prompt list | ground_truth string |
|---|---|
[
{
"content": "Evaluate the limit: \\[ \\lim_{x\\to \\sqrt{2}}\\left(\\frac{e^{x^2}+e^2(1-x^2)}{[\\ln(x^2-3\\sqrt{2}x+5)]^2}\\right) \\]",
"role": "user"
}
] | 2e^2 |
[
{
"content": "Determine the kernel of the homomorphism \\(\\phi: \\mathbb{Z}[x] \\to \\mathbb{R}\\) given by \\(\\phi(f(x)) = f(\\sqrt{2})\\). Express the kernel as an ideal in \\(\\mathbb{Z}[x]\\).",
"role": "user"
}
] | (x^2 - 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... | 3.8e+3 m/s |
[
{
"content": "Determine whether the function defined by the equation \\( xy = \\cot{xy} \\) is differentiable. If it is differentiable, find the correct expression for \\( \\frac{dy}{dx} \\).",
"role": "user"
}
] | -\dfrac{y}{x} |
[
{
"content": "Calculate the limit \\( \\lim_{n\\to\\infty} n^2 \\log \\left(\\cos \\frac 1n\\right) \\) without using l'Hospital's rule, Taylor series, or derivatives.",
"role": "user"
}
] | -\dfrac{1}{2} |
[
{
"content": "How many of the following numbers are factors of 34 or multiples of 7?\n\n1, 2, 3, 4, 8, 14, 17, 29, 56, 91",
"role": "user"
}
] | 6 |
[
{
"content": "Consider the sequence of random variables \\(X_1, X_2, \\ldots\\) given by \\(X_n = \\exp(-n^2(Z - \\frac{1}{n}))\\), where \\(Z\\) is a uniformly distributed random variable on the interval \\([0,1]\\). Does this sequence converge almost surely? If so, to what random variable does it converge?",
... | 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... | <img>/cimages/multimages/16/jag6914779224617488346.jpg</img> |
[
{
"content": "Consider the boundary value problem \\(x^2y''-2xy'+2y=0\\) with the boundary conditions \\(y(1)+\\alpha y'(1)=1\\) and \\(y(2)+\\beta y'(2)=2\\). Determine the values of \\(\\alpha\\) and \\(\\beta\\) for which the problem has a unique solution. Choose the correct option:\n\n1. \\(\\alpha=-1,\\bet... | 1 |
[
{
"content": "Find the limit of the complex sequence \\( \\{b_n\\} \\) defined by \\( b_{n+1}=\\frac{i\\,b_n}{i+b_n} \\) as \\( n \\) approaches infinity.",
"role": "user"
}
] | 0 |
[
{
"content": "Given two lines, \\(l_1\\) and \\(l_2\\), and two points, \\(A_1\\) on \\(l_1\\) and \\(A_2\\), construct the center \\(O\\) of the rotational homothety with a ratio \\(k \\neq 1\\) that maps \\(l_1\\) to \\(l_2\\) and \\(A_1\\) to \\(A_2\\).",
"role": "user"
}
] | O |
[
{
"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... | 7523.3kg |
[
{
"content": "Given that \\( \\lim_{x\\to2}\\frac{\\ xf(x)-4}{x-2}=3, \\) find the value of \\( \\lim _{ x\\rightarrow 2 }{ \\frac { f\\left( x \\right) -2 }{ \\sqrt { x-1 } -1 } }. \\)",
"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... | 14.99 m/s |
[
{
"content": "In an isosceles triangle, the tangent of the base angle is \\( \\frac{3}{4} \\). Calculate the tangent of the angle formed between the median and the angle bisector drawn to the lateral side.",
"role": "user"
}
] | \dfrac{1}{13} |
[
{
"content": "A production line has a yield of 90%, with the daily variation in the proportion of defective units having a standard deviation of 0.03. On a particular day, a sample of 120 units is inspected, and 18 defectives are found. Assuming a Beta prior distribution, estimate the proportion of defective un... | 0.127 |
[
{
"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... | 1.545e+19 electrons/cm^3 |
[
{
"content": "Evaluate the infinite series \\( \\sum_{n=1}^\\infty \\frac{n^2}{2^n} \\).",
