prompt list | ground_truth string |
|---|---|
[
{
"content": "Let $X(t)$ be the price of JetCo stock at time $t$ years from the present. Assume that $X(t)$ follows a geometric Brownian motion with zero drift and volatility $\\sigma = 0.4/yr$. If the current price of JetCo stock is 8.00 USD, calculate the probability that the price will be at least 8.40 USD s... | 0.377 |
[
{
"content": "Let $p$ be a prime number. Determine the number of abelian groups of order $p^5$, up to isomorphism.",
"role": "user"
}
] | 7 |
[
{
"content": "Prove that the series \\( \\sum_{n=1}^\\infty x_n \\sin(nx) \\) converges uniformly if and only if \\( nx_n \\to 0 \\) as \\( n \\to \\infty \\), where \\( x_n \\) is a decreasing sequence with \\( x_n > 0 \\) for all \\( n = 1, 2, \\ldots \\).",
"role": "user"
}
] | nx_n \to 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... | -9e8 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... | 1.27e-8 cm |
[
{
"content": "Determine whether the series \\( \\sum_{n=0}^\\infty \\int_{[-1,1]^2} \\left(1-\\frac{x^2+y^2}{2}\\right)^{n/2}\\lambda_2(d(x,y)) \\) is finite, where \\( \\lambda_2 \\) is the 2-dimensional Lebesgue measure.",
"role": "user"
}
] | \infty |
[
{
"content": "Find the maximum value of the expression \\[ P = \\frac{3}{a^2+1} - \\frac{1}{b^2+1} + \\frac{2}{c^2+1} \\] given that \\( a, b, c > 0 \\) and \\( abc + b + c = a \\).",
"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... | 50.92J |
[
{
"content": "For which values of \\( x \\in \\mathbb{R} \\) is the integral \\( \\int_0^\\infty e^{-t}t^{x-1} \\, dt \\) convergent?",
"role": "user"
}
] | (0, \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... | 0.1767 |
[
{
"content": "How many triangles are in the figure to the right? [asy]\ndefaultpen(linewidth(0.7));\npair hexcoords (real over, real upover)\n{\n\nreturn dir(0)*over+dir(60)*upover;\n}\nreal r = 0.3;\nint i,j;\nfor(i=0;i<=2;++i)\n\n{\n\nfor(j=0;j<=2-i;++j)\n\n{\n\ndraw(hexcoords(i,j)--hexcoords(i+1,j));\n\ndraw... | 16 |
[
{
"content": "Find all functions \\( f: \\mathbb{N} \\rightarrow \\mathbb{N} \\) such that \\( f(x+f(y)) = f(x) + y \\) for all \\( x, y \\in \\mathbb{N} \\).",
"role": "user"
}
] | f(x) = x |
[
{
"content": "Find all polynomials \\( P(x) \\) with integer coefficients such that for all positive integers \\( m \\) and \\( n \\), the following divisibility condition holds: \\( m+n \\mid P^{(m)}(n) - P^{(n)}(m) \\).",
"role": "user"
}
] | P(x) = c |
[
{
"content": "Let $x,$ $y,$ and $z$ be positive real numbers. Find the minimum value of\n\\[\\frac{4z}{2x + y} + \\frac{4x}{y + 2z} + \\frac{y}{x + z}.\\]",
"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... | $50,544.17 |
[
{
"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... | 4443.6 m |
[
{
"content": "Determine whether there exists a convex quadrilateral and a point $P$ inside it such that the sum of the distances from $P$ to the vertices is greater than the perimeter of the quadrilateral.",
"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... | {
"Production cost": [24800, 24800, 24800, 24800, 24800, 24800, 24800, 24800, 24800, 24800, 24800, 24800],
"Other payments": [44000, 44000, 44000, 44000, 44000, 44000, 44000, 44000, 44000, 44000, 44000, 44000],
"Total Payments": [68800, 68800, 68800, 68800, 68800, 68800, 68800, 68800, 68800, 68800, 68800, 68800]
} |
[
{
"content": "Determine if there exist positive integers $x$, $y$, and $k$ such that $2^{2x} - 2^{y + 1} = 2^{k}$. Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Given positive integers $a, b, c$ that are pairwise relatively prime, determine whether there exist positive integers $x, y, z$ such that $a x + b y = cz$ and $ax, by, cz$ are pairwise relatively prime.",
"role": "user"
}
] | Yes |
[
{
"content": "Determine the set of values of $k$ for which the polynomial equation \\(3x^{4}+4x^3-12x^2+k=0\\) has four real roots.",
"role": "user"
}
] | [0, 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... | 9.83e-11 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... | 1504 MW |
[
{
