prompt list | ground_truth string |
|---|---|
[
{
"content": "Calculate the limit: \\[ \\lim_{n\\to\\infty}\\frac{(2n+1)(2n+3)\\cdots(4n+1)}{(2n)(2n+2)\\cdots(4n)} \\]",
"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... | $66,059 |
[
{
"content": "In triangle $ABC$, $O$ is the circumcenter and $D$ is the midpoint of side $BC$. Given that $\\overrightarrow{AO} \\cdot \\overrightarrow{AD} = 4$ and $BC = 2\\sqrt{6}$, determine the length of $AD$.",
"role": "user"
}
] | \sqrt{2} |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x\\rightarrow 0^{+}}\\frac{\\int_{1}^{+\\infty}\\frac{e^{-xy} - 1}{y^3}dy}{\\ln(1+x)}. \\]",
"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... | 225820 V/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... | $13,000 |
[
{
"content": "Consider a complete graph $K_9$. Determine the number of spanning trees such that vertex 1 has a degree of 4.",
"role": "user"
}
] | 143360 |
[
{
"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... | 5465.5 g |
[
{
"content": "Determine whether the Lebesgue measure of the set \\( \\{t \\in \\mathbb{R} \\mid \\sum_{i=1}^N \\beta_i e^{\\alpha_i t} = 0 \\} \\) is zero, given that \\( \\{\\beta_i\\}_{i=1}^N \\) and \\( \\{\\alpha_i\\}_{i=1}^N \\) are non-zero complex numbers.",
"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... | 3120 Pa |
[
{
"content": "Suppose $x$ and $y$ are elements in a group $G$ where the order of $y$ is not equal to $1$, $yx = xy^2$, and the order of $x$ is $3$. Determine the order of $y$.",
"role": "user"
}
] | 7 |
[
{
"content": "Let $G$ be a finite group with two generators $x$ and $y$ such that $|x|=2$, $|y|=2$, and $|xy|=4$. Determine the isomorphism class of the quotient group $G/Z(G)$. Provide your answer as the name of the group or its standard notation.",
"role": "user"
}
] | C_2 \times C_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... | 25.83 atm |
[
{
"content": "Determine the values of $k$ for which the improper integral \\( \\int_{2}^{\\infty} \\frac{1}{x^k\\ln(x)}\\,dx \\) converges.",
"role": "user"
}
] | k > 1 |
[
{
"content": "Consider the function \\( g(x) \\) defined as:\n\\[\ng(x) =\n \\begin{cases}\n1, & \\text{if $x$ belongs to $[-1,1]$} \\\\\n0, & \\text{otherwise}\n\\end{cases}\n\\]\n\nand the function \\( f(x) \\) defined as:\n\\[\nf(x)=\\lim_{h \\to 0} \\frac{\\int_{x-h}^{x+h} g(y) \\,dy}{2h}.\n\\]\n\nFind th... | \dfrac{1}{2} |
[
{
"content": "Given the function \\( f(x) = e^{-1/x^2} \\) with \\( f(0) = 0 \\), determine \\( f'(0) \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Evaluate the infinite series \\( \\sum_{n=1}^{\\infty} \\frac{2n}{(n+1)!} \\). Determine the sum of this series.",
"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... | 13.28 m/s |
[
{
"content": "Evaluate the contour integral \\( \\frac{(1-|a|^2)}{\\pi} \\int_{\\gamma} \\frac{|dz|}{|z+a|^2} \\), where \\( a \\in \\mathbb{C} \\) with \\( |a| < 1 \\) and \\( \\gamma \\) is the unit circle \\( |z| = 1 \\) traced counterclockwise. Use the theorem: \\( \\int_{\\gamma} f |dz| = \\int_{a}^{b} f(\... | 2 |
[
{
"content": "Let $H$ be a Hilbert space, and let $T:H \\to H$ be a bounded self-adjoint operator satisfying the equation $T^3 + T^2 + T + I = 0$. Determine the spectrum $\\sigma(T)$.",
"role": "user"
}
] | \{-1\} |
[
{
"content": "Compute the contour integral \\( \\int_{|z|=3}\\frac{e^{-z}}{z^2}dz \\) using residue calculus.",
"role": "user"
}
] | -2\pi i |
[
{
"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... | 3.375e+20 |
[
{
"content": "Determine the number of solutions to the equation \\(16 \\sin^3x = 14 + (\\sin x + 7)^{\\frac{1}{3}}\\) in the interval \\([0, 4\\pi]\\).",
"role": "user"
}
] | 2 |
[
{
"content": "Use the summation convention to simplify the expression \\( g^{1}_{11} + g^{1}_{12} + g^{1}_{21} + g^{1}_{22} + g^{2}_{11} + g^{2}_{12} + g^{2}_{21} + g^{2}_{22} \\) and determine the value of \\( n \\) necessary for this simplification.",
