Context:
Evaluate the integral \( \int_0^{2\pi}\frac{\cos 3x}{5-4\cos x}dx \) using complex integration techniques.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the domain of convergence for the series \( \sum_{n = 1}^{\infty} \frac{\cos(nz)}{n^2} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{-\infty}^{\infty}\frac{\sin(t)}{t}f(t)dt \) given that the Fourier transform of \( f \) is \( \hat{f}(w)=\frac{w}{1+w^4} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the values of $x$ for which $\lim_{n \to \infty}\cos(nx)=1$. Provide your answer in terms of $x$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the surface area of the cone defined by $x^2 + y^2 = z^2$ that lies above the $xy$-plane and beneath the cylinder $y^2 + z^2 = 16$. Provide your answer in terms of exact values or expressions.
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Context:
Evaluate the limit: $$\lim\limits_{n\to\infty} \int_0^n \left( 1- \frac{x}{n}\right)^n \log \left(2+\cos\left( \frac{x}{n}\right)\right)\, dx$$
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Context:
Evaluate the integral \( \int_0^{\pi} [\cot(x)] \, dx \), where \([\cdot]\) denotes the greatest integer function.
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Context:
Since counting the numbers from 1 to 100 wasn‘t enough to stymie Gauss, his teacher devised another clever problem that he was sure would stump Gauss. Defining $\zeta_{15} = e^{2\pi i/15}$ where $i = \sqrt{-1}$, the teacher wrote the 15 complex numbers $\zeta_{15}^k$ for integer $0 \le k＜ 15$ on the board. Then, he told Gauss:

On every turn, erase two random numbers $a, b$, chosen uniformly randomly, from the board and then write the term $2ab - a - b + 1$ on the board instead. Repeat this until you have one number left. What is the expected value of the last number remaining on the board?
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Context:
For what values of \( \gamma \in \mathbb{C} \) do there exist non-singular matrices \( A, B \in \mathbb{C}^{n \times n} \) such that \( AB = \gamma BA \)?
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Context:
Given a Poisson random variable \( Z \) with parameter \( \mathbb{E}Z \) and another random variable \( G \) that converges in distribution to \( Z \) with an error bound \( d_{TV}(G,Z) \leq c \), where \( c \) is a constant and \( d_{TV} \) is the total variation distance, find an upper bound for the probability \( P( |G-Z| > t) \) for a fixed real \( t > 0 \).
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Context:
Determine the points on the helix parameterized by $x=\cos(t)$, $y=\sin(t)$, $z=t$ where the function $f(x,y,z)$, with partial derivatives $f_x=\cos(t)$, $f_y=\sin(t)$, and $f_z=t^2$, can take an extreme value.
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Context:
For any integer $k\ge1$, let $p(k)$ be the smallest prime which does not divide $k$. Define the integer function $X(k)$ to be the product of all primes less than $p(k)$ if $p(k)>2$, and $X(k)=1$ if $p(k)=2$. Let $\{x_n\}$ be the sequence defined by $x_0=1$, and $x_{n+1}X(x_n)=x_np(x_n)$ for $n\ge0$. Find the smallest positive integer, $t$ such that $x_t=2090$.
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Context:
Find the number of subgroups isomorphic to $\mathbb{Z}_{10}\oplus\mathbb{Z}_{10}$ in the group $A=\mathbb{Z}_{600}\oplus\mathbb{Z}_{200}\oplus\mathbb{Z}_{100}\oplus\mathbb{Z}_{4}$.
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Context:
Find a quadratic polynomial \( B(x) \in (\mathbb{Z}/3\mathbb{Z})[x] \) such that \( \bar{B}(x) = x^3 + 2x^2 + x + 2 \in (\mathbb{Z}/3\mathbb{Z})[x]/(x^2 + 2x + 2) \).
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Context:
Calculate the flux of the vector field \( \nabla f \) across the boundary of the region \( S \) bounded between the spheres centered at the origin with radii 1 and 2, and in the first octant, using Green's theorem. Here, \( f(x,y,z) = \frac{1}{x^2+y^2+z^2} \).
