inclusionAI/Ring-lite-rl-data huggingface dataset:
There are 25 different types of coupons, each equally likely to be drawn. If you draw 10 coupons, what is the expected number of different types of coupons you will have? Provide your answer to two decimal places.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8.38}
Zhao is studying for his upcoming calculus test by reviewing each of the 13 lectures, numbered Lecture 1, Lecture 2, ..., Lecture 13. For each n, he spends 5n minutes on Lecture n. Which lecture is he reviewing after 4 hours?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
In a department store, 10 suitcases and 10 keys are delivered separately, and each key opens one and only one suitcase. What is the expected number of attempts required to match each key to its corresponding suitcase, considering both successful and unsuccessful attempts?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{65}{2}}
Determine the cardinality of the set of sequences of rational numbers that converge to 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
Given $\frac{x^2+y^2}{x^2-y^2}+\frac{x^2-y^2}{x^2+y^2}=k$, express $\frac{x^8+y^8}{x^8-y^8}+\frac{x^8-y^8}{x^8+y^2}$ in terms of k.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{k^4+24k^2+16}{4k^3+16k}$}
A circle $C_0$ is inscribed in an equilateral triangle $XYZ$ of side length 112. Then, for each positive integer $n$, circle $C_n$ is inscribed in the region bounded by $XY$, $XZ$, and an arc of circle $C_{n-1}$, forming an infinite sequence of circles tangent to sides $XY$ and $XZ$ and approaching vertex $X$. If these circles collectively have area $m\pi$, find $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
1176
}
What is the product of real numbers $a$ which make $x^2+ax+1$ a negative integer for only one real number $x$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-8
}
Find the limit of the sequence \(\left\{\left(T_n\right)^{\frac{1}{n}}\right\}\) where \(T_n=\frac{(n+1)^{n+1}}{(n+2)(n+3) \ldots(n+n+1)}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{e}{4}}
Calculate the tensor product \( \mathbb{Z}/2 \otimes_{\mathbb{Z}} \mathbb{Z} \) considered as \( \mathbb{Z} \)-modules. What is the result?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}/2\mathbb{Z}}
We know about a convex pentagon that each side is parallel to one of its diagonals. What can be the ratio of the length of a side to the length of the diagonal parallel to it?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{5} - 1}{2}}
The edges of $K_{2017}$ are each labeled with $1,2,$ or $3$ such that any triangle has sum of labels at least $5.$ Determine the minimum possible average of all $\\dbinom{2017}{2}$ labels.\n \n(Here $K_{2017}$ is defined as the complete graph on 2017 vertices, with an edge between every pair of vertices.)\n \n[i]Proposed by Michael Ma[\/i]\n
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$2 - \frac{1}{2017}$}
Calculate the quadratic variation of the process \( Y_t = \int_0^t \text{sign}(X_s) \, ds \), where \( X_t \) is a Brownian motion and \( \text{sign}(x) \) is the sign function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the limit \( \lim_{n \to + \infty} \frac{f(n)}{n!} \), where \( f(n) \) is the number of permutations \( \sigma \) of \( \{1, 2, \ldots, n\} \) such that \( \sigma(i) \neq i \) for all \( i = 1, 2, \ldots, n \) and there are no \( 1 \leq i < j \leq n \) such that \( \sigma(i) = j \) and \( \sigma(j) = i \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{-3/2}}
Evaluate the function \( P(n) = \sum_{i=1}^n i^{10} \) at \( n = -2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Compute \( \alpha^{53} \) in the finite field \( GF(2^6) \) with the primitive polynomial \( h(x) = 1 + x^2 + x^3 + x^5 + x^6 \), where \( \alpha \) is the generator representing the class of \( x \). Note that \( 53 \equiv -11 \pmod{64} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\alpha^3 + 1}
