prompt list | ground_truth string |
|---|---|
[
{
"content": "Evaluate the limit: $$\\lim\\limits_{x\\to 0}\\frac{1}{x^3}\\int_0^x\\frac{t^2}{t^4+1}dt$$",
"role": "user"
}
] | \dfrac{1}{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... | 104.14 m |
[
{
"content": "Given that the equation \\( \\cos(ax^2 + bx + c) = -1 \\) has two distinct real roots at \\( x=1 \\) and \\( x=-1 \\), where \\( a, b, c \\in (2,5) \\), find the maximum value of \\( \\frac{ac}{b^2} \\).",
"role": "user"
}
] | 1 |
[
{
"content": "Calculate the limit of the ratio of the Lebesgue measure of the set $S_n$ to the Lebesgue measure of the cube $[0,a]^n$ as $n$ approaches infinity.",
"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... | 18 years and 2 months |
[
{
"content": "Evaluate the integral \\( \\int_{-\\infty}^{\\infty}\\frac{x^4}{x^6+64}dx \\) using the residue theorem.",
"role": "user"
}
] | \dfrac{\pi}{3} |
[
{
"content": "Evaluate the limit: $$\\lim _{x\\rightarrow0}\\frac{1-\\cos\\left(\\frac{1-\\cos x \\cos 2x}{x^2}-\\frac {5}{2}\\right)\\cos2x}{x^2}$$",
"role": "user"
}
] | 2 |
[
{
"content": "Let \\( a(n) \\) be a sequence defined as follows: \\( a(0)=\\frac{1}{2} \\), and for any natural number \\( n \\), \\( a(n+1)=a(n)+\\frac{a(n)^2}{2013} \\). Determine the smallest natural number \\( k \\) such that \\( a(k)>1 \\).",
"role": "user"
}
] | 2014 |
[
{
"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... | Nonroutine technologies |
[
{
"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... | -116 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... | 50.1% |
[
{
"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... | 10.42% |
[
{
"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... | 0.78 |
[
{
"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... | 784800 N.m |
[
{
"content": "Determine whether the following convergence in distribution holds for a sequence of i.i.d. random variables $(X_i)_{i \\geq 1}$, each with mean 0 and variance 1: $$\\left(\\frac{1}{\\sqrt{n}}\\sum\\limits_{i=1}^{n^a t}{X_i}\\right)_t \\xrightarrow{d} 0$$ for $0<a<\\frac{1}{2}$. Justify 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... | 98.75 |
[
{
"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... | -16.66 |
[
{
"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... | 23 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... | 3.64 m/s |
[
{
"content": "Compute the sum of all possible values for $x,$ where $(x, y)$ is a solution to the system of equations \\[\\begin{aligned} x &= y^2 - 6y + 5, \\\\ y &= |x|. \\end{aligned}\\]",
"role": "user"
}
] | 2 |
[
{
"content": "Solve for \\( n \\) in the equation:\n\\[ \\frac{\\sin\\left(\\frac{90^\\circ}{2^n}\\right)}{\\underbrace{\\sqrt{2+\\sqrt{2+\\sqrt{2+\\ldots\\sqrt{2}}}}}_{n \\text{ radicals}}} = \\frac{-1+\\sqrt{2}}{2} \\]",
"role": "user"
}
] | 2 |
[
{
"content": "Determine the limit of the sequence \\( x_n = n\\log n\\left(\\frac{\\log(n+2)}{\\log (n+1)}-1\\right) \\).",
"role": "user"
}
] | 1 |
[
{
"content": "Calculate the sum of the series \\(\\sum \\limits_{k=0}^n {n\\choose k}\\left(\\frac{1}{3}\\right)^{k-n}\\) using the Euler transform of a series.",
"role": "user"
}
] | 4^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... | $173.51 |
[
{
"content": "In a regular tetrahedron \\(ABCD\\) with edge length 2, a point \\(P\\) is located on edge \\(AB\\) such that \\(AP < 1\\). A cut is made through point \\(P\\) perpendicular to edge \\(AB\\) and extends through the faces of the tetrahedron. The cut stops when the length of the cut on face \\(ABD\\... | 1 |
[
{
"content": "Solve the inequality: \\[ \\frac{1}{2}\\log_{\\frac{1}{2}}(x-1) > \\log_{\\frac{1}{2}}\\left(1-\\sqrt[3]{2-x}\\right) \\]",
