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... | 2.35e-11 M |
[
{
"content": "Let $a,$ $b,$ and $c$ be nonnegative real numbers such that $a^2 + b^2 + c^2 = 1.$ Find the maximum value of\n\\[2ab \\sqrt{2} + 2bc.\\]",
"role": "user"
}
] | \sqrt{3} |
[
{
"content": "Find the operator norm \\( \\|A\\|_o \\) of the matrix \\( A = \\begin{pmatrix} 0 & 1 & 1 \\\\ 1 & 0 & 1 \\\\ 1 & 1 & 0 \\end{pmatrix} \\). The operator norm is defined as \\( \\|A\\|_o = \\sup \\|Ax\\|_2 \\), where the supremum is taken over all \\( x \\in \\mathbb{R}^3 \\) with \\( \\|x\\|_2 = 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... | Product B |
[
{
"content": "Determine the region of convergence for the series \\( \\sum\\limits_{n=0}^{\\infty} \\frac{z^n}{a^n+b^n} \\), where \\( a \\) and \\( b \\) are real parameters with \\( 0<a<b \\).",
"role": "user"
}
] | |z| < 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... | $570,000 |
[
{
"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... | 9.98% |
[
{
"content": "Given that \\(\\sin \\alpha + \\sin \\beta = 2 \\sin (\\alpha + \\beta)\\) with \\(\\alpha + \\beta \\neq 2 \\pi n\\) where \\(n \\in \\mathbb{Z}\\), find \\(\\tan \\frac{\\alpha}{2} \\tan \\frac{\\beta}{2}\\).",
"role": "user"
}
] | \dfrac{1}{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... | $970.75 |
[
{
"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... | 0.926 W/m K |
[
{
"content": "Evaluate the product: \\( \\prod_{n=1}^{9} \\sin(20n - 10^\\circ) \\).",
"role": "user"
}
] | \dfrac{1}{256} |
[
{
"content": "Find a non-constant polynomial in \\( \\mathbb{Z}/8\\mathbb{Z}[x] \\) that is a unit.",
"role": "user"
}
] | 1 + 4x |
[
{
"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... | Inner sides of the plates will have equal and opposite charges q = (Q1 - Q2)/2. Outer sides will have same charge (Q1+Q2)/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... | Progressive Tax:
Income $5,000: Tax = 10% * $5,000 = $500
Income $15,000: Tax = 10% * $10,000 + 20% * ($15,000 - $10,000) = $2,000
Proportional Tax:
Income $5,000: Tax = 10% * $5,000 = $500
Income $15,000: Tax = 10% * $15,000 = $1,500 |
[
{
"content": "Find the variance of a binomial distribution $B(n,p)$ where $n$ is a random variable following another binomial distribution $B(m,q)$. In other words, calculate $Var(B(B(m,q),p))$. Provide your answer in terms of $m$, $q$, and $p$.",
"role": "user"
}
] | m p q (1 - p q) |
[
{
"content": "Solve the trigonometric inequality: \\( \\sin^{-1} \\frac{2}{x} > \\cos^{-1} \\frac{2}{x} \\). Determine the range of values for \\( x \\).",
"role": "user"
}
] | [2, 2\sqrt{2}) |
[
{
"content": "Let $f(x)=x+5$ and let $g(x)=x^2+1$. Let $p(x)=g(x)+f(x)$ and let $q(x)=g(x)-f(x)$. Find $p(x)\\cdot q(x)$.",
"role": "user"
}
] | x^4+x^2-10x-24 |
[
{
"content": "Find the constant \\(a\\) such that the primitive function \\(F(x)\\) of \\(f(x) = x^3 - 4a^2x\\) has a minimum value of 0 and satisfies \\(F(2) = 4\\).",
"role": "user"
}
] | \sqrt{2} |
[
{
"content": "If $\\lim_{x \\to \\infty} \\frac{\\ln g(x)}{\\ln f(x)} = 0$ and $f(x) \\to \\infty$ as $x \\to \\infty$, does it follow that $\\lim_{x \\to \\infty} \\frac{g(x)}{f(x)} = 0$? Answer yes or no and justify your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Solve the linear programming problem using the simplex method, starting with the basic feasible solution where $x_1=4$ and $x_2=0$. Maximize $z = -x_1 + 2x_2$ subject to the constraints:\n\n\\[ 3x_1 + 4x_2 = 12 \\]\n\\[ 2x_1 - x_2 \\leq 12 \\]\n\\[ x_1 \\geq 0, \\quad x_2 \\geq 0 \\]",
