prompt list | ground_truth string |
|---|---|
[
{
"content": "Determine whether the sequence of functions \\( f_n(x) = \\left(x - \\frac{1}{n}\\right)^2 \\) converges pointwise on the interval \\([0,1]\\). If it does, find the limit function.",
"role": "user"
}
] | x^2 |
[
{
"content": "Given the differential equation $y'(t) = \\frac{t}{4} - \\frac{1}{y^2 + 1}$ with the initial condition $y(0) = 1$, and the function $x(t) = 1 - \\frac{t}{2}$, approximate $|x(t) - y(t)|$ for $|t| < \\frac{1}{2}$ using a corollary of Gronwall's theorem.",
"role": "user"
}
] | \dfrac{t^3}{48} |
[
{
"content": "For a finite group \\( G \\), determine the maximum proportion of irreducible rational-valued characters that can take the values 0 or \\( \\pm 1 \\).",
"role": "user"
}
] | 1 |
[
{
"content": "Evaluate the limit \\(\\lim_{n\\to\\infty}\\left(H_n^2 - 2\\sum_{i=1}^n \\frac{H_i}{i} + \\sum_{k=1}^n \\frac{1}{k^2}\\right)\\), where \\(H_i\\) is the \\(i\\)-th harmonic number.",
"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... | 80 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... | An infinite uniformly charged sheet |
[
{
"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... | 84.4% |
[
{
"content": "Calculate the limit of the function \\( f(x) = \\left(\\frac{x^2+1}{x}\\right)^{1/\\ln(x)} \\) as \\( x \\to \\infty \\).",
"role": "user"
}
] | 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... | -0.9375h |
[
{
"content": "How many integers between 0 and 8 inclusive have an inverse modulo 9?",
"role": "user"
}
] | 6 |
[
{
"content": "Let \\( f \\in \\mathbb{C}[x] \\) be a monic polynomial of prime degree \\( p \\) such that \\( f \\) is not co-prime with any of its derivatives \\( f^{(k)} \\) for \\( 1 \\leq k < p \\). Prove or disprove that there exists \\( a \\in \\mathbb{C} \\) such that \\( f(x) = (x-a)^p \\).",
"role"... | (x - a)^p |
[
{
"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... | 0.5 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... | 3.72 A |
[
{
"content": "Determine the number of real solutions for the equation \\(x^2 - x\\sin x - \\cos x = 0\\) on the interval \\((-\\infty, \\infty)\\).",
"role": "user"
}
] | 2 |
[
{
"content": "Find the limit of the expression \\( L = \\lim_{x \\to 0} \\frac{x\\int_0^x{f(x-t)dt}}{\\int_0^x tf(x-t)dt} \\) given that \\( f(x) \\) is continuous and \\( \\lim_{x \\to 0} \\frac{f(x)}{x} = 1 \\).",
"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... | 2.64 s |
[
{
"content": "Given that the function \\( f(x) \\) satisfies the condition \\( f\\left(\\frac{a+2b}{3}\\right)=\\frac{f(a)+2f(b)}{3} \\) for any real numbers \\( a \\) and \\( b \\), find the value of \\( f(2022) \\) if \\( f(1)=5 \\) and \\( f(4)=2 \\).",
"role": "user"
}
] | -2016 |
[
{
"content": "Determine the greatest common divisor (GCD) of all integers of the form \\( n^{13} - n \\) for integer values of \\( n \\).",
"role": "user"
}
] | 2730 |
[
{
"content": "Find the residue of the function \\( f(z) = \\frac{z}{8-z^3} \\) at \\( z = 2 \\).",
"role": "user"
}
] | -\dfrac{1}{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... | -2.736 Pa, opposite to the direction of the flow of water |
[
{
"content": "Find the multiplicative inverse of the polynomial $2x+1$ in the quotient ring $F/(m(x))$, where $m(x) = x^2 + x + 2$ and $F = \\mathbb{Z}/3\\mathbb{Z}$. Provide your answer as a polynomial in $F/(m(x))$. ",
"role": "user"
}
] | x + 2 |
[
{
