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... | I_2(s)+10Cu^{2+}(aq)+6H_2O(l)\rightarrow 2IO_3^-(aq)+10Cu^+(aq)+12H^+(aq) |
[
{
"content": "Given that \\( f(x) \\) is a continuous function and \\( f(0) = 2 \\), and the functional equation \\( f(x) - f\\left(\\frac{x}{23}\\right) = \\frac{x}{23} \\) holds, find the value of \\( f(484) \\).",
"role": "user"
}
] | 24 |
[
{
"content": "For which values of \\( p \\) does the triangle inequality \\( ||x + y||_p \\leq ||x||_p + ||y||_p \\) hold true for the \\( \\ell^p \\) norm in \\( \\mathbb{R}^n \\)? Here, \\( ||x||_p = \\left( \\sum_{k=1}^n |x_k|^p \\right)^{1/p} \\) for \\( p \\in (0, \\infty) \\) and \\( ||x||_\\infty = \\sup... | [1, \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... | $6,786 U |
[
{
"content": "Given two permutations \\(\\sigma = (13624)(587)(9)\\) and \\(\\tau=(15862)(394)(7)\\) in \\(S_9\\), determine the number of permutations \\(\\alpha\\) such that \\(\\alpha\\sigma\\alpha^{-1} = \\tau\\). How many such \\(\\alpha\\) are there?",
"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... | 8.07 |
[
{
"content": "Find a parametrization of the surface given by the equation \\(x^3 + 3xy + z^2 = 2\\) with \\(z > 0\\). Use this parametrization to determine the equation of the tangent plane at the point \\((x, y, z) = \\left(1, \\frac{1}{3}, 0\\right)\\).",
"role": "user"
}
] | 4x + 3y = 5 |
[
{
"content": "How many polynomials \\( P \\) with integer coefficients and degree at most 5 satisfy the inequality \\( 0 \\le P(x) < 120 \\) for all \\( x \\in \\{0,1,2,3,4,5\\} \\)?",
"role": "user"
}
] | 86400000 |
[
{
"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... | $720,000 |
[
{
"content": "Find the smallest integer greater than 1 that, when divided by \\(\\frac{3}{5}, \\frac{5}{7}, \\frac{7}{9}, \\frac{9}{11}\\), results in a mixed number with fractional parts \\(\\frac{2}{3}, \\frac{2}{5}, \\frac{2}{7}, \\frac{2}{9}\\), respectively.",
"role": "user"
}
] | 316 |
[
{
"content": "Calculate the surface area of the surface generated when the curve defined by \\(8y^2 = x^2 - x^4\\) is revolved around the x-axis, for \\(-1 \\leq x \\leq 1\\).",
"role": "user"
}
] | \dfrac{\pi}{2} |
[
{
"content": "Consider the system of equations given by:\n\n$$ \\prod_{i=1}^n p_i = \\prod_{i=1}^n (1-p_i)= \\frac{1}{2^n} $$\n\nwhere $p_i \\in (0,1)$ for $i = 1, 2, \\ldots, n$. Determine if the only solution to this system is $p_1 = p_2 = \\ldots = p_n = \\frac{1}{2}$ for all $n \\in \\mathbb{N}$. Is this st... | True |
[
{
"content": "Evaluate the surface integral \\( \\iint_{S} [ \\nabla \\phi \\times \\nabla \\psi] \\cdot \\mathbf{n} \\, dS \\) where \\( \\phi=(x+y+z)^2 \\) and \\( \\psi=x^2-y^2+z^2 \\). The surface \\( S \\) is the curved surface of the hemisphere \\( x^2+y^2+z^2=1 \\) with \\( z \\ge 0 \\).",
"role": "u... | -2\pi |
[
{
"content": "Determine whether the arithmetic genus of a projective plane curve changes when the base field is extended from $K$ to its algebraic closure $\\bar{K}$. Provide a justification for your answer.",
"role": "user"
}
] | No |
[
{
"content": "Determine the smallest number of sets required to generate the power set of a finite set $A$ where $|A| = n$.",
"role": "user"
}
] | \lceil \log_2 n \rceil |
