prompt list | ground_truth string |
|---|---|
[
{
"content": "Given three probability groups with proportions \\( p_1(\\theta):p_2(\\theta):p_3(\\theta)=(2-\\theta):(2+3\\theta):(2-\\theta) \\) and sample sizes \\( n_1=25, n_2=55, n_3=20 \\), find the log-likelihood function and determine the maximum likelihood estimate (MLE) of \\( \\theta \\).",
"role"... | \dfrac{26}{49} |
[
{
"content": "Find a cubic polynomial \\( g(x) \\) such that \\( g(x) - 2 \\) is divisible by \\( (x+1)^2 \\) and \\( g(x) \\) is divisible by \\( x^2 \\).",
"role": "user"
}
] | 4x^3 + 6x^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... | $45,620 |
[
{
"content": "Determine whether the integral \\( \\int_0^1 (1-x)\\ln(x) \\, dx \\) converges. If it does, find its value.",
"role": "user"
}
] | -\dfrac{3}{4} |
[
{
"content": "Evaluate the limit \\( \\lim_{z\\rightarrow 0}\\frac{\\ln(\\sin (x+z))-\\ln(\\sin x)}{z} \\) without using L'Hopital's rule, where \\( x\\in\\left(0,\\dfrac{\\pi}{2}\\right) \\).",
"role": "user"
}
] | \cot x |
[
{
"content": "Given a function \\( f \\) defined for all real numbers, it is known that for every \\( a < x < b \\), the inequality \\( \\min (f(a), f(b)) < f(x) < \\max (f(a), f(b)) \\) is satisfied. Does it follow that \\( f \\) is monotonous? Answer with 'Yes' or 'No' and justify your answer.",
"role": "... | 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... | $22,400 |
[
{
"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... | 1348.73 |
[
{
"content": "Find the number \\( b \\) such that \\( 0 \\leq b \\leq 85 \\) and \\( b \\equiv (200^{80} + 1)^{20} \\pmod{86} \\).",
"role": "user"
}
] | 21 |
[
{
"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... | $8,295.02 |
[
{
"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... | 3.6e-8 M |
[
{
"content": "Let $M$ be a topological manifold that is homeomorphic to the interior of a compact manifold with boundary. If $f \\colon M' \\to M$ is a finitely-sheeted covering, is $M'$ also homeomorphic to the interior of a compact manifold with boundary?",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate the limit:\n\\[ \\lim_{k \\to \\infty, p \\to \\infty} \\frac{a_{k,p}}{b_{k,p}} \\]\nwhere\n\\[\na_{k,p} = \\sum_{n=1}^{k} (2^{n}-1) + \\sum_{n=1}^{k}{(2^{2n}-2^n-1)} + \\sum_{n=1}^{k}{(2^{3n}-2^{2n}-2^n-1)} + \\sum_{n=1}^{k}{(2^{4n}-2^{3n}-2^{2n}-2^n-1)} + \\ldots + \\sum_{n=1}^{k}{(2^{p... | 0 |
[
{
"content": "Given the elliptic curve \\( y^2 \\equiv x^3 + 17x + 5 \\pmod{59} \\) and the point \\( P = (4, 14) \\), calculate the point \\( 8P \\) on the curve.",
"role": "user"
}
] | (16, 40) |
[
{
"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... | Actual Results | Flexible Budget | Variance | Flexible Budget Sales Volume Variance | Static Budget
---|---|---|---|---|
Units | 26,300 | 26,300 | 0 | 5,700 F | 20,600
Sales Revenue | $557,900 | $604,900 | $47,000 U | $131,100 F | $473,800
Variable costs | 113,700 | 78,900 | 34,800 U | 17,100 U | 61,800
Fixed costs | 134,000 | 143,000 | 9,000 F | 0 | 143,000
Operating Income | $310,200 | $383,000 | $72,800 U | $114,000 F | $269,000 |
[
{
"content": "In how many ways can 7 distinct balls be packed into 16 boxes, where 7 of the boxes are unlabeled and green, and 9 of the boxes are unlabeled and yellow?",
"role": "user"
}
] | 14214 |
[
{
"content": "Evaluate the integral \\( \\int_0^{\\pi/2} (\\cos^4 \\theta + \\sin^4 \\theta) \\, d\\theta \\) without using polar or rectangular coordinates.",
"role": "user"
}
] | \dfrac{3\pi}{8} |
[
{
