prompt list | ground_truth string |
|---|---|
[
{
"content": "Two positive acute angles $A$ and $B$ satisfy the equations $3\\cos^2A+2\\cos^2B=4$ and $\\dfrac{3\\sin A}{\\sin B}=\\dfrac{2\\cos B}{\\cos A}$. Find the value of $A+2B$. \\( \\boxed{\\text{Answer}} \\)",
"role": "user"
}
] | \dfrac{\pi}{2} |
[
{
"content": "Compute \\( \\lim\\limits_{n\\to \\infty} \\int\\limits_0^1 x^{2019} \\{nx\\} \\, dx, \\) where \\( \\{a\\} \\) denotes the fractional part of the real number \\( a \\).",
"role": "user"
}
] | \dfrac{1}{4040} |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x\\to 0} \\left(\\frac{\\arctan(x)}{x}\\right)^{1/x^2} \\]",
"role": "user"
}
] | e^{-1/3} |
[
{
"content": "Let $(B_t)$ be a Brownian motion on $\\mathbb{R}$. For a fixed $x >0$, let $\\tau$ be a stopping time defined by \\( \\tau = \\inf \\{t \\geq 0 : |B_t| = x \\} \\). Compute, for $\\lambda \\in \\mathbb{R}$, the expectation \\( \\mathbb{E} \\bigg(\\exp \\bigg\\{ \\lambda B_\\tau -\\frac{\\lambda ^2... | 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... | 8 minutes |
[
{
"content": "In trapezoid $ABCD$, $\\overline{AB}$ and $\\overline{CD}$ are perpendicular to $\\overline{AD}$, with $AB+CD=BC$, $AB<CD$, and $AD=7$. What is $AB \\cdot CD$? Express your answer in decimal form.",
"role": "user"
}
] | 12.25 |
[
{
"content": "Given the infinite series \\( \\alpha = \\frac{5}{2!\\cdot 3} + \\frac{5\\cdot 7}{3!\\cdot 3^2} + \\frac{5\\cdot 7 \\cdot 9}{4!\\cdot 3^3} + \\cdots \\), find the value of \\( \\alpha^2 + 4\\alpha \\).",
"role": "user"
}
] | 23 |
[
{
"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... | Scanner Data |
[
{
"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... | 20.05% |
[
{
"content": "If $f(x)$ and $g(x)$ are polynomials over $\\mathbb{Z}$ and $f(n)$ divides $g(n)$ for all integers $n$, does it follow that $f(x)$ divides $g(x)$ in $\\mathbb{Z}[x]$? Justify your answer.",
"role": "user"
}
] | No |
[
{
"content": "Determine which of the following integral solutions is correct for \\( \\int{\\frac{x^2+1}{x^4-x^2+1}}dx \\):\n\n1. \\( \\arctan(x-\\frac{1}{x})+C \\)\n2. \\( \\arctan(\\frac{x}{1-x^2})+C \\)",
"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... | 4.10e+3 |
[
{
"content": "Evaluate the surface integral \\( \\int \\int_S z \\, dS \\) where \\( S \\) is the surface of the hemisphere defined by \\( x^2 + y^2 + z^2 = a^2 \\) with \\( z \\geq 0 \\).",
"role": "user"
}
] | \pi a^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... | Payback and Discounted Payback |
[
{
"content": "Evaluate the limit: $$\\lim_{{x\\to\\infty}} \\left(\\frac{e^x-x^2}{e^x-5x}\\right)^{\\log x}$$",
"role": "user"
}
] | 1 |
[
{
"content": "Given a subgroup $G \\leq S_p$, where $p$ is a prime number and $|G| < p$, does the action of $G$ on the set $\\{1, 2, \\ldots, p\\}$ necessarily have a fixed point?",
"role": "user"
}
] | Yes |
[
{
"content": "Determine whether the limit \\( \\lim_{n\\to\\infty}\\frac{n!}{\\prod_{i=1}^n(2i-1)} \\) equals zero.",
"role": "user"
}
] | 0 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | Copper |
[
{
"content": "In an alphabet with $n > 1$ letters, a word is defined as any finite sequence of letters where any two consecutive letters are different. A word is called \"good\" if it is not possible to delete all but four letters from it to obtain a sequence of the form $a a b b$, where $a$ and $b$ are differe... | 2n + 1 |
[
{
"content": "Evaluate the limit: \n\\[\n\\lim_{x \\rightarrow 0} \\frac{x^3 + \\sin(2x) - 2\\sin(x)}{\\arctan(x^3) - \\arctan(x)^3}\n\\]",
"role": "user"
}
] | \dfrac{1}{4} |
[
{