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Context:
Calculate the limit \( \lim\limits_{n \to \infty} \int_0^2 \frac{nx^{\frac{n}{2}}}{(1+x^n)} \, dx \) using the convergence theorem.
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Context:
Find the limit of the difference between the prime zeta function $P(s)$ and the natural logarithm of the Riemann zeta function $\ln(\zeta(s))$ as $s$ approaches 1. Given that $M = \gamma + \sum_{n=2}^{\infty} \mu(n) \frac{\ln(\zeta(n))}{n}$, $P(s) = \sum_{n=1}^{\infty} \mu(n) \frac{\ln(\zeta(sn))}{n}$, where $M$ is Mertens' constant, $\gamma$ is the Euler-Mascheroni constant, $\mu(s)$ is the Möbius function, and $\zeta(s)$ is the Riemann zeta function.
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Context:
Find \( \frac{dy}{dz} \) when \( y = \cot^{-1}(f(x)) \) and \( z = \cos^{-1}(\sqrt{1-x^4}) \), where \( f(x) = \frac{2\sqrt{1+x^2}-5\sqrt{1-x^2}}{5\sqrt{1+x^2}+2\sqrt{1-x^2}} \).
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Context:
Determine the median area of a random n-gon formed inside a unit square. The square is bounded by the coordinates (0,0), (0,1), (1,0), and (1,1). Random x and y coordinates are chosen in the interval [0,1] for each of the n points, and these points are randomly connected to form a cycle. What is the median area of this polygon?
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Context:
Given the cumulative distribution function \( F(x,y) = \frac{(x \cdot y)^2}{4} \) for \( 0 < x < 2 \) and \( 0 < y < 1 \), find the probability \( P(X+Y < 1) \).
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Context:
In the ring \( \mathbb{Z}_n = \{0, 1, \ldots, n-1\} \) under addition and multiplication modulo \( n \), suppose \( a, b, c, x \in \mathbb{Z}_n \) are nonzero and the cyclic order \( R(a, b, c) \) holds. Under what conditions does \( R(ax, bx, cx) \) hold?
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Context:
Determine the dimension of the vector space of multilinear forms \( \Delta: V^n \rightarrow K \), where \( V \) is a finite-dimensional vector space over the field \( K \).
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Context:
Consider a system of polynomials $f_1, \ldots, f_m \in \mathbb{F}[x_1, \ldots, x_n]$ each of degree $d$, where $n = O(1)$. If this system has a solution in the algebraic closure $\bar{\mathbb{F}}$, what is the minimum degree of extension of $\mathbb{F}$ required to guarantee a solution? Express your answer in terms of $m$ and $d$.
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Context:
Determine if there exists a function \( f(x) \) such that \( f(f(x)\cos x) = f(x)\sin x \) for all \( x \).
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Context:
Evaluate the integral: \[ \int_0^{ + \infty } \frac{x \ln x}{(x^2 + 1)^2} \, \mathrm{d}x \]
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Context:
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.
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Context:
 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?
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Context:
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?
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Context:
Determine the cardinality of the set of sequences of rational numbers that converge to 1.
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Context:
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.
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Context:
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 $m$.
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Context:
What is the product of real numbers $a$ which make $x^2+ax+1$ a negative integer for only one real number $x$?
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Context:
Find the limit of the sequence \(\left\{\left(T_n\right)^{\frac{1}{n}}\right\}\) where \(T_n=\frac{(n+1)^{n+1}}{(n+2)(n+3) \ldots(n+n+1)}\).
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Context:
Calculate the tensor product \( \mathbb{Z}/2 \otimes_{\mathbb{Z}} \mathbb{Z} \) considered as \( \mathbb{Z} \)-modules. What is the result?
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Context:
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?
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Context:
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
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Context:
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.
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Context:
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{}.


Context:
Evaluate the function \( P(n) = \sum_{i=1}^n i^{10} \) at \( n = -2 \).