"content": "Let $f, g : \\mathbb{R} \\rightarrow \\mathbb{R}$ be two differentiable functions such that $f'(x) = g(x)$, $g'(x) = f(x)$, $f(0) = 1$, and $g(0) = 0$. Define the function $F(x) = f^2(x) + g^2(x)$. Which of the following statements is correct?\n\n(a) $F(x) = 1 - \\sin(x)$.\n(b) $F$ is the constant... | C |
[
{
"content": "Determine the value of \\(k\\) such that the polynomial \\(x^{2n} + 1 + (x+1)^{2n}\\) is not divisible by \\(x^2 + x + 1\\) if and only if \\(n\\) is divisible by \\(k\\). Choose the correct answer from the following options:\n\nA. 2\nB. 3\nC. 6\nD. None of the above",
"role": "user"
}
] | B |
[
{
"content": "Determine whether the following statement is true or false: Every non-measurable subset of a Borel set with positive Lebesgue measure is contained in a null set.",
"role": "user"
}
] | B |
[
{
"content": "Determine the value of $a$ such that the following integral equation holds:\n\n$$\\int \\limits^{\\infty }_{0} \\frac{dx}{a^2 + \\left(x - \\frac{1}{x}\\right)^2} = \\frac{\\pi}{5050}$$\n\nwhere $a^2 \\geq 0$. Provide your answer as a single value of $a$.",
"role": "user"
}
] | 2525 |
[
{
"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... | 2.70% |
[
{
"content": "Suppose $(a_n)$ is a sequence of positive real numbers such that \\(\\lim\\limits_{n\\to\\infty} \\frac{a_na_{n+2}}{a_{n+1}^2}=L^2\\) for some non-negative real number \\(L\\). Determine \\(\\lim\\limits_{n\\to\\infty} \\sqrt[n^2]{a_n}\\) in terms of \\(L\\).",
"role": "user"
}
] | L |
[
{
"content": "Determine the radius of convergence of the series \\( \\sum_{n=0}^\\infty \\left(\\frac{z^2-1}{z^2+1}\\right)^n \\) using the ratio test.",
"role": "user"
}
] | 1 |
[
{
"content": "Determine whether the inequality \\(| \\lambda | \\leq \\max_j \\left(\\sum_{i=1}^n | A_{ij} |\\right)\\) holds for any real square matrix \\(A\\) and its eigenvalue \\(\\lambda\\).",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate the integral \\( \\int_0^4 r(\\sqrt{25-r^2}-3) \\, dr \\).",
"role": "user"
}
] | \dfrac{26}{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... | 6.7e-6 |
[
{
"content": "Let $\\{N(t): t \\geq 0\\}$ be a Poisson process with rate $1$, and let $T_1 < T_2 < \\cdots$ denote the arrival times. Determine the probability density function of the random variable $Y = \\frac{T_1}{T_3}$. Provide your answer in terms of $y$. ",
"role": "user"
}
] | 2(1 - y) |
[
{
"content": "Solve the exponential equation: \\[ 2^{2x} = 7 \\cdot 2^{x+\\sqrt{x-1}} + 8 \\cdot 2^{2\\sqrt{x-1}} \\] subject to the condition \\( x - 1 \\geq 0 \\).",
"role": "user"
}
] | 5 |
[
{
"content": "Calculate the limit: \\[ \\lim_{x\\rightarrow \\infty} e^{x \\ln (\\sin(1/x)+\\cos(1/\\sqrt{x}))}. \\]",
"role": "user"
}
] | \sqrt{e} |
[
{
"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... | 5625 |
[
{
"content": "Consider a right angle $X O Y$ with distances $O A = a$ and $O B = b$ on the side $O Y$, where $a < b$. From a point $M$ on the line $O X$ such that $O M = x$, determine the angle at which the segment $A B$ is viewed. Find the value of $x$ that maximizes this angle.",
"role": "user"
}
] | \sqrt{ab} |
[
{
"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... | 6.50 m/s^2 |
[
{
"content": "Determine the formula for the number of solutions to the equation \\(x^3 \\equiv 1 \\pmod{mn}\\), where \\(m\\) and \\(n\\) are distinct prime numbers. Consider the maximum number of solutions possible for \\(n\\) or \\(m\\) individually, which is 3. What is the limit to the number of solutions fo... | 9 |
[
{
"content": "Let $T$ and $S$ be diagonalizable linear operators on $\\mathbb{R}^n$ such that $TS = ST$. If $E$ is an eigenspace of $T$, is the restriction of $S$ to $E$ also diagonalizable? 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... | <img>/cimages/multimages/16/cms_82360341849460055052.jpg</img> |