"content": "Let $I_0 = [a,b]$ with $b > a$. Consider a measurable set $A \\subset I_0$ such that for all distinct rational numbers $p$ and $q$, the sets $(\\{p\\} + A) \\cap (\\{q\\} + A)$ are disjoint. Determine the measure $\\mu(A)$.",
"role": "user"
}
] | 0 |
[
{
"content": "Calculate the volume of a torus formed by rotating a circle of radius \\( b \\) around an axis located at a distance \\( a \\) (where \\( a > b \\)) from the circle's center.",
"role": "user"
}
] | 2\pi^2 a b^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... | 0.12 s |
[
{
"content": "Find a subgradient \\( g \\) of the function \\( f(x) = ||x||_2^2 \\).",
"role": "user"
}
] | 2x |
[
{
"content": "Compute the limit: $$\\lim_{n\\to\\infty} \\left(\\frac{n!}{n^n}\\right)^{\\frac{1}{n}}.$$",
"role": "user"
}
] | \dfrac{1}{e} |
[
{
"content": "Given an idempotent matrix \\(X_{m\\times m}\\) with \\(\\text{rank}(X) = n < m\\), compute the rank of the matrix \\(I_m - X\\).",
"role": "user"
}
] | m - 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... | $69.54 |
[
{
"content": "A right hexagonal prism has a height of 3 feet and each edge of the hexagonal bases is 6 inches. What is the sum of the areas of the non-hexagonal faces of the prism, in square feet?\n\n[asy]import three;\n\ncurrentprojection=orthographic(1/2,-1,1/2);\n\ndraw((0,0,0)--(1,0,0)--(1.5,0.86,0)--(1,1.7... | 9 |
[
{
"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... | interdependent |
[
{
"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... | 6.00 |
[
{
"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... | S = \dfrac{(\sqrt{21})v^2}{78.4} |
[
{
"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... | 18.92e4 J |
[
{
"content": "Verify the correctness of the integral equation: $$ \\frac{2}{\\pi \\sigma_k} \\int_{-\\infty}^{\\infty} \\frac{\\sin^2\\left(\\frac{\\sigma_k}{2}(v_gt-x)\\right)}{(v_gt-x)^2} dx =1 $$ using the known identity $\\int_{-\\infty}^{\\infty} \\frac{\\sin^2(x)}{x^2} dx = \\pi$. Provide a detailed verif... | 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... | 173720 |
[
{
"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... | 15.24% |
[
{
"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... | 2e-3 M |
[
{
"content": "Let \\((X, \\le)\\) be a finite lattice. Suppose \\(f, g: (X, \\le) \\to (X, \\le^{-1})\\) are order-isomorphisms. Are \\(f\\) and \\(g\\) necessarily the same function?",
"role": "user"
}
] | No |
[
{
"content": "Evaluate the limit: \\[\\lim_{n \\to \\infty}\\left( \\frac{1^{a+1}+2^{a+1}+3^{a+1}+...+n^{a+1}}{n^{a+1}}-\\frac{n}{a+2} \\right)\\] where \\(a \\in \\mathbb{R}\\) and \\(a > 0\\).",
"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... | 0.01 m |
[
{
"content": "Let \\(\\alpha, \\beta, \\gamma\\) be real numbers satisfying \\(\\alpha + \\beta + \\gamma = 2\\) and \\(\\alpha \\beta \\gamma = 4\\). Determine the minimum value of \\(|\\alpha| + |\\beta| + |\\gamma|\\).",
"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... | Common StockSharesAmountPaid-in CapitalRetained EarningsTotal Stockholders' Equity
Balances as of December 31, 2014 7,000 $15 $17,000 $44,000 $166,000
a. 1,600 $15$22,400 $- $46,400
b. $44,000 $44,000
c. $(7,000) $(7,000)
Balances as of December 31, 2015 8,600 $15.00 $39,400 $81,000 $249,400 |
[
{
"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... | 6ClO_2(aq)+3H_2O(l) o 5ClO_3^-(aq)+Cl^-(aq)+6H^+(aq) |
[
{
"content": "Calculate the surface integral \\(\\iint_R xy^2 \\, ds\\) where \\(R\\) is the upper half of the circle \\(x^2+y^2=25\\).",
"role": "user"
}
] | 0 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 3.5 g KIO3 |
[
{
"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... | $340 |
[
{
"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. Measure the unstretched length of the spring.