[
{
"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... | 280.8 kg m/s |
[
{
"content": "Suppose $S$ and $T$ are two linear maps from a vector space $V$ to itself, and let $v \\neq 0$ be an eigenvector associated with eigenvalue $\\lambda$ for the composition $TS$. If $S(v) = 0$, prove or disprove that $\\lambda = 0$, and consequently, both $TS$ and $ST$ have only the zero eigenvalue.... | 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... | 68.83 |
[
{
"content": "Find the positive value of \\( a \\) such that the remainder of \\( x^3 - x^2 + ax \\) when divided by \\( x+a \\) is twice the remainder when divided by \\( x-2a \\).",
"role": "user"
}
] | \dfrac{2}{17} |
[
{
"content": "Is it possible to construct an error correction code in the finite field $F_8$ that can correct $n$ errors, where each error is an increment or decrement by 1?",
"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... | 3.58 s |
[
{
"content": "Calculate the flux of the vector field \\( \\mathbb{v} = (x^2-2xz, -2xy, z^2-x) \\) downwards through the paraboloid \\( z = 1 - x^2 - y^2 \\) for \\( x \\geq 0 \\) and \\( z \\geq 0 \\).",
"role": "user"
}
] | \dfrac{2}{3} |
[
{
"content": "Given complex numbers $z_1, z_2, \\ldots, z_n \\in \\mathbb{C}$, not necessarily distinct, if they satisfy the condition $\\sum_{i=1}^n z_i^k = 0$ for all positive integers $k$, determine whether it must be true that $z_1 = z_2 = \\ldots = z_n = 0$. Provide a justification for your answer.",
"... | 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... | 19,514 J |
[
{
"content": "Given a matrix $A$ with a null trace, determine the minimum number of linear measurements, defined as $A\\vec{x}_i \\cdot \\vec{h}_j$ for specified $\\vec{x}_i$ and $\\vec{h}_j$, necessary to uniquely determine the matrix $A$. Provide your answer as a single integer.",
"role": "user"
}
] | 3 |
[
{
"content": "Let $F$ be a field, and $V$ be an $F$-vector space. Define the ring $R = F \\oplus V$ with multiplication given by $xy = 0$ for $x, y \\in V$. Determine if the Krull dimension of $R$ is zero.",
"role": "user"
}
] | 0 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 1.062e-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... | u = sqrt(g(d + k + h)) |
[
{
"content": "Determine the number of integer values of \\( k \\) for which the system of equations \\( \\begin{cases} a^2+b^2=kc(a+b), \\\\ b^2+c^2 = ka(b+c), \\\\ c^2+a^2=kb(c+a) \\end{cases} \\) has a solution other than \\( a = b = c = 0 \\) in the set of real numbers.",
"role": "user"
}
] | 1 |
[
{
"content": "Determine the cardinality of the set of real-valued continuous functions \\( f \\) defined on \\([0,1]\\) such that \\( f(x) \\) is rational whenever \\( x \\) is rational.",
"role": "user"
}
] | \mathfrak{c} |
[
{
"content": "Determine whether the following statement is true: For all $a, b \\in \\mathbb{F}_9$, the product $a \\cdot b$ is a square if and only if $a \\cdot \\overline{b}$ is a square.",
"role": "user"
}
] | True |
[
{
"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... | P0= 21132(1-1.06^{-7})/.06 + $528300/1.06^7=$469316 |
[
{