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Context:
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 \( 53 \equiv -11 \pmod{64} \).
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Context:
Calculate the expected value \( E[X(u)X(u+v)X(u+v+w)] \) for a Brownian motion \( X(t) \), where \( 0 < u < u+v < u+v+w \).
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Context:
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?
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Context:
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{}.


Context:
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.
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Context:
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 $k < n$.
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Context:
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 \[ X = \frac{(Z_1 + Z_2)^2}{(Z_1 - Z_2)^2} \]
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Context:
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 \( \mathbb{Z}_3 \).
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Context:
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
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Context:
Calculate the expected number of coin tosses required to first observe the pattern "HTTH" when tossing a fair coin repeatedly.
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Context:
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.
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Context:
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 $AP$?
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Context:
Calculate \( \mathop{\lim }\limits_{{n \rightarrow  \infty }}\frac{1}{n}\mathop{\sum }\limits_{{k = 0}}^{{n - 1}}k{\int }_{\frac{k}{n}}^{\frac{k - 1}{n}}\sin \left( {\pi {x}^{2}}\right) {dx} \) .
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Context:
Determine the order of contact between the curves \(y=x^3+x^6\) and \(y=x^3+x^5\) at the point \((0,0)\).
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Context:
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.
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Context:
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 \(\overrightarrow{OQ}\)?
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Context:
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.
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Context:
Determine the splitting field of the polynomial \(p(x) = x^4 + x^3 + x - 1\) over \(\mathbb{F}_3\).
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Context:
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?
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Context:
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 $CH=HE$.
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Context:
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.
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Context:
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.
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Context:
Evaluate the iterated integral by changing to polar coordinates: \[ \int_0^{1/2}\int_0^{\sqrt{1-x^2}}xy\sqrt{x^2+y^2}\,dy\,dx \]
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Context:
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.
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Context:
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 \( n \ge 2 \).
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Context:
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?
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Context:
 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?
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Context:
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
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Context:
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.
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Context:
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 \(y = -1\).
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Context:
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.
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Context:
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.
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Context:
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.
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Context:
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 \(\lim_{n\to\infty}{(\frac{a_{n+1}}{a_n})}^{1/n}=e\).
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Context:
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 \( \sin x \).
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Context:
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 \( M \).
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Context:
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 $m = 20 , a = 4 , b = 1$.
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Context:
Determine the values of $x$ for which the integral \( \int_0^1 \frac{e^{xy}-1}{y} \, dy \) converges.
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Context:
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 \((0,\sqrt{\frac{\pi}{2}},-1)\).
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Context:
Calculate the value of \( \lim_{n\to +\infty}E[Y_{n+1}Y_{n}] \) for the sequence \( \{Y_n\} \).
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Context:
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.
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Context:
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?
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Context:
Calculate the area of the region enclosed by the curve \((x+y)^4 = x^2y\) using the coordinate transformation \(x = r\cos^2 t, y = r\sin^2 t\).
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Context:
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 \( T_6(2) \).
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Context:
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{}.


Context:
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{}.


Context:
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.
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Context:
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 \( A \).
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Context:
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{}.


Context:
Evaluate the integral \( \int_{S_r} \frac{e^{iz}-1}{z^2} dz \) as \( r \rightarrow 0 \), where \( S_r = re^{it} \) and \( t \in [0, \pi] \).
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Context:
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{}.


Context:
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.
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Context:
Evaluate the limit: $$\lim\limits_{n\to\infty}\sum\limits_{r=0}^\infty \left(\frac{r}{n}\right)^n$$
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Context:
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.
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Context:
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?
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Context:
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.
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Context:
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?
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Context:
Find the smallest positive integer \( n \) such that \( 5000 n \log(n) \leq 2^{n/2} \).
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Context:
Evaluate the limit: \[ \lim_{n\to\infty} \sin^2{\left(\frac{1}{n^{48}}\right)}\left((n+\frac{1}{n^3})^{100}-(n^{32}+n^{10}+1)^3-n^{100}\right). \]
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Context:
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?