"content": "Given the sequence \\(\\{a_{n}\\}\\) which satisfies the recurrence relation \\(3 a_{n+1} + a_{n} = 4\\) for \\(n \\geq 1\\), with the initial condition \\(a_{1} = 9\\), let \\(S_{n}\\) be the sum of the first \\(n\\) terms of the sequence. Determine the smallest integer \\(n\\) such that the ineq... | 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... | 2.398 mH |
[
{
"content": "Evaluate the integral: $$\\int_{-1}^{1} \\frac{\\sin x}{6+x^2} \\, dx.$$",
"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... | Consolidated Worksheet
Revenues: 480,000
Expenses: 347,000
Equity in subsidiary earnings: 0
Net income: 133,000
Retained earnings January 1, 2012: 765,000
Net income (above): 133,000
Dividends paid: 84,000
Retained earnings, December 31, 2012: 814,000
Current assets: 172,000
Investment in Spaulding Inc.: 0
Buildings (n... |
[
{
"content": "Given that \\(\\alpha\\) and \\(\\beta\\) are acute angles, and the equations \\(3 \\sin^2 \\alpha + 2 \\sin^2 \\beta = 1\\) and \\(3 \\sin 2\\alpha - 2 \\sin 2\\beta = 0\\) hold, find the value of \\(\\alpha + 2\\beta\\).",
"role": "user"
}
] | \dfrac{\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... | The magnitude of the acceleration is #sqrt(26)/2 m/s^2#, and the direction is #arctan(-1/5)# with the x-axis. |
[
{
"content": "In a kingdom, the king plans to build 25 new towns on 13 uninhabited islands, ensuring that each island has at least one town. Direct ferry connections will be established between any pair of towns located on different islands. What is the minimum number of ferry connections needed?",
"role": ... | 222 |
[
{
"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... | 0.103 M |
[
{
"content": "Calculate the product \\( p = 1 \\cdot 2^{\\frac{1}{2}} \\cdot 4^{\\frac{1}{4}} \\cdot 8^{\\frac{1}{8}} \\cdot 16^{\\frac{1}{16}} \\cdot \\ldots \\).",
"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... | 3.4 eV |
[
{
"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... | 229 nm |
[
{
"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... | Pb(NO_3)_2(s) + Delta rarr PbO(s) + 2NO_2(g)uarr + 1/2O_2(g)uarr |
[
{
"content": "Determine the number of real solutions to the equation \\( e^{x-1} = \\frac{1}{\\ln x} \\).",
"role": "user"
}
] | 1 |
[
{
"content": "Determine for which of the following initial conditions the solution to the differential equation \\( y' = 2\\sqrt{|y-1|} \\) exists and is unique:\n\n1. \\( y(0) = 1 \\)\n2. \\( y(1) = 2 \\)\n3. \\( y(2) = 1 \\)",
"role": "user"
}
] | 2 |
[
{
"content": "Consider a triangle $ABC$ where $M$ is the midpoint of side $BC$. Given that $\\angle MAC = \\angle ABC$ and $\\angle BAM = 105^{\\circ}$, find the measure of $\\angle ABC$. Solve the equation $$(4+\\sqrt{3})\\cos2x = 2 + \\sin2x$$ to find the value of $x = \\angle ABC$. Provide your answer in deg... | 30 |
[
{
"content": "Unit circles $O_1$ and $O_2$ are tangent to each other and line $L$. Circle $O_n$ is tangent to $O_1$, $O_{n-1}$, and $L$. Find the radius of $O_{20}$.",
"role": "user"
}
] | \dfrac{1}{361} |
[
{
"content": "Let $N \\geq 2$ be an even natural number and define the set $A_N=\\{k=(k_1,k_2) \\in \\mathbb{N} \\times \\mathbb{N}:1<k_2<k_1\\leq N\\}$. Consider the set $U_N=\\{(k,l): k \\in A_N, l \\in A_N, \\angle(k,l)>0\\}$, where $\\angle(k,l)$ is the angle between two vectors $k$ and $l$ in the plane. De... | 0 |
[
{
"content": "Determine whether the condition \\( \\lim_{n \\to \\infty} a_n = 0 \\) implies that \\( \\lim_{n \\to \\infty} \\frac{1}{n} \\sum_{j=0}^n |a_j| = 0 \\). Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Determine the projective dimension of the ring $A = \\mathbb{Z}_l[T,T^{-1}]$, where $l$ is a prime number and $\\mathbb{Z}_l$ is the ring of $l$-adic integers.",
"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... | 86.18 |
[
{
"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... | 92164 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... | Fe^{2+}_{(aq)} + C_2O^{2-}_4{(aq)} + 2H_2O_{(l)} \rightarrow FeC_2O_4 \cdot 2H_2O_{(s)} |
[
{
"content": "Determine whether the statement is true: For every natural number $x \\geq 2$, $x \\leq 2^{\\pi(x)}$, where $\\pi(x)$ is the prime-counting function.",
"role": "user"
