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... | Market demand of goods and services and the number of resources left with the organization |
[
{
"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... | 13.716% |
[
{
"content": "Find the value of \\( \\lim_{x\\rightarrow \\infty}\\left[\\left(x^5+7x^4+2\\right)^c-x\\right] \\) for a certain value of \\( c \\) such that the limit is finite and non-zero.",
"role": "user"
}
] | \dfrac{1}{5} |
[
{
"content": "Evaluate \\( \\lim_{n\\to \\infty}\\int_{0}^{1}\\frac{n\\cos{x}}{1+n^2x^2}dx \\) using convergence theorems.",
"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... | 1.11e-4 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... | 0.32 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... | 1.63 m |
[
{
"content": "Let $V$ be a vector space over a field $F$ with dimension $n$, and let $W$ be a subspace of $V$ with dimension $m < n$. Define $E = \\{T: V \\to V \\mid T \\text{ restricted to } W \\text{ is zero}\\}$. Determine the dimension of $E$ as a subspace of $L(V,V)$. \\( \\text{Express your answer in ter... | n(n - 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... | 5.2e-20 m/s |
[
{
"content": "Given the functional equation \\( f(x) + f(y) = f(x+y) \\), with \\( f'(0) = 3 \\) and \\( f(5) = 3 \\), find \\( f'(5) \\).",
"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... | 4.34 g |
[
{
"content": "Find a formula for the sum \\(\\sum^n_{i = 0} {n\\choose i} F_{i+m+2}\\) using the generating function of the Fibonacci numbers.",
"role": "user"
}
] | F_{2n + m + 2} |
[
{
"content": "Evaluate the integral \\( \\int_1^2 xe^{\\sin (4-x^2)}dx \\) in terms of \\( k \\), where \\( \\int_0^3 e^{\\sin x}dx=k \\). Choose the correct answer from the following options:\n\n(a) \\(-\\frac{k}{2}\\) \n\n(b) \\(-\\frac{k}{3}\\) \n\n(c) \\(\\frac{k}{6}\\) \n\n(d) \\(\\frac{k}{3}\\) \n\n(e) \\... | E |
[
{
"content": "Consider the function \\(f(z) = z^5 + z^3 + 3z + 1\\). Determine the number of zeros this function has within the open unit disc \\(\\{z : |z| < 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... | Vs = 343 m/s - (523 Hz/554 Hz)(343 m/s + Vo) |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x\\to 0}\\frac{\\sinh x^4-x^4}{(x-\\sin x)^4} \\]",
"role": "user"
}
] | 216 |
[
{
"content": "Determine the radius of convergence for the series \\(\\sum_{n=1}^{\\infty} (-1)^n x^n \\ln(1+1/n)\\).",
"role": "user"
}
] | 1 |
[
{
"content": "Calculate the residue of the function \\( f(z) = z \\sin\\left(\\frac{1}{1-z}\\right) \\) at \\( z = 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... | The structure of C4H4 with 4 sp2 carbons is a cyclic structure with alternating single and double bonds. |
[
{
"content": "Calculate the flux \\( \\iint \\vec{F} \\cdot \\hat{n} \\, ds \\) of the vector field \\( \\vec{F} = 4x \\hat{i} - 2y^2 \\hat{j} + z^2 \\hat{k} \\) through the surface \\( S = \\{(x,y,z) : x^2 + y^2 = 4, z = 0, z = 3\\} \\).",
"role": "user"
}
] | 84\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... | $28,480 |
[
{
"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... | 5.89e-10 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... | 155 |
[
{
"content": "Determine the number of natural number solutions to the equation \\(x_1 + x_2 + x_3 + x_4 + x_5 + x_6 = 24\\) subject to the condition \\(x_1 + x_2 + x_3 > x_4 + x_5 + x_6\\).",
"role": "user"
}
] | 15312 |
[
{
"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... | Fall: 270, Winter: 390, Spring: 189, Summer: 351 |