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Context:
Evaluate the expression \( \frac{\int_0^\pi x^3\ln(\sin x)\,dx}{\int_0^\pi x^2\ln(\sqrt{2}(\sin x))\,dx} \).
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Context:
Determine the cardinality of the quotient set of the equivalence relation defined on the set $\mathbb{Z}^{\mathbb{N}}$ of all infinite sequences of integers, where $f \equiv g$ if and only if $((f(2n) = g(2n)) \wedge (f(n) \cdot g(n) > 0 \vee f(n) = 0 = g(n)))$ for $f, g \in \mathbb{Z}^{\mathbb{N}}$. What is the power of this quotient set?
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Context:
Jill and Kelly are doing spring cleaning. Jill takes an exponentially distributed time with a mean of 30 minutes to clean the kitchen, and Kelly takes an exponentially distributed time with a mean of 40 minutes to clean the bathroom. The first one to finish will then rake leaves, which takes an exponentially distributed time with a mean of 60 minutes. If the second person finishes their task, they will join the first person, and the raking will be completed at twice the original rate. What is the expected time until all chores are completed?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose the prime numbers $p$ and $q$ satisfy $q^2+3p=197p^2+q$.Write $\frac{p}{q}$ as $l+\frac{m}{n}$, where $l,m,n$ are positive integers , $m<n$ and $GCD(m,n)=1$. Find the maximum value of $l+m+n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many nonempty subsets of ${1, 2, \dots, 15}$ are there such that the sum of the squares of each subset is a multiple of $5$?
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Context:
In a finite abelian group \( G \), two subgroups \( H \) and \( K \) both have an index of 3 in \( G \). Determine the index of the intersection \( H \cap K \) in \( G \).
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Context:
You have 100 chips and are offered a game with a winning probability of \( p = 0.66 \) and a payout of 1:1. Determine the optimal number of chips to bet in this game.
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Context:
Given a multiset of symbols \((A, A, B, B, B, C, D, D)\), find the nth permutation of the symbols when sorted in alphabetical order.
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Context:
Find the smallest number that rounds to 0.7 after rounding to the nearest thousandth, then to the nearest hundredth, and finally to the nearest tenth. Provide your answer as a decimal.
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Context:
Determine the non-negative integer value(s) of $a$ for which the following limit equals $\frac{1}{4}$:
\[\lim_{x \to 1 }\left\{ \frac{-ax+\sin(x-1)+a}{x+\sin(x-1)-1} \right\}^{\frac{1-x}{1-\sqrt{x}}}\]
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Context:
Let $f(z) = u + iv$ be an entire function. If the Jacobian matrix \( \begin{bmatrix} u_{x}(a) & u_{y}(a) \\ v_{x}(a) & v_{y}(a) \\ \end{bmatrix} \) is symmetric for all complex numbers $a$, determine the form of $f(z)$. What is $f(z)$?
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Context:
The New York Public Library requires patrons to choose a 4-digit Personal Identification Number (PIN) to access its online system.  (Leading zeros are allowed.)  The PIN is not allowed to contain either of the following two forbidden patterns:\n* A digit that is repeated 3 or more times in a row.  For example, 0001 and 5555 are not PINs, but 0010 is a PIN.   \n* A pair of digits that is duplicated.  For example, 1212 and 6363 are not PINs, but 1221 and 6633 are PINs.  \nHow many distinct possible PINs are there?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a town of 9,000 people, 20 people have heard a rumor by 8 am. The spread of the rumor follows the logistic equation \(\frac{dP}{dt}=kP\left(1-\frac{P}{C}\right)\). By 12 noon, 500 people have heard the rumor. Determine how long it takes for half the population to have heard the rumor. Provide your answer in hours, rounded to one decimal place.