Calculate the expected value \( E[X(u)X(u+v)X(u+v+w)] \) for a Brownian motion \( X(t) \), where \( 0 < u < u+v < u+v+w \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Ari the Archer is shooting at an abnormal target. The target consists of $100$ concentric rings, each of width $1$, so that the total radius of the target is $100$. The point value of a given ring of the target is equal to its area (so getting a bull's eye would be worth $\pi$ points, but hitting on the outer ring would give $199\pi$ points). Given that Ari hits any point on the target uniformly at random, what is his expected score?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
13333π/100
}
Evaluate the integral \( \int\int_R y^{-N(x)} \, dA \), where \( R \) is the unit square \([0, 1] \times [0, 1]\) in the plane, \( \mu \) is the usual Lebesgue measure on the real Cartesian plane, and \( N(x) \) is the function that assigns to each real number \( x \) in the unit interval the positive integer that indicates the first place in the decimal expansion of \( x \) after the decimal point where the first \( 0 \) occurs. If there are two expansions, use the expansion that ends in a string of zeroes. If \( 0 \) does not occur, let \( N(x) = \infty \). For example, \( N(0.0) = 1, N(0.5) = 2, N(1/9) = \infty \), and \( N(0.4763014 \dots) = 5 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Determine the number of invariant subspaces of a transformation $T$ that maps $v \mapsto J_{\lambda,n}v$, where $J_{\lambda,n}$ is a Jordan block. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n+1}
Find the maximum possible dimension of a subspace in $V_n = \{(a_1,a_2,\ldots,a_n) \mid a_i \in \mathbb{GF}(2)\}$, given $k$ linearly independent vectors that are not part of this subspace, where $k < n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n - 1}
Given that \(Z_1, Z_2, \ldots, Z_n\) are independent identically distributed standard Gaussian random variables with mean 0 and variance 1, find the distribution of \[ X = \frac{(Z_1 + Z_2)^2}{(Z_1 - Z_2)^2} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{F(1,1)}
In the group \( G = \mathbb{Z}_3 \ltimes (\mathbb{Z}_5 \times \mathbb{Z}_5) \), where \( \phi: \mathbb{Z}_3 \rightarrow \operatorname{Aut}(\mathbb{Z}_5 \times \mathbb{Z}_5) \), determine the number of homomorphisms \( \phi_i \) induced by the elements \( \bar{0}, \bar{1}, \bar{2} \) of \( \mathbb{Z}_3 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
方程 $x+\frac{x}{\sqrt{x^{2}-1}}=\frac{35}{12}$ 的所有实数根的和为 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{47}
Calculate the expected number of coin tosses required to first observe the pattern "HTTH" when tossing a fair coin repeatedly.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Determine the values of \( \alpha \in (0,1) \) for which the operator \( A: L_2[0,1]\rightarrow L_2[0,1] \), defined by \((Ax)(t) = \int_0^1 \frac{x(s)}{|t-s|^{\alpha}}ds\), is compact.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0,1)}
Triangle $ABC$ satisfies $AB=28$, $BC=32$, and $CA=36$, and $M$ and $N$ are the midpoints of $\\overline{AB}$ and $\\overline{AC}$ respectively.  Let point $P$ be the unique point in the plane $ABC$ such that $\\triangle PBM\\sim\\triangle PNC$.  What is $AP$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
Calculate \( \mathop{\lim }\limits_{{n \rightarrow  \infty }}\frac{1}{n}\mathop{\sum }\limits_{{k = 0}}^{{n - 1}}k{\int }_{\frac{k}{n}}^{\frac{k - 1}{n}}\sin \left( {\pi {x}^{2}}\right) {dx} \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\dfrac{1}{\pi}\)}
Determine the order of contact between the curves \(y=x^3+x^6\) and \(y=x^3+x^5\) at the point \((0,0)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find the largest integer \( N \) such that there exists a permutation \( p \) on the set of residues modulo 17 where \( p, p^2, \ldots, p^{N-1} \) are all dispersive and \( p^N \) is connective.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
The plane \(4x-3y+8z=5\) intersects the cone \(z^2=x^2+y^2\) in an ellipse. Let \(P\) and \(Q\) be the highest and lowest points on the ellipse. If \(O\) denotes the origin, what is the angle between the vectors \(\overrightarrow{OP}\) and \(\overrightarrow{OQ}\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{2}}