[
{
"content": "$12! = 47a001600$, for some digit $a$. What is the value of $a$?",
"role": "user"
}
] | a=9 |
[
{
"content": "Derive the result that if $n \\epsilon_M \\leq 0.1$ and $\\epsilon_i \\leq \\epsilon_M$ for $i = 1,2,...,n$, then $(1+\\epsilon_1)(1+\\epsilon_2)\\cdots(1+\\epsilon_n) = 1 + \\eta$ with $\\eta \\leq 1.06n \\epsilon_M$. Specifically, show that $\\sum_{k=1}^n \\frac{(n-1)!}{k!(n-k)!(10n)^{k-1}} \\le... | 1.06 |
[
{
"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... | 2.6e-14 g CO2 |
[
{
"content": "Let $H_0^1$ be the space of all functions $g:[0,1]\\to \\mathbb{R}$ such that $g'$ is in $L^2[0,1]$ and $g(0)=g(1)=0$. Given that $H_0^1$ is a Hilbert space, and let $f\\in C[0,1]$. Suppose $u\\in C^{(2)}[0,1] \\cap H_0^1$ satisfies the equation \\( \\int_0^1 u'v'\\,dx = \\int_0^1 fv\\,dx \\) for ... | -u'' = f |
[
{
"content": "A right triangle has leg lengths of 8 and 15 units. What is the length, in units, of the median to the hypotenuse? Express your answer as a common fraction.",
"role": "user"
}
] | \frac{17}{2} |
[
{
"content": "Calculate the limit: $$\\lim_{(x,y)\\rightarrow (\\infty,\\infty)}(x^2+y^2)e^{-(x+y)}$$",
"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... | 1.00 m or 4.00 m |
[
{
"content": "Let $f: \\mathbb{R} \\to \\mathbb{R}$ be a differentiable function. Suppose there exists a bounded sequence $\\{x_n\\}$ such that $x_n \\neq x_m$ for $n \\neq m$ and $f(x_n) = 0$ for all $n$. Prove that there exists a point $x_0$ such that $f(x_0) = 0$ and $f'(x_0) = 0$. ",
"role": "user"
}
... | x_0 |
[
{
"content": "Given a function $f \\in L^2(K)$, where $K \\subseteq \\mathbb{R}^n$ is compact, is it possible to construct a sequence of continuous functions $f'_n$ that converges to $f$ pointwise almost everywhere, with the property that $f'_n(x) \\geq f'_m(x)$ whenever $n \\geq m$?",
"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... | 413 million |
[
{
"content": "Let $p$ be a prime number and $k$ a positive integer. Let $d$ be the smallest positive integer such that $p^k$ divides $d!$. Determine whether $d$ is necessarily a multiple of $p$. Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 984.097 g |
[
{
"content": "Given that $X$ follows a uniform distribution $Uniform(1, 9.9)$ and, conditionally on $X = x$, $Y$ is normally distributed $Normal(1.4, x^2)$, find the covariance between $X$ and $Y$, denoted as $Cov(X, Y)$. Calculate $E[XY]$ as part of your solution.",
"role": "user"
}
] | 0 |
[
{
"content": "In a non-constant arithmetic progression (AP), the 7th, 11th, and 13th terms are also three consecutive terms of a geometric progression (GP). Find the value of \\(\\frac{a}{c}\\), where \\(a\\), \\(b\\), and \\(c\\) are the respective terms of the GP.",
"role": "user"
}
] | 4 |
[
{
"content": "Determine the range of values of $R$ for which the inequality \\(\\left|\\frac{1}{R^2 e^{2i\\theta}-1}\\right|\\le\\frac{1}{R^2+1}\\) holds true.",
"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... | -20 N 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... | Normal: 18.67%
Expansion: 22.40%
Recession: 14.00% |
[
{
"content": "Calculate the probability \\( \\mathbb{P}\\left(\\inf_{t\\in[\\frac{1}{2},1]}W_t>0\\right) \\) for a standard Wiener process \\((W_t)_{t\\in[0,1]}\\).",
"role": "user"
}
] | \dfrac{1}{4} |
[
{
"content": "Determine the number of nonisomorphic magmas of order 3.",
"role": "user"
}
] | 3330 |
[
{
