left(\\frac{\\sin(n)+7}{15}\\right)^{n^2} \\) converges. If it converges, find its limit.",
"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... | 250Hz and 249Hz |
[
{
"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... | 381000 J |
[
{
"content": "Find all integers \\( n \\) such that \\( 2\\sigma(n) = 3n \\), where \\( \\sigma(n) \\) is the sum of divisors of \\( n \\).",
"role": "user"
}
] | 2 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 1.25*10^-18 N |
[
{
"content": "Consider the group homomorphism \\( \\phi:M_2(\\mathbb{R}) \\to \\mathbb{R} \\) given by \\( \\phi(A) = \\text{trace}(A) \\). The kernel of \\( \\phi \\) is isomorphic to which of the following groups? \n\n(a) \\( M_2(\\mathbb{R})/ \\{A \\in M_2(\\mathbb{R}):\\phi(A)=0\\} \\) \n(b) \\( \\mathbb{R... | 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... | $\frac{n_l}{n_l+1}||\mu_l - p||_2^2 < \frac{n_k}{n_k - 1}|| \mu_k-p ||_2^2 $ |
[
{
"content": "Determine the norm of the linear functional $L: E \\rightarrow \\mathbb{R}$, where $E = \\{f \\in C([a,b]); f(a) = f(b) = 0\\}$ and $L(f) = \\int_a^b f(x) \\, dx$. The norm is given by $\\|L\\| = \\sup_{f \\in E, f \\neq 0} \\frac{|L(f)|}{\\|f\\|}$, with $\\|f\\| = \\int_a^b |f(x)| \\, dx$. How do... | 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... | $605,000 |
[
{
"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... | -1.16 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 2.103 kJ/K |
[
{
"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... | 3.37e-6 J |
[
{
"content": "Given the axiom \\(\\lnot a \\implies a\\), determine whether \\(a\\) is a valid conclusion or if there is a contradiction in the axiom.",
"role": "user"
}
] | a |
[
{
"content": "Calculate the limit \\( \\lim_{(x,y)\\to(0,0)} \\frac{xy(1-\\cos(x^2+y^2))}{(x^2+y^2)^{\\frac{5}{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... | Y = \frac{a - b(R - r)}{1 - x} |
[
{
"content": "Determine whether it is possible to arrange 1968 line segments on a plane such that each segment has both of its endpoints strictly inside other line segments.",
"role": "user"
}
] | Yes |
[
{
"content": "Given the function \\(f_X(x) = \\begin{cases}4x^3 & 0<x<1\\\\ 0 & \\text{otherwise}\\end{cases}\\) and the transformation \\(y = -\\log(x)\\), find the cumulative distribution function (CDF) \\(F_Y(y)\\).",
"role": "user"
}
] | 1 - e^{-4y} |
[
{
"content": "Consider the sequence $(x_n)_n$ defined by $x_0 = 1$, $x_n = 2x_{n−1} + 1$ if $n$ is odd, and $x_n = 3x_{n−1} + 2$ if $n$ is even. Determine the correct limit of $\\lim_{n\\to \\infty} \\dfrac{\\log(x_n)}{n}$ from the following options: \\[(a) \\text{ does not exist} \\quad (b) \\dfrac{\\log6}{2} ... | B |
[
{
"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... | 3237 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... | 2,003.2 kJ |
[
{
"content": "Evaluate the limit: $$\\lim\\limits_{x \\to 0}\\left(\\frac{e^2}{(1+ 4x )^{\\frac1{2x}}}\\right)^{\\frac1{3x}}$$",
"role": "user"
}
] | e^{\frac{4}{3}} |
[
{
"content": "Determine the interval of convergence for the series \\( \\sum_{n=1}^{\\infty} \\frac{n!x^n}{1\\cdot 3\\cdot 5\\cdots (2n-1)} \\).",
"role": "user"
}
] | (-2, 2) |
[
{
"content": "Let $X$ be a finite set, and let $F \\langle X \\rangle$ be the free group on the set $X$. Suppose there is a surjective homomorphism \\( \\varphi : F \\langle X \\rangle \\longrightarrow G \\). Can you conclude that the group $G$ is finitely generated?",
"role": "user"
}
] | Yes |
[
{
"content": "Given the function \\(f(x) = \\sin(\\ln x) - 2\\cos(\\ln x)\\), it has a local minimum at \\(x = e^\\lambda\\) and a local maximum at \\(x = e^\\mu\\). Determine the value of \\(\\sqrt{5}\\cos(\\mu - \\lambda)\\).",
"role": "user"
}
] | -\sqrt{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... | 13.45% |
[
{
"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... | -187.05 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... | 4.42e23 molecules |
[
{
"content": "Evaluate the definite integral \\( \\int_{-\\frac{\\pi}{2}}^{\\frac{\\pi}{2}} \\cos(\\cos x) \\, dx \\).",
"role": "user"
}
] | \pi J_0(1) |
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