In the country of Distant, a province is considered large if more than 7% of the country's population resides in it. For each large province, there are two other provinces with smaller populations such that their combined population exceeds that of the large province. Determine the minimum number of provinces that can exist in the country of Distant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Determine the splitting field of the polynomial \(p(x) = x^4 + x^3 + x - 1\) over \(\mathbb{F}_3\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{F}_{3^2}}
Consider an archipelago along the southern shore of an endless sea, consisting of an infinite number of islands. Each island is connected to the shore by a bridge, forming an infinite chain of bridges. During a strong earthquake, each bridge independently has a probability of 0.5 of being destroyed. What is the probability that, after the earthquake, it will still be possible to travel from the first island to the shore using the remaining bridges?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}}
Find the angle $\angle A$ of a triangle $ABC$, when we know it's altitudes $BD$ and $CE$ intersect in an interior point $H$ of the triangle and $BH=2HD$ and $CH=HE$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$45^{\circ}$

}
Determine the number of elements in the symmetric group $S_8$ that commute with the cycle composition $(123)(456)$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Determine the range of values for \( \alpha \) such that the function \( c(u_1,u_2) = 1 + \alpha(1- 2u_1)(1- 2u_2) \), where \( u_1, u_2 \in (0,1) \), is a valid copula density.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-1, 1]}
Evaluate the iterated integral by changing to polar coordinates: \[ \int_0^{1/2}\int_0^{\sqrt{1-x^2}}xy\sqrt{x^2+y^2}\,dy\,dx \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{19}{480}}
Consider the base 27 number \( n = ABCDEFGHIJKLMNOPQRSTUVWXYZ \), where each letter represents its position in the alphabet. Find the remainder when \( n \) is divided by 100.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Evaluate the limit \( \displaystyle \lim_{n\to\infty}\frac{I_n}{n^n} \), where the sequence \( I_n \) is defined as \( I_n = \int_0^n x^n\arctan\left(x^\frac{1}{n}\right)\,dx \) for \( n \ge 2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{4}}
In a three-layered pyramid made of 14 identical cubes, each cube is assigned a natural number. The numbers on the cubes in the bottom layer are all different, and the number on any other cube is the sum of the numbers on the four adjacent cubes from the layer below. What is the smallest number divisible by four that can be assigned to the topmost cube?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64}
Joe Quigley writes the following expression on the board for his students to evaluate:$$2\times 4+3-1$$However, his students have not learned their order of operations so they randomly choose which operations to perform first, second, and third. How many different results can the students obtain?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Two points $A$ and $B$ are selected independently and uniformly at random along the perimeter of a unit square with vertices at $(0,0)$, $(1,0)$, $(0,1)$, and $(1,1)$. The probability that the $y$-coordinate of $A$ is strictly greater than the $y$-coordinate of $B$ can be expressed as $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $100m+n$.

Proposed by Rajiv Movva
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
716
}
Evaluate the limit \( \lim_{h\to 0} \frac{1}{h} \int_0^h\left|\sin\left(\frac{1}{t}\right)\right| dt \) if it exists.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{\pi}}
Given the differential equation \((xy^3 + x^2y^7)\frac{dy}{dx} = 1\) and the initial condition \(y(1/4) = 1\), find the value of \(\frac{dy}{dx}\) when \(y = -1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{16}{5}}
Calculate the relative homology group $H_n(M, M - \{x\})$, where $M$ is a closed manifold and $x$ is a point in $M$. Provide your answer in terms of known homology groups or other topological invariants.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}}
Given a Gaussian random vector $X = (Y, X, Z)$ with mean $\mu = (1, 2, 4)^T$ and covariance matrix:

$$\Sigma = \begin{pmatrix} 2 & 3 & 1\\ 3 & 5 & 2\\ 1 & 2 & 6 \end{pmatrix}$$