"role": "user"
}
] | (2, 10) |
[
{
"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... | 3.03 ft/second |
[
{
"content": "Let \\(a_n\\) be a sequence of real numbers with a finite Cesaro limit, i.e., \\(\\frac{a_1+ \\cdots + a_n}{n} \\rightarrow c\\) for some real number \\(c\\). If \\(M_n= \\max\\{a_1, \\cdots, a_n\\}\\), does \\(\\frac{M_n}{n} \\rightarrow 0\\)? Justify your answer.",
"role": "user"
}
] | No |
[
{
"content": "Compute the limit: \\[ \\lim_{x\\to\\infty} \\left( \\frac{x^2+x+1}{3x^2+2x+1} \\right) ^{\\left( \\frac{3x^2+x}{x-2} \\right)} \\]",
"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... | C: 60.00%, H: 4.476%, O: 35.52% |
[
{
"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... | 1.21e-3 N |
[
{
"content": "Determine whether the function \\( f: \\mathbb{N} \\rightarrow \\mathbb{N} \\) defined by \\( f(x) = ((x \\bmod 9) + 1) \\cdot 10^{\\lfloor \\frac{x}{9} \\rfloor} \\) has an inverse that maps elements from its image back to its domain.",
"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... | -53.64% |
[
{
"content": "Evaluate the limit: \\[ \\lim_{t \\to \\infty} t^{1/2} \\int_{0}^{t^{-0.25}} e^{-\\frac{1}{2} x^2} \\, dx \\]",
"role": "user"
}
] | \infty |
[
{
"content": "In how many ways can 4 people be selected from 100 people sitting at a circular table such that no two selected people are adjacent?",
"role": "user"
}
] | 3460375 |
[
{
"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... | $68.41 |
[
{
"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... | 70642.62 |
[
{
"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... | 0.33N |
[
{
"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... | $8455 |
[
{
"content": "In triangle \\(ABC\\), angle \\(A\\) is the largest angle. Points \\(M\\) and \\(N\\) are symmetric to vertex \\(A\\) with respect to the angle bisectors of angles \\(B\\) and \\(C\\) respectively. If \\(\\angle MAN = 50^\\circ\\), find the measure of \\(\\angle A\\).",
"role": "user"
}
] | 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... | 42.34 |
[
{
"content": "Determine the size of the group of linear functions of the form \\( f(x) = ax + b \\) under composition, where \\( a \\neq 0 \\) and \\( a, b \\in \\mathbb{F}_q \\). Here, \\( q = p^n \\) for some prime \\( p \\).",
"role": "user"
}
] | q(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... | M=A'C + 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... | 1060.8 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... | 1.717e15 m/s^2 |
[
{
"content": "How many nonzero terms are in the expansion of $$(x+4)(2x^2+3x+9)-3(x^3-2x^2+7x)?$$",
"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... | $405.70 |
[
{
"content": "In the binomial expansion of \\(\\left(\\sqrt{x}+\\frac{1}{2 \\sqrt[4]{x}}\\right)^{n}\\), the coefficients of the first three terms form an arithmetic sequence. Determine the number of terms in the expansion where the exponent of \\(x\\) is an integer.",
"role": "user"
}
] | 3 |
[
{
"content": "Evaluate the infinite series \\( \\sum_{n=1}^\\infty \\frac{2^{f(n)} + 2^{-f(n)}}{2^n} \\), where \\( f(n) = \\left[ \\sqrt{n} + \\frac{1}{2} \\right] \\) and \\( [\\cdot] \\) denotes the greatest integer function.",
"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... | 0.294N |
[
{
"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... | 24.57 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... | 2574 |
[
{
"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... | Triangle Abe |
[
{
"content": "Evaluate the contour integral \\( \\oint_C \\frac{z}{z-1} \\, dz \\) where \\( C \\) is a circle with radius 4 centered at the origin.",
"role": "user"
