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... | 4.92 years |
[
{
"content": "Evaluate the limit of the integral over the semicircle as \\( R \\to \\infty \\):\n\n\\[ \\lim_{R\\to\\infty}\\int_{\\Gamma_R}\\frac{e^{izt}}{\\pi\\cos(iz)}dz \\]\n\nwhere \\( t > 0 \\) and \\( \\Gamma_R \\) is the upper semicircle of radius \\( R \\) in the complex plane.",
"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... | 24.5 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... | 0.197 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... | 27.52% |
[
{
"content": "Determine the number of pairs $(a,b)$ of real numbers such that $10, a, b, ab$ is an arithmetic progression.",
"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... | 2100 kW |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 9.58% |
[
{
"content": "Determine whether the sequence of functions \\( g_n(x) = \\begin{cases} \\left(1+\\frac{x^2}{n}\\right)^{-n} & 0\\leq x\\leq \\sqrt{n} \\\\ 0 & \\text{otherwise}\\end{cases} \\) converges uniformly to a limit function on \\([0, \\infty)\\). If so, identify the limit function.",
"role": "user"
... | e^{-x^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... | 620 g Cl_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... | -246 J |
[
{
"content": "Evaluate the integral \\( \\int_C \\frac{1}{z^2(z^2-16)} \\, dz \\), where \\( C \\) is the contour \\( |z| = 1 \\).",
"role": "user"
}
] | 0 |
[
{
"content": "In a rhombus \\(ABCD\\), circles \\(\\Gamma_{B}\\) and \\(\\Gamma_{C}\\) are centered at \\(B\\) and \\(C\\) and pass through \\(C\\) and \\(B\\) respectively. Let \\(E\\) be an intersection point of circles \\(\\Gamma_{B}\\) and \\(\\Gamma_{C}\\). The line \\(ED\\) intersects circle \\(\\Gamma_{B... | 60^\circ |
[
{
"content": "Consider the infinite system of equations given by:\n\\[\np(k+2)q(k+2) - 3p(k+1)q(k+1) + 2p(k)q(k) = 0\n\\]\nfor every natural number \\(k\\), where \\(p(k)\\) and \\(q(k)\\) are natural numbers greater than 2. Does there exist a pair of integer sequences \\(p(k)\\) and \\(q(k)\\) that satisfy thi... | 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... | 18.9 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... | 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... | 2650 min |
[
{
"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... | 5.653 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $51.16 |
[
{
"content": "In triangle $ABC,$\n\\[(b \\sin C)(b \\cos C + c \\cos B) = 42.\\]Compute the area of triangle $ABC.$",
"role": "user"
}
] | 21 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | C2H3O3 |
[
{
"content": "Determine the value of the infinite series \\( \\sum_{n=1}^{\\infty} \\frac{n}{(n-1)!} \\). Verify if the value is \\(2e\\).",
"role": "user"
}
] | 2e |
[
{
"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... | 0.93 sec |
[
{
"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... | 4.8 cm^3 |
[
{
"content": "Determine whether a circle can be continuously embedded into the real line. Provide a yes or no answer.",
"role": "user"
}
] | No |
[
{
"content": "Determine whether every simply connected open set in \\(\\mathbb{R}^2\\) is homeomorphic to an open ball.",
"role": "user"
}
] | Yes |
[
{
"content": "In a right square prism \\( P-ABCD \\) with side edges and base edges both equal to 4, calculate the total length of all curves on its surface that are 3 units away from vertex \\( P \\).",
"role": "user"
}
] | \dfrac{9\pi}{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... | -957 J |
[
{
"content": "Find the values of $b$ for which the integral $$\\int_{0}^{1}\\frac{1}{x^b(1+x)}\\,dx$$ converges.",
"role": "user"
}
] | (-\infty, 1) |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 6.2% |
[
{
"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... | 6504.78 m/s |
[
{
"content": "Find the length of the arc of the curve defined by the parametric equations \\( x=3(2 \\cos t-\\cos 2t) \\) and \\( y=3(2 \\sin t-\\sin 2t) \\) for \\( 0 \\leq t \\leq 2\\pi \\).",
"role": "user"
}
] | 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... | 6.7 J |