2. Hang different masses from the spring and measure the stretched length.
3. Calculate the change in length for each mass.
4. Use the formula F = kx to calculate the spring constant k for each mass, where F is the force applied by the mass (mass * gravity) and x is the change in length.
5. Calculate the average k and the standard deviation of k.
Possible sources of error: human error, measurement error with the meter stick, altitude. |
[
{
"content": "Determine whether \\( \\lim \\sup_{n\\to\\infty}\\frac{m_n}{n} = 1 \\), where \\( m_n \\) is the largest integer such that there exists a subset \\( T \\subseteq \\{0,1\\}^n \\) with \\(|T|=n\\) and for distinct \\( x, y \\in T \\), the Hamming distance \\( d^H_n(x,y) \\geq m_n \\).",
"role": ... | 1 |
[
{
"content": "Calculate the expected value of the random variable $Y = \\min(X, 3)$, where $X$ is a random variable with the density function \\( g(x) = \\begin{cases} 0 & \\text{if } x < 0 \\\\ e^{-x} & \\text{if } x \\geq 0 \\end{cases} \\).",
"role": "user"
}
] | 1 - e^{-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... | 14.0 g |
[
{
"content": "Using Fubini's theorem, evaluate the limit \\( \\lim_{n \\rightarrow \\infty}\\int_{0}^{n}\\frac{\\sin x}{x}dx \\). Given that \\( \\int_{0}^{\\infty}e^{-tx}dx = \\frac{1}{t} \\) for \\( t > 0 \\), show that this limit equals \\( \\frac{\\pi}{2} \\).",
"role": "user"
}
] | \dfrac{\pi}{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... | 158.9 kg |
[
{
"content": "The number $x$ satisfies $5x^2 + 4 = 3x + 9$. Find the value of $(10x - 3)^2$.",
"role": "user"
}
] | 109 |
[
{
"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... | (1.732*10^4i + 1.002*10^4j) 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... | $\sigma^2 (1 - \frac{d+1}{N})$ |
[
{
"content": "For a function $f:\\mathbb{R}^n \\rightarrow \\mathbb{R}$ defined as $f(x)=x^T A x$, where $A$ is a positive definite matrix, determine if the maximum value of $f(x)$ over the set of vectors with norm $t$ is an increasing function of $t$.",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate the integral \\( \\int_{0}^{2\\pi} e^{\\sin (e^{i \\theta} )} \\, d\\theta \\).",
"role": "user"
}
] | 2\pi |
[
{
"content": "Let $A \\subseteq B \\subseteq B(H)$ be an inclusion of $C^*$-algebras where $H$ is a Hilbert space. Assume the following conditions:\n\n- $B$ is a von Neumann algebra with $A'' = B$.\n- The inclusion $A \\subseteq B(H)$ is non-degenerate.\n- $B$ contains $\\operatorname{id}_H$, but $A$ does not.\... | 0 |
[
{
"content": "Evaluate the double integral \\( \\int_0^1 \\int_0^{\\sqrt{1-x^2}} e^{\\sqrt{x^2 + y^2}} \\, dy \\, dx \\) by converting to polar coordinates.",
"role": "user"
}
] | \dfrac{\pi}{2} |
[
{
"content": "Find the coefficient of $x^{12}$ in the expansion of the expression: $(x^5+x^6+x^7+\text{...})^2 \\cdot (1+x+x^2+\\text{...})^8$.",
"role": "user"
}
] | 55 |
[
{
"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... | 1.87 m/s |
[
{
"content": "If every function from a topological space X to itself is continuous, must the space X be equipped with the discrete topology?",
"role": "user"
}
] | No |
[
{
"content": "Calculate the flux of the vector field \\( F = \\{z, y^2, x\\} \\) through the surface defined by \\( x^2 + y^2 + z^2 \\le 25 \\) and \\( y^2 + z^2 \\le 9 \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Find the minimum value of the expression \\(x^5 + y^5 + z^5 - 5xyz\\) where \\(x, y,\\) and \\(z\\) are positive numbers. You may use the inequality:\n\\[ (t_1\\cdot t_2\\cdot t_3\\cdots t_n)^{\\frac{1}{n}} \\leq \\frac{t_1+t_2+t_3+\\cdots +t_n}{n} \\]",
