prompt list | ground_truth string |
|---|---|
[
{
"content": "Suppose $Y_k = \\sum_{i=1}^k X_i$ for $k = 1, \\ldots, n$ are jointly Gaussian random variables. Are the random variables $X_1, \\ldots, X_n$ jointly Gaussian?",
"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... | fiscal stimulus |
[
{
"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... | 11.89 degrees |
[
{
"content": "Given that \\((1+\\sin t)(1+\\cos t)=\\frac{5}{4}\\) and \\((1-\\sin t)(1-\\cos t)=\\frac{m}{n}-\\sqrt{k}\\), where \\(k, m,\\) and \\(n\\) are positive integers with \\(m\\) and \\(n\\) relatively prime, find the value of \\(k+m+n\\).",
"role": "user"
}
] | 27 |
[
{
"content": "Let \\( S \\) be the sum of all positive real roots of the equation \\( x^{2 \\sqrt{2}}=\\sqrt{2}^{2^{x}} \\). Determine which of the following options is correct:\n(A) \\( S<\\sqrt{2} \\)\n(B) \\( S=\\sqrt{2} \\)\n(C) \\( \\sqrt{2}<S<2 \\)\n(D) \\( 2 \\leqslant S<6 \\)\n(E) \\( S \\geqslant 6 \\)... | D |
[
{
"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... | $9163 |
[
{
"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... | 8.6 mV |
[
{
"content": "Define $f(0)$ such that the function $f(x) = (1+x^2)^{1/\\tan(x)}$ is continuous on the interval $(-\\pi/2, \\pi/2)$. Determine the value of $f(0)$. \\( \\boxed{} \\)",
"role": "user"
}
] | 1 |
[
{
"content": "Determine whether the following statement is true or false: If $P(x,y)$ and $Q(x,y)$ are homogeneous polynomials of degree $n$ and $m$ respectively, and $f(x,y)=\\frac{P(x,y)}{Q(x,y)}$ is defined as 0 when $Q(x,y)=0$, then $f(x,y)$ is continuous at $(0,0)$ if $m>n$. Provide your answer as either \... | False |
[
{
"content": "Let $X$ and $Y$ be continuous random variables with a joint probability density function given by \\( f(x,y) = 3y \\) for \\( 0 \\le x \\le y \\le 1 \\). Find the probability density function of the random variable \\( T = XY \\).",
"role": "user"
}
] | 3(1 - \sqrt{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... | 1. Heating silicon dioxide with coke at high temperature: SiO2 + 2C -> Si + 2CO
2. Reacting silicon with chlorine gas: Si + 2Cl2 -> SiCl4
3. Purifying silicon tetrachloride via distillation.
4. Reducing silicon tetrachloride to produce ultrapure silicon. |
[
{
"content": "Find the number of different complex numbers $z$ with the properties that $|z|=1$ and $z^{6!}-z^{5!}$ is a real number.",
"role": "user"
}
] | 1440 |
[
{
"content": "Find all functions \\( f: \\mathbb{R} \\rightarrow \\mathbb{R} \\) such that for all real numbers \\( x \\) and \\( y \\):\n\n\\[ f\\left(x^{2}-y^{2}\\right)=x f(x)-y f(y) \\]",
"role": "user"
}
] | f(x) = kx |
[
{
"content": "Let $f:[2,4] \\rightarrow \\mathbb{R}$ be a function such that $-2 \\leq f(x) \\leq 2$ for all $x$ and $\\int_{2}^{4}f(x)dx = 0$. Determine the maximum value of $\\int_{2}^{4} \\frac{f(x)}{x^2}dx$. \\(\\text{Express your answer as a single number.}\\)",
"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... | $59109 |
[
{
"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... | 13.27 mm |
[
{
"content": "Find the minimum value of\n\\[3x^2 + 12y^2 + 27z^2 - 4xy - 6xz - 12yz - 8y - 24z\\]over all real numbers $x,$ $y,$ and $z.$",
"role": "user"
}
] | -14 |
[
{
"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... | 54000 |
[
{
"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... | 7.35e-8 C/m |
[
{
"content": "How many perfect squares are factors of 180?",
"role": "user"
}
] | 4 |
[
{
"content": "Determine if the linear system represented by the matrix \\[ \\begin{matrix} c^2-c & 0 & c & c^2 \\\\ 0 & 1-c^2 & c^2 & -c \\\\ 0 & c^2-1 & c & c \\\\ c^2-c & 0 & c & 2c \\\\ \\end{matrix} \\] can have infinitely many solutions.",
