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... | 289 m |
[
{
"content": "Evaluate the limit \\( \\lim\\limits_{b \\to -\\infty} \\ln\\left(\\left(\\frac{b-m}{b}\\right)^b\\right) \\), where \\( m \\) is a constant.",
"role": "user"
}
] | -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... | 1.39e8 Newton |
[
{
"content": "Calculate the limit: $$\\lim_{n\\to\\infty}\\sum\\limits_{i=1}^n\\frac{3i^2+2i-1}{3^i}$$",
"role": "user"
}
] | \dfrac{11}{2} |
[
{
"content": "Determine whether the $n$-th singular homology group of an oriented, compact, and connected $n$-dimensional smooth manifold with boundary, denoted as $H_n(M)$, is trivial.",
"role": "user"
}
] | 0 |
[
{
"content": "Determine whether there exists a linear code where a word with weight greater than \\(\\lfloor\\frac{d-1}{2}\\rfloor\\) can be a unique coset leader.",
"role": "user"
}
] | Yes |
[
{
"content": "Determine the number of steps required to construct the centroid of 10 points using a compass with a fixed opening.",
"role": "user"
}
] | 45 |
[
{
"content": "Given the modular exponentiation equations \\(m^5 \\equiv 7 \\pmod{7769}\\) and \\(m^7 \\equiv 252 \\pmod{7769}\\), find the value of \\(m\\).",
"role": "user"
}
] | 6 |
[
{
"content": "Calculate the volume of the tetrahedron with vertices at the points \\(A_{1}(1, 2, 0)\\), \\(A_{2}(3, 0, -3)\\), \\(A_{3}(5, 2, 6)\\), and \\(A_{4}(8, 4, -9)\\). The height is dropped from the vertex \\(A_{4}\\) onto the face \\(A_{1} A_{2} A_{3}\\).",
"role": "user"
}
] | 34 |
[
{
"content": "Determine the radius of convergence for the power series \\(\\sum_{n=1}^{\\infty} \\frac{n^{n}}{n!}z^{n}\\).",
"role": "user"
}
] | \dfrac{1}{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... | -18*10^{12} volt |
[
{
"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... | Mary |
[
{
"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... | q1 = q2 |
[
{
"content": "Evaluate the contour integral \\( \\int_\\gamma \\frac{1}{z^2 - 2z} \\, dz \\), where \\( \\gamma \\) is the circle of radius 1 centered at 2, traveled once counterclockwise.",
"role": "user"
}
] | \pi i |
[
{
"content": "In the symmetric group $S_{13}$, determine the number of permutations $\\tau$ that satisfy the equation $\\tau\\alpha\\tau^{-1}=\\gamma$, where $\\alpha = (1,2,4,8)(3,5,7,9,11,13)(6,12)$ and $\\gamma = (1,2,3,4)(5,6,7,8,9,10,11)(12,13)$. Provide your answer as a single integer.",
"role": "user... | 0 |
[
{
"content": "Suppose $F$ is a field and $a, b \\in F$ with relatively prime integers $m, n \\geq 1$ such that $a^m = b^n$. Determine if there exists some $c \\in F$ such that $c^{mn} = a^m = b^n$. 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... | 4.45 ^0 C |
[
{
"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... | 176*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... | 4.90e-20 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... | 1.22 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... | 8286 |
[
{
"content": "Calculate the expected value of the function \\( F(x) = x^2 1_{(0,1/2)}(x) + \\frac{1}{2} 1_{(1/2,2)}(x) + 1_{[2, \\infty)}(x) \\) using the formula \\( E(X) = \\int_{0}^{\\infty} (1 - F(x)) \\, dx \\).",
"role": "user"
}
] | \dfrac{29}{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... | 0.134 |
[
{
"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... | $226,539.73 |
[
{