"content": "Given the relation $x = Ky$ where $K$ is a fixed scalar, and a function $f$ with its partial derivative $f'(x) = \\frac{\\partial f}{\\partial x}(Ky)$, find the expression for $\\frac{\\partial f}{\\partial y}(x)$ in terms of $x$.",
"role": "user"
}
] | K f'(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... | 4.30e14 m/s^2 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 13.28 years |
[
{
"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... | 8.33e-3 |
[
{
"content": "Evaluate the complex contour integral \\( J = \\oint_{C}\\frac{1}{\\cos z+1}dz \\) where \\( C: |z|=5 \\). Use the residue theorem to find the value of the integral.",
"role": "user"
}
] | 0 |
[
{
"content": "Solve the functional equation \\(f(xy+x)+f(y)=f(xy+y)+f(x)\\) for all real functions \\(f:\\mathbb{R}\\to\\mathbb{R}\\). Use the theorem that states if \\(f:\\mathbb{R}\\to\\mathbb{R}\\) satisfies the Hosszu functional equation \\(f(x+y-xy)+f(xy)=f(x)+f(y)\\) for all \\(x,y \\in \\mathbb R\\), the... | f(x) = A(x) + a |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n \\to \\infty} \\left(1+\\frac{a}{n^2}\\right)^\\frac{n}{b} \\] where \\(a\\) and \\(b\\) are constants.",
"role": "user"
}
] | 1 |
[
{
"content": "Find the smallest integer \\( n \\) such that the remainder estimate \\( |R_n| \\le \\frac{M}{(n+1)!} (x-a)^{n+1} \\), where \\( M \\) is the maximum value of \\( |f^{(n+1)}(z)| \\) on the interval between \\( a \\) and the indicated point, satisfies \\( |R_n| \\le \\frac{1}{1000} \\) for the func... | 10 |
[
{
"content": "Find an integer $n$ such that $0\\leq n<398$ and $n$ is a multiplicative inverse to 7 modulo 398.",
"role": "user"
}
] | 57 |
[
{
"content": "Compute the integral \\( \\int_0^{2\\pi} -e^{\\cos(t)}[\\sin(t)\\cdot \\cos(\\sin(t))+\\cos(t)\\cdot \\sin(\\sin(t))] \\,dt \\).",
"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... | SL1: $2,045.45
SL2: $2,550 |
[
{
"content": "Determine the minimum distance $d(C)$ of a linear code $C$ with a parity-check matrix $H$ over $\\Bbb F_2$. The first seven rows of $H$ are the identity matrix $I_7$, and the remaining rows consist of all weight-$3$ vectors in $\\{0,1\\}^7$. The rank of $H$ is 7. What is the minimum distance $d(C)... | 4 |
[
{
"content": "Calculate the expected value $E(X_1X_2^2)$ for the joint probability density function given by \\( f(x_1,x_2) = 8x_1x_2 \\) for \\( 0 < x_1 < x_2 < 1 \\).",
"role": "user"
}
] | \dfrac{8}{21} |
[
{
"content": "In a right-angled triangle, medians are drawn from the acute angles to the opposite sides. If the maximum acute angle between these medians can be expressed as \\( \\tan^{-1}\\left(\\frac{p}{q}\\right) \\), where \\( p \\) and \\( q \\) are relatively prime positive integers, find \\( p+q \\).",
... | 7 |
[
{
"content": "Calculate \\( \\lim_{\\lambda \\to \\infty} \\frac{A}{B} \\), where \\( A = \\int_\\lambda^\\infty \\frac{x}{\\lambda} e^{-x^2/2} \\, dx \\) and \\( B = \\int_\\lambda^\\infty e^{-x^2/2} \\, dx \\).",
"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... | 91.1 g |
[
{
"content": "Compute the expected value of the surplus water when water flows in and out of a dam. The daily inflow is uniformly distributed on $[0, 2]$ liters, and the daily outflow is uniformly distributed on $[0, 1]$ liters, independently of the inflow. Let $X$ be the inflow and $Y$ be the outflow, with $X ... | \dfrac{1}{2} |