"role": "user"
}
] | 2 |
[
{
"content": "Find \\( \\lim_{n\\to\\infty} \\lvert x_n \\rvert ^\\frac{1}{n} \\) given that the sequence \\( x_{n+1}=3x_n + \\frac{2}{x_n^2} \\) with \\( x_1=1 \\).",
"role": "user"
}
] | 3 |
[
{
"content": "Solve the differential equation: \\[ f'''(x) = f(x)f'(x)f''(x) \\]",
"role": "user"
}
] | f(x) = Ax + 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... | 6164.38 N/C |
[
{
"content": "Can the inverse of a reducible unitary matrix be expressed in block triangular form with the same block structure as the original matrix?",
"role": "user"
}
] | Yes |
[
{
"content": "Determine whether the expression \\(4^{1536} - 9^{4824}\\) is divisible by 35.",
"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... | cognitive dissonance |
[
{
"content": "For any integer \\( p \\ge 3 \\), determine the largest integer \\( r \\) such that \\((x-1)^r\\) is a factor of the polynomial \\(2x^{p+1}-p(p+1)x^2+2(p^2-1)x-p(p-1)\\).",
"role": "user"
}
] | 3 |
[
{
"content": "Is it always true that two matrices with the same characteristic polynomial and the same number of linearly independent eigenvectors for each eigenvalue are similar?",
"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... | 134 L |
[
{
"content": "Let $f: \\mathbb{R} \\to \\mathbb{R}$ be a function that is bounded on bounded intervals. Given that $\\lim\\limits_{x \\to +\\infty} [f(x+1) - f(x)] = \\ell$, prove that $\\lim\\limits_{x \\to +\\infty} \\frac{f(x)}{x} = \\ell$. Provide a rigorous justification for your answer.",
"role": "use... | \ell |
[
{
"content": "Evaluate the integral \\( \\int_{-\\pi/2}^{\\pi/2} \\frac{\\cos^3 x}{e^x+1} \\, dx \\).",
"role": "user"
}
] | \dfrac{2}{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... | 11 g |
[
{
"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... | 13.55% |
[
{
"content": "Compute the limit: \\[ \\lim_{n\\to\\infty}\\left(\\frac{a_{n+1}}{\\sqrt[n+1]{b_{n+1}}}-\\frac{a_n}{\\sqrt[n]{b_n}}\\right), \\] given that \\((a_n)_{n\\geq 1}\\) and \\((b_n)_{n\\geq 1}\\) are positive real sequences satisfying \\[ \\lim_{n\\to\\infty}\\frac{a_{n+1}-a_n}{n}=a\\in \\mathbb{R}_{>0}... | \dfrac{a e}{2 b} |
[
{
"content": "Given points \\( A, B, \\) and \\( C \\) in the plane that do not lie on the same line, if \\( \\vec{OA} + \\vec{OB} + \\vec{OC} = 0 \\), and \\( |\\vec{OA}| = \\sqrt{6} \\), \\( |\\vec{OB}| = 2 \\), and \\( |\\vec{OC}| = \\sqrt{14} \\), find the area of triangle \\( ABC \\).",
"role": "user"
... | 3\sqrt{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... | bipolar |
[
{
"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... | 1038.8 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... | 4.09 mL |
[
{
"content": "Does the property that \"closed bounded subsets of \\(\\Bbb{R}\\) are compact\" imply the least upper bound property of \\(\\Bbb{R}\\)? Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate the limit: $$\\lim_{x\\rightarrow 0}\\frac{\\int_{\\cos x}^{1}\\arctan t^2\\ dt}{\\tan^2 x}$$",
"role": "user"
}
] | \dfrac{\pi}{8} |
[
{
"content": "Determine the norm of the integral operator \\( A: (C^1[0,1];|||\\cdot|||)\\rightarrow(C^1[0,1];||\\cdot||_{\\infty}) \\) defined by \\( Af(x)=\\int_0^x f(t)dt \\), where \\( |||f|||= ||f||_\\infty+||Af||_\\infty \\). What is \\( ||A||? \\)",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "Evaluate the sum of the Fourier coefficients \\( \\sum_{n=-\\infty}^\\infty \\hat{g}(n) \\) for the function \\( g(x) = \\cos(\\pi (x + 2010)) \\), where \\( \\hat{g}(n) \\) is the \\( n \\)-th Fourier coefficient of \\( g(x) \\).",
"role": "user"
}
] | 1 |
[
{
"content": "Evaluate the limit \\(\\displaystyle\\lim_{(x,y)\\rightarrow (0,0)} \\frac{\\tan(x^2+y^2)}{\\arctan(\\frac{1}{x^2+y^2})}\\).",