"content": "Solve the equation \\(2a^x = b^x + c^x\\) for \\(x\\), given that \\(a, b, c\\) are strictly positive real numbers and \\(c = \\sqrt{ab}\\).",
"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... | 0.0708 |
[
{
"content": "Consider the infinite sequence of functions $f_n$, where $f_n(x)=ng_{[0,1/n]}(x)$ for all real numbers $x$ and for each $n = 1,2,3, \\dots$. Determine which of the following statements is true:\n\n(A) For every $x$, the sequence of numbers $(f_n(x))_{n=1,2,\\ldots\\infty}$ has a limit in the space... | D |
[
{
"content": "What is $\\left(\\frac{7}{8}\\right)^3 \\cdot \\left(\\frac{7}{8}\\right)^{-3}$?",
"role": "user"
}
] | 1 |
[
{
"content": "Let $E$ be a subset of the unit circle $S^1$ with positive Lebesgue measure. Define $\\Gamma$ as the group of roots of unity: $\\Gamma = \\{z \\in \\mathbb{C} \\mid z^n = 1 \\text{ for some } n \\geq 0\\}$. Consider the set $\\Gamma E = \\{z\\zeta \\mid z \\in \\Gamma, \\zeta \\in E\\}$. If $E$ ha... | 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... | 2.1390% |
[
{
"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... | 0.0224 V |
[
{
"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... | 0.0307 mm |
[
{
"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... | 11.460 |
[
{
"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... | 5.36% |
[
{
"content": "Suppose $A$, $B$, and $C$ are three independent exponential random variables with parameters $1$, $2$, and $3$, respectively. Calculate the probability $P(A < B < C)$. \\( \\boxed{} \\)",
"role": "user"
}
] | \dfrac{1}{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... | 2.27 Ci |
[
{
"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... | 539.98 nm |
[
{
"content": "Evaluate the integral \\( \\iiint_W (x^2 + y^2 + z^2) \\,\\mathrm{d}x\\,\\mathrm{d}y\\,\\mathrm{d}z \\) where \\( W \\) is the region defined by \\( x \\ge 0, \\ y \\ge 0, \\ z \\ge 0, \\ 0 \\le x + y + z \\le a \\) with \\( a > 0 \\).",
"role": "user"
}
] | \dfrac{a^5}{20} |
[
{
"content": "Compute the residue of the function \\( \\frac{z}{e^z - 1} \\) at its singularity.",
"role": "user"
}
] | 0 |
[
{
"content": "Find the value(s) of $a$ for which the matrix $A = \\begin{pmatrix} 3&0&0\\\\ 0&a&a-2\\\\ 0&-2&0 \\end{pmatrix}$ is not diagonalizable.",
"role": "user"
}
] | 4 |
[
{
"content": "Given two real-valued sequences $\\{x_n\\}$ and $\\{y_n\\}$, if $\\lim_{n\\to\\infty}(x_n^3 - y_n^3) = 0$, determine whether it is necessarily true that $\\lim_{n\\to\\infty}(x_n - y_n) = 0$. Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "From a group of seven persons, seven committees are formed such that any two committees have exactly one member in common, and each person is in exactly three committees. Determine which of the following statements is true about the sizes of the committees:\n\n(a) At least one committee must have ... | B |
[
{
"content": "Determine the maximum possible cardinality of a topology generated by a countable basis, assuming the axiom of choice.",
"role": "user"
}
] | 2^{\aleph_0} |
[
{
"content": "Determine the range of the function \\( f(x) = 4 \\cos \\left( \\frac{\\pi}{3} \\sin \\left( x^{2} + 6x + 10 - \\sin x \\right) \\right) \\).",
"role": "user"
}
] | [2, 4] |
[
{
"content": "Evaluate \\( \\lim_{n \\to \\infty} \\frac{d(n, > \\sqrt{n})}{n} \\), where \\( d(n, > \\sqrt{n}) \\) is the number of divisors of \\( n \\) that are greater than \\( \\sqrt{n} \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Given the differential equation \\( z' = 5 - \\cos(tz) \\) for \\( 2 \\le t \\le 4 \\) with the initial condition \\( z(2)=1 \\), find an upper bound for \\(|z''(t)|\\).",