"role":... | 6 |
[
{
"content": "Is it possible for matrices $A$ and $B$ to have different ranks while satisfying the equation $A A^T = B B^T$?",
"role": "user"
}
] | No |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 90 degrees |
[
{
"content": "Given matrices $A$, $P$, and $D$, where $PP^{-1} = I$, determine if the equation $A^n = D^n$ can be derived from $A^n = PD^nP^{-1}$. Provide a justification for your answer.",
"role": "user"
}
] | No |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $350 (unfavorable) |
[
{
"content": "Two fair 6-sided dice are rolled, and the sum of the outcomes is noted. If the dice are rolled again, in how many ways can the new sum be less than the previous sum?",
"role": "user"
}
] | 575 |
[
{
"content": "Determine the Krull dimension of the localization $S^{-1}A$, where $A$ is a commutative reduced ring and $S$ is the set of all non-zerodivisors in $A$. 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... | 1272 kW |
[
{
"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... | 258 grams/mole |
[
{
"content": "Find the remainder when the number \\(2^{2^{2^{\\cdot^{\\cdot^{2}}}}}\\), where 2 appears 2016 times, is divided by 2016.",
"role": "user"
}
] | 1024 |
[
{
"content": "Determine the number of triplets \\((m, n, p)\\) where \\(p\\) is a prime number that satisfy the equation \\(m^2 - 3mn + p^2n^2 = 12p\\).",
"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... | \frac{\sqrt{6}-6}{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... | -$5498 |
[
{
"content": "Evaluate the limit: \\[ \\lim _{x\\to 0^+}\\left(\\left(\\left(1+x\\right)^x-1\\right)^x\\right) \\] as \\( x \\) approaches 0 from the positive numbers.",
"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... | 3.604e-8 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... | 7 degrees Celsius |
[
{
"content": "Given three exponentially distributed random variables $X$, $Y$, and $Z$ with means $E[X] = \\frac{1}{\\lambda_X}$, $E[Y] = \\frac{1}{\\lambda_Y}$, and $E[Z] = \\frac{1}{\\lambda_Z}$, calculate the probability $P(X > \\max\\{Y,Z\\})$. Assume independent exponential service times with a mean of 5 f... | \dfrac{1}{3} |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n \\to \\infty} e^{-n} \\sum_{k = 0}^n \\frac{n^k}{k!} \\]",
"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... | $65,140,000 |
[
{
"content": "Let $A_1, A_2, A_3$ be three distinct points on a circle $S$. For any point $A_1'$ on $S$, the points $A_2', A_3', A_1''$ are chosen on $S$ such that $A_1'A_2', A_2'A_3', A_3'A_1''$ are perpendicular to $A_1A_2, A_2A_3, A_3A_1$, respectively. If $A_1'$ is chosen such that $A_1'$ and $A_1''$ coinci... | 90^\circ |
[
{
"content": "Side $CD$ of rectangle $ABCD$ measures 12 meters, as shown. Each of the three triangles with a side along segment $CD$ is an equilateral triangle. What is the total area of the shaded regions? Express your answer in simplest radical form. [asy]\nimport olympiad; size(150); defaultpen(linewidth(0.8... | 8\sqrt{3} |
[
{
"content": "Calculate the limit: \\[ \\lim_{x\\to 0}\\frac{\\ln(1+x^{2018})-\\ln^{2018}(1+x)}{x^{2019}}. \\]",
"role": "user"
}
] | 1009 |
[
{
"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... | 2.87 degrees East of North |
[
{