"role": "user"
}
] | 0 |
[
{
"content": "In the Poincaré disc model of hyperbolic geometry, consider a triangle where all interior angles are equal. Does this imply that the lengths of its sides, measured as the geodesics between the vertices, are all equal?",
"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... | US: 4 cars; Japan: 2/3 of a car |
[
{
"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... | $120,100 |
[
{
"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... | 435 |
[
{
"content": "What is the product of the two integer values for $x$ for which $|x^2 - 16|$ is a prime number?",
"role": "user"
}
] | -9 |
[
{
"content": "Given $g(1) = g'(1) = 4$ and the function defined by the integral \\( f(x) = \\int_0^x (x - t^2)g(t) \\, dt \\), find the value of \\( f''(1) \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Determine the range of real numbers \\(a\\) such that the inequality \\[ \\sin 2 \\theta - (2 \\sqrt{2} + \\sqrt{2} a) \\cdot \\sin \\left(\\theta + \\frac{\\pi}{4}\\right) - \\frac{2 \\sqrt{2}}{\\cos \\left(\\theta - \\frac{\\pi}{4}\\right)} > -3 - 2a \\] holds for all \\(\\theta \\in \\left[0, \... | (3, \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... | 1108.38 |
[
{
"content": "Given a Lie algebra $\\frak{g}$ in $\\mathbb{R}^3$ spanned by $X, Y, Z$ with the relations $[X,Y]=Y$ and $[X,Z]=Y+Z$, determine the commutator $[Y, Z]$. \\(\\text{Express your answer in terms of } X, Y, \\text{ and } Z.\\)",
"role": "user"
}
] | 0 |
[
{
"content": "For $n \\geq 3$, does the group $K$, generated by the group of diagonal matrices with positive entries and the group of permutation matrices, contain any subgroup $H$ of $\\mathrm{GL}_n(\\mathbf{R})$ where each element is a matrix with non-negative entries?",
"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... | $17 |
[
{
"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 mph |
[
{
"content": "Given a function \\(f\\colon \\Bbb R^2\\setminus\\{(0,0)\\}\\rightarrow \\Bbb R\\) which satisfies the equation \\(x \\frac{\\partial f}{\\partial y} -y \\frac{\\partial f}{\\partial x} = f\\), prove that the function is zero on \\(\\mathbb{R}^2\\setminus\\{(0,0)\\}\\).",
"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... | Year | Depreciation Expense | Accumulated Depreciation | Net Book Value
---- | ------------------ | ------------------------ | --------------
2016 | $3,300 | $3,300 | $35,700
2017 | $9,900 | $13,200 | $25,800
2018 | $13,200 | $26,400 | ... |
[
{
"content": "Find the infimum of the set \\(\\left\\{ \\frac{ | B \\cap \\{1,\\ldots,n\\}|}{n} : n \\in \\mathbb{N} \\right\\}\\) where \\(B = \\{ \\lfloor n \\ln (n+2) \\rfloor : n = 1,2,3,... \\}\\).",
"role": "user"
}
] | 0 |
[
{
"content": "Let $N$ denote the number of $7$ digit positive integers have the property that their digits are in increasing order. Determine the remainder obtained when $N$ is divided by $1000$. (Repeated digits are allowed.)",
"role": "user"
}
] | 435 |
[
{
"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... | 3.31e4 J |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $492,000 |
[
{
"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... | 83.2 m |
[
{
"content": "Determine whether the constant in the integral identity \\( \\int_{\\mathbb{R}^n} f(x) \\, dx = \\int_0^\\infty \\int_{\\mathbb S^{n-1}} f(r\\omega) \\, d\\sigma(\\omega) \\, r^{n-1} dr \\) depends on \\(n\\), \\(\\pi\\), or is independent of these variables.",
"role": "user"
}
] | 1 |
[
{
"content": "A convex 1000-sided polygon has 1000 vertices, and there are an additional 500 points inside the polygon such that no three of these 500 points are collinear. The polygon is triangulated so that all 1500 points are vertices of the triangles, and these triangles have no other vertices. Determine th... | 1998 |
[
{