"role": "user"
}
] | Yes |
[
{
"content": "Determine the function \\( y_{max}(x) \\) that represents the maximum y-coordinate achievable at \\( x = x_f = x_0 + \\Delta x \\) on the graph, given the movement rules: starting at \\((x_0, 1 - x_0)\\), descend by an arbitrary amount \\(\\Delta y\\), move along the line to the origin \\((0, 0)\\... | 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... | -188.20 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... | Fr_87^223 -> Ra_88^223 + e_-1^0 + nu_0^0 |
[
{
"content": "Determine the degree of the minimal polynomial of $\\cos \\frac{\\pi}{48}$ over $\\mathbb{Q}$, given that it is a root of the $96$-th cyclotomic polynomial.",
"role": "user"
}
] | 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... | $5.01 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n \\to \\infty} \\sum_{k=0}^n \\frac{1}{{n \\choose k}} \\] for \\( n \\geq 1 \\).",
"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... | m_{Al} C_{Al} (T_1 - T_3) = (m_{water} C_{water} + m_{Cu} C_{Cu}) (T_3 - T_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... | 1.48 |
[
{
"content": "Find the point $P(x, y, z)$ on the curve $z^2 = x^2 + y^2$ that is closest to the point $A(2, 4, 0)$. What is the distance from point $P$ to the origin?",
"role": "user"
}
] | \sqrt{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... | 89 m |
[
{
"content": "Determine the value of $k$ such that the limit \\( \\lim_{x\\rightarrow0} \\frac{(e^{x^2}-x^2-1)(\\cos(x)-1)}{x^k} \\) is finite and non-zero.",
"role": "user"
}
] | 6 |
[
{
"content": "Let \\( \\sum _{n=-\\infty}^\\infty a_nz^n \\) be the Laurent series expansion of \\( f(z)=\\dfrac{1}{2z^2-13z+15} \\) in the annulus \\( \\frac{3}{2}<|z|<5 \\). Find the value of \\( \\frac{a_1}{a_2} \\).",
"role": "user"
}
] | 5 |
[
{
"content": "Let $g(x) > 0$ be a function for $x \\in (a, b)$ such that \\( g(x_1 + x_2) = g(x_1) + g(x_2) \\) for all $x_1, x_2 \\in (a, b)$. Must $g(x)$ be of the form $g(x) = cx$ for some constant $c$? Provide a justification for your answer.",
"role": "user"
}
] | g(x) = cx |
[
{
"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... | 10.6 s |
[
{
"content": "Let \\( A \\) be an \\( n \\times n \\) Hermitian matrix such that \\( A^5 + A^3 + A = 3I_n \\). Determine whether it is true that \\( A = I_n \\).",
"role": "user"
}
] | A = I_n |
[
{
"content": "Compute the integral \\( \\int_{|z|=2 } \\frac{z^3}{(z+1)^3} dz \\) using the Cauchy Integral formula.",
"role": "user"
}
] | -6\pi i |
[
{
"content": "Let $G$ be a group of order $p^3$, where $p$ is a prime number. If $|Z(G)|=p$, determine the number of conjugacy classes of $G$.",
"role": "user"
}
] | p^2 + p - 1 |
[
{
"content": "Evaluate the integral \\( \\int_{7}^{10}\\sqrt{-40+14x-x^2}dx \\).",
"role": "user"
}
] | \dfrac{9\pi}{4} |
[
{
"content": "Calculate the limit: $$\\lim_{n \\to \\infty} \\frac{\\frac{\\sin(1)}{1} + \\frac{\\sin(2)}{2} + \\frac{\\sin(3)}{3} + \\cdots + \\frac{\\sin(n)}{n}}{n}$$",
"role": "user"
}
] | 0 |
[
{
"content": "Find the natural number solutions to the equation: \\[ 25^{x+1} + 4^{y-2} - 2 \\cdot 5^{x+2} - 2^{y} + 29 = 0 \\]",
"role": "user"
}
] | (0, 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... | 12 m^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... | 1.1 * 10^3 s |
[
{