[
{
"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... | $51,170.45 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 9.33e-32 M |
[
{
"content": "Evaluate the expression \\( \\int_{-1}^{1} f(x) \\, dx + \\int_{0}^{4} f^{-1}(x) \\, dx \\) where \\( f(x) = x^3 + 3x + 4 \\).",
"role": "user"
}
] | \dfrac{23}{4} |
[
{
"content": "Given segments $a$ and $b$, construct a segment $x$ such that $x = \\sqrt{a^2 + ab + b^2}$. Provide a geometric construction for this segment.",
"role": "user"
}
] | 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... | $12,800 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n \\to \\infty}\\left(\\frac{12^n-6^n}{3^n-6^n} +\\frac{9^n-3^n+\\cos(3^n)}{(-1)^n+9^n}\\right) \\]",
"role": "user"
}
] | -\infty |
[
{
"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... | 177.8 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... | 4285.7 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... | $12.42 |
[
{
"content": "Determine the number of homomorphisms from the dihedral group \\(\\mathbb{D}_5\\) to the alternating group \\(\\mathbb{A}_4\\). Additionally, provide an example of a non-trivial homomorphism.",
"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... | Mother's genotype: hhee
Mother's gametes: he
Father's genotype: HhEe
Father's gametes: HE, He, hE, he
Offspring genotypes and phenotypes:
HhEe: curly hair, long eyelashes
Hhee: curly hair, short eyelashes
hhEe: straight hair, long eyelashes
hhee: straight hair, short eyelashes
Probability ratio: 1:1:1:1 |
[
{
"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... | $6,105.01 |
[
{
"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... | 2.26 moles |
[
{
"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... | -0.75 |
[
{
"content": "Given that $x^{x^4} = 4$, find the value of $x^{x^2} + x^{x^6}$.",
"role": "user"
}
] | 18 |
[
{
"content": "Calculate the absolute value of the line integral \\( \\int \\vec{F} \\cdot d\\vec{r} \\) for the vector field \\( \\vec{F} \\) such that \\( \\nabla \\times \\vec{F} = x\\hat{i} - y\\hat{j} \\), where \\( C \\) is the closed curve forming the boundary of the surface \\( S \\) of the cone \\( z = ... | 0 |
[
{
"content": "Calculate the Legendre symbol \\( \\left(\\frac{83}{127}\\right) \\).",
"role": "user"
}
] | -1 |
[
{
"content": "A cylindrical barrel with radius $4$ feet and height $10$ feet is full of water. A solid cube with side length $8$ feet is set into the barrel so that the diagonal of the cube is vertical. The volume of water thus displaced is $v$ cubic feet. Find $v^2$.\n[asy] import three; import solids; size(5c... | 384 |
[
{
"content": "Calculate the sum of the series \\( \\sum\\limits_{b=0}^\\infty (b+k+1)2^{-b-k-1}\\binom{b+k}{k} \\).",
"role": "user"
}
] | 2(k + 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... | 12.1455 |
[
{
"content": "Evaluate the limit \\( \\lim_{n\\to\\infty}\\frac{x_n}{\\sqrt n} \\), given that \\( x_1 > 0 \\), \\( x_{n+1} = x_n + \\frac{1}{x_n} \\) for \\( n \\in \\mathbb{N} \\), and \\( \\{x_n\\} \\) is a monotonically increasing sequence with \\( x_n \\geq 2 \\) for \\( n \\geq 2 \\).",
"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... | {eq}s(t) = \frac{t^4}{12} - \frac{3t^3}{2} + \frac{5t^2}{2} + \frac{115t}{6}{/eq} |
[
{
"content": "Find the number of integers $n$ such that $2000 \\leq n \\leq 2010$ and $7$ divides $\\left\\lfloor \\frac{\\sqrt{2}}{4}(1+\\sqrt{2})^n \\right\\rfloor$. Provide your answer as a single integer.",