"content": "Let \\( f: \\mathbb{R}^+ \\to \\mathbb{R}^+ \\) be a \\( C^\\infty \\) function such that \\( f^{(k)}(x) > 0 \\) for all \\( k \\in \\mathbb{N} \\) and \\( f(m) \\) is an integer for all natural numbers \\( m \\). Does \\( f(x)/f'(x) \\) necessarily remain bounded on \\([1, \\infty)\\)? Provide a ... | Yes |
[
{
"content": "Let \\( A \\in \\text{Mat}_{n,n}(\\mathbb{C}) \\) be a diagonalizable matrix with distinct eigenvalues \\( u_1, \\ldots, u_k \\). Prove that \\((A-u_1I_n) \\cdots (A-u_k I_n) = 0\\).",
"role": "user"
}
] | 0 |
[
{
"content": "A coin is tossed until two consecutive heads appear. The probability of the coin landing heads, denoted by \\( p \\), follows a beta distribution with parameters \\( p \\sim \\beta(6, 8) \\). What is the expected number of coin tosses required to achieve two consecutive heads?",
"role": "user"... | \dfrac{52}{5} |
[
{
"content": "Let $X, Y, Z \\sim \\mathcal{N}(0,1)$ be independent standard normal random variables. Calculate the probability $\\mathbb{P}(X > Z \\mid X > Y)$. \\( \\)",
"role": "user"
}
] | \dfrac{2}{3} |
[
{
"content": "Let $M$ be a von Neumann algebra in $B(H)$, and assume $\\\\zeta\\\\$ is a cyclic vector for $M$. Does the orthogonal projection of $H$ onto the subspace $\\\\mathbb{C}\\\\zeta$ belong to $M$?",
"role": "user"
}
] | No |
[
{
"content": "Given that the $p$-th, $q$-th, and $r$-th terms of an arithmetic progression (AP) are $a$, $b$, and $c$ respectively, and the $p$-th, $q$-th, and $r$-th terms of a geometric progression (GP) are also $a$, $b$, and $c$ respectively, find the value of the expression $a^b b^c c^a - a^c b^a c^b$. Assu... | 0 |
[
{
"content": "Can the center of a nonabelian group of order $2p$, where $p$ is an odd prime, have an order of $2$ or $p$? Provide a justification for your answer.",
"role": "user"
}
] | No |
[
{
"content": "Calculate the limit \\(\\lim_{x\\rightarrow +\\infty}\\left[x\\left(4\\arctan\\left(\\frac{x+1}{x}\\right)-\\pi\\right)\\right]\\) without using L'Hospital's rule or Taylor expansions.",
"role": "user"
}
] | 2 |
[
{
"content": "Let $X_k$, $k \\geq 0$, be iid random variables with pmf $p_j = P(X_1 = j)$ for $1 \\leq j \\leq m$, where $\\sum_{j = 1}^{m} p_j = 1$. Let $T = \\min\\{n > 0 : X_n = X_0\\}$ denote the first time we observe the initial value again. Compute $E[T]$. \\(\\text{Express your answer in terms of } p_j.\... | m |
[
{
"content": "Determine the order of the pole in the expansion of \\( \\frac{z - \\sin z}{z^3} \\) around \\( z = 0 \\).",
"role": "user"
}
] | 0 |
[
{
"content": "For a natural number $b > 3$, determine the largest common divisor of the set $A = \\{a^2 - 1 \\mid a \\geq b, 2 \\nmid a, 3 \\nmid a\\}$. What is this largest common divisor?",
"role": "user"
}
] | 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... | ideal |
[
{
"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... | Semi-annual period end|Discount to be amortized in every period|Unamortized discount|Bonds carrying value
Jan. 1, 2009| |9,477|165,523
Jun. 30, 2009|1,579.50|7,897.50|167,102.50
Dec. 31, 2009|1,579.50|6,318|168,682
Jun. 30, 2010|1,579.50|4,738.50|170,261.50
Dec. 31, 2010|1,579.50|3,159|171,841
Jun. 30, 2011|1,579.50|1,... |
[
{
"content": "Let $\\vec{a}$ and $\\vec{b}$ be unit vectors. If $\\vec{c}$ is a vector such that $\\vec{c} + (\\vec{c} \\times \\vec{a}) = \\vec{b}$, find the maximum value of $| (\\vec{a} \\times \\vec{b}) \\cdot \\vec{c} |$ in the form $\\frac{A}{10}$. Determine the value of $A$.",
"role": "user"
}
] | 5 |
[
{
"content": "Given the sets \\( A = \\{(x, y) \\mid x^{2}-3xy+4y^{2}=\\frac{7}{2}\\} \\) and \\( B = \\{(x, y) \\mid kx + y=2\\} \\) with \\( k>0 \\), determine the value of \\( k \\) such that the intersection \\( A \\cap B \\) contains exactly one element.",
"role": "user"
}