"content": "Let $K \\subset L$ be two fields with rings of integers $\\mathcal{O}_K$ and $\\mathcal{O}_L$. If a prime $p$ is totally ramified in $\\mathcal{O}_K$, is it true that $p$ is also ramified in $\\mathcal{O}_L$? Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Let $X$, $Y$, and $Z$ be linear spaces, and let $T: X \\times Y \\rightarrow Z$ be a bilinear mapping. Suppose $(e_i)_{i \\in I}$ is a basis for $X$ and $(f_j)_{j \\in J}$ is a basis for $Y$, such that $(T(e_i, f_j))_{i \\in I, j \\in J}$ forms a basis for $Z$. If $(e'_i)_{i \\in I}$ and $(f'_j)_{... | Yes |
[
{
"content": "Evaluate the limit of the sequence \\(a_n\\), defined as follows:\n\n\\[ a_n = \\frac{1^k + 2^k + \\ldots + n^k}{n^k} - \\frac{n}{k+1} \\]\n\nwhere \\(k\\) is a positive integer. Simplify or evaluate the following limit without using Big O notation:\n\n\\[\\lim_{n\\to\\infty} \\left( \\frac{\\left... | \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... | 12.46% |
[
{
"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... | 1.809 s |
[
{
"content": "Can the interior and closure of a non-measurable subset of $[0,1]$ have the same Lebesgue measure?",
"role": "user"
}
] | No |
[
{
"content": "Evaluate the integral \\( \\iint\\limits_{D} e^{-(x^2+y^2)} \\, dx \\, dy \\) over the region \\( D \\) bounded by the positive \\( x \\) and \\( y \\) axes.",
"role": "user"
}
] | \dfrac{\pi}{4} |
[
{
"content": "The width, length, and height of a rectangular prism are each increased by $10\\%$. What is the percent increase in the volume of the prism? Express your answer to the nearest whole number.",
"role": "user"
}
] | 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... | 9.5% |
[
{
"content": "Evaluate \\( \\Gamma\\left(-\\frac{1}{2}\\right) \\) directly from the definition \\( \\Gamma(s) = \\int_0^\\infty e^{-t}t^{s-1}dt \\), without using the known values \\( \\Gamma\\left(\\frac{1}{2}\\right) = \\sqrt{\\pi} \\) and \\( \\Gamma(s+1) = s\\Gamma(s) \\).",
"role": "user"
}
] | -2\sqrt{\pi} |
[
{
"content": "Given a bounded self-adjoint operator \\( A = \\begin{pmatrix}3 & 0 \\\\ 0 & 1\\end{pmatrix} \\) on \\( \\mathbb{R}^2 \\) and a vector \\( \\psi = \\begin{pmatrix}1 \\\\ 1\\end{pmatrix} \\), find the spectral measure \\( \\mu_\\psi \\) such that \\( (\\psi, f(A)\\psi) = \\int_{\\sigma(A)} f(\\lamb... | 10 |
[
{
"content": "Given the joint probability density function \\( f(x,y) = 9x^2y^2 \\) for \\( 0 < y < 1 \\) and \\( -y < x < y \\), find the probability \\( P(X > Y^2) \\).",
"role": "user"
}
] | \dfrac{1}{6} |
[
{
"content": "Determine the values of \\( n \\) for which \\( (\\phi_{(1\\,2\\,4\\,3)})^n = e \\), where \\( \\phi_{(1\\,2\\,4\\,3)} = (1\\,2\\,4\\,3)x(3\\,4\\,2\\,1) \\) and \\( x \\in \\text{Inn}(S_5) \\).",
"role": "user"
}
] | 4 |
[
{
"content": "Determine whether the Gaussian prime $\\mathfrak{p} = 5 + 8i$ can be expressed as the sum of two squares in the ring of Gaussian integers $\\mathbb{Z}[i]$. Provide your answer as either \"Yes\" or \"No\".",
"role": "user"
}
] | No |
[
{
"content": "Determine the number of elements in the kernel of a non-trivial, non-injective group homomorphism $g: D_4 \\to \\mathbb{Z}_2$, where $D_4$ is the dihedral group of order 8 and $\\mathbb{Z}_2$ is the cyclic group of order 2.",
"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... | 196 |
[
{
"content": "Consider the ordinary differential equation $$(3y^2-x)f(x+y^2)+2y(y^2-3x)f(x+y^2)y'=0$$ where $f$ is an arbitrary continuously differentiable function. If this differential equation is exact, determine which of the following statements is true:\n\n1) $xf'(x)-3f(x)=0$\n2) $xf'(x)+3f(x)=0$\n3) $f'(x... | 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... | C9H12 |
[
{
"content": "The real number $x$ satisfies $x^2 - 5x + 6 < 0.$ Find all possible values of $x^2 + 5x + 6.$",