[
{
"content": "Determine whether the fundamental solution to the Laplacian on \\(\\mathbb{R}^n\\) is locally integrable.",
"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... | 11.80% |
[
{
"content": "Determine whether there exists a continuous function whose Fourier series converges everywhere, but not uniformly to the function itself.",
"role": "user"
}
] | Yes |
[
{
"content": "Determine whether there exists a group such that the number of its subgroups exceeds the number of elements in the group.",
"role": "user"
}
] | Yes |
[
{
"content": "Let $T$ be a linear operator on an inner product space $V$. Suppose there exists a positive integer $k$ such that $T^k(v) = 0$ for some vector $v \\in V$, and $T^*T$ is self-adjoint. Does it follow that $T(v) = 0$? Answer with \"Yes\" or \"No\" and justify your answer.",
"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... | Sales: $110,440
Expenses:
Labor (F): $12,600
Labor (V): $19,800
Cost of Sales (V): $39,600
Supplies (V): $3,520
Energy (V): $2,882
Promotion (V): $1,430
Maintenance (F): $2,100
Maintenance (V): $1,320
Property taxes (F): $840
Depreciation (F): $1,050
Rent (F): $892.5
Insurance (F): $656.25
Subtotal: $86,690.75
Net inco... |
[
{
"content": "Identify the incorrect statement about holomorphic functions from the following options:\n\n(1) If $f(z)$ is holomorphic, then $f(\\bar{z})$ is not holomorphic.\n(2) If $f(\\mathbb{R}) \\subseteq \\mathbb{R}$ and $f$ is non-constant, then $f(\\mathbb{R})$ is not open in $\\mathbb{R}$.\n(3) The map... | 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... | $188,410 |
[
{
"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... | 22.62Nm |
[
{
"content": "Evaluate the summation \\( S = \\sum_{r=0}^{1010} \\binom{1010}{r} \\sum_{k=2r+1}^{2021}\\binom{2021}{k} \\).",
"role": "user"
}
] | 2^{3030} |
[
{
"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... | 214.94g |
[
{
"content": "Let $G$ be a compact connected real Lie group. Is the set of torsion elements $T$ always dense in $G$?",
"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... | 5,100 units |
[
{
"content": "Evaluate the line integral \\( \\oint \\frac{\\partial F}{\\partial n} \\, ds \\) around the circle \\( x^2 + y^2 = 1 \\), where \\( F(x,y) = x^2 - y^2 \\) and \\( \\frac{\\partial F}{\\partial n} \\) is the directional derivative of \\( F \\) along the outer normal. Use first principles to solve ... | 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... | 8.99% |
[
{
"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... | AM radio wave < FM radio wave < yellow light < green light < X ray |
[
{
"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... | $134,350 |
[
{
"content": "In a finite-dimensional vector space $V$, suppose $\\lambda_j, j=1,\\cdots,p$ are the eigenvalues of a linear operator $A$ with multiplicities $m_1,\\cdots,m_p$. Let $k_j$ be the smallest index such that $\\text{Ker}(A-\\lambda_j)^{k_j} = \\text{Ker}(A-\\lambda_j)^{k_j+1}$. Is it true that the sub... | Yes |
[
{
"content": "Given the differential equation of the curves \\(c(y+c)^2 = x^3\\) where 'c' is an arbitrary constant, the differential equation is \\(12y(y')^2 + ax = bx(y')^3\\). Find the value of \\(a+b\\).",
"role": "user"
}
] | 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... | 0.021 |
[
{
"content": "Let $\\theta_1,\\dots,\\theta_N$ be complex numbers with absolute value 1. Assume that for any natural number $r\\geq 1$, the sum \\(\\theta_1^r+\\dots+\\theta_N^r\\) is a constant independent of $r$. Does it follow that $\\theta_1=\\dots=\\theta_N=1$? Answer yes or no and justify your answer.",
... | Yes |
[
{
"content": "Does there exist a generating combination of turns for a Rubik's Cube, such that all possible states of the cube can be achieved through repeated application of this combination?",
"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... | {eq}F = - \frac{{6k{P_1}{P_2}}}{{{X^4}}}{/eq} |
[
{
"content": "Find the smallest positive integer \\( n \\) such that both of the following equations hold true: \\( \\left(\\frac{\\sqrt{2}}{2}+\\frac{\\sqrt{2}}{2}i\\right)^{n}=1 \\) and \\( \\left(\\frac{\\sqrt{2}}{2}-\\frac{\\sqrt{2}}{2}i\\right)^{n}=1 \\).",