"content": "Let \\(f = u + iv\\) be an entire function such that \\(u_x v_y - u_y v_x = 1\\) throughout the complex plane. Prove that \\(f(z) = az + b\\), where \\(a\\) and \\(b\\) are constants and \\(|a| = 1\\).",
"role": "user"
}
] | f(z) = az + b |
[
{
"content": "Compute the remainder when \\(67!\\) is divided by \\(71\\).",
"role": "user"
}
] | 12 |
[
{
"content": "A set $S$ is constructed as follows. To begin, $S = \\{0,10\\}$. Repeatedly, as long as possible, if $x$ is an integer root of some nonzero polynomial $a_{n}x^n + a_{n-1}x^{n-1} + \\dots + a_{1}x + a_0$ for some $n\\geq{1}$, all of whose coefficients $a_i$ are elements of $S$, then $x$ is put into... | 9 |
[
{
"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... | 1960 J/kg (m) |
[
{
"content": "Given the quartic polynomial equation \\(x^4 - x^3 - 1 = 0\\) with roots \\(\\alpha, \\beta, \\gamma, \\delta\\), find the exact value of \\(\\alpha^6 + \\beta^6 + \\gamma^6 + \\delta^6\\).",
"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... | Deep Valley Foods
Direclt labor budget for March
Total Smoking Packing
Estimated production units (pounds) 400,000 400,000 400,000
Direct labor hours per pound 0.0726 0.053 0.0196
Total direct labor hours required 29,040 21,200 7,840
Direct labor rate per hour $16 $9
Total estimated direct labor cost $409,760 $339,200 $70,560 |
[
{
"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... | 0.49 m/s |
[
{
"content": "Consider a power series \\( f(z) = \\sum_{n \\geq 0} a_n z^n \\), which has a positive finite radius of convergence \\( R > 0 \\). If \\( f \\) cannot be holomorphically extended to a larger disk, does it imply that \\( f \\) is unbounded within the open disk \\( D(0,R) \\)? Provide a justificatio... | No |
[
{
"content": "Find the domain of the function \\( g(x) = \\sqrt{\\frac{(e^x+1)^x + \\arcsin(e^x +1)}{x- \\sqrt{x^2 +1}}} \\).",
"role": "user"
}
] | \emptyset |
[
{
"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... | Goods and services are needed by the public, but are not profitable to produce; The good or product is needed or produced by the military; The Constitution requires the government to produce a service. |
[
{
"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... | 227.273 V |
[
{
"content": "Determine the values of $n$ for which the following statement is true: If $H$ is a normal subgroup of $O(n)$ and contains an element $h$ that acts as the negative identity on a one-dimensional subspace and as the identity on its orthogonal complement, then $H$ must be equal to $O(n)$. What are the... | n \geq 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... | 645 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... | 950 L |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | d < e < a = b = 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... | 60.26 deg |
[
{
"content": "Calculate the limit: $$\\lim_{n \\to \\infty}\\sum\\limits_{k=1}^n \\left(\\dfrac {n-k}{n}\\right)^{tk}$$ where $t\\in \\mathbb{R}$. \\(t\\) is a real number.",
"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... | 1.484e-7 N towards the +x axis |
[
{
"content": "Find the total curvature of the curve $C$ in $\\Bbb{R}^2$ given by $(t-\\sin t,1-\\cos t)$ for $0 \\le t \\le 2 \\pi$. Express your answer as a single numerical value.",
"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... | 0.074 kg |
[
{
"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... | $125,000 |
[
{