"role": "user"
}
] | 6 |
[
{
"content": "Let $\\psi:\\mathbb{R}\\to\\mathbb{R}$ be a convex function such that $\\psi'(0)<0$. Determine whether the point at which $\\psi$ attains its minimum is positive.",
"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... | 2.9e6 |
[
{
"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... | 20.49% |
[
{
"content": "Find the limit of the sequence \\(a_n=\\sqrt[n+1]{(n+1)!}-\\sqrt[n]{n!}\\) as \\(n\\) approaches infinity, without using Stirling's approximation formula.",
"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... | 3 years |
[
{
"content": "Let \\( g:(\\alpha,\\beta)\\rightarrow\\mathbb{R} \\) be a twice differentiable function such that\n\\[\ng(x)=\\frac{x^2g''(x)+xg'(x)}{x^2+1},\n\\]\nand \\(0<\\alpha<\\beta\\). Given that \\(g(\\alpha)=g(\\beta)=0\\), determine if \\(g(x)=0\\) for all \\(x\\in(\\alpha,\\beta)\\).",
"role": "us... | g(x) = 0 |
[
{
"content": "Find the area of the region in the plane bounded by the curve \\(x^2 - 2x + 4y^2 - 8y + 1 = 0\\).",
"role": "user"
}
] | 2\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... | $149.98 |
[
{
"content": "Calculate the difference $A - B$, where $A$ is the number of 6-digit tickets (from 000000 to 999999) that are considered lucky, and $B$ is the number of tickets that are considered medium. A ticket is lucky if the sum of its first three digits equals the sum of its last three digits. A ticket is m... | 0 |
[
{
"content": "Jar A has exactly four red buttons and eight blue buttons. Carla then removes the same number of red buttons as blue buttons from Jar A and places them in an empty Jar B. Jar A now has $\\frac{2}{3}$ of its original number of buttons. If Carla were now to randomly choose a button from Jar A and... | \frac{1}{8} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $17,781 |
[
{
"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... | 8.94e-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... | Assets
Cash 2,900
Accounts receivable 2,200
Supplies 1,800
Equipment (net) 12,000
Total Assets 18,900
Liabilities and Equity
Accounts payable 3,300
Unearned service revenue 5,000
Common stock 10,000
Retained earnings (Including all income statement items and dividends) 600
Liabilities + Equity 18,900 |
[
{
"content": "Find the remainder when \\(5^{5555}\\) is divided by \\(10,000\\) using modular arithmetic and the method of exponentiation by squaring.",
"role": "user"
}
] | 8125 |
[
{
"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... | 1400 |
[
{
"content": "Calculate the y-coordinate of the centroid of the upper half of a unit semicircle using the formula:\n\\[ \\frac{1}{l(s)}\\int y \\, ds \\]\nwhere \\( l(s) = \\frac{\\pi}{2} \\) is the arc length of the semicircle. Verify that the y-coordinate of the centroid is \\( \\frac{4}{3\\pi} \\).",
"ro... | \dfrac{4}{3\pi} |
[
{
"content": "Let $f$ and $g$ be two real-valued continuously differentiable functions with domain $[a,b]$. If $f(x)g(x)=0$ for all $x$ in $[a,b]$, must either $f$ or $g$ be identically zero on $[a,b]$? Answer \"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... | 0.054 mol O |
[
{
"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... | -$7,676.09 |
[
{
"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... | 119.4 atm |
[
{
"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... | $475,473.76 |
[
{