"content": "Let \\( \\Phi: \\mathbb{R}^{n-1} \\to \\mathbb{R} \\) be a vector function. Suppose there exists a set \\( S_1 \\subset \\mathbb{R}^{n-1} \\) on which the gradient \\( G = \\nabla\\Phi \\) is unbounded. Does this necessarily imply that there exists a set \\( S_2 \\) on which the Hessian determinan... | 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... | 74.3333 mL |
[
{
"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... | 0.57(3.0 {eq}\hat i {/eq} + 6.7 {eq}\hat j {/eq}) |
[
{
"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... | 2.84s |
[
{
"content": "Evaluate the integral: $$\\int_0^\\infty \\frac{1}{(1+x^{2015})(1+x^2)}\\,dx.$$",
"role": "user"
}
] | \dfrac{\pi}{4} |
[
{
"content": "Determine the values of the real number \\( k \\) for which there exists a \\( t > 2 \\) such that the inequality \\( f\\left( k \\log_{2} t \\right) + f\\left( \\left( \\log_{2} t \\right)^2 - \\log_{2} t - 2 \\right) < 0 \\) holds, given that \\( f(x) \\) is a monotonic function on \\( \\mathbb{... | (-\infty, 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... | $2100 |
[
{
"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... | 139 g/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... | There are four potential products. See attached image. |
[
{
"content": "Determine the number of tangents to the curve \\(x^{\\frac{3}{2}} + y^{\\frac{3}{2}} = a^{\\frac{3}{2}}\\) that are equally inclined to the coordinate axes.",
"role": "user"
}
] | 1 |
[
{
"content": "Find a power series expression \\( \\sum_{n=0}^\\infty A_n z^n \\) for the function \\( \\frac{1}{z^2 - \\sqrt{2}z + 2} \\), and determine its radius of convergence.",
"role": "user"
}
] | \sqrt{2} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 24.75% |
[
{
"content": "Given the function \\( f(x) = \\frac{1}{2^x - 1} + a \\) for all \\( x \\neq 0, x \\in \\mathbb{R} \\), such that \\( f(-x) = -f(x) \\), determine the value of \\( a \\).",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "How many positive integers less than 1000 are congruent to 6 (mod 11)?",
"role": "user"
}
] | 91 |
[
{
"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... | 6.81e-3 |
[
{
"content": "Evaluate the integral \\( \\frac{1}{2\\pi i}\\int_{\\gamma}\\frac{q'(\\phi)}{q(\\phi)} d\\phi \\) where \\( q(z) \\) is a polynomial with a zero of multiplicity 2 at \\( z_0 \\), and \\( \\gamma \\) is a simple \\( C^1 \\) curve that encircles \\( z_0 \\).",
"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... | 0.2396 M |
[
{
"content": "Suppose $f:E\\rightarrow E'$ is a linear embedding between two locally convex topological vector spaces. Given an open, convex subset $A\\subseteq E$ with $0\\in A$, determine if there exists an open, convex set $A'\\subseteq E'$ such that $f(A)=f(E)\\cap A'$.",
"role": "user"
}
] | Yes |
[
{
"content": "In the figure, point $O$ is the center of the circle, the measure of angle $RTB$ is 28 degrees, and the measure of angle $ROB$ is three times the measure of angle $SOT$. What is the measure of minor arc $RS$, in degrees? [asy]\nunitsize(1.2cm);\ndefaultpen(linewidth(.7pt)+fontsize(8pt));\ndotfacto... | 68^\circ |
[
{
"content": "Let $P$ be an invertible $n \\times n$ matrix. Consider the vector space $L$ consisting of $n \\times n$ matrices $X$ such that $\\operatorname{tr}(PX) = 0$. Determine the dimension of the vector space $L$. Express your answer as a single integer.",
"role": "user"
}
] | n^2 - 1 |
[
{
"content": "Find the limit, if it exists: \\[ \\lim_{x \\to -\\infty} \\frac{\\sqrt{4x^6 - x}}{x^3 + 2} \\]",
"role": "user"
}
] | -2 |
[
{