"role": "user"
}
] | 8 |
[
{
"content": "Suppose X and Y are continuous random variables with joint density function f(x, y). What is the probability that X equals Y?",
"role": "user"
}
] | 0 |
[
{
"content": "What is the order of the global truncation error at the end of an integration using the RK4 method with a step size of 2 seconds, from $t = 0$ seconds to $t = 258492$ seconds?",
"role": "user"
}
] | 4 |
[
{
"content": "Find the point $P$ on the $x$-axis such that its orthogonal projection onto the line $r: x + y = z - 1 = 0$ is the point $M = (1, -1, 1)$. What are the coordinates of point $P$?",
"role": "user"
}
] | (2, 0, 0) |
[
{
"content": "Let $g(x)$ be a continuous and twice differentiable function satisfying $g(x)>0$, $g'(x)>0$, and $g''(x)<0$ for all $x\\in[2,9]$. If $g^{-1}(x)$ exists, determine the value of $g^{-1}(7)-2g^{-1}(8)+g^{-1}(9)$. Provide your answer as a single value.",
"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... | 1.22 rev/s |
[
{
"content": "Given the polynomial \\( W(x) = -7x^5 + a_4x^4 + a_3x^3 + a_2x^2 + a_1x + a_0 \\), where \\( W(1) = -2 \\), \\( W(2) = -4 \\), \\( W(3) = -6 \\), \\( W(4) = -8 \\), and \\( W(5) = -10 \\), find the constant term \\( a_0 \\).",
"role": "user"
}
] | 840 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 2.82e7 m/s |
[
{
"content": "For what value of \\( p \\) is the function \\( w \\) in the fractional Sobolev space \\( H^{s}(\\mathbb{R}) \\) for some \\( s > 3/2 \\), given that \\( w \\) is analytic for \\( x<0 \\), behaves as \\( e^{x} \\) as \\( x\\rightarrow -\\infty \\), \\( w \\sim |x|^p \\) as \\( x\\rightarrow 0^- \\... | p > 1 |
[
{
"content": "Use Stokes' theorem to evaluate the line integral \\(\\int_C (y-z)\\,dx + (z-x)\\,dy + (x-y)\\,dz\\) where \\(C\\) is the circle of radius 2 in the \\(xy\\)-plane centered at the origin, oriented counterclockwise when viewed from above.",
"role": "user"
}
] | -8\pi |
[
{
"content": "Determine whether there exist any integral solutions for the equation \\(x^2 - 2 = y^p\\) where \\(p \\geq 3\\).",
"role": "user"
}
] | Yes |
[
{
"content": "Find a prime factor of \\( \\frac{3^{41} - 1}{2} \\) without using prime factorization.",
"role": "user"
}
] | 83 |
[
{
"content": "In a cafe with 123 individuals, the sum of their ages is 3813 years. Is it always possible to select a group of 100 people from them whose combined age is at least 3100 years?",
"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... | 0.32 m/s |
[
{
"content": "Evaluate the limit: $$\\lim_{t\\rightarrow0}\\frac{\\delta\\theta t(1-t)^{-\\theta-1}}{(1-t)^{-\\theta}-1} \\ ,$$ where $\\theta \\geq 0$ and $\\delta \\geq 1$. ",
"role": "user"
}
] | \delta |
[
{
"content": "Evaluate the integral: \\[ \\int_0^\\pi \\left(\\frac{\\sin{2x}\\sin{3x}\\sin{5x}\\sin{30x}}{\\sin{x}\\sin{6x}\\sin{10x}\\sin{15x}}\\right)^2 \\, dx \\]",
"role": "user"
}
] | 7\pi |
[
{
"content": "Evaluate the limit: $$\\lim_{n\\to\\infty}\\left[\\frac{\\left(\\frac{3n}{e}\\right)^{3n}\\cdot \\sqrt{6\\pi n}}{n^n\\cdot \\left(\\frac{n}{e}\\right)^{n}\\cdot \\sqrt{3\\pi n}}\\right]^{\\frac{1}{n}}$$",
"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... | 3.85e-11 m/s^2 |
[
{
"content": "Bob plays a game where, for some number $n$, he chooses a random integer between 0 and $n-1$, inclusive. If Bob plays this game for each of the first four prime numbers, what is the probability that the sum of the numbers he gets is greater than 0?",
"role": "user"
}
] | \frac{209}{210} |
[
{
"content": "Find the coefficient of \\(x^{1998}\\) in the expansion of \\(\\prod_{n=1}^{1996}(1+nx^{3^n})\\).",
"role": "user"
}
] | 0 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{N \\to \\infty} \\frac{\\sum\\limits_{k=1}^{d} {\\varphi(N) \\choose k} {d-1 \\choose k-1}}{(\\varphi(N)+d)^d} \\] where \\( \\varphi(N) \\) is a function that tends to infinity as \\( N \\) increases.",
"role": "user"
}
] | \dfrac{1}{d!} |
[
{
"content": "Calculate the sum \\( S_n = \\sum^{2n-1}_{r=1} 2f\\left(\\frac{r}{2n}\\right) \\) where \\( f(x) = \\frac{2^x}{2^x + \\sqrt{2}} \\).",