"role": "user"
}
] | (20,30) |
[
{
"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... | 10819.11 |
[
{
"content": "Determine the group \\( I := \\langle a, b, c \\ | \\ a^2, b^2, c^2, (ab)^2, (ac)^3, (bc)^7, (abc)^{19}, (abcbc)^{25} \\rangle \\).",
"role": "user"
}
] | 1 |
[
{
"content": "Evaluate the limit: $$\\lim_{n \\to \\infty} \\frac{n^3(\\sqrt{2}+(-1)^n)^n}{3^n}$$",
"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... | 161 m/s |
[
{
"content": "Evaluate the limit: $$\\lim_{n \\to \\infty} 2^n\\arcsin\\left(\\dfrac{1}{2^n}\\right).$$",
"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... | 240 mg |
[
{
"content": "Find the limit of the expression: $$\\lim_{n\\to\\infty}{\\frac{1}{n} {\\sum_{k=3}^{n}{\\frac{3}{k^2-k-2}}}}$$",
"role": "user"
}
] | 0 |
[
{
"content": "Provide an example of a group that is not abelian but has the property that all of its subgroups are normal.",
"role": "user"
}
] | Q_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... | 2.43 |
[
{
"content": "Compute the expectation \\( E[e^{tX}] \\) where \\( X \\sim \\mathcal{N}(0, 1) \\). Use the method of completing the square in the exponent to set up and solve the integral.",
"role": "user"
}
] | e^{\frac{t^2}{2}} |
[
{
"content": "Consider the initial value problem (IVP) given by the heat equation $u_t = u_{xx}$ for $x \\in \\mathbb{R}$ and $t > 0$, with the initial condition:\n\\[ u(x,0) = \\begin{cases} 1, & x \\in [0,1] \\\\ 0, & \\text{otherwise} \\end{cases} \\]\nDetermine the value of \\( \\lim_{t \\to 0^+} u(1,t) \\)... | \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... | 1.3e-13 C |
[
{
"content": "Determine whether $S = \\operatorname{SL}(n, \\mathbb{R}) = \\{ A \\in M_{n\\times n}(\\mathbb{R}) \\mid \\det(A) = 1 \\}$ is a regular submanifold of the $n^2$-dimensional $C^\\infty$ manifold $M = M_{n\\times n}(\\mathbb{R})$. Is $S$ a regular submanifold?",
"role": "user"
}
] | Yes |
[
{
"content": "Given four groups, each containing four unique people, determine the total number of ways to form new groups by selecting one person from each group. For example, if the groups are:\n\nGroup 1: \\(A, B, C, D\\)\nGroup 2: \\(H, I, J, K\\)\nGroup 3: \\(M, N, O, P\\)\nGroup 4: \\(X, Y, Z, T\\)\n\nHow... | 256 |
[
{
"content": "Find the remainder when the binomial coefficient \\( {}^{72}C_{36} \\) is divided by 73 using Fermat's Little Theorem.",
"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... | 1809.6kgm^2s^-1 |
[
{
"content": "Find the sum of the series \\( \\frac{1}{1!} + \\frac{1+2}{2!} + \\frac{1+2+3}{3!} + \\ldots \\). Use the properties of exponential functions to derive your answer.",
"role": "user"
}
] | \dfrac{3e}{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... | 32.199 m/s |
[
{
"content": "Determine the number of solutions to the equation \\(2\\tan^{-1}|x|\\cdot\\ln|x|=1\\).",
"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... | Account Name | Debit | Credit
FICA Social Tax ($28,000 x 6.2%) | $1,736 |
FICA Medicare Tax ($28,000 x 1.45%) | 406 |
Federal Income Tax | 3,800 |
Medical Insurance | 678 |
Union Dues | 190 |
Salaries and Wages Payable | $6,810 | |
[
{
"content": "Is it possible to arrange 1000 line segments on a plane such that each segment has both of its endpoints strictly inside other segments?",
"role": "user"
}
] | Yes |
[
{
"content": "Given an open and connected set $U$ in $\\mathbb{R}^n$ and a compact set $K$ contained in $U$, determine whether there exists a compact and connected set $K' \\subset U$ such that $K \\subset K'$.",
"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... | 35.73% |
