prompt list | ground_truth string |
|---|---|
[
{
"content": "Calculate the limit: \\[ \\lim_{n\\to+\\infty} \\frac{2^n \\cdot (\\sqrt{4^n+2n} - 2^n)}{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... | 4.2g |
[
{
"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... | 8.2 years |
[
{
"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... | -106eV |
[
{
"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... | 28.6 degrees Celsius |
[
{
"content": "Determine the radius of convergence for the power series $$\\sum_{n=0}^{\\infty}z^n(5+e^{i\\pi n})^n.$$",
"role": "user"
}
] | \dfrac{1}{6} |
[
{
"content": "Evaluate the integral \\( \\int_{0}^{4}\\sqrt{9t^2+t^4}\\,dt \\).",
"role": "user"
}
] | \dfrac{98}{3} |
[
{
"content": "Determine if the equation \\( \\sin\\left(\\frac{x+y}{\\sqrt{2}}\\right) = \\frac{y-x}{\\sqrt{2}} \\) can be expressed as a function of \\(x\\).",
"role": "user"
}
] | Yes |
[
{
"content": "Find the 4x4 matrix $A$ with entries $a_{ij} \\in \\{1, -1\\}$ that maximizes the determinant $\\det(A)$. What is the maximum value of $\\det(A)$?",
"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... | W=\frac{1}{2}mV^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... | 56497.17 N/C |
[
{
"content": "Determine the dimension of the quotient space Span $\\mathcal{C} / \\sim$, where $\\mathcal{C}$ is the collection of all finite, closed, and connected intervals in the real numbers $\\mathbb{R}$, and $X \\sim \\tilde{X}$ if and only if $E(X) = E(\\tilde{X})$. The evaluation map $E$ is defined as $... | 1 |
[
{
"content": "Evaluate the integral \\( \\int_0^{\\infty} e^{-x} \\sum_{k=0}^{\\infty} \\frac{(-1)^k (2015x)^{2k}}{((2k)!)^2} \\, dx \\).",
"role": "user"
}
] | \cos(2015) |
[
{
"content": "Given the differential equation \\( y'(x) = y(x) + \\int_0^1 y \\, dx \\) with the initial condition \\( y(0) = 1 \\), find the value of \\( y\\left(\\ln\\frac{11-3e}{2}\\right) \\).",
"role": "user"
}
] | 4 |
[
{
"content": "Determine the value of $n$ such that the limit \\( \\lim_{x\\rightarrow 0}\\frac{x^n\\sin^{n}(x)}{x^{n}-(\\sin x)^{n}} \\) is a non-zero finite number, given that \\( n \\neq 0 \\).",
"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... | 3.82e-20 J |
[
{
"content": "Find the limit of the sequence \\(x_n = \\log\\left(1+\\frac{1}{n}\\right)\\sqrt[n]{n!}\\) as \\(n\\) approaches infinity, given that \\(n!\\sim \\sqrt{2\\pi n}\\left(\\frac{n}{e}\\right)^n\\).",
"role": "user"
}
] | \dfrac{1}{e} |
[
{
"content": "Find the 10th derivative of \\( f(x) = \\ln(2+x^2) \\) at \\( x=0 \\) using a Taylor series expansion. Use the properties of the Taylor series and the relation \\( f^{(n)}(0) = a_n \\cdot n! \\) to simplify the process.",
"role": "user"
}
] | 22680 |
[
{
"content": "Can an entire function with a pole at infinity of order $n$ be expressed as a polynomial of degree greater than $n$?",
"role": "user"
}
] | No |
[
{
"content": "Given the solution to the differential equation \\(x(t) = Ae^{k_1t} + Be^{-k_2t}\\), where \\(k_1 > 0\\) and \\(-k_2 < 0\\), find the relationship between the initial conditions \\(x(0) = x_0\\) and \\(\\dot{x}(0) = y_0\\) that ensures \\(x(t) \\to 0\\) as \\(t \\to \\infty\\).",
"role": "user... | y_0 = -k_2 x_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... | 2.96 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... | $1,570.33 |
[
{
"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... | 2.75e+23 |
[
{
"content": "Evaluate the limit: $$ \\lim_{x\\to0}\\left(\\sqrt{\\frac{1}{x} + \\sqrt{\\frac{1}{x} + \\sqrt{\\frac{1}{x}}}} - \\sqrt{\\frac{1}{x} - \\sqrt{\\frac{1}{x} + \\sqrt{\\frac{1}{x}}}}\\,\\right) $$",
"role": "user"
}
] | 1 |
[
{
"content": "Find the limit of the sequence \\(x_n\\) defined by \\[x_n = \\left(\\frac {n}{n+\\ln(n)}\\right)^{\\ln(n)}\\].",
"role": "user"
}
] | 1 |
[
{