"role": "user"
}
] | 2n - 1 |
[
{
"content": "Let \\((1+x+x^{2})^{150}=\\sum_{k=0}^{300} c_{k} x^{k}\\), where \\(c_{0}\\), \\(c_{1}\\), \\ldots, \\(c_{300}\\) are constants. Find \\(\\sum_{k=0}^{100} c_{3k}\\).",
"role": "user"
}
] | 3^{149} |
[
{
"content": "Solve the exponential equation for \\( n \\): \\[ 3\\sqrt{5} \\cdot 2^n = (1+\\sqrt{5})^n - (1-\\sqrt{5})^n \\]",
"role": "user"
}
] | 4 |
[
{
"content": "Let $A$ and $B$ be two $n \\times n$ matrices such that $\\operatorname{rank}(A) = n$ and $\\operatorname{rank}(B) = n-1$. Determine if it is true that $\\operatorname{rank}(AB) = \\operatorname{rank}(BA) = n-1$. 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... | 0.55 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... | 1.2 in |
[
{
"content": "Find the smallest integer \\(x\\) such that \\(\\gcd(50 + x^2, 4x + 2) = 6\\).",
"role": "user"
}
] | 4 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 20.05 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... | {eq}v(s) = \frac{s^6}{6} - \frac{s^4}{4} + 12 s + \text{c}{/eq} |
[
{
"content": "Find the maximum value of \\( \\frac{2}{\\sin{2x}} - \\frac{x}{2} \\) such that the equation \\( \\left(\\frac{2}{\\sin{2x}} - \\frac{x}{2}\\right)^2 = 3 - k^2 \\) has real roots.",
"role": "user"
}
] | \sqrt{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... | $27,408.71 |
[
{
"content": "Determine the value of $x$ at which the function \\( f(x) = \\int_{-10}^{x^2+2x} e^{t^2}\\,dt \\) reaches its minimum. Use the Chain Rule in your solution.",
"role": "user"
}
] | -1 |
[
{
"content": "Let $X$ be a topological manifold, and consider a finite collection $\\{Y_i : Y_i \\subset X\\}$ of pairwise disjoint submanifolds without boundary, each equipped with the subspace topology. If the union $\\bigcup_i Y_i$ is an open submanifold in $X$, does it necessarily follow that at least one $... | Yes |
[
{
"content": "Let $f$ be a real-valued function on some open subset of $\\mathbb{R}$, defined as algebraic if there exists a polynomial $P \\in \\mathbb{R}[x,y]$ such that $P(x,f(x))=0$ identically on the domain of $f$. Assume $f$ has an inverse function $g$ on some open subset of $\\mathbb{R}$ such that $f(g(y... | 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... | co-adaptation |
[
{
"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... | 0.09 or 9% |
[
{
"content": "Consider an autonomous ordinary differential equation (ODE) given by \\( \\dot x = f(x) \\), where \\( x \\in \\mathbb{R}^n \\) and \\( x_0 \\) is an equilibrium point. Determine whether the following statement is true: If all eigenvalues of the Jacobian matrix \\( \\frac{\\partial f}{\\partial x}... | True |
[
{
"content": "A fifth degree polynomial $P(x)$ with integral coefficients takes on values $0, 1, 2, 3, 4$ at $x = 0, 1, 2, 3, 4$, respectively. Which of the following is a possible value for $P(5)$? A) $5$ B) $24$ C) $125$ D) None of the above",
"role": "user"
}
] | C |
[
{
"content": "Given that the function $f$ satisfies the equation $f(xy) = f(x) + f(y)$ for any positive $x$ and $y$, and that $f\\left(\\frac{1007}{1}\\right) = 1$, find the value of $f(2007)$. ",
"role": "user"
}
] | \log_{1007} 2007 |
[
{
"content": "Steve has an isosceles triangle with base 8 inches and height 10 inches. He wants to cut it into eight pieces that have equal areas, as shown below. To the nearest hundredth of an inch what is the number of inches in the greatest perimeter among the eight pieces? [asy]\nsize(150);\ndefaultpen(line... | 22.21 |
[
{
"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... | (0, -7.04 m, 0) |
[
{
"content": "Given that \\(a, b, c\\) are the roots of the polynomial \\(x^3 + 20x^2 + x + 5\\), compute the value of \\((a^2+1)(b^2+1)(c^2+1)\\).",
"role": "user"
}
] | 225 |
[
{
"content": "If $C^2[0,1]$ is a subspace of $L^2[0,1]$, does it follow that if a sequence $f_n \\in C^2[0,1]$ converges to $f$ in the $L^2[0,1]$ topology, then $f$ must also be in $C^2[0,1]$?",
"role": "user"
}
] | No |
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