"content": "Given the matrix \\( A = \\begin{bmatrix} 1 & 0 & 0 \\\\ i & \\frac{-1 + i\\sqrt{3}}{2} & 0 \\\\ 0 & 1 + 2i & \\frac{-1 - i\\sqrt{3}}{2} \\end{bmatrix} \\), find the trace of \\( A^{102} \\).",
"role": "user"
}
] | 3 |
[
{
"content": "Let \\(x_n\\) be the \\(n\\)th positive solution to the equation \\(x = \\tan(x)\\). Determine \\(\\lim_{n\\to\\infty}(x_n - x_{n-1})\\).",
"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... | 0.176 mol/L |
[
{
"content": "Determine the number of connected components in the set \\(\\{t\\in \\Bbb{R}: f \\text{ is differentiable at } t\\}\\), where the function \\(f: \\Bbb{R} \\to \\Bbb{R}\\) is defined as follows:\n\n\\[f(t) = \\begin{cases}\n\\frac{|t|}{2}, & t <-2,\\\\\n\\frac{|t+3|}{2} + \\frac{1}{2}, & -2\\le t<-... | 5 |
[
{
"content": "Solve the equation: \\[ \\sin\\left(\\frac{x}{x-1}\\right) + 2 \\tan^{-1}\\left(\\frac{1}{x+1}\\right) = \\frac{\\pi}{2} \\]",
"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... | 1.99e+20 N |
[
{
"content": "Find the least positive residue of \\(3^{83} \\pmod{3600}\\).",
"role": "user"
}
] | 27 |
[
{
"content": "Evaluate the second mixed partial derivative \\( \\frac{\\partial^2 f}{\\partial x \\partial y}(0,0) \\) for the function \\( f(x,y) = \\begin{cases} \\frac{xy(x^2-y^2)}{x^2+y^2} & \\text{if }(x,y) \\neq (0,0) \\\\ 0 & \\text{otherwise} \\end{cases} \\).",
"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... | 848.13 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... | 3.35 mol air |
[
{
"content": "Evaluate the triple integral \\( \\int \\int \\int (x^2 + y^2 + z^2) \\, dx \\, dy \\, dz \\) over the region \\( R \\) bounded by the planes \\( x+y+z = 1 \\), \\( y = 0 \\), and \\( z = 0 \\).",
"role": "user"
}
] | \dfrac{1}{20} |
[
{
"content": "Evaluate the integral \\( \\int_D z^n \\overline{z^m} \\, dx \\, dy \\) over the domain \\( D = \\{ z \\in \\mathbb{C} : |z| < 1 \\} \\) and show that it equals zero when \\( n \\neq m \\).",
"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... | 110 J |
[
{
"content": "Let $f(x)$ be a third-degree polynomial with real coefficients satisfying \\[|f(1)|=|f(2)|=|f(3)|=|f(5)|=|f(6)|=|f(7)|=12.\\]Find $|f(0)|$.",
"role": "user"
}
] | 72 |
[
{
"content": "Determine if the partial derivative \\( \\frac{\\partial u}{\\partial x} \\) is discontinuous along the line \\( x=2 \\) for the given solution of the PDE. If a discontinuity exists, find the value of \\( y \\) at which it occurs.",
"role": "user"
}
] | 2 |
[
{
"content": "Let non-negative real numbers \\( a, b, c \\) satisfy \\( a+b+c=1 \\). Determine the maximum value of \\( S = \\sqrt{4a+1} + \\sqrt{4b+1} + \\sqrt{4c+1} \\).",
"role": "user"
}
] | \sqrt{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... | 193 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... | 6.29e5 N/C |
[
{
"content": "Verify that for \\( x \\in \\mathbb{N} \\), the Gamma function \\( \\Gamma(x) = \\int_0^\\infty t^{x-1} \\mathrm{e}^{-t}\\,\\mathrm{d}t \\) satisfies \\( \\Gamma(x) = (x-1)! \\).",
"role": "user"
}
] | \Gamma(x) = (x-1)! |
[
{
"content": "Let $x$ and $y$ be positive real numbers. Find the minimum value of\n\\[\\left( x + \\frac{1}{y} \\right) \\left( x + \\frac{1}{y} - 2018 \\right) + \\left( y + \\frac{1}{x} \\right) \\left( y + \\frac{1}{x} - 2018 \\right).\\]",
"role": "user"
}
] | -2036162 |
[
{
"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... | 4547200 Js |
[
{