}
] | 2\pi i |
[
{
"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... | 16.56 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... | Month 1: Budgeted Overhead $14,400, Actual Overhead $14,700, Variance $300
Month 2: Budgeted Overhead $16,800, Actual Overhead $17,400, Variance $600 |
[
{
"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... | Silver fluoride |
[
{
"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... | April: $326,662; May: $466,662; June: $586,672 |
[
{
"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... | 4.84e-6 Parsec |
[
{
"content": "Find the maximum value of the expression \\((\\sin 2x + \\sin 3y + \\sin 4z)(\\cos 2x + \\cos 3y + \\cos 4z)\\).",
"role": "user"
}
] | \dfrac{9}{2} |
[
{
"content": "Determine the dimension of the $R$-module $N$ over the ring $R = \\mathbb{R}[T]$, where $N$ is generated by elements $u$ and $v$ with the relations $Tu = v$ and $T^{3}u = v$. What is the dimension of $N$?",
"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... | 2703.5953 kJ |
[
{
"content": "Given that \\( a \\) and \\( b \\) are unequal real numbers, and \\((\\sqrt[3]{a} + \\sqrt[3]{b})^{3} = a^{2}b^{2}\\), find the value of \\((3a + 1)(3b + 1) - 3a^{2}b^{2}\\).",
"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... | 64.7oC |
[
{
"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... | 8.01% |
[
{
"content": "Let $f(x)$ be a function that satisfies the functional equation $(x+y)f(x+y)=xf(x)+yf(y)$. Given that $f(2x)=f(x)$, $f(3x)=f(x)$, and $f(nx)=f(x)$ for all positive integers $n$, is $f(x)$ necessarily a constant function?",
"role": "user"
}
] | Yes |
[
{
"content": "Determine if there exists a non-integral number X, not between -1 and 1, such that the sequence of distances from X^n to the nearest integer converges to 0 as n approaches infinity.",
"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... | Unknown |
[
{
"content": "Find the function \\( f(x) \\) that satisfies the equation \\( f(x) + f(y) = f \\left( \\frac{x+y}{1-xy} \\right) \\) and the condition \\( f'(0) = 5 \\).",
"role": "user"
}
] | 5 \arctan x |
[
{
"content": "When $(x\\sqrt{x^3})^4$ is simplified, what is the exponent of $x$?",
"role": "user"
}
] | 10 |
[
{
"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... | G = \dfrac{a-c}{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... | $1,059.73 |
[
{
"content": "Evaluate the contour integral \\( \\int_{\\mathcal{C}} \\frac{1}{z^3(z+4)} \\, dz \\), where \\( \\mathcal{C} \\) is the circle \\( |z + 2| = 3 \\) described in the anti-clockwise direction 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... | 1.33e6 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... | 409.62 K |
[
{
"content": "Calculate the limit: \\[ \\lim_{n \\to \\infty} \\left( \\frac{1}{n+1} + \\frac{1}{n+2} + \\frac{1}{n+3} + \\cdots + \\frac{1}{2n} \\right) \\]",
"role": "user"
}
] | \ln 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... | 1.15 g |
[
{
"content": "Find the minimum value of the function \\( f(x,y,z) = x^2 + y^2 + z^2 \\) subject to the constraints \\( x^2 + y^2 + z^2 - 5 = 0 \\) and \\( x^2 + y^2 + z^2 - 2x - 3 = 0 \\) using the Lagrange multiplier method.",
"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... | 2.5 m |
[
{
"content": "Determine the range of values for \\(a \\in \\mathbb{R}\\) such that the integral \\(\\int_0^\\infty \\frac{\\arctan(5x)\\,dx}{x^a}\\) converges.",
"role": "user"
}
] | (1, 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... | 15.4% |
[
{
"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... | 0.36 g |
[
{