"content": "Let $V$ be an inner product space and $B$ be a closed subspace of $V$. Determine whether $(B^\\bot)^\\bot = B$ is true.",
"role": "user"
}
] | (B^\bot)^\bot = B |
[
{
"content": "Let \\( f: [0, a] \\to \\mathbb{R} \\) be a continuous and one-to-one function with \\( f(0) = 0 \\). Verify that the following equation holds: \\[ \\int_0^{f(a)} f^{-1}(t)\\,dt + \\int_0^a f(t)\\,dt = af(a). \\]",
"role": "user"
}
] | af(a) |
[
{
"content": "Given the quartic equation \\( x^4 - mx^2 - 4 = 0 \\), find the value of \\( m \\) such that \\( \\sqrt[4]{3-2\\sqrt{2}} + \\sqrt[4]{3+2\\sqrt{2}} \\) is a root.",
"role": "user"
}
] | 4 |
[
{
"content": "A dragon has 40 piles of gold coins, with each pile containing a different number of coins. After plundering a city, the number of coins in each pile is multiplied by either 2, 3, or 4. What is the minimum number of distinct piles of coins that could result?",
"role": "user"
}
] | 14 |
[
{
"content": "Find all values of $x$ that satisfy \\[\\frac{x^2}{x+1} \\ge \\frac{2}{x-1} + \\frac{5}{4}.\\]",
"role": "user"
}
] | (-1, 1) \cup [3, \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... | 7.185% |
[
{
"content": "Let \\( f: \\mathbb{N}^{*} \\to \\mathbb{N}^{*} \\) be a function satisfying the equation: \\[ \\frac{f(x+1)f(x)-2x}{f(x)} = \\frac{2f^2(x)}{x+f(x)} - 1. \\] Determine all such functions \\( f \\).",
"role": "user"
}
] | f(x) = x |
[
{
"content": "Find all real functions \\( f: \\mathbb{R} \\rightarrow \\mathbb{R} \\) such that \\((x+y)(f(x)-f(y))=f(x^2)-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... | 25 Earth days |
[
{
"content": "Let \\( f, g \\in L^2 \\). Find the limit \\( \\lim_{t \\to \\infty} \\frac{\\|f+tg\\|_2-\\|f\\|_2}{t} \\).",
"role": "user"
}
] | \|g\|_2 |
[
{
"content": "In triangle ABC, the lengths of the medians from vertices A, B, and C are 13, 14, and 15, respectively. Calculate the area of triangle ABC.",
"role": "user"
}
] | 112 |
[
{
"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... | 0.308 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... | 21 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... | -36.92% |
[
{
"content": "Evaluate the integral \\( \\int_S (x \\, dy \\, dz + y \\, dz \\, dx + z \\, dx \\, dy) \\) over the surface \\( S \\) of a unit cube using the Gauss divergence theorem.",
"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... | 1.30e10 N/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... | 8.719 |
[
{
"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... | 161387.43 |
[
{
"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, -9) |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n\\to\\infty} n \\cdot f^n\\left(\\frac{1}{n}\\right) \\] where \\( f(x) = \\sin(x) \\) and \\( f^n \\) denotes the \\( n \\)-th iteration of the function \\( f \\).",
"role": "user"
}
] | 1 |
[
{
"content": "Let $X \\subset \\mathbb{R}$ be a Lebesgue measurable set with $|X| < 2^{\\aleph_0}$. Determine whether $X$ must be a null set.",
"role": "user"
}
] | Yes |
[
{
"content": "The sequence \\(\\{a_n\\}\\) consists of distinct positive numbers, and their reciprocals form an arithmetic sequence. Evaluate the expression:\n\\[\n\\frac{a_1 a_2 + a_2 a_3 + \\cdots + a_{2014} a_{2015}}{a_1 a_{2015}}\n\\]",
"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... | 12.8e-19 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... | 0.833 m |
[
{
"content": "Let $S$ be a Cohen-Macaulay local ring, $I$ an ideal of $S$, and $R = S/I$. If $I$ is generated by $\\dim S - \\dim R$ elements, is $R$ necessarily Cohen-Macaulay?",
"role": "user"
}
] | Yes |
[
{
"content": "Consider a graph $G$ where each vertex represents a 3-element subset of the set $\\{1, 2, 3, 4, 5, 6\\}$. Two vertices, corresponding to subsets $A$ and $B$, are connected by an edge if they share exactly one element, and the cost of the edge is equal to the shared element. Determine the minimum c... | 34 |