"role": "user"
}
] | 5 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 1.72 A |
[
{
"content": "Find the value of \\(x\\) that satisfies the equation \\(x = \\ln(x+1)\\).",
"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... | (2186.72, 2733.41) |
[
{
"content": "Given the integral representation of the arccosine function:\n\\[ \\arccos(x) = \\int_{1}^{x} -\\frac{1}{\\sqrt{1 - t^2}} \\, \\mathrm{d}t \\]\nIf \\(\\theta\\) is a known angle such that:\n\\[ \\theta = \\int_{1}^{x} -\\frac{1}{\\sqrt{1 - t^2}} \\, \\mathrm{d}t \\]\nDetermine if it is possible to... | x = \cos(\theta) |
[
{
"content": "What is the sum of $1+2+4+8+16+ \\cdots + 1024$?",
"role": "user"
}
] | 2047 |
[
{
"content": "A polynomial function \\( f(x) \\) with real coefficients has a remainder of \\( 15 \\) when divided by \\( x-3 \\) and a remainder of \\( 2x+1 \\) when divided by \\( (x-1)^2 \\). Determine the remainder when \\( f(x) \\) is divided by \\( (x-3)(x-1)^2 \\). Express the remainder as \\( ax^2+bx+c ... | 2x^2 - 2x + 3 |
[
{
"content": "Given the equations \\( \\frac{1}{a} + \\frac{1}{b} + \\frac{1}{a+x} = 0 \\), \\( \\frac{1}{a} + \\frac{1}{c} + \\frac{1}{a+y} = 0 \\), and \\( \\frac{1}{a} + \\frac{1}{x} + \\frac{1}{y} = 0 \\), where \\(a \\neq 0\\), \\(b \\neq 0\\), and \\(c \\neq 0\\), find the value of \\(a+b+c\\).",
"rol... | 0 |
[
{
"content": "Evaluate the limit: \\[ l = \\lim_{x\\to 0}\\frac{e^{\\sin x}-e^{\\tan x}}{e^{\\sin 2x}-e^{\\tan 2x}} \\]",
"role": "user"
}
] | \dfrac{1}{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... | 2,840 |
[
{
"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... | $222,972 |
[
{
"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... | arctan(B/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... | The final answer is an image. |
[
{
"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 atm |
[
{
"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... | $747.26 |
[
{
"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... | lower |
[
{
"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... | 110.9 nm |
[
{
"content": "Calculate the variance of the Laplace distribution using its moment generating function.",
"role": "user"
}
] | 2b^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... | Variable cost: $1.30 per mile, Fixed cost: $2,300 |
[
{
"content": "Let $A$ be a Banach algebra, and suppose $e \\in A$ such that $e^2 = e$ and $eAe$ is a division algebra. Define the operators $T_e: A \\to A$ by $T_e(a) = eae$ and $S_e: Ae \\to Ae$ by $S_e(ae) = eae$. If $S_e$ is not a weakly compact operator, is $T_e$ also not a weakly compact operator?",
"r... | 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... | MV^2/2D |
[
{
"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... | 6.31 m |
[
{
"content": "Find the minimum value of \\((\\alpha+5-3|\\cos \\beta|)^2+(\\alpha-2|\\sin \\beta|)^2.\\)",
"role": "user"
}
] | 2 |
[
{
"content": "Let $A_1 A_2 \\dotsb A_{11}$ be a regular 11-gon inscribed in a circle of radius 2. Let $P$ be a point such that the distance from $P$ to the center of the circle is 3. Calculate the value of $PA_1^2 + PA_2^2 + \\dots + PA_{11}^2$. \\(\\boxed{\\text{Answer}}\\)",
"role": "user"
}
] | 143 |
[
{
"content": "Determine the value of the limit: $$\\lim_{n\\to \\infty}n\\left(e-\\sum_{k=0}^{n}\\frac{1}{k!}\\right)$$ where $e$ is defined as $\\sum_{k=0}^{\\infty}\\frac{1}{k!}$. Is the limit equal to $0$?",
"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... | 0.247 |
[
{
"content": "Transform the quadratic form \\( F(x) = x_1^2 - 3x_3^2 - 2x_1x_2 + 2x_1x_3 - 6x_2x_3 \\) into its normal form over the real number field.",