"content": "Suppose $f(x) = \\int g(u)e^{2 \\pi iux}du$ for almost every $x \\in \\mathbb{R}$. Can we conclude that $g$ is the Fourier transform of $f$? Justify your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Determine whether any connected and simply connected open subset of \\(\\mathbb{R}^2\\) can be expressed as a countable union of convex open subsets of \\(\\mathbb{R}^2\\).",
"role": "user"
}
] | Yes |
[
{
"content": "Find the degree of the field extension $\\mathbb{Q}(c,d)/\\mathbb{Q}$, where $c$ is a root of $x^4 - 3$ and $d$ is the fourth root of 5. Determine whether this extension is a Galois extension.",
"role": "user"
}
] | 16 |
[
{
"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... | $38,953 |
[
{
"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... | external validity |
[
{
"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... | 72.97 |
[
{
"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... | 33.24 |
[
{
"content": "Let \\( S = \\{ \\tan(k) : k = 1, 2, \\dots \\} \\). Find the set of limit points of \\( S \\) on the real line.",
"role": "user"
}
] | \mathbb{R} |
[
{
"content": "Inside the tetrahedron \\(ABCD\\), there is a point \\(O\\) such that the lines \\(AO, BO, CO,\\) and \\(DO\\) intersect the faces \\(BCD, ACD, ABD,\\) and \\(ABC\\) at the points \\(A_{1}, B_{1}, C_{1},\\) and \\(D_{1}\\) respectively. Given that \\(\\frac{AO}{OA_{1}}=\\frac{BO}{OB_{1}}=\\frac{CO... | 3 |
[
{
"content": "Determine the largest real number \\( k \\) such that the inequality \\[ \\frac{a^{2}}{b} + \\frac{b^{2}}{a} \\geq 2 \\frac{a^{k+1} + b^{k+1}}{a^{k} + b^{k}} \\] holds for all positive real numbers \\( a \\) and \\( b \\).",
"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... | $1,272.76 |
[
{
"content": "Determine the minimum number of points necessary to approximate the integral \\(\\int_{0}^{1} \\sin(x^2\\pi)\\,dx\\) with an error below \\(10^{-8}\\) using the Composite Simpson method, considering the maximum value of the fourth derivative of \\(\\sin(x^2\\pi)\\).",
"role": "user"
}
] | 175 |
[
{
"content": "Suppose that n observations are chosen at random from a continuous and uniform probability density function fY(y). What is the probability that the second to last observation recorded is the smallest number in the entire sample?",
"role": "user"
}
] | \dfrac{1}{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... | 1,5-heptadiene |
[
{
"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... | 1.89e12 m/s^2 |
[
{
"content": "Evaluate the line integral \\( \\int_C F \\cdot ds \\) for the vector field \\( F(x, y, z) = \\left(4x\\left(1+\\frac{z^2}{8}\\right), \\frac{\\pi}{3}\\cos\\left(\\frac{\\pi}{3}y\\right), \\frac{x^2z}{2}-e^{\\frac{z}{3}}\\right) \\) along the curve \\( C \\) parameterized by \\( \\gamma(t) = (e^{t... | 12 - 3e^2 |
[
{
"content": "Determine whether the set of polynomials \\( \\{ P \\in \\mathbb{Q}(X) \\mid P(1) = P(3) = 0 \\} \\) forms an ideal in \\( \\mathbb{Q}[X] \\). If it does, find the monic generator of this ideal.",
"role": "user"
}
] | X^2 - 4X + 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... | 1/(2pi) * sqrt(4kqQ/mR^3) |
[
{
"content": "Does there exist a unique cyclic subgroup of order \\( n \\) in the quotient group \\( G = (\\mathbb{Q}/\\mathbb{Z}, +) \\)? Justify 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... | 21.67 m/s^2 |
[
{