}
] | True |
[
{
"content": "In the binomial expansion of \\((1 + x)^n\\), where \\(n\\) is a positive integer and \\(p > 0\\) is a real number with \\(q = 1 - p\\), find the value of the following sum:\n\n\\[ \\sum\\limits_{r=0}^n r^2 \\binom nr p^{n-r}q^r \\]\n\nChoose the correct expression for the sum from the options bel... | D |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 44.27 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... | 139.3 V |
[
{
"content": "Evaluate the limit \\( \\lim_{x\\to0}\\frac{\\ln(\\cos(4x))}{\\ln(\\cos(3x))} \\) without using L'Hopital's rule.",
"role": "user"
}
] | \dfrac{16}{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... | 40.4 mA |
[
{
"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... | 24.7 min |
[
{
"content": "Simplify $\\displaystyle\\frac{1-i}{2+3i}$, where $i^2 = -1.$",
"role": "user"
}
] | -\frac{1}{13} - \frac{5}{13}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... | 9.76 mm |
[
{
"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... | 8.87 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... | Ba>Sr>Ca>Mg |
[
{
"content": "Find all continuous functions \\( f: \\mathbb{R} \\to \\mathbb{R} \\) that satisfy the equation \\[ f\\left(\\frac{x+y}{2}\\right) = \\frac{f(x) + f(y)}{2} \\] for all real numbers \\( x \\) and \\( y \\).",
"role": "user"
}
] | f(x) = ax + 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... | $301,000 |
[
{
"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... | 108.909 amu |
[
{
"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... | $310,410 |
[
{
"content": "Determine the value of \\( a+b \\) such that the function \\( f(x) = x^2|x-2| + a|x+b| \\) is differentiable at every point.",
"role": "user"
}
] | -6 |
[
{
"content": "Find the volume of the solid bounded by the surfaces $z = y^2$ and $z = 2 - x^2$. Provide the setup for the integral and calculate the volume.",
"role": "user"
}
] | 2\pi |
[
{
"content": "Consider a holomorphic map $f: \\mathbb{D} \\to \\mathbb{C}$ on the unit disk $D_1(0)$ with $f(0) = 0$. Determine if there exists an $r > 0$ such that for all such $f$, the disk $D_r(0)$ is contained in $f(\\mathbb{D})$. Provide a justification for your answer.",
"role": "user"
}
] | No |
[
{
"content": "Evaluate $\\left\\lceil{\\frac32}\\right\\rceil^2+\\left\\lceil{\\left(\\frac32\\right)^2}\\right\\rceil$.",
"role": "user"
}
] | 7 |
[
{
"content": "Evaluate the limit without using L'Hopital's rule: \\[ \\lim_{x\\to0}\\frac{\\sqrt{\\cosh{(3x^2)}}\\cdot e^{4x^3}-1}{x^2\\tan(2x)} \\]",
"role": "user"
}
] | 2 |
[
{
"content": "Evaluate the improper integral \\(\\int_0^\\infty \\left( \\frac{x^{10}}{1+x^{14}} \\right)^{2} \\, dx\\) by rewriting it as a limit: \\(\\lim \\limits_{b \\to \\infty} \\int_0^b \\left(\\frac{x^{10}}{1+x^{14}} \\right)^{2} \\, dx\\). Find the value of this integral.",
"role": "user"
}
] | \dfrac{\pi}{28} |
[
{
"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... | propanamine |
[
{
"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... | -1.006 kg-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... | 2.0e+2 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... | 6.67e6 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... | $23,420 |
[
{
"content": "If the sequence of partial sums of a sequence \\((x_n)\\) divided by \\(n\\) converges, does it imply that \\((x_n)\\) is bounded? Answer yes or no.",
"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... | 3.47 m/s^2 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 39.88N |
[
{
"content": "Let $(M, \\omega)$ be a connected closed symplectic manifold of dimension $2n$. Assume there exist smooth covering maps $\\phi_k:M\\to M$ such that $\\phi_k^* \\omega=\\sqrt[n]{k}\\omega$ for all $k\\geq 1$. Determine if $n=1$.",
"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... | 530,000 |
[
{
"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... | $2,500 |
[
{
"content": "Evaluate the limit \\(\\lim_{n\\to\\infty} \\sqrt[n]{(n+1)(n+2)\\dots(2n)}.\\)",
"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... | 0.347 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... | 6269.4kg |