[
{
"content": "Let \\( x, y, z \\) be positive integers. The sets are defined as \\( A = \\{3(x-y)(y-z)(z-x), xyz, 2(y^2z^2 + z^2x^2 + x^2y^2)\\} \\) and \\( B = \\{(x-y)^3 + (y-z)^3 + (z-x)^3, x+y+z, x^4+y^4+z^4\\} \\). If \\( A = B \\), find the value of \\( x^3 + y^3 + z^3 \\).",
"role": "user"
}
] | 36 |
[
{
"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... | $40,774 |
[
{
"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... | $90,000 |
[
{
"content": "Evaluate the limit: \\( \\lim_{n \\to \\infty}\\int_{0}^{1} nx^n e^{x^2} \\, dx \\).",
"role": "user"
}
] | e |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 126,000 |
[
{
"content": "Compute the Euler characteristic of the $n$-sphere $S^n$ for $n \\ge 2$ using the standard triangulation of the $(n+1)$-simplex.",
"role": "user"
}
] | 1 + (-1)^n |
[
{
"content": "Evaluate the limit: $$\\lim_{{n\\to \\infty}} \\left(10\\sqrt[n]{a_n}+5\\sqrt[n]{b_n}\\right),$$ where $$a_n=\\int_0^1 (1-x^2)^n \\,dx$$ and $$b_n=\\int_0^1 (1-x^3)^n \\,dx$$ for $n \\in \\mathbb{N}$.\"",
"role": "user"
}
] | 15 |
[
{
"content": "Evaluate the integral: \\[ \\int_0^{2\\pi} e^{\\cos x} \\cos(\\sin x) \\, dx \\]",
"role": "user"
}
] | 2\pi |
[
{
"content": "Calculate \\( \\lim_{a \\rightarrow 0^+} \\int_{D_a} \\frac{|x|}{y(1+y^2)} \\, dx \\, dy \\) where \\( D_a=\\{(x,y) \\in \\mathbb{R}^2: \\frac{x^2}{2} \\le y \\le 2x^2 ; a \\le \\sqrt{x^2+y^2} \\le \\frac{1}{a} \\} \\) without using polar coordinates.",
"role": "user"
}
] | \dfrac{3\pi}{4} |
[
{
"content": "Let $H = H_1 \\times H_2$ be a closed subgroup of a second-countable locally compact Hausdorff group $G = G_1 \\times G_2$, with $H_i \\leq G_i$. Let $\\chi = \\chi_1 \\otimes \\chi_2$ be a unitary character of $H$, where $\\chi_i$ is a character of $H_i$. Consider the induced representation $I(\\... | 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... | Virus capsids are symmetrical primarily for two reasons:
1. Ensuring the stability of the structure.
2. Ensuring each protein subunit is exposed to an identical environment as its counterparts, conserving energy. |
[
{
"content": "Find the general term of the sequence \\(\\{x_n\\}\\) defined by the recurrence relation \\(x_{n+2} = \\frac{x_{n+1}^2 - 4}{x_n}\\), with initial terms \\(x_1 = 6\\) and \\(x_2 = 4\\).",
"role": "user"
}
] | 8 - 2n |
[
{
"content": "Determine the smallest integer $n$ such that $n! > 10^{17}$ without using a calculator or the gamma function.",
"role": "user"
}
] | 19 |
[
{
"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... | 17.2 m/s |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to 0^+} \\left(1-a^x\\right)^{\\frac{1}{x}} \\quad (0 < a < 1) \\]",
"role": "user"
}
] | 0 |
[
{
"content": "Evaluate the integral \\( \\int_0^k \\frac{\\cos x}{1+\\sin^2 x} \\, dx \\) such that the result is \\( \\frac{\\pi}{4} \\). Determine the value of \\( k \\).",
"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... | chained |
[
{
"content": "Is it possible to have a group automorphism of the additive group of real numbers that fixes a subset of real numbers but is not the identity? The subset may be infinite.",
"role": "user"
}
] | Yes |
[
{
"content": "Determine the number of points on the elliptic curve given by \\(x^3 + x + 1 = y^2\\) over the finite field \\(\\mathbb{F}_{125}\\).",
"role": "user"
}
] | 108 |
[
{
"content": "Compute the index of the unit circle with respect to the vector field given by the system of differential equations:\n\\[\\begin {cases}\n\\frac{dx}{dt} = 2x^2-1\\\\\n\\frac{dy}{dt} = 2xy\n\\end{cases}\\]",
"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... | y=e^(x^2/2-x)-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... | 2.99 m |
[
{