"content": "In $\\Delta ABC$, where $AB = 2017$ units, point $O$ is the midpoint of $AB$. Point $D$ lies inside the triangle such that $\\angle DAP = \\angle DBQ$. If $\\Delta APD$ is similar to $\\Delta BQD$, find the difference $OP - OQ$. Express your answer as a single numerical value.",
"role": "user"... | 0 |
[
{
"content": "Calculate the radical of the ideal \\((4,x)\\) in the ring \\(\\mathbb{Z}[x]\\). Verify if the following calculation is correct:\n\\[\nr(4,x) = r(4) \\cap r(x) = r(2^2) \\cap r(x) = r(2) \\cap r(x) = r(2,x) = (2,x).\n\\]",
"role": "user"
}
] | (2, x) |
[
{
"content": "Find the supremum of the set \\( A = \\{ \\cos(10^n) \\mid n \\in \\mathbb{N} \\} \\) to five decimal places.",
"role": "user"
}
] | 1.00000 |
[
{
"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... | 53 N |
[
{
"content": "Let $X$ be a projective variety with two morphisms $f:X\\rightarrow Y$ and $g:X\\rightarrow Z$ with irreducible fibers of positive dimension. Assume that $Pic(X) = f^{*}Pic(Y)\\oplus g^{*}Pic(Z)$. If $D$ is a divisor on $X$, we can write $D = f^{*}D_Y + g^{*}D_Z$, where $D_Y$ and $D_Z$ are divisor... | Yes |
[
{
"content": "Given a curve \\(C\\subset\\mathbb{R}^2\\), determine the number of angles \\(\\theta\\) such that \\(C\\) can be parameterized as a function from the rotated \\(X\\) axis with angle \\(\\theta\\).",
"role": "user"
}
] | 2 |
[
{
"content": "Given the matrix $A=\\begin{bmatrix} 3 & 0 & 0 & 2\\\\ 0 & 3 & 2 & 0\\\\ 0 & 2 & 3 & 0\\\\ 2 & 0 & 0 & 3 \\end{bmatrix}$, determine if the trace of the linear transformation $T : \\mathbb{R}^{4 \\times 4} \\rightarrow \\mathbb{R}^{4 \\times 4}$ defined by $T(X)=AX$ can be directly computed from $A... | Yes |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 1.98e-2 H |
[
{
"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... | |q| = 2q_0\sqrt{2} |
[
{
"content": "Evaluate the limit using rationalization: $$\\lim_{t\\to0}\\frac{(1-(1-t)^{1/2})(1-(1-t)^{1/3})\\cdots(1-(1-t)^{1/n})} {t^{n-1}}$$",
"role": "user"
}
] | \dfrac{1}{n!} |
[
{
"content": "For all integers $n$ greater than 1, define $a_n =\n\\dfrac{1}{\\log_n 2002}$. Let $b = a_2 + a_3 + a_4 + a_5$ and $c=\na_{10} + a_{11} + a_{12} + a_{13} + a_{14}$. Find $b - c.$",
"role": "user"
}
] | -1 |
[
{
"content": "Given the real numbers $a, x, y$ that satisfy the equation:\n$$\nx \\sqrt{a(x-a)}+y \\sqrt{a(y-a)}=\\sqrt{|\\lg (x-a)-\\lg (a-y)|},\n$$\nfind the value of the algebraic expression $\\frac{3 x^{2}+x y-y^{2}}{x^{2}-x y+y^{2}}$. \\(\\boxed{\\text{Your answer here}}\\)",
"role": "user"
}
] | \dfrac{1}{3} |
[
{
"content": "In the context of integral domains, if $R$ is a free $\\mathbb{Z}$-module and $I$ is a non-zero ideal of $R$, do $R$ and $I$ have the same rank as $\\mathbb{Z}$-modules? Answer yes or no.",
"role": "user"
}
] | Yes |
[
{
"content": "Find the extrema of the implicit function defined by the equation \\( x^2 + y^2 - z^2 = 0 \\).",
"role": "user"
}
] | (0, 0, 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... | 0.66 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... | 144.58 |
[
{
"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... | $1,100,000 |
[
{
"content": "Calculate the expected value of the determinant of a $3 \\times 3$ Vandermonde matrix with nodes sampled from a standard normal distribution.",
"role": "user"
}
] | 0 |
[
{
"content": "Fill the integers from 1 to \\(2016^2\\) into a \\(2016 \\times 2016\\) grid, ensuring each cell contains a unique number. Draw a vector from the center of one cell to the center of another if the number in the first cell is smaller than the number in the second cell. Given that the sum of the num... | 0 |