[
{
"content": "In an electromagnetic plane wave, the electric field is given in phasor form as \\(\\vec{E}=E_{x 0} e^{-\\gamma z} \\hat{x}\\), where \\(\\gamma\\) is the propagation constant. The wave propagates in the positive \\(z\\) direction through a conducting medium with conductivity \\(\\sigma=0.2 \\math... | 0.126 |
[
{
"content": "Using the Dirichlet convolution where \\( f = \\mu \\ast \\mu \\) (Möbius function), find \\( f(24) \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Given that $a, b, c$ are positive integers satisfying the equation $2(a^3 + b^3 + c^3) = 3(a + b + c)^2$, find the maximum possible value of $a + b + c$ such that the sum is even.",
"role": "user"
}
] | 12 |
[
{
"content": "Is it possible for a generating set of a group to have fewer subsets than the number of subgroups of the group?",
"role": "user"
}
] | Yes |
[
{
"content": "$ABCDEFGH$ shown below is a cube. Find $\\sin \\angle GAC$.\n\n[asy]\n\nimport three;\n\ntriple A,B,C,D,EE,F,G,H;\n\nA = (0,0,0);\n\nB = (1,0,0);\n\nC = (1,1,0);\n\nD= (0,1,0);\n\nEE = (0,0,1);\n\nF = B+EE;\n\nG = C + EE;\n\nH = D + EE;\n\ndraw(B--C--D);\n\ndraw(B--A--D,dashed);\n\ndraw(EE--F--G--... | \frac{\sqrt{3}}{3} |
[
{
"content": "Given the expression \\(\\frac{1}{(1-x)^4(1+2x^3+x^6)^4}\\), simplify it and find the sum of the coefficients of \\(x^{20}\\) and \\(x^{21}\\) in the resulting expansion.",
"role": "user"
}
] | 3829 |
[
{
"content": "Find the value of \\( \\lim _{x\\to \\infty }\\left(1+\\frac{4}{\\ln x}\\right)^{\\ln\\left(x^2+x\\right)} \\).",
"role": "user"
}
] | e^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... | 1165 m/s |
[
{
"content": "Given a continuous, smooth curve $\\mathcal{C}$ of length $\\ell$ in $\\mathbb{R}^2$, at each point on the curve, draw a line segment of length $d$ normal to the curve and centered at the point. Assuming no two line segments intersect, is the area covered by all the line segments equal to $d \\cdo... | d \cdot \ell |
[
{
"content": "Determine the radius of curvature for the plane curve \\(x^3 + y^3 = 12xy\\) at the point \\((0, 0)\\).",
"role": "user"
}
] | 6 |
[
{
"content": "Let $\\Omega$ be a bounded open set in $\\mathbb{R}^n$, and let $f$ be a function defined on $\\Omega$ which is positive almost everywhere. Let $m(\\Omega)$ denote the Lebesgue measure of $\\Omega$. If there exists a positive constant $M$ such that for any subset $V$ of $\\Omega$ with $m(V) \\geq ... | No |
[
{
"content": "Evaluate the integral \\( \\int_0^\\infty \\frac{\\sin^2(x)}{x^2} \\, dx \\).",
"role": "user"
}
] | \dfrac{\pi}{2} |
[
{
"content": "Let $S$ be a subset of $\\{0,1\\}^n$ such that any two elements of $S$ are at least Hamming distance 5 apart. Find an upper bound on the size of the automorphism group of $S$.",
"role": "user"
}
] | 2 \cdot 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... | 1090000 |
[
{
"content": "Evaluate the double integral \\( \\iint_D x^2 y^4 \\, dx \\, dy \\) where \\( D \\) is the region bounded by the curves \\( y = x^4 \\) and \\( y = x^{1/8} \\).",
"role": "user"
}
] | \dfrac{31}{667} |
[
{
"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... | 21.86 m/s |
[
{
"content": "Given $X$ and $Y$ are independent standard normal random variables, i.e., $X, Y \\sim \\mathcal{N}(0,1)$, and $Z = \\sqrt{X^2 + Y^2}$, find the probability density function (PDF) of $Z$.",
"role": "user"
}
] | z e^{-z^2/2} |
[
{
"content": "Allie and Betty play a game where they take turns rolling a standard die. If a player rolls $n$, she is awarded $f(n)$ points, where \\[f(n) = \\left\\{\n\\begin{array}{cl} 6 & \\text{ if }n\\text{ is a multiple of 2 and 3}, \\\\\n2 & \\text{ if }n\\text{ is only a multiple of 2}, \\\\\n0 & \\tex... | 32 |
[
{
"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... | -(2.0 m/s) i -(4.0 m/s) j |
[
{
"content": "Given $a, b, c \\in \\mathbb{R}$ such that $a+b+c = 7$, $a^2+b^2+c^2 = 23$, and $\\frac{1}{a+1}+\\frac{1}{b+1}+\\frac{1}{c+1} = 31$, find the value of $a^3+b^3+c^3$. \\( \\boxed{\\ } \\)",
"role": "user"
}
] | \dfrac{307}{31} |
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