"content": "Determine whether every diagonalizable matrix can be diagonalized using an orthogonal basis of eigenvectors, even over the complex numbers. Answer with 'Yes' or 'No'.",
"role": "user"
}
] | No |
[
{
"content": "Evaluate the limit \\( \\lim\\limits_{n\\to \\infty}\\frac{1\\times3+3\\times5+\\dots+(2n-1)\\times(2n+1)}{n^2} \\) using the Stolz-Cesaro theorem.",
"role": "user"
}
] | \infty |
[
{
"content": "Compute the marginal probability density function of \\( Y \\) for the joint probability distribution function: \\[ f_{XY}(x,y) = \\begin{cases} 2e^{-y} & \\text{if } -y< x < y \\ \\text{ and }\\ 0 < y < \\infty ,\\\\ 0 & \\text{otherwise.} \\end{cases} \\]",
"role": "user"
}
] | 4ye^{-y} |
[
{
"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... | $20400 |
[
{
"content": "Find a linear operator on \\( \\mathbb{C} \\) (considered as a vector space over \\( \\mathbb{R} \\)) such that the range of the operator is a proper subset of its kernel.",
"role": "user"
}
] | 0 |
[
{
"content": "Verify the following identity for any uniformly bounded function $f$:\n\nLet $\\mathcal{F}$ be the Fourier transform, $P$ the law of a random variable $X$, and $\\varphi$ the characteristic function of $X$. Show that:\n\\[ \\int_{\\mathbb{R}} f \\ast \\mathcal{F}^{-1}\\left[\\frac{1}{\\varphi}(-\\... | f(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... | 56.65 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... | $72.73 |
[
{
"content": "Determine the number of ordered pairs \\((m, n)\\) of positive integers that satisfy the equation \\(4^m - 3^n = 1\\).",
"role": "user"
}
] | 1 |
[
{
"content": "Calculate the work done by the vector field \\( \\mathbf{F}(x,y,z)=(2x\\sin(\\pi y)-e^z)\\mathbf{i}+(\\pi x^2 \\cos (\\pi y)-3e^z)\\mathbf{j}-xe^z \\mathbf{k} \\) along the curve \\( C \\) from \\((0,0,0)\\) to \\((1,1,\\ln(2))\\), where \\( C \\) is defined by the surfaces \\( z=\\ln(1+x) \\) and... | -\dfrac{13}{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... | $11,457.07 |
[
{
"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 m |
[
{
"content": "An equilateral triangle has an area of $64\\sqrt{3}$ $\\text{cm}^2$. If each side of the triangle is decreased by 4 cm, by how many square centimeters is the area decreased?",
"role": "user"
}
] | 28\sqrt{3} |
[
{
"content": "Let $N$ be the number of ways to write $2010$ in the form $2010 = a_3 \\cdot 10^3 + a_2 \\cdot 10^2 + a_1 \\cdot 10 + a_0$, where the $a_i$'s are integers, and $0 \\le a_i \\le 99$. An example of such a representation is $1\\cdot 10^3 + 3\\cdot 10^2 + 67\\cdot 10^1 + 40\\cdot 10^0$. Find $N$.",
... | 202 |
[
{
"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^-7 mole/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... | the cat that strongly resembles Pluto |
[
{
"content": "Find the largest solution to \\[\\lfloor x \\rfloor = 5 + 100 \\{ x \\},\\]where $\\{x\\} = x - \\lfloor x \\rfloor.$",
"role": "user"
}
] | 104.99 |
[
{
"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... | 55.9 g/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... | 28.04 months |
[
{
"content": "Evaluate the triple integral \\( \\int\\limits_0^5 \\int\\limits_{\\sqrt{25-x^2}}^{-\\sqrt{25-x^2}} \\int\\limits_{\\sqrt{25-x^2-z^2}}^{-\\sqrt{25-x^2-z^2}} \\frac{1}{x^2+y^2+z^2} \\,\\mathrm dy\\,\\mathrm dz\\,\\mathrm dx \\) by converting to spherical coordinates.",
"role": "user"
}
] | 10\pi |
[
{
"content": "Consider the infinite product \\(G=\\prod_{n=1}^{\\infty} \\frac{2\\tan^{-1}(v_{n})}{\\pi}\\), where \\(v_{n}\\) is an increasing, monotonic sequence of natural numbers. Determine if there exists a sequence \\(v_{n}\\in\\mathbb{N}\\) such that \\(G>1\\) given that \\(G\\) is nonzero.",