"content": "Determine whether the derivative of the function \\(f(x)\\), defined as\n\n\\[f(x)=\\begin{cases} x^{2}\\sin\\left(\\frac{5}{x}\\right) & \\text{for }x\\neq0;\\\\ 0 & \\text{for }x=0 \\end{cases}\\]\n\nexists at \\(x=0\\) and, if it does, find its value.",
"role": "user"
}
] | 0 |
[
{
"content": "Let $X$ and $Y$ be random variables defined on a complete probability space. If $X$ is measurable with respect to the completion of the sigma-algebra generated by $Y$, does it imply that $X$ is almost surely measurable with respect to the sigma-algebra generated by $Y$?",
"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... | 1.2 L |
[
{
"content": "Consider the differential equation \\(\\frac{dy}{dx}+10y=f(x), x>0\\), where \\(f(x)\\) is a continuous function such that \\(\\lim_{x\\to \\infty}f(x)=1\\). Find the value of \\(\\lim_{x\\to \\infty}y(x)\\).",
"role": "user"
}
] | \dfrac{1}{10} |
[
{
"content": "Determine whether the statement \\( \\liminf\\left(\\frac{a_{n}+1}{a_{n}}\\right)^{n}\\leq e \\) is true or false, where \\( \\{a_{n}\\} \\) is a sequence of positive real numbers.",
"role": "user"
}
] | False |
[
{
"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... | 152 |
[
{
"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... | 68.8 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... | $3.15 |
[
{
"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... | 5e6 N/C, vertically down |
[
{
"content": "A frustum of a right circular cone is formed by cutting a small cone off of the top of a larger cone. If a particular frustum has an altitude of $24$ centimeters, the area of its lower base is $225\\pi$ sq cm and the area of its upper base is $25\\pi$ sq cm, what is the altitude of the small cone ... | 12 |
[
{
"content": "Given that $g(x) > 0$ for all $x \\in I$, can we conclude that $\\lim_{x\\to x_0} f(x) = L$?",
"role": "user"
}
] | No |
[
{
"content": "Evaluate the contour integral \\( \\int_{|z|=4} \\tan z \\, dz \\).",
"role": "user"
}
] | -4\pi i |
[
{
"content": "Evaluate the limit: \\[ \\lim_{(x,y)\\to(0,0)} \\frac{6(1-\\cos(xy))}{x^2y\\sin(2y)} \\]",
"role": "user"
}
] | \dfrac{3}{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... | 1410.7 m |
[
{
"content": "Determine the largest integer \\( n \\) such that \\( 2^n \\) divides \\( 3^{2008} - 1 \\).",
"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... | -1.4 m/s |
[
{
"content": "Determine the smallest positive integer \\(a\\) such that there exists a positive odd integer \\(n\\) for which \\(2001\\) divides \\(55^n + a \\cdot 32^n\\).",
"role": "user"
}
] | 436 |
[
{
"content": "Consider the set \\(\\mathcal{S}_X\\) of trigonometric polynomials \\(f(t) = \\sum_{|k| \\leq X} c_k e^{2\\pi i kt}\\) on the circle \\(\\mathbb{T} = \\mathbb{R}/\\mathbb{Z}\\) with degree \\(\\leq X\\), such that \\(f(0) = 1\\) and \\(c_0 = 0\\). Define\n\\[ M_X(f) = \\sup_{\\mathbb{T} \\setminus... | 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... | 2NO_3^- + H_2 (g) + 2H_2O (l) -> 2NO (g) + 2OH^- (aq) |
[
{
"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... | $257,000 |
[
{
"content": "Determine whether the statement \"If $\\det(M-N^2)=0$, then $M=N^2$\" is true or false.",
"role": "user"
}
] | B |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $380,000 |
[
{
"content": "Let \\( v \\) be a random unit vector in \\( \\mathbb{R}^3 \\). Calculate the expected length of the projection of \\( v \\) onto a given plane.",
"role": "user"
}
] | \dfrac{\pi}{4} |
[
{