"role": "user"
}
] | y_1^2 - y_2^2 |
[
{
"content": "Find the minimum value of \\(2x + 3y + 4z\\) given that \\(x, y, z \\in \\mathbb{R}\\) are non-negative and satisfy the inequality \\(z(x+y+z) \\geq 4 - xy\\).",
"role": "user"
}
] | 8 |
[
{
"content": "Let $S$ be the set of positive real numbers. Let $f : S \\to \\mathbb{R}$ be a function such that\n\\[f(x) f(y) = f(xy) + 2005 \\left( \\frac{1}{x} + \\frac{1}{y} + 2004 \\right)\\]for all $x,$ $y > 0.$\n\nLet $n$ be the number of possible values of $f(2),$ and let $s$ be the sum of all possible ... | \frac{4011}{2} |
[
{
"content": "Find a change of variable that will transform the quadratic form \\(x_1^2-x_3^2-4x_1x_2+4x_2x_3\\) into a sum of squares. Express the quadratic form in terms of the new variables.",
"role": "user"
}
] | y_1^2 - y_2^2 |
[
{
"content": "Determine whether a region in the complex plane can be both closed and a star domain.",
"role": "user"
}
] | Yes |
[
{
"content": "Determine the number of real solutions to the equation \\( \\sin(\\cos(\\sin x)) = \\cos(\\sin(\\cos x)) \\) for \\( x \\in [0, \\pi] \\).",
"role": "user"
}
] | 2 |
[
{
"content": "Find an explicit expression for the determinant of the matrix $A_n$ defined by the elements:\n\\[ a_{i,j} = \\begin{cases} 1 & \\text{if } i=j \\\\ -j & \\text{if } i = j+1 \\\\ i & \\text{if } i = j-1 \\\\ 0 & \\text{otherwise} \\end{cases} \\]",
"role": "user"
}
] | n! |
[
{
"content": "Find the number of compositions of 50 into exactly 5 parts, where the first and last parts are odd, the other parts are even, and no part is greater than 20. Use generating functions to determine the number of such compositions and simplify your answer to an integer.",
"role": "user"
}
] | 5875 |
[
{
"content": "Determine whether a divergent sequence can have a convergent sequence of its squares. Answer 'yes' or 'no'.",
"role": "user"
}
] | yes |
[
{
"content": "Solve the congruence equation: \\(636^{369} \\equiv x \\pmod{126}\\).",
"role": "user"
}
] | 90 |
[
{
"content": "Determine the order of the automorphism \\( \\sigma \\) in the Galois group \\( \\text{Gal} \\left( \\frac{\\mathbb{Q}(\\sqrt[3]{3},\\sqrt{-3})}{\\mathbb{Q}} \\right) \\), where \\( \\sigma(\\sqrt[3]{3}) = \\omega\\sqrt[3]{3} \\) and \\( \\sigma(\\sqrt{-3}) = -\\sqrt{-3} \\).",
"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... | 7813.76 |
[
{
"content": "A sports conference has 14 teams in two divisions of 7. How many games are in a complete season for the conference if each team must play every other team in its own division twice and every team in the other division once?",
"role": "user"
}
] | 133 |
[
{
"content": "Calculate the second-order directional derivative \\(D_{\\mathbf{v}_0 \\mathbf{u}_0}\\mathbf{f}(\\mathbf{x}_0)\\) of the vector-valued function \\(\\mathbf{f}(\\mathbf{x}) = (x_1 + x_2)^2\\mathbf{e}_1 + x_1\\sin(x_2)\\mathbf{e}_2\\) at the point \\(\\mathbf{x}_0 = \\mathbf{e}_1 + \\pi\\mathbf{e}_2... | -\mathbf{e}_2 |
[
{
"content": "Given two linear functions \\( f(x) \\) and \\( g(x) \\) such that the graphs \\( y = f(x) \\) and \\( y = g(x) \\) are parallel lines, not parallel to the coordinate axes. If the minimum value of the function \\( (f(x))^{2} + 2 g(x) \\) is 5, find the minimum value of the function \\( (g(x))^{2} ... | -7 |
[
{
"content": "Given that \\( z \\) is an imaginary number and \\( z + \\frac{3}{2z} \\) is a real root of the equation \\( x^2 - 2ax + 1 - 3a = 0 \\), determine the range of values for the real number \\( a \\).",
"role": "user"
}
] | \dfrac{1}{3} |
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