"role": "user"
}
] | -2 |
[
{
"content": "Calculate the flux of the vector field \\( \\mathbf{F}(x,y,z) = \\mathbf{i} + \\mathbf{j} + \\mathbf{k} \\) across the surface \\( \\sigma \\), which is the portion of the cone \\( z = \\sqrt{x^2 + y^2} \\) between the planes \\( z = 3 \\) and \\( z = 6 \\), oriented by downward unit normals.",
... | -27\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... | 0.015 kNm |
[
{
"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... | The fundamental frequency is 288.4 Hz. The frequency of the first overtone is 576.8 Hz. The frequency of the second overtone is 865.2 Hz. |
[
{
"content": "Determine whether the Cartier dual of an abelian étale finite group scheme $G$ is étale, given that the characteristic of the field $k$ is $p > 0$ and the dimension of $k[G]$ is not divisible by $p$.",
"role": "user"
}
] | Yes |
[
{
"content": "In a country where gold and platinum sand are valuable, gold can be exchanged for platinum and vice versa at an exchange rate determined by natural numbers $g$ and $p$. Specifically, $x$ grams of gold sand is equivalent to $y$ grams of platinum sand if $xp = yg$. Initially, the banker has one kilo... | Yes |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n \\to \\infty} \\frac{(n+1)^{\\sqrt{n+1}}}{n^{\\sqrt{n}}} \\]",
"role": "user"
}
] | 1 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | -0.97 m/s |
[
{
"content": "Find the value of \\( n \\) that is the closest solution to the equation:\n\\[\n\\frac{(4.554 \\times 10^{9})!(4.6 \\times 10^9 - n)!}{(4.554 \\times 10^{9} -n)!(4.6 \\times 10^9)!} \\approx 0.997\n\\]",
"role": "user"
}
] | 0 |
[
{
"content": "Determine which of the following two inequalities is correct when applying the generalized Hölder's inequality with measure $d\\mu(x) = w(x)^p$:\n\n1. $$ \\left(\\int_{\\mathbb{R}^n}|T(\\overrightarrow{f})(x)w(x)|^pdx\\right)^{\\frac{1}{p}} \\leq \\left(\\int_{\\mathbb{R}^n}|\\prod_{i=1}^{n}f_{i}(... | 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... | 5.884e+26 formula units |
[
{
"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... | 800,000 |
[
{
"content": "Compute the value of the expression \\(2\\arctan{2} + \\arcsin{\\frac{4}{5}}\\).",
"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... | $\dfrac{d^2 x^\alpha}{d\tau^2} =-\eta^{\alpha \mu}\left(g_{\nu\mu,\beta\alpha}-\dfrac{1}{2}g_{\beta\nu,\mu\alpha}\right)x^{\alpha}\dfrac{dx^\beta}{d\tau}\dfrac{dx^\nu}{d\tau}$ |
[
{
"content": "Derive the gradient of \\( \\beta = x^TA^Tx \\) with respect to \\( x \\), where \\( x \\) is an \\( n \\times 1 \\) vector and \\( A \\) is an \\( n \\times n \\) matrix that does not depend on \\( x \\).",
"role": "user"
}
] | (A + A^\top)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... | 11.32 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... | $1,699.46 |
[
{
"content": "Determine whether the limit \\( \\lim_{n\\rightarrow\\infty }nz^{n!n} = 0 \\) holds for \\(|z|<1\\).",
"role": "user"
}
] | 0 |
[
{
"content": "Determine the value of \\(a\\) for which the line \\(y = \\frac{3}{2}x - 2\\) is tangent to the curve \\(y^2x + a = x^2 + y^2\\).",
"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... | $699,000 |
[
{
"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... | 0.31 dm^3 |
[
{
"content": "If $x^2+x+4 = y - 4$ and $x = -7$, what is the value of $y$?",
"role": "user"
}
] | 50 |
[
{
"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% |