[
{
"content": "Given the system of equations:\n\\[\\begin{align*}\na+b+c&=0,\\\\\nab+ac+bc&=-1,\\\\\na^3+b^3+c^3&=19.\n\\end{align*}\\]\nFind the value of $abc$. \\(\\boxed{\\,\\,}\\)",
"role": "user"
}
] | \dfrac{19}{3} |
[
{
"content": "Find the inverse Laplace transform of the function given by the integral: \\[ f(t) = \\int_t^\\infty \\frac{e^{-u}}{u} \\, du \\]",
"role": "user"
}
] | \dfrac{u(t - 1)}{t} |
[
{
"content": "A group of 10 Caltech students go to Lake Street for lunch. Each student eats at either Chipotle or Panda Express. In how many different ways can the students collectively go to lunch?",
"role": "user"
}
] | 1024 |
[
{
"content": "Solve the differential equation \\( \\frac{dy}{dx} = \\frac{6}{x+y} \\) with the initial condition \\( y(0) = 0 \\). Find the value of \\( y \\) when \\( x+y = 6 \\).",
"role": "user"
}
] | 6 \ln 2 |
[
{
"content": "The center of a circle has coordinates $(6,-5)$. The circle is reflected about the line $y=x$. What are the $x,y$ coordinates of the center of the image circle? State the $x$ coordinate first.",
"role": "user"
}
] | (-5, 6) |
[
{
"content": "Find $100m+n$ if $m$ and $n$ are relatively prime positive integers such that \\[ \\sum_{\\substack{i,j \\ge 0 \\\\ i+j \\text{ odd}}} \\frac{1}{2^i3^j} = \\frac{m}{n}. \\]",
"role": "user"
}
] | 504 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to 0} \\frac{a^{\\tan x} - a^{\\sin x}}{\\tan x - \\sin x} \\]",
"role": "user"
}
] | \ln 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... | q_1 = 0.716q
q_2 = 0.089q |
[
{
"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... | Sales (7950 units) $206,700, Variable expenses $143,100, Contribution margin $63,600, Fixed expenses $56,000, Net Operating Income $7,600 |
[
{
"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... | 0.277 mol |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 16 m/s |
[
{
"content": "Let $R$ be a commutative integral domain with $I$, $J$, and $K$ as three ideals, where $I$ is a non-zero finitely generated ideal. Determine whether the equality $IJ = IK$ implies that $J = K$. Provide a proof or counterexample.",
"role": "user"
}
] | J = K |
[
{
"content": "Let $A$ and $B$ be square matrices such that $A^2 = A$, $B^2 = B$, and $AB = BA$. Find the determinant of the matrix $A - B$. Express your answer as a single number.",
"role": "user"
}
] | 0 |
[
{
"content": "In propositional logic, consider the following definition of a formula: Every propositional variable is a formula. If $a$ is a formula, then so is $\\neg a$. If $a$ and $b$ are formulae, then so are $(a \\star b)$ for any connective $\\star$. Nothing else is a formula, and all formulae can be deri... | 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... | 0.03 |
[
{
"content": "Evaluate the integral \\( \\int_0^{\\pi}\\frac{1}{5+3\\cos{x}}dx \\) using the substitution \\( t=\\tan{\\frac{x}{2}} \\).",
"role": "user"
}
] | \dfrac{\pi}{4} |
[
{
"content": "Find the maximum area of a rectangle DACB, where points C and B lie on the curve \\( y = \\frac{8}{1+x^2} \\) and points A and D are on the x-axis. Points B and C are symmetric with respect to the y-axis. Determine the maximum possible area of this rectangle.",
"role": "user"
}
] | 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... | ClCH2CHClCOOH |
[
{
"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... | Oberon: Recognizes no gain, stock basis is $75,000.