[
{
"content": "Determine the radius of convergence of the power series \\( \\sum_{n\\geq 1} n^n z^{n!} \\).",
"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... | $33,000 |
[
{
"content": "In a triangle $\\Delta ABC$, it is given that \\( \\frac{2\\cos A}{a} +\\frac{\\cos B}{b} +\\frac{2\\cos C}{c}=\\frac{a}{bc} +\\frac{b}{ca} \\). Find the angle $A$.",
"role": "user"
}
] | 90^\circ |
[
{
"content": "In an equilateral triangle $ABC$, points $D$ and $E$ are located on segment $AB$ such that $AD = DE = EB$. A point $F$ is on segment $BC$ such that $CF = AD$. Determine the value of the sum of angles $\\widehat{CDF}$ and $\\widehat{CEF}$. \\( \\text{Express your answer in degrees.} \\)",
"role... | 30 |
[
{
"content": "The first term of an arithmetic sequence is 1, another term of the sequence is 91 and all of the terms of the sequence are integers. How many distinct arithmetic sequences meet these three conditions?",
"role": "user"
}
] | 12 |
[
{
"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... | 50% type A, 37.5% type AB, and 12.5% type B |
[
{
"content": "Find the remainder when \\(2^{1990}\\) is divided by 1990.",
"role": "user"
}
] | 1024 |
[
{
"content": "Determine the Galois group of the field extension \\(\\mathbb{Q}(x,y)\\), where \\(x\\) is a nontrivial cube root of unity and \\(y\\) is the cube root of 2. Identify the specific group of order 6 to which it is isomorphic.",
"role": "user"
}
] | S_3 |
[
{
"content": "Evaluate the integral \\( \\lim_{n\\to \\infty} \\int_0^{\\infty} \\frac{\\cos(nx)}{1+x^2} \\, dx \\) without using complex analysis, specifically the residue theorem.",
"role": "user"
}
] | 0 |
[
{
"content": "Determine the radius of convergence of the power series \\( \\operatorname{f}(x) = \\sum_{n \\ge 0} a_n x^n \\) given that \\( a_i \\ge 0 \\) and \\( a_{n+1} + a_{n+2} + \\cdots < 1/n! \\).",
"role": "user"
}
] | \infty |
[
{
"content": "Evaluate the limit: \\[ \\lim_{R\\rightarrow \\infty} \\int_{C_R} \\frac{6z^6 + 5z^5}{z^6 + z^5 + 10}dz, \\] where \\( C_R=\\{z \\in \\mathbb{C} : |z|=R \\} \\) for \\( R>0 \\).",
"role": "user"
}
] | -2\pi i |
[
{
"content": "Find the natural number \\( n \\) such that \\( \\sqrt[n]{22-10\\sqrt7} = 1 - \\sqrt7. \\)",
"role": "user"
}
] | 3 |
[
{
"content": "Evaluate the definite integral: $$\\int_{0}^{\\frac{\\pi}{2}} \\cot\\left(\\frac{x}{2}\\right) (1-\\cos^4x)\\,dx.$$",
"role": "user"
}
] | \dfrac{25}{12} |
[
{
"content": "Given a set with 25 elements, determine the minimum number of subsets of size 5 required such that every possible subset of size 3 is included in at least one of these subsets of size 5.",
"role": "user"
}
] | 240 |
[
{
"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... | 5774.89 |
[
{
"content": "Calculate the integral \\( \\int_{-\\infty}^ \\infty dx \\int_{-\\infty}^ \\infty e^{-x^2-y^2} dy \\). Use a method different from the standard approach to evaluate this integral.",
"role": "user"
}
] | \pi |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 36.36 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... | 320,250N |
[
{