"content": "Given that \\(\\lim_{n \\rightarrow \\infty} \\int_{0}^{a_{n}} x^{n} \\, dx = 2\\) and \\(a_n > 0\\), find \\(\\lim_{n \\rightarrow \\infty} a_n\\).",
"role": "user"
}
] | 1 |
[
{
"content": "Consider a Markov chain $Z_k$ with 6 states, where the probability of moving from one node to a neighboring node is $1/2$. For instance, the probability of moving from node 1 to node 2 is $1/2$, and the probability of moving from node 1 to node 6 is $1/2$. Given that $P(Z_0=1)=1$, meaning we start... | 4 |
[
{
"content": "Determine whether a function over a field of characteristic 2 can be both even and odd.",
"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... | $37,880,554.31 |
[
{
"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... | 0.724 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... | 4,583,231 |
[
{
"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... | 95,000 |
[
{
"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... | 1.54e+3 m/s |
[
{
"content": "Find the maximum value of\n\\[\\cos \\theta_1 \\sin \\theta_2 + \\cos \\theta_2 \\sin \\theta_3 + \\cos \\theta_3 \\sin \\theta_4 + \\cos \\theta_4 \\sin \\theta_5 + \\cos \\theta_5 \\sin \\theta_1,\\]over all real numbers $\\theta_1,$ $\\theta_2,$ $\\theta_3,$ $\\theta_4,$ and $\\theta_5.$",
... | \frac{5}{2} |
[
{
"content": "Evaluate the integral \\( \\int_{-\\infty}^\\infty e^{-\\frac{1}{2}x^2} \\, dx \\).",
"role": "user"
}
] | \sqrt{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... | 0.936J |
[
{
"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... | $25.64 |
[
{
"content": "Find all rational numbers \\( \\frac{p}{q} \\) such that:\n1. \\( 0 < \\frac{p}{q} < 1 \\)\n2. \\( p \\) and \\( q \\) are coprime\n3. \\( pq = 25! \\)",
"role": "user"
}
] | 256 |
[
{
"content": "Given that \\( \\lim_{x\\to-\\infty} x\\,f(x)=0 \\) and \\( \\lim_{x\\to-\\infty} x\\,f''(x)=0 \\), determine if \\( \\lim_{x\\to-\\infty} x\\,f'(x)=0 \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Evaluate the limit: $$\\lim_{x\\to\\infty} x^2\\left(\\sqrt[7]{\\frac{x^3 + x}{1 + x^3}} - \\cos\\frac{1}{x}\\right)$$",
"role": "user"
}
] | \dfrac{9}{14} |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n\\to \\infty} \\left( \\sum_{r=0}^n \\frac {2^r}{5^{2^r}+1}\\right) \\]",
"role": "user"
}
] | \dfrac{1}{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... | 10^4 |
[
{
"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... | 42.55 m |
[
{
"content": "Determine whether it is possible to find a vector field on a surface in \\(\\mathbb{R}^3\\) that has a zero with index 0.",
"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... | -6400 |
[
{
"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... | 7.6 x 10^3 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... | 6.26e-26 |
[
{
"content": "Evaluate the surface integral \\( \\iint_S \\mathbf{F} \\cdot d\\mathbf{S} \\) for the portion of the cylinder \\( x^2 + y^2 = 1 \\) between the planes \\( z = 1 \\) and \\( z = 4 \\).",
"role": "user"
}
] | 6\pi |
[
{
"content": "Determine the determinant of a square matrix \\( A \\) of size \\( n \\times n \\) with the property that if \\( a \\neq b \\), each row is either identical to the first row or the sum of the first row and a scalar multiple of the second row.",
"role": "user"
}
] | 0 |
[
{
"content": "Determine whether an $n \\times n$ matrix $\\mathbf{A}$ is symmetric if it is diagonalizable and has orthogonal eigenvectors forming a basis of $\\mathbb{R}^n$. Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Find the sum of the infinite series: \\( \\sum_{n=1}^{\\infty} \\frac{n^2}{2^n} \\).",
"role": "user"
}
] | 6 |
[
{
"content": "Evaluate the limit of the expression \\((x^{\\frac{1}{x}}-1)^{\\frac{1}{\\ln x}}\\) as \\(x\\) approaches infinity.",
"role": "user"
}
] | \dfrac{1}{e} |
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