Puck: Recognizes a gain of $250,000, stock basis is $20,000, OSI securities basis is $250,000.
OSI: No gain or loss, land basis is $75,000, wine and cellar basis is $270,000. |
[
{
"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... | The magnitude of the electric field in the dielectric space is \( E_1 = \frac{\Delta V}{t+\epsilon_r (d-t)} \) and in the intervening space is \( E_2 = \frac{\epsilon_r \Delta V}{t+ \epsilon_r (d-t)} \) |
[
{
"content": "Three circles are given by the equations: \\( S_1: x^2+y^2+4y-1=0 \\), \\( S_2: x^2+y^2+6x+y+8=0 \\), and \\( S_3: x^2+y^2-4x-4y-37=0 \\). These circles touch each other. Let \\( P_1, P_2, P_3 \\) be the points of contact, and \\( C_1, C_2, C_3 \\) be the centers of the circles. Find the ratio \\(... | \dfrac{2}{5} |
[
{
"content": "Determine the number of perfect squares between $1$ and $N$ that can be expressed as $\\sqrt{N^2 + k^2}$, where $k$ ranges from $1$ to $\\left\\lfloor \\frac{N}{2} \\right\\rfloor$. Provide your answer as a single integer.",
"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... | 18.25% |
[
{
"content": "Find the limit of the expression \\(\\frac{{n/2 \\choose np}}{n \\choose np}\\) as \\(n\\) approaches infinity, where \\(p<\\frac{1}{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... | 318 min |
[
{
"content": "Solve for $x$: $\\left(\\frac{1}{4}\\right)^{2x+8} = (16)^{2x+5}$.",
"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... | 2.04e10 J |
[
{
"content": "Find the ideal in the ring of Gaussian integers $\\mathbb{Z}[i]$ generated by the elements $5 + 2i$ and $3 + 8i$. Express your answer in the form of a principal ideal if possible.",
"role": "user"
}
] | (1) |
[
{
"content": "Paula invests $\\$10,\\!000$ at the start of a 5 year period at an interest rate of $10\\%$. At the end of those 5 years, how much is her investment worth if the interest is compounded quarterly? Express your answer rounded to the nearest cent.",
"role": "user"
}
] | \$16,\!386.16 |
[
{
"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... | Sales: $4,000,000
Cost of goods sold: $2,000,000
Gross profit: $2,000,000
Administrative expenses: $400,000
Selling expenses: $600,000
Earnings before interest and taxes: $1,000,000
Interest expense: $110,000
Earnings before taxes: $890,000
Income tax: $427,200
Net Income: $462,800
Earnings per share: $9.26 |
[
{
"content": "Find the distribution of the random variable \\( Y = X^2 \\), where \\( X \\sim N(0,1) \\).",
"role": "user"
}
] | \chi^2(1) |
[
{
"content": "Consider a partially ordered set \\( P \\). If \\( P \\) contains antichains of arbitrarily large size, does it necessarily contain an infinite antichain? Answer 'yes' or 'no' and justify your answer.",
"role": "user"
}
] | No |
[
{
"content": "Find the derivative with respect to \\( a \\) of the solution to the initial value problem at \\( a = 0 \\):\n\n\\[ x' = x^2 - x\\cos t, \\quad x(0) = a \\]",
"role": "user"
}
] | e^{-\sin t} |
[
{
"content": "Compute \\( \\lim_{n \\to \\infty} \\int_{0}^{\\pi/3} \\frac{1}{1+\\tan^n(x)}\\,dx \\).",
"role": "user"
}
] | \dfrac{\pi}{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... | 4.11% |
[
{
"content": "Let $f(x)=x^2-2x$. What is the value of $f(f(f(f(f(f(-1))))))$?",
"role": "user"
}
] | 3 |
[
{
"content": "Determine the pushout of the diagram \\( \\mathbf{1} \\stackrel{f}{\\longleftarrow} \\mathbf{2} \\stackrel{g}{\\longrightarrow} \\mathbf{1} \\), where \\( \\mathbf{1} \\) and \\( \\mathbf{2} \\) denote the discrete finite categories of one and two objects, respectively, and both \\( f \\) and \\( ... | \mathbf{1} |
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