"content": "Calculate the expected residual service time for an arbitrary job in a queueing system with two types of arrivals. Type A arrivals occur at a rate of \\(\\lambda_A = 1/20\\) per second and have a service time of exactly 8 seconds. Type B arrivals occur at a rate of \\(\\lambda_B = 1/30\\) per seco... | 5 |
[
{
"content": "Let $R$ be a simply-connected compact region in the plane with boundary $\\partial R$ given by a closed curve $\\gamma: [0,1] \\to \\mathbb{R}^2$. Can $\\gamma$ be chosen to be weakly simple, meaning there exists a sequence of simple closed curves $\\gamma_1, \\gamma_2, \\ldots$ that converge to $... | Yes |
[
{
"content": "Let $U$ be an orthonormal set in a Hilbert space $H$ and let $x \\in H$ be such that $\\| x \\| = 2$. Consider the set \\(E=\\{ u\\in U: |\\langle x, u \\rangle| \\geq \\frac{1}{4} \\}\\). Determine the maximal possible number of elements in $E$. \\(\\boxed{\\text{Answer}}\\)",
"role": "user"
... | 64 |
[
{
"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... | 20100 J |
[
{
"content": "Given that \\(\\lim_{x\\to \\infty}xf(x^2+1) = 2\\), find \\(\\lim_{x\\to 0}\\dfrac{2f'(1/x)}{x\\sqrt{x}}\\).",
"role": "user"
}
] | -2 |
[
{
"content": "Calculate the Fourier coefficient $\\hat{s}(-1)$ for the 1-periodic signal $s : \\mathbb{R}/\\mathbb{Z} \\to \\mathbb{C}$ defined by the equation $s(t) = (2\\cos(\\pi t))^{16}$.",
"role": "user"
}
] | 11440 |
[
{
"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... | 20.54% |
[
{
"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... | 656.35 nm |
[
{
"content": "Determine whether the following proposition is true or false: Let $V$ and $W$ be non-zero vector spaces over a field $F$, and let $S_v \\subset V$. If every mapping $f: S_v \\to W$ can be uniquely extended to a homomorphism $g: V \\to W$, then $S_v$ is a basis of $V$. Provide a justification for y... | True |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to \\infty} \\left[ \\frac{x^2 - \\ln^2(\\cosh(x))}{x + \\ln(\\cosh(x))} \\right]. \\]",
"role": "user"
}
] | \ln 2 |
[
{
"content": "In a regular tetrahedron \\( ABCD \\), points \\( E \\) and \\( F \\) are located on edges \\( AB \\) and \\( AC \\) respectively, such that \\( BE = 3 \\), \\( EF = 4 \\), and \\( EF \\) is parallel to face \\( BCD \\). Calculate the area of \\( \\triangle DEF \\).",
"role": "user"
}
] | 2\sqrt{33} |
[
{
"content": "Find an upper bound, independent of \\( p \\), for the sum \\( S_p = \\sum_{j=1}^p \\frac{p+1}{p-j+1} \\frac{1}{2^j} \\).",
"role": "user"
}
] | \dfrac{5}{3} |
[
{
"content": "In an isosceles triangle, the incircle has a radius of 3. A smaller circle, with a radius of 2, is tangent to the incircle and the two congruent sides of the triangle. Determine the length of the base of the triangle.",
"role": "user"
}
] | 3\sqrt{6} |
[
{
"content": "Let $f(x) = g(x) \\cdot |x-1| \\cdot |x-2| \\cdot |x-3|^2 \\cdot |x-4|^3$, where $g(x) = x^3 + ax^2 + bx + c$. If $f(x)$ is differentiable for all $x$ and $f'(3) + f'''(4) = 0$, find the third factor of $g(x)$ given that $x-1$ and $x-2$ are factors and $f'(3) = 0$.",
"role": "user"
}
] | x - 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... | 29.80% |
[
{
"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... | CaCl_2(l) +Deltararr Ca(l) + Cl_2(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... | 5182 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... | 72.2 g |
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