"content": "Find the square matrix $A$ of minimal order such that the vector function \\( \\vec{X}_0(t) = \\vec{W}(t) + \\vec{V}e^{2t} \\) satisfies the differential equation \\( \\vec{X}_0'(t) = A \\cdot \\vec{X}_0(t) \\), where:\n\n\\[ \\vec{W}(t) = \\begin{pmatrix} \\sin(t) + \\cos(t) \\\\ \\cos(t) - \\sin... | 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... | Number of Setups |
[
{
"content": "Compute the operator norm \\( \\|T\\| \\) of the integral operator \\( T: C[0,1] \\rightarrow C[0,1] \\) defined by \\( Tf(x) = \\int_0^1\\frac{f(y)}{1+x^2+y^2}dy \\), where the norm \\( \\|f\\|_\\infty = \\sup_{x\\in[0,1]}|f(x)| \\).",
"role": "user"
}
] | \dfrac{\pi}{4} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 0.19 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... | 22.2 m |
[
{
"content": "Find the limit of the sequence \\(a_n = \\int_n^{n+1} f(x) \\, dx\\), where \\(f: [0, +\\infty) \\to \\mathbb{R}\\) is a continuous function satisfying \\(0 < f(x+1) < f(x)\\) for all \\(x \\geq 0\\) and \\(\\lim_{x \\to \\infty} f(x) = 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... | {\displaystyle \text{component-x: }V_{ox}=V_{o}\;cos\theta\\ \text{component-y:}V_{oy}=V_{o}\;sin\theta} |
[
{
"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... | 6.04e-5 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... | 5.57 kg*m/s |
[
{
"content": "For $n \\geq 1$, the $n$th triangular number is defined as $T_n = 1+2+\\cdots+n$. Let $C$ be the least positive integer such that $CT_n + 1$ is a triangular number for all positive integers $n$. Determine the number of factors of $C$. \\(\\text{Express your answer as an integer.}\\)",
"role": ... | 3 |
[
{
"content": "Calculate the volume of the solid formed by rotating the region bounded by the curves $x^2 - y = 0$ and $x + y = 0$ about the line $y = 1$.",
"role": "user"
}
] | \dfrac{\pi}{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... | Year (a) Hours Used (b) Depreciation Rate (c) Depreciation Expense (d = b x c)
13,000 $20.25 $60,750
21,200 $20.25 $24,300
31,300$20.25 $26,325
42,700$20.25 $54,675
52,500$20.25 $50,625
61,300$20.25 $26,325 |
[
{
"content": "Let $f: X \\to Y$ be a proper, flat morphism of projective schemes where $Y$ is an irreducible, non-singular surface. Assume there exists a Zariski open subset $U$ of $Y$ whose complement is of codimension 2 such that the total space $f^{-1}(U)$ is non-singular and $f: f^{-1}(U) \\to U$ has geomet... | Yes |
[
{
"content": "Suppose $r$ is a real number for which\n\\[\\left\\lfloor r + \\frac{19}{100} \\right\\rfloor + \\left\\lfloor r + \\frac{20}{100} \\right\\rfloor + \\left\\lfloor r + \\frac{21}{100} \\right\\rfloor + \\cdots + \\left\\lfloor r + \\frac{91}{100} \\right\\rfloor = 546.\\]Find $\\lfloor 100r \\rflo... | 743 |
[
{
"content": "Evaluate the infinite series: \\( \\sum_{n=1}^{\\infty}\\frac{n^2}{n!} \\).",
"role": "user"
}
] | 2e |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n\\to \\infty} \\left( \\lim_{x\\to0} \\left(1+\\tan^2(x)+\\tan^2(2x)+ \\cdots + \\tan^2(nx)\\right)^{\\frac{1}{n^3x^2}} \\right) \\]",
"role": "user"
}
] | e^{\frac{1}{3}} |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n \\to \\infty} \\left\\{ \\frac{n!}{\\prod_{k=1}^{n} H_k} \\right\\}^{\\frac{1}{n}} \\] where \\( H_k = \\sum_{i=1}^{k} \\frac{1}{i} \\).",
"role": "user"
}
] | \infty |
[
{