Calculate the conditional variance $\operatorname{Var}(Y \mid X, Z)$. Use the formula for conditional variance in a multivariate Gaussian distribution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{5}{26}}
Consider the Sturm-Liouville problem given by the differential equation:
\[ y'' + \lambda y = 0, \]
with boundary conditions:
\[ y(0) = c\pi y'(\pi) - y(\pi) = 0, \]
where \( c > 1 \). Determine the number of negative eigenvalues for this problem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Calculate the limit \(\lim_{n\to\infty} \frac {{a_{n+1}}^{1/{n+1}}}{{a_n}^{1/n}}\) for the sequence \((a_n)_{n \geq 1}\) defined by \(a_1 >0 \) and \(a_n = 2a_{n-1}*\frac{(n-1)^{(n-1)}}{(n-1)!}\), using the fact that \(\lim_{n\to\infty}{(\frac{a_{n+1}}{a_n})}^{1/n}=e\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{e}}
Calculate the accuracy of the approximation for the integral \( \int_0^1 \sqrt{x}\sin(x)\,dx \) using the first three terms of the Taylor series expansion of \( \sin x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{42840}}
In the decimal representation of the even number \( M \), only the digits \( 0, 2, 4, 5, 7, \) and \( 9 \) are used, and digits may repeat. The sum of the digits of the number \( 2M \) is 31, and the sum of the digits of the number \( M / 2 \) is 28. Determine all possible values for the sum of the digits of the number \( M \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29}
There are $n \geq 2$ classes organized $m \geq 1$ extracurricular groups for students. Every class has students participating in at least one extracurricular group. Every extracurricular group has exactly $a$ classes that the students  in this group participate in. For any two extracurricular groups, there are no more than $b$ classes with students participating in both groups simultaneously.

Find the minimum value of $n$ when $m = 20 , a = 4 , b = 1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Determine the values of $x$ for which the integral \( \int_0^1 \frac{e^{xy}-1}{y} \, dy \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, \infty)}
Evaluate the line integral \(\int_C(2ye^{x^2-z}\cos(y^2)-9xy^2)dy+(12z-e^{x^2-z}\sin(y^2))dz+(2xe^{x^2-z}\sin(y^2)-3y^3)dx\), where \(C\) is the broken line path from \(A(0, \sqrt{\pi}, 3)\) to \(B(0, \sqrt{\frac{\pi}{2}}, -1)\) passing through the points \((0, \sqrt{\pi}, 3)\), \((1,3,5)\), and \((0,\sqrt{\frac{\pi}{2}},-1)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e - 48}
Calculate the value of \( \lim_{n\to +\infty}E[Y_{n+1}Y_{n}] \) for the sequence \( \{Y_n\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the least positive integer  $n$  such that no matter how  $10^n$  is expressed as the product of any two positive integers, at least one of these two integers contains the digit 0.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
A fair coin is flipped repeatedly until three consecutive tails (TTT) appear. What is the expected number of heads that will be observed before this sequence occurs?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Calculate the area of the region enclosed by the curve \((x+y)^4 = x^2y\) using the coordinate transformation \(x = r\cos^2 t, y = r\sin^2 t\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{210}}
Given the differential equation \( f'(x) = \frac{1}{2}f(x) \) with \( f(0) = 1 \) and \( |f(x)| \leq 6 \) for all \( x \in [-3, 3] \), determine the maximum error in approximating \( f(2) \) using the 6th degree MacLaurin polynomial \( T_6(2) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{840}}
Given two distributions \( p \) and \( q \) over the same finite set \( X \) with a total variation distance \( TV(p,q) = \epsilon \), and a function \( f: X \to \mathbb{R} \) such that \( 0 \leq \sum_{x \in X} f(x) p(x) \leq 1 \) and \( 0 \leq \sum_{x \in X} f(x) q(x) \leq 1 \), determine the best upper bound for \( \Big|\sum_x f(x)(p(x)-q(x))\Big| \) as a function of \( \epsilon \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Calculate the flux \( \iint \vec{F} \cdot d\vec{S} \) of the vector field \( \vec{F}(x, y, z) = x\hat{i} + y\hat{j} + z^4\hat{k} \) through the part of the cone \( z = \sqrt{x^2 + y^2} \) that lies below the plane \( z = 1 \), with the surface oriented downward.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{3}}