"role": "user"
}
] | 37 |
[
{
"content": "Determine the number of solutions to the equation \\(3\\log_{3}(3^x+2x)=3^x+2\\).",
"role": "user"
}
] | 0 |
[
{
"content": "Determine the angle \\( \\theta \\) in degrees, where \\( 0 \\leq \\theta < 360^\\circ \\), for the complex number sum \\( z = \\text{cis } 75^\\circ + \\text{cis } 83^\\circ + \\text{cis } 91^\\circ + \\dots + \\text{cis } 147^\\circ \\), when expressed in the form \\( z = r \\, \\text{cis } \\th... | 111 |
[
{
"content": "In $\\triangle ABC$, let $H$ be the orthocenter, $BC = a$, and the circumradius of $\\triangle ABC$ be $R$. Given that $a = 2AH - 2R$, determine the value of $\\sin A$. \\( \\text{Express your answer in terms of known constants or values.} \\)",
"role": "user"
}
] | \dfrac{3}{5} |
[
{
"content": "Evaluate the integral \\( \\int_{0}^{\\sqrt{\\frac{\\pi}{2}}}\\int_{x}^{\\sqrt{\\frac{\\pi}{2}}}\\sin(y^2) \\, \\, \\mathrm{d}y \\, \\, \\mathrm{d}x \\).",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "Let $q \\in \\mathbb{C}$ be a primitive $n$th root of unity, for some $n > 1$. Consider the $n^2 \\times n^2$ matrix \\( M = \\left( q^{ki+lj} \\right)_{(k,l),(i,j)} \\) indexed by all pairs \\((k,l), (i,j) \\in \\{0,\\cdots, n-1\\}^2\\) using lexicographic order. Are these matrices $M$ always inv... | Yes |
[
{
"content": "Determine whether a morphism between integral curves induces a map between the corresponding function fields.",
"role": "user"
}
] | Yes |
[
{
"content": "Consider the integer sequence \\(152, a_{1}, a_{2}, \\cdots\\) that satisfies the equation \\(a_{1} \\cdot a_{2} + a_{2} \\cdot a_{3} + \\cdots + a_{n-1} \\cdot a_{n} = \\frac{(n-1) \\cdot n \\cdot (n+1)}{3}\\) for \\(n = 2, 3, \\cdots\\). Determine the number of such sequences. Choose from the fo... | C |
[
{
"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... | $18,241.19 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 14 m |
[
{
"content": "Suppose \\( y = f(x) \\) is a function defined on the domain and range \\(\\mathbf{R}\\), with its inverse function denoted as \\( y = f^{-1}(x) \\). Given that \\( f(x) + f(1 - x) = 2 \\) holds for all \\( x \\in \\mathbf{R} \\), find the value of \\( f^{-1}(x - 2) + f^{-1}(4 - x) \\).",
"rol... | 1 |
[
{
"content": "Let \\( f: \\mathbb{R} \\rightarrow \\mathbb{R} \\) be a non-constant, three times differentiable function such that \\( f\\left(1+\\frac{1}{n}\\right) = 1 \\) for all integers \\( n \\). Determine the value of \\( f''(1) \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Evaluate the limit: $$\\lim_{x\\to\\infty}\\left(\\sqrt{9x^6+7x^3+5x}-\\sqrt{9x^6+6x^2+3x}\\right)$$",
"role": "user"
}
] | \dfrac{7}{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... | 4.37*10^4 V |
[
{
"content": "Consider quadratic functions \\( f(x) = ax^2 + bx + c \\) such that \\( a < b \\) and \\( f(x) \\geq 0 \\) for all \\( x \\). Determine the minimum value of the expression \\( \\frac{a + b + c}{b - a} \\).",
"role": "user"
}
] | 3 |
[
{
"content": "Let \\( f(z) \\) be a holomorphic function on the open unit disk such that \\( \\lim_{z \\to 1} f(z) \\) exists. If \\( f(z) = \\sum_{i=1}^{\\infty} a_i z^i \\) is its Taylor series around 0, determine the radius of convergence of \\( f(z) \\).",
"role": "user"
}