] | \dfrac{1}{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... | 16.84 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... | 34800 |
[
{
"content": "Determine whether every bilinear symmetric form on a finite-dimensional vector space admits a self-adjoint map representation. Specifically, given a bilinear symmetric form \\( Q(x,y) \\) on a vector space \\( V \\), does there exist a linear self-adjoint map \\( K \\) such that \\((K(x),y) = Q(x,... | Yes |
[
{
"content": "In triangle $ABC$, let $I$, $O$, and $H$ be the incenter, circumcenter, and orthocenter, respectively. Given that $AI = 11$ and $AO = AH = 13$, find the distance $OH$. \\[ \\text{Proposed by Kevin You} \\]",
"role": "user"
}
] | 10 |
[
{
"content": "In a graph, determine if it is always possible to find two distinct paths that share only their endpoint. Answer 'yes' or 'no'.",
"role": "user"
}
] | no |
[
{
"content": "Let $X$ be a Banach space and $D \\subset X^{*}$ be a norm-dense subset. Suppose $(x_{n})_{n} \\subset X$ is a bounded sequence such that $\\lim_{n\\to \\infty}\\langle x_{n},x^{*}\\rangle$ exists for all $x^{*} \\in D$. Is the sequence $(x_{n})_{n}$ necessarily $\\sigma(X,X^{*})$-convergent?",
... | No |
[
{
"content": "Determine the order of magnitude of the function \\( \\sinh \\left( \\frac{1}{\\epsilon} \\right) \\) as \\( \\epsilon \\) tends to zero, expressing your answer in terms of an exponential function.",
"role": "user"
}
] | e^{1/\epsilon} |
[
{
"content": "Determine whether there exists a function \\( f : \\mathbb{R} \\rightarrow \\mathbb{R} \\) such that the first and second derivatives \\( f' \\) and \\( f'' \\) exist, but the third derivative \\( f''' \\) does not exist.",
"role": "user"
}
] | Yes |
[
{
"content": "Let \\( f \\in L_1([0,1], \\lambda) \\) be a function such that \\( \\int_0^1 |f|^2 \\, dx \\leq M \\). Does there exist a function \\( g \\in L_1([0,1], \\lambda) \\) such that \\( \\|f-g\\|_{L_1} < \\epsilon \\) and \\( \\int_0^1 |g|^2 \\, dx > M \\)?",
"role": "user"
}
] | Yes |
[
{
"content": "Find all possible values of \\( a \\) such that the sum of the distances from eighteen consecutive natural numbers to \\( a \\) is 1005, and the sum of the distances from these same eighteen numbers to \\( a^2 \\) is 865.",
"role": "user"
}
] | -\dfrac{7}{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... | 1973 k |
[
{
"content": "Find the value of $a \\in \\mathbb{R}$, where $a \\neq 1$, such that \\[ \\lim_{n \\to \\infty} \\frac{1^a + 2^a + \\cdots + n^a}{(n+1)^{a-1} \\left[ (na+1) + (na+2) + \\cdots + (na+n) \\right]} = \\frac{1}{60}. \\]",
"role": "user"
}
] | 7 |
[
{
"content": "Two points are drawn on each side of a square with an area of 81 square units, dividing the side into 3 congruent parts. Quarter-circle arcs connect the points on adjacent sides to create the figure shown. What is the length of the boundary of the bolded figure? Express your answer as a decimal... | 30.8 |
[
{
"content": "If $\\mathbf{A}^{-1} = \\begin{pmatrix} 4 & 0 \\\\ 1 & -1 \\end{pmatrix}$ and $\\mathbf{B}^{-1} = \\begin{pmatrix} 2 & -1 \\\\ -1 & 3 \\end{pmatrix},$ then find the inverse of $\\mathbf{A} \\mathbf{B}.$",
"role": "user"
}
] | \begin{pmatrix} 7 & 1 \\ -1 & -3 \end{pmatrix} |
[
{
"content": "Evaluate the limit: \\[ \\lim _{x\\to 0^+}\\left(\\frac{\\frac{e^{-\\frac{1}{x}}}{x^3} +e^{-\\frac{1}{\\sqrt{x}}}}{e^{-\\frac{1}{x}}\\ln x}\\right) \\]",
"role": "user"
}
] | -\infty |
[
{
"content": "Seven committees are formed from a group of seven persons. Each pair of committees shares exactly one member in common, and each person is a member of exactly three committees. Determine which of the following statements is true regarding the sizes of the committees:\n\n(a) At least one committee ... | 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... | 0.490 grams |
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