"content": "Determine the range of values of \\(\\beta\\) for which the integral \\(\\int_{e}^{4} \\frac{dt}{\\ln^\\beta(\\ln^2(t))}\\) converges.",
"role": "user"
}
] | (-\infty, 1) |
[
{
"content": "Is it possible for Nikolai to choose a chessboard polygon \\(F\\) such that any congruent chessboard polygon has at least one green cell but at most 2020 green cells, making it impossible to paint the grid cells to meet this challenge?",
"role": "user"
}
] | Yes |
[
{
"content": "Consider the initial value problem \\(u'(t) = Au(t), t > 0\\) with \\(u(0) = u_0\\), where \\(u_0\\) is a given vector in \\(\\mathbb{R}^2\\) and \\(A = \\begin{bmatrix} 1 & -2 \\\\ 1 & a \\end{bmatrix}\\). Find the range of values of \\(a\\) such that \\(|u(t)| \\to 0\\) as \\(t \\to \\infty\\)."... | (-2, -1) |
[
{
"content": "Determine the type and order of the singularity that the function \\( f(z) = \\frac{z}{\\cos(z) - 1} \\) has at \\( z = 0 \\).",
"role": "user"
}
] | 1 |
[
{
"content": "What is the area, in square units, of the triangle bounded by $y = 0$, $y = x + 4$ and $x + 3y = 12$?",
"role": "user"
}
] | 32 |
[
{
"content": "Determine whether there exists a computable function \\( f:\\mathbb{N}\\to\\mathbb{Q} \\) such that:\n\n1. For all \\( m \\in \\mathbb{N} \\), \\( 0 \\leq f(m) \\).\n2. \\( \\lim_{m\\to \\infty} f(m) = x_K \\), where \\( x_K = \\sum_{n\\in K} 2^{-n} \\) and \\( K \\) is the Halting problem.",
... | 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... | $6,947.90 |
[
{
"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... | (43.75, 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... | f. Identify potential capital investments.
b. Project investments' cash flows.
g. Screen/analyze investments.
a. Budget capital investments.
d. Make investments.
c. Perform post-audits.
e. Use feedback to reassess investments already made. |
[
{
"content": "The inverse of $f(x) = \\frac{2x-1}{x+5}$ may be written in the form $f^{-1}(x)=\\frac{ax+b}{cx+d}$, where $a$, $b$, $c$, and $d$ are real numbers. Find $a/c$.",
"role": "user"
}
] | -5 |
[
{
"content": "Determine the convergence domain of the series \\( \\sum_{n=1}^{\\infty} \\frac{(-1)^{n-1}}{n3^n(x-5)^n} \\).",
"role": "user"
}
] | (-\infty, \frac{14}{3}) \cup [\frac{16}{3}, \infty) |
[
{
"content": "In a game, three players, $A$, $B$, and $C$, compete to earn the highest score. The score of each player is represented by a Poisson distributed random variable: $A$ with parameter $\\lambda_A = 2$, $B$ with $\\lambda_B = 3$, and $C$ with $\\lambda_C = 4$. The scores are independent. Given that th... | \dfrac{40}{81} |
[
{
"content": "Determine the values of \\( n \\) for which the integral \\( \\iiint_E \\frac{1}{(x^2+y^2+z^2)^{n/2}} \\, dV \\), where \\( E \\) is the region bounded by spheres of radii \\( r \\) and \\( R \\), has a limit as \\( r \\to 0^+ \\). The integral evaluates to \\( 4\\pi\\left(\\frac{R^{3-n}}{3-n}-\\f... | n < 3 |
[
{
"content": "Calculate the integral \\( \\iiint _V \\frac{\\sin x-\\sin y}{xy+1} \\, dx\\,dy\\,dz \\) over the region \\( V \\) determined by the surfaces \\( x^2 = y \\), \\( y^2 = x \\), \\( z = 0 \\), and \\( z = 1 \\). Use symmetry to simplify the calculation.",
"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... | 14.92 mL |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 5.83% |
[
{
"content": "Let $f: \\Bbb R \\rightarrow \\Bbb R$ be a differentiable function with $f'$ uniformly continuous. Suppose $\\lim\\limits_{x\\rightarrow \\infty}f(x)=L$ for some $L$. Does $\\lim\\limits_{x\\rightarrow \\infty}f'(x)$ exist?",