[
{
"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... | 1060 K |
[
{
"content": "Calculate the flux of the vector field \\( \\mathbf{F} = 3\\mathbf{i} + 2\\mathbf{j} + 3\\mathbf{k} \\) across the part of the plane \\( 1x + 3y + z = 3 \\) that lies in the first octant, oriented upwards. The surface is defined by \\( 0 \\leq x \\leq 3 - 3y \\) and \\( 0 \\leq y \\leq \\frac{1}{3... | 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... | 4.17e-4 L or 0.417 ml |
[
{
"content": "Let $V \\subset H$ be Hilbert spaces with inner products $(\\cdot,\\cdot)_V$ and $(\\cdot,\\cdot)_H$. Suppose $V$ is dense in $H$ and both spaces are separable. If $(\\cdot,\\cdot)_{V_2}$ and $(\\cdot,\\cdot)_{H_2}$ are different inner products with norms equivalent to the original inner products,... | Yes |
[
{
"content": "Find a function \\( f(x) \\) that satisfies the functional equation \\( f(x \\cdot y) = f(x) + f(y) \\) for all \\( x, y > 0 \\).",
"role": "user"
}
] | \ln x |
[
{
"content": "Consider three different types of computer components with lifetimes following exponential distributions with parameters 1, 2, and 3, respectively. The probability density functions are given by:\n\n\\[ f_1(x_1) = e^{-x_1}, \\quad f_2(x_2) = \\frac{1}{2}e^{-\\frac{x_2}{2}}, \\quad f_3(x_3) = \\fra... | \dfrac{1}{374} |
[
{
"content": "Suppose $f:(0,1) \\to \\mathbb{R}$ is a continuously differentiable function with a finite number of zeros, and $f'$ changes sign at exactly two of these zeros. Does this guarantee that for any $y \\in \\mathbb{R}$, the equation $f(x) = y$ has at most three solutions? Provide a justification for y... | No |
[
{
"content": "Consider a hyperkähler manifold $X$ and its associated twistor space $T = S^2 \\times X$ with the tautological complex structure. For each point $x \\in X$, there is a holomorphic map $u_x: S^2 \\to T$ given by $u_x(\\theta) = (\\theta, x)$. Is every holomorphic map $u: S^2 \\to T$ that satisfies ... | Yes |
[
{
"content": "Evaluate the volume of the solid below the plane \\(z = 1+y\\), bounded by the coordinate planes and the vertical plane \\(2x + y = 2\\). Use the triple integral:\n\\[ V=\\int_{x=0}^1\\int_{y=0}^{2-2x}\\int_{z=0}^{1+y}\\,\\mathrm dz\\mathrm dy\\mathrm dx \\]\nVerify the result by changing the orde... | \dfrac{5}{3} |
[
{
"content": "Find the function \\( f(x) \\) that satisfies the functional equation \\( f\\left(\\frac{x}{y}\\right) = \\frac{f(x)}{f(y)} \\) for all \\( x, y \\) such that \\( f(y) \\neq 0 \\), and the derivative condition \\( f'(1) = 2 \\).",
"role": "user"
}
] | x^2 |
[
{
"content": "What is the simply connected cover of the Lie group $SO(2,\\mathbb{R})$?",
"role": "user"
}
] | \mathbb{R} |
[
{
"content": "Does the proof of the equivalence between the two definitions of a well-founded set require the full axiom of dependent choice? Answer yes or no.",
"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... | 8.9 m |
[
{
"content": "Consider a $3 \\times 3$ matrix $A$ with eigenvalues $-1, -1, 1$. Let $f(x) = x^2$. Determine if the algebraic multiplicity of the eigenvalue $1$ in the matrix $A^2$ is 3.",
"role": "user"
}
] | Yes |
[
{
"content": "Let $G = (\\mathbb{Z/2Z})^n$ be an additive group, with $n > 1$. Suppose we have $2^n$ nonzero elements $v_i \\in G$, such that $\\sum_{i=1}^{2^n} v_i = 0$. Does there exist a subset $J \\subset \\{1, \\dots, 2^n\\}$ with $|J| = 2^{n-1}$ satisfying $\\sum_{j \\in J} v_j = 0$?",
"role": "user"
... | Yes |
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