] | 1 |
[
{
"content": "Evaluate the integral \\( \\int_{C} \\frac{1}{\\sin(z)} \\, dz \\) where \\( C \\) is the circle centered at the origin with radius 1, using complex analysis methods.",
"role": "user"
}
] | 2\pi i |
[
{
"content": "Determine whether the following statement is true or false: Every infinite set contains a countably infinite subset.",
"role": "user"
}
] | 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... | 48 days |
[
{
"content": "Evaluate the integral \\( \\oint_{\\gamma} \\frac{1}{\\cos z} \\, dz \\) along a simple, closed path \\( \\gamma \\) that contains \\(-2\\pi\\) and \\(2\\pi\\) in its interior but excludes all other multiples of \\(\\pi\\).",
"role": "user"
}
] | 0 |
[
{
"content": "Determine whether the following limit diverges:\n\\[ \\lim_{n \\to +\\infty} \\frac{n+o(n)}{\\sqrt{n+o(n)} - \\sqrt{n+o(n)}} \\cdot \\cos(1) \\]",
"role": "user"
}
] | \infty |
[
{
"content": "Determine whether connected smooth orientable 2-manifolds can have torsion elements in their fundamental group.",
"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... | 1.8e-4 |
[
{
"content": "Determine the exact smallest distance between the graphs of the functions \\( y_1 = \\sin x \\) and \\( y_2 = 2 + \\sin x \\).",
"role": "user"
}
] | 2 |
[
{
"content": "Assume $f:G\\to K$ is a group homomorphism, and $S\\subset G$ is a subset of elements such that the restriction $f|_S$ is the trivial homomorphism. Let $N\\leq G$ denote the smallest normal subgroup that contains $S$. Prove or disprove that $f$ is trivial on the entire $N$. \\( \\text{True or Fals... | 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... | MnO2(s) + C(s) -> Mn(s) + CO2(g) |
[
{
"content": "Chris and Nathan order a giant cookie together. If Chris eats $\\frac{2}{7}$ of the cookie and Nathan eats $\\frac{3}{5}$ of the cookie, what fraction of the original cookie is left?",
"role": "user"
}
] | \frac{4}{35} |
[
{
"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... | 196250 N/m^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... | $10,818,000 |
[
{
"content": "Evaluate the integral \\( \\int_{C}\\frac{dz}{z^2+4} \\) where \\( C \\) is a simple closed contour oriented counter-clockwise, enclosing \\( z = 2i \\) but not enclosing \\( z = -2i \\).",
"role": "user"
}
] | \dfrac{\pi}{2} |
[
{
"content": "Given sequences of positive numbers $a_n$ and $b_n$ such that \\( \\lim_{n\\to\\infty}a_n^n = a \\) and \\( \\lim_{n\\to\\infty}b_n^n = b \\), where \\( a, b \\in (0, \\infty) \\), determine the limit \\( \\lim_{n\\to\\infty}(pa_n + qb_n)^n \\) for nonnegative numbers \\( p \\) and \\( q \\) with ... | a^p b^q |
[
{
"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... | 372.8J |
[
{
"content": "Evaluate the triple integral \\( \\iiint_V xyz \\, dxdydz \\), where \\( V \\) is the volume in \\( \\mathbb{R}^3 \\) defined by the inequalities \\( x^2+y^2+z^2\\leq1, \\quad x\\geq0, \\quad y\\geq0, \\quad z\\geq0. \\) Use spherical coordinates to perform the integration.",
"role": "user"
... | \dfrac{1}{48} |
[
{
"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... | $42.22 |
[
{
"content": "The quadratic $x^2-4x-14=3x+16$ has two solutions. What is the sum of these solutions?",
"role": "user"
}
] | 7 |
[
{
"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... | $\mathbf{g} = \frac{\partial \phi}{\partial \mathbf{x}} = \left[3+3ur+u^2 r^2\right] \frac{f x_k}{r^4} \mathbf{x} - \left[ u+\frac{1}{r} \right] \frac{f}{r} \mathbf{e}_k $ with $\phi = \frac{\partial f}{\partial x_k} = - \left[ u+\frac{1}{r} \right] \frac{f}{r} x_k$ |