Let $X_1, X_2, \ldots$ be a sequence of random variables such that $P(X_n = \frac{k}{n}) = \frac{1}{n}$ for $k = 1, 2, \dots, n$. Determine the limit distribution of $X_n$ as $n \to \infty$. Express your answer in terms of a standard probability distribution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{U(0,1)}
If $A$ is a $4 \times 4$ matrix with $\det(A) = 2$, find $\det\left(-\frac{1}{2} \operatorname{adj}(2A)\right)$. \( \operatorname{adj}(A) \) denotes the adjugate of matrix \( A \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2048}
Given the change of variables \(x = u + v\) and \(y = uv^2\), which transforms \(f(x,y)\) into \(g(u, v)\), compute the value of \(\frac{\partial^2 g}{\partial v \partial u}\) at the point \((u, v) = (1, 1)\). Assume that at this point, \(\frac{\partial f}{\partial y} = \frac{\partial^2 f}{\partial x^2} = \frac{\partial^2 f}{\partial y^2} = \frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial y \partial x} = 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Evaluate the integral \( \int_{S_r} \frac{e^{iz}-1}{z^2} dz \) as \( r \rightarrow 0 \), where \( S_r = re^{it} \) and \( t \in [0, \pi] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\pi}
Given two lightbulbs, the first has a lifetime $X \sim \exp(1)$ months and the second has a lifetime $Y \sim \exp(3)$ months. The first lightbulb is turned on, and after one month, the second lightbulb is turned on (even if the first one has burnt out by then). Given that the first lightbulb lasts more than one month, calculate the expected time from turning on the first lightbulb until one of the two lightbulbs burns out.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{5}{4}}
Evaluate the integral of the function \( f(x, y) = 1 + xy \) over the region \( D = \{(x, y) \in \mathbb{R}^2 : 1 \leq x^2 + y^2 \leq 2 \text{ and } y \geq 0\} \). Use polar coordinates to perform the integration.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3\pi}{2}}
Evaluate the limit: $$\lim\limits_{n\to\infty}\sum\limits_{r=0}^\infty \left(\frac{r}{n}\right)^n$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Determine the spectrum of the $C^*$-algebra formed by the direct sum $C(K) \oplus C(K')$, where $K$ and $K'$ are compact Hausdorff spaces.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{K \sqcup K'}
Identify the missing conjugacy class in the list of conjugacy classes of the alternating group $A_7$: $
\mathrm{id}$, $(123)$, $(12345)$, $(12)(34)$, $(123)(456)$, $(1234)(56)$. Note that the conjugacy class of 7-cycles splits into two in $A_7$. Given that $A_7$ has 9 conjugacy classes, which one is missing?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(123)(45)(67)}
Evaluate the integral \( \int_{|z|=2} \frac{dz}{z\sqrt{1-\frac{1}{z^2}}} \) over the circle \(|z|=2\), considering the main branch of the square root.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi i}
A black witch's hat is in the classic shape of a cone on top of a circular brim. The cone has a slant height of $18$ inches and a base radius of $3$ inches. The brim has a radius of $5$ inches. What is the total surface area of the hat?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70π
}
Find the smallest positive integer \( n \) such that \( 5000 n \log(n) \leq 2^{n/2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{39}
Evaluate the limit: \[ \lim_{n\to\infty} \sin^2{\left(\frac{1}{n^{48}}\right)}\left((n+\frac{1}{n^3})^{100}-(n^{32}+n^{10}+1)^3-n^{100}\right). \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{99}
Let $ABCD$ be a square with side length $2015$. A disk with unit radius is packed neatly inside corner $A$ (i.e. tangent to both $\overline{AB}$ and $\overline{AD}$). Alice kicks the disk, which bounces off $\overline{CD}$, $\overline{BC}$, $\overline{AB}$, $\overline{DA}$, $\overline{DC}$ in that order, before landing neatly into corner $B$. What is the total distance the center of the disk travelled?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
10065
}
Evaluate the expression \( \frac{\int_0^\pi x^3\ln(\sin x)\,dx}{\int_0^\pi x^2\ln(\sqrt{2}(\sin x))\,dx} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3\pi}{2}}