"content": "Determine the value of \\( f(0) + 2\\lambda \\) if the function \\[ f(x) = \\lim_{n\\to\\infty} \\frac{(1-\\cos(1-\\tan(\\frac{\\pi}{4}-x)))(1+x)^n + \\lambda\\sin((n-\\sqrt{n^2-8n})x)}{x^2(1+x)^n + x}, \\quad x\\ne0 \\] is continuous at \\( x=0 \\). Assume the value of \\( f(0) \\) is unknown.",
... | 3 |
[
{
"content": "Functions that aren't invertible can be made invertible by restricting their domains. For example, the function $x^2$ is invertible if we restrict $x$ to the interval $[0,\\infty)$, or to any subset of that interval. In that case, the inverse function is $\\sqrt x$. (We could also restrict $x^2$ t... | (-\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... | 124g |
[
{
"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... | 6.55% |
[
{
"content": "In the ring of Gaussian integers \\(\\mathbb{Z}[i]\\), factor the element 2 into irreducible elements.",
"role": "user"
}
] | (1 + i)(1 - i) |
[
{
"content": "Determine whether every slowly changing function $f:[0,\\infty)\\to\\mathbb R\\setminus\\{0\\}$ has a constant sign for sufficiently large $x$. Specifically, is it true that for every slowly changing $f$, there exists an $N$ such that for every $x,y>N$, we have $f(x)f(y)>0$?",
"role": "user"
... | Yes |
[
{
"content": "For the sequence $u_n$ defined by $u_1=1$, $u_2=2$, and $u_{n+1}=3u_n-u_{n-1}$, let $v_n=\\sum_{k=1}^{n}\\cot^{-1}(u_k)$. Determine the value of $\\lim_{n\\to\\infty} v_n$. \\( \\text{Express your answer as a single number.} \\)",
"role": "user"
}
] | \dfrac{\pi}{2} |
[
{
"content": "Find the total number of non-similar triangles that can be formed such that all the angles of the triangles are integers.",
"role": "user"
}
] | 2700 |
[
{
"content": "Let $f$ be an entire function on $\\mathbb{C}$ such that $|f(z)| \\le 100\\log|z|$ for each $z$ with $|z| \\ge 2$. If $f(i) = 2i$, find the value of $f(1)$. \\( \\text{(Note: An entire function is a function that is holomorphic over the entire complex plane.)} \\)",
"role": "user"
}
] | 2i |
[
{
"content": "At constant temperature, the pressure of a sample of gas is inversely proportional to its volume. I have some oxygen in a 2.28 liter container with a pressure of 5 kPa. If I move all of it to a 5.7 liter container at the same temperature, what will the new pressure be in kPa?",
"role": "user"
... | 2 |
[
{
"content": "Determine whether every normal space is regular if the T1 axiom (every singleton set is closed) is not assumed. Provide a true or false answer.",
"role": "user"
}
] | False |
[
{
"content": "Given the matrix differential equation:\n\\[ \\frac{d}{dt} A(t) = \\begin{pmatrix} 2 & 5 \\\\ 1 & -2 \\end{pmatrix}A(t) \\]\nwith the initial condition \\( A(0) = I \\), find the determinant \\( \\det(A(t)) \\) at \\( t = 1, 10, \\) and \\( 100 \\).",
"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... | 5550 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... | 4.896% |
[
{
"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... | 2.84 m/s |
[
{
"content": "Let \\( X \\) be a compact manifold and \\( \\mathscr{D}^\\prime(X \\times X) \\) be the space of tempered distributions on \\( X \\times X \\). Given two test functions \\( \\varphi, \\psi \\in \\mathscr{D}(X) \\), an element \\( T \\in \\mathscr{D}^\\prime(X \\times X) \\) can be evaluated at th... | Yes |
[
{
"content": "Calculate the value of \\( 7^{7^{7^{7^{7^{7^7}}}}} \\mod 100 \\).",
"role": "user"
}
] | 43 |
[
{
"content": "Given that \\( f \\) is a differentiable function with \\( f(0) = 0 \\) and satisfies the equation \\[ \\frac{f(4h) + f(2h) + f(h) + f\\left(\\frac{h}{2}\\right) + f\\left(\\frac{h}{4}\\right) + f\\left(\\frac{h}{8}\\right) + \\cdots}{h} = 64 \\] for all \\( h \\neq 0 \\), compute \\( f'(0) \\) wi... | 8 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.