[
{
"content": "Determine whether every $\\mathbb{R}$-algebra homomorphism $h: \\mathbb{R}[V] \\to \\mathbb{R}$ is of the form $ev_x$ for some point $x \\in V$, where $ev_x(f) = f(x)$ for $f \\in \\mathbb{R}[V]$. Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Determine the values of $p$ for which the function \\( \\rho(x, y) = \\int_a^b \\lvert x(t) - y(t) \\rvert^p \\, \\mathrm{d} t \\) is a metric on the set of continuous functions on \\([a, b]\\).",
"role": "user"
}
] | (0, 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... | h20 |
[
{
"content": "Is there an algorithm that allows any element \\( \\alpha \\in L(x,y) \\) to be expressed as \\( \\sum_{\\alpha} x_{\\alpha} h_{\\alpha} \\), where \\( h_{\\alpha} \\) is the Hall basis of the free Lie algebra \\( L(x,y) \\)?",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate the integral \\( \\displaystyle \\int_{0}^{\\infty} \\dfrac{x^{49}}{(1+x)^{51}}dx \\).",
"role": "user"
}
] | \dfrac{1}{50} |
[
{
"content": "Determine whether there exists a set of numbers $S = \\{s_1, s_2, \\ldots, s_n\\}$ such that the average of the numbers in $S$ equals the sum of the proper divisors common to all elements of $S$. In other words, can a \"perfect set\" exist where:\n\n$$\\frac{s_1+s_2+\\cdots+s_n}{n} = \\sum_{d\\mid... | Yes |
[
{
"content": "Let \\( T : \\mathbb{R}^7 \\to \\mathbb{R}^7 \\) be a linear transformation with nullity \\( N(T) = 2 \\). Determine the minimum possible rank of \\( T^2 \\).",
"role": "user"
}
] | 3 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 18.1 degrees Celsius |
[
{
"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... | $2,380,340.74 |
[
{
"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... | 19.7 m/s^2 |
[
{
"content": "Suppose $f(x)$ and $g(x)$ are functions defined on the real numbers such that $f(x)g(x) \\geq 0$ for all $x$. If $\\lim \\limits _{x \\to c} f(x)g(x)$ exists for some real number $c$, does it necessarily follow that either $\\lim \\limits _{x \\to c} f(x)$ or $\\lim \\limits _{x \\to c} g(x)$ exis... | 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... | upward sloping volatility smile |
[
{
"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... | 222,407.37 |
[
{
"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... | sp3d2 |
[
{
"content": "Prove that for any complex number $z \\in \\mathbb{C}$, the following inequality holds:\n\\[ \\bigg\\vert\\frac{1-e^{iz}}{z^2}\\bigg\\vert \\leq \\frac{2}{\\big\\vert z\\big\\vert^2} \\]",
"role": "user"
}
] | \frac{2}{|z|^2} |
[
{
"content": "What is the probability that the greatest common divisor of two natural numbers, $a$ and $b$, is twice a prime number? Denote the set of prime numbers by $P$. Express your answer as a numerical probability.",
"role": "user"
}
] | 0.0687 |
[
{
"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... | 6.4e-5 m^3 |
[
{
"content": "Find the Taylor series for the function \\( f(x) = x^3 \\cdot \\ln{\\sqrt{x}} \\) around the point \\( a = 1 \\). Also, determine the radius of convergence of this series.",
"role": "user"
}
] | 1 |
[
{
"content": "Find an ordered pair $(u,v)$ that solves the system: \\begin{align*} 5u &= -7 - 2v,\\\\ 3u &= 4v - 25 \\end{align*}",
"role": "user"
}
] | (-3,4) |
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