"role":... | No |
[
{
"content": "Find the value of the positive constant \\( A \\) such that the graph of the function \\( f(x) = A\\sin x \\) intersects the graph of its derivative perpendicularly.",
"role": "user"
}
] | \sqrt{2} |
[
{
"content": "Triangle $ABC$ has side-lengths $AB = 12, BC = 24,$ and $AC = 18.$ The line through the incenter of $\\triangle ABC$ parallel to $\\overline{BC}$ intersects $\\overline{AB}$ at $M$ and $\\overline{AC}$ at $N.$ What is the perimeter of $\\triangle AMN?$\n$\\textbf{(A)}\\ 27 \\qquad \\textbf{(B)}\\ ... | 30 |
[
{
"content": "Given a simple graph on $n$ vertices, determine the maximum number of graphs that are isomorphic to the given graph. Is it $\\Theta(n^2!)$ or $\\Theta(n!)$?",
"role": "user"
}
] | \Theta(n!) |
[
{
"content": "Determine the number of distinct elements in the set \\( \\left\\{ \\left( \\frac{2+i}{2-i} \\right) ^n : n \\in \\mathbb{N} \\right\\} \\).",
"role": "user"
}
] | \infty |
[
{
"content": "Given the sequences \\(a_n=\\int_0^1 (1-x^2)^n dx\\) and \\(b_n=\\int_0^1 (1-x^3)^n dx\\) for \\(n \\in \\mathbb{N}\\), find the value of \\(L=\\lim_{n\\to \\infty} \\left(10 \\sqrt[n]{a_n} + 5\\sqrt[n]{b_n}\\right)\\).",
"role": "user"
}
] | 15 |
[
{
"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... | 6.24 N |
[
{
"content": "If $n$ is the smallest positive integer for which there exist positive real numbers $a$ and $b$ such that\n\\[(a + bi)^n = (a - bi)^n,\\]compute $\\frac{b}{a}.$",
"role": "user"
}
] | \sqrt{3} |
[
{
"content": "Given a set $X$ and $n$ subsets $S_1, S_2, \\ldots, S_n$ of $X$, each of cardinality $n$, can you always totally order each $S_i$ such that each $k$-th transverse set $T_k$ contains exactly $n$ distinct elements of $X$ for all $1 \\leq k \\leq n$? Here, $T_k$ is defined as the set containing the $... | Yes |
[
{
"content": "Evaluate the triple integral: \\[ \\int_{-1}^1 \\int_{0}^{2} \\int_{-2}^2 \\left[1+x^2 z^2 + x y \\cos(x^4 + y^4 + z^4) \\right]\\;\\operatorname d\\!z \\;\\operatorname d\\!y \\operatorname d\\!x \\]",
"role": "user"
}
] | \dfrac{208}{9} |
[
{
"content": "Find all functions \\( f: \\mathbb{N_{0}} \\rightarrow \\mathbb{N_{0}} \\) that satisfy the equation \\( f(f(n)) + f(n) = 2n + 3k \\) for all \\( n \\in \\mathbb{N_{0}} \\), where \\( k \\) is a fixed natural number.",
"role": "user"
}
] | f(n) = n + k |
[
{
"content": "Given the function \\( w = z \\tan^{-1}\\left(\\frac{x}{y}\\right) \\), calculate the value of \\( \\frac{\\partial^2 w}{\\partial x^2} + \\frac{\\partial^2 w}{\\partial y^2} + \\frac{\\partial^2 w}{\\partial z^2} \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Determine the values of \\( \\alpha \\) for which the series \\( \\sum_{n = 1}^{\\infty}\\big(2^{n^{-\\alpha}} - 1\\big) \\) converges.",
"role": "user"
}
] | \alpha > 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... | 26.34 cm |
[
{
"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/(42*pi)*10^(-2) F |
[
{
"content": "Evaluate the expression \\( \\frac{J_{6,3}}{J_{5,2}} \\), where \\( J_{m, n} = \\int_{0}^{\\pi/2} \\cos^m x \\cos nx \\, dx \\).",
"role": "user"
}
] | \dfrac{2}{3} |
[
{
"content": "Find the positive integer $a$ such that for the function \\( f(x) = (60073 - x^{10})^{1/10} \\), it holds that \\( f'(2) = \\frac{1}{f'(a)} \\).",
"role": "user"
}
] | 3 |
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