[
{
"content": "Given real numbers $a$ and $b$, and sequences $(x_n)_n$ and $(y_n)_n$ such that $x_n > 0$ and $y_n > 0$ for all $n \\in \\mathbb{N}$, if the series $\\sum_{n=0}^{\\infty} (ax_n + by_n)$ converges to a real number $c$, determine whether it is true that $\\sum_{n=0}^{\\infty} ax_n + \\sum_{n=0}^{\\i... | false |
[
{
"content": "Consider a matrix $E_0$ with non-negative entries. Define a sequence of matrices $(E_n)$ where $E_{n+1}$ is obtained from $E_n$ by performing the following operations:\n\n1. Divide each entry by the sum of all entries in its column to make the matrix column-stochastic.\n2. Divide each entry by the... | \frac{1}{5} |
[
{
"content": "Evaluate the limit:\n\n$$\\lim_{c \\rightarrow 0} \\iint_{R} \\frac{1}{(x^2+y^2)^{3/4}}\\,dA$$\n\nwhere the region \\( R \\) is defined as:\n\n$$R=\\{(x,y): x^2+y^2 \\leq 1\\} \\setminus ([-c,c] \\times [0,c])$$",
"role": "user"
}
] | 4\pi |
[
{
"content": "Compute \\( E\\left((B_t - 1)^2 \\int_0^t (B_s + 1)^2 \\, dB_s\\right) \\), where \\((B_t)\\) is a standard Brownian motion for \\( t \\geq 0 \\).",
"role": "user"
}
] | t^2 - 2t |
[
{
"content": "In a metric space with a bounded metric and the Heine-Borel property, is any closed set compact?",
"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... | 9756.10 |
[
{
"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... | ParticularsDebit ($)Credit ($)Income tax expense127,228Deferred tax asset10,506Income tax payable121,380Deferred tax liability16,354 |
[
{
"content": "Consider two sequences of random variables, $\\{Y_1^N\\}_N$ and $\\{Y_2^N\\}_N$, defined on the same probability space. Suppose that $Y_1^N \\leq Y_2^N$ almost surely and that both sequences converge in distribution to $Y_1^*$ and $Y_2^*$, respectively. If the joint distribution of $(Y_1^N, Y_2^N)... | 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... | 27.1 m |
[
{
"content": "A cube with edge length 3 is divided into 27 unit cubes. The numbers 1, 2, ..., 27 are distributed arbitrarily over the unit cubes, with one number in each cube. Determine the maximum number of odd row sums that can be formed from the 27 possible row sums, where there are nine such sums of three i... | 24 |
[
{
"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... | t_1-t_0=\frac{1}{a}\sqrt{\frac{a}{g}}\left[\arctan\left(v_1 \sqrt{\frac{a}{g}}\right)-\arctan\left(v_0 \sqrt{\frac{a}{g}}\right)\right] |
[
{
"content": "Calculate \\( g'(0) \\) for the function \\( g(x) = \\int_{0}^{2^x} \\sin(t^2) \\, dt \\).",
"role": "user"
}
] | \ln(2) \sin(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... | 0.58% |
[
{
"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... | Prepare the same certain number of antacid tablets for three or more trials.
Prepare the three or more quantities of hydrochloric acid of different concentrations.
Mix the tablets and hydrochloric acid for the first trial and start the timer at the same time.
Stop the timer when the expected change or changes occur and take note of the time.
Measure the quantity of hydrochloric acid that has been consumed.
Repeat the same steps for the other trails.
Collect the data and graph it to determine the trend of reaction rate and increasing concentration of hydrochloric acid. |
[
{
"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... | Two disadvantages of producing a photograph before 1839 were the long exposure times required, which meant subjects had to remain still for extended periods and often keep their eyes shut, and the risk of mercury poisoning for the photographer. |
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