"content": "Calculate the third central moment of a Poisson distribution using its moment generating function.",
"role": "user"
}
] | \lambda |
[
{
"content": "Find the interval of convergence for the series \\(\\sum_{n=1}^{\\infty} \\sin\\left(\\frac{1}{n}\\right)\\tan\\left(\\frac{1}{n}\\right)x^n.\\)",
"role": "user"
}
] | [-1, 1] |
[
{
"content": "A function \\( f \\) is even. It is known that there are four values \\( x_1, x_2, x_3, x_4 \\) satisfying \\( f\\left(\\frac{x+1}{3-x}\\right) = f(x) \\). What is the value of \\( x_1 + x_2 + x_3 + x_4 \\)?\n\nA) 0\nB) 2\nC) 4\nD) 6",
"role": "user"
}
] | D |
[
{
"content": "Determine whether the following two linear programs are equivalent:\n\n1. \\( \\min\\limits_{a\\in A} \\left( \\max\\limits_{b\\in B} f(a, b) + \\max\\limits_{c\\in C} f(a, c) \\right) \\)\n\n2. \\( \\begin{align} \\min\\limits_{a\\in A} & \\quad (z_1 + z_2) \\\\\n\\text{subject to}& \\quad z_1\\g... | 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... | WACC = (wd)(Kd)(1-Tc)+(wp)(Kp)+(wc)(Kc) |
[
{
"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... | 1600 |
[
{
"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... | Southern hemisphere |
[
{
"content": "Given two positive coprime integers $a$ and $d$, does Dirichlet's theorem on arithmetic progressions imply that there are infinitely many prime numbers of the form $a + nd$ if $a = -1$?",
"role": "user"
}
] | Yes |
[
{
"content": "Find all continuous and bounded functions \\( g: \\mathbb{R} \\rightarrow \\mathbb{R} \\) that satisfy the functional equation \\( 4g(x) = g(x+1) + g(x-1) + g(x+\\pi) + g(x-\\pi) \\) for every \\( x \\in \\mathbb{R} \\).",
"role": "user"
}
] | g(x) = C |
[
{
"content": "Find the smallest integer \\( n \\) such that there exists a solution to the following system of equations:\n\\[\n\\sin x_1 + \\sin x_2 + \\cdots + \\sin x_n = 0,\n\\]\n\\[\n\\sin x_1 + 2 \\sin x_2 + \\cdots + n \\sin x_n = 100.\n\\]",
"role": "user"
}
] | 20 |
[
{
"content": "Determine the type of singularity at the point $z=0$ for the function $f(z)= \\sin\\bigg(\\frac{1}{\\cos(\\frac{1}{z})}\\bigg)$. Choose from the following options:\n1) Removable singularity\n2) Pole\n3) Essential singularity\n4) Non-isolated singularity",
"role": "user"
}
] | D |
[
{
"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... | 3.05 g KCl |
[
{
"content": "Evaluate the infinite series: \\[ \\sum_{n=1}^{\\infty}\\frac{1}{(2n-1)^2} \\] Choose the correct answer from the following options: \\(\\text{a) } \\frac{\\pi^2}{2} \\quad \\text{b) } \\frac{\\pi^2}{4} \\quad \\text{c) } \\frac{\\pi^2}{6} \\quad \\text{d) } \\frac{\\pi^2}{8}\\)",
"role": "use... | D |
[
{
"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... | 233.48 N |
[
{
"content": "Let \\( f: \\mathbb{N} \\rightarrow \\mathbb{N} \\) be a bijective function. Determine whether \\( \\lim_{n \\rightarrow \\infty} f(n) = \\infty \\).",
"role": "user"
}
] | \infty |
[
{
"content": "Prove that the space \\( c = \\{ f: \\mathbb{N} \\to \\mathbb{C} \\mid \\lim_{n \\to \\infty} f(n) \\text{ exists} \\} \\), equipped with the norm \\( ||f||_\\infty = \\sup_{n \\in \\mathbb{N}} |f(n)| \\), is a Banach space.",
"role": "user"
}
] | c |
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