"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... | 36 m/h |
[
{
"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... | C_3H_6O |
[
{
"content": "Calculate the remainder when \\(177^{20^{100500}}\\) is divided by 60.",
"role": "user"
}
] | 21 |
[
{
"content": "Determine the radius of convergence, \\( R \\), for the series \\( \\sum_{n=2}^{\\infty} \\frac{(-1)^n z^{2n}}{(2n-1)!} \\).",
"role": "user"
}
] | \infty |
[
{
"content": "Evaluate the triple integral \\( \\int \\int \\int_{V} xyz \\, dV \\), where \\( V \\) is the volume of the tetrahedron with vertices at \\((0,0,0)\\), \\((1,0,0)\\), \\((0,1,0)\\), and \\((0,0,1)\\).",
"role": "user"
}
] | \dfrac{1}{720} |
[
{
"content": "Solve the integral equation: \\[ \\int_0^x e^{-x} u(y)\\, dy = e^{-x} + x - 1. \\]",
"role": "user"
}
] | u(x) = x e^x |
[
{
"content": "Calculate the greatest common divisor (gcd) of the complex numbers $3+i$ and $1-i$. Provide your answer in the form of a complex number.",
"role": "user"
}
] | 1 - i |
[
{
"content": "Find all integer values of $n$ such that $n^8 + n + 1$ is a prime number.",
"role": "user"
}
] | 1 |
[
{
"content": "Determine the minimum number of subspaces required to cover an entire vector space over a finite field.",
"role": "user"
}
] | q + 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... | 0.7m |
[
{
"content": "Find a number $a$, with $a < 2000$, that is not divisible by 7 and satisfies the congruence $a \\equiv 7^{3003} \\pmod{m}$, given that $\\phi(m) = 1000$. Use Euler's Theorem, which implies $7^{1000} \\equiv 1 \\pmod{m}$, to solve this without a calculator.",
"role": "user"
}
] | 1454 |
[
{
"content": "Find the maximum value of the function \\( f(x,y,z) = y \\) subject to the constraints \\( y^2 + x^2 + z^2 = 3 \\) and \\( y + x + z = 1 \\).",
"role": "user"
}
] | \dfrac{5}{3} |
[
{
"content": "Let \\( f : \\mathbb{R} \\to \\mathbb{R} \\) be a periodic and continuous function with period \\( T \\), and let \\( F : \\mathbb{R} \\to \\mathbb{R} \\) be an antiderivative of \\( f \\). Prove that \\[ \\int \\limits_0^T \\left[F(nx)-F(x)-f(x)\\frac{(n-1)T}{2}\\right]dx=0. \\]",
"role": "us... | 0 |
[
{
"content": "Determine the class corresponding to the Bott projection \\( b(z)=\\frac{1}{1+|z|^2}\\begin{pmatrix} 1 & \\bar{z} \\\\ z & |z|^2 \\end{pmatrix} \\) in \\( K_1(C(S^1)) \\) under the suspension isomorphism \\( K_1(A)\\cong K_0(SA) \\) for any \\( C^* \\)-algebra \\( A \\).",
"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... | 5.06% |
[
{
"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... | 0.36N |
[
{
"content": "Is it true that every smooth curve \\( \\gamma:[a,b]\\to \\mathbb{R}^n \\), where \\( \\gamma \\in C^1 \\), has a length given by \\( L(\\gamma)=\\int_{a}^b\\|\\gamma'(t)\\|\\,dt \\)?",
"role": "user"
}
] | Yes |
[
{
"content": "Simplify $\\frac{3}{\\sqrt[5]{16}}+\\frac{1}{\\sqrt{3}}$ and rationalize the denominator. The result can be expressed in the form $\\frac{a^2\\sqrt[5]{b}+b\\sqrt{a}}{ab}$, where $a$ and $b$ are integers. What is the value of the sum $a+b$?",
"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... | $120 |
[
{
"content": "If $p: \\Bbb R \\rightarrow \\Bbb R$ is a real polynomial with a right inverse $q$, must $q$ also be a left inverse of $p$? Provide a justification for your answer.",
"role": "user"
}
] | No |
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