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Context:
Determine the period of the real-valued function $f(x)$ that satisfies the equation $f(x) + f(x + 4) = f(x + 2) + f(x + 6)$ for all $x \in \mathbb{R}$ using elementary algebraic manipulations.
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Context:
Find the flux of the vector field $$\mathbf{F}=(-y,x,z)$$ across the surface S that is part of the paraboloid $$z=x^{2}+y^{2}$$ that lies between the planes $z=2$ and $z=5$, using cylindrical coordinates.
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Context:
Given the function $f(x) = \frac {|x|+x}{2}+1$, find the range of $x$ that satisfies the inequality $f(1-x^2) > f(2x)$.
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Context:
Let \(a > 0\) be a real number, and consider the sequence \(\{x_n\}\) defined by \(x_1 = 1\) and \(ax_n = x_1 + x_2 + \cdots + x_{n+1}\) for all \(n \geq 1\). Determine the minimum value of \(a\) such that \(x_n > 0\) for every \(n \geq 1\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the volume of the solid obtained by revolving the area under the first arc of the cycloid, given by the parametric equations \(x(t) = a(t - \sin t)\) and \(y(t) = a(1 - \cos t)\), around the \(y\)-axis. Use the formula \( V = \pi \int [f(x)]^2 \, dx \) to find the volume.
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Context:
Evaluate the integral \( \int_m^{n}\frac{1}{\frac{b}{x -\frac{1}{2b}}-\lfloor{b/x}\rfloor} \, dx \) where \( b, m, n, \) and \( x \) are natural numbers such that \( m < n < b \).
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Context:
Determine the minimum number of dimensions $n$ for the system described by the ordinary differential equation $\dot{\vec{x}} = A\vec{x}$, where the first coordinate of the solution vector $\vec{x}(t)$ is given by $x_1(t) = 5t^2\cos(3t) + 2\sin(3t) - e^{2t}\sin(3t) + 4t$, and $A$ is an $n \times n$ matrix.
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Context:
In a deck of cards numbered from 1 to 200, calculate the expected number of cycles formed when performing a card-flipping process. What is the expected number of cycles?
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Context:
Find the ideal $J$ of $\mathbb{Z}[x]$ such that the quotient ring $\mathbb{Z}[x]/J$ is isomorphic to $\mathbb{Z}/3\mathbb{Z}$.
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Context:
Determine the constraint on the parameter $a$ such that the function \( f(x) = \frac{x^2 + 2x + a}{x^2 + 4x + 3a} \) is surjective.
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Context:
Find all integer triples \((x, y, z)\) with each integer greater than 1 such that:
\[ x \mid yz - 1, \quad y \mid zx - 1, \quad z \mid xy - 1. \]
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Context:
Consider the graph of the function $f(x) = x^3 - 3x$. How many squares can be formed by selecting four distinct points from this function's graph, given the following conditions: The vertices of the square, labeled $M$, $N$, $P$, and $Q$ in clockwise order, satisfy that $M$ and $P$ lie between the local extremums $A(-1, 2)$ and $B(1, -2)$, $N$ lies to the right of $B$, $Q$ lies to the left of $A$, and the square has its center at the origin $O(0, 0)$. Determine the number of such squares.
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Context:
Let $S$ be a set of 2020 distinct points in the plane. Define \( M = \{ P : P \text{ is the midpoint of } XY \text{ for some distinct points } X, Y \text{ in } S \} \). Determine the minimum possible number of points in the set \( M \).
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Context:
In a soccer game, how many ways can the first 12 goals be scored such that the difference in scores between the two teams never exceeds 2 at any point during the game?
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Context:
Find the last four digits of the number \(2^{1965}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given two independent normally distributed random variables, X and Y, with means \( E[X] = 1 \) and \( E[Y] = 10^{-3} \), and variances \( \text{Var}(X) = (0.12)^2 \) and \( \text{Var}(Y) = (0.05 \times 10^{-3})^2 \), find the variance of the random variable \( Z = \frac{X}{Y} \). Assume that X and Y are independent. What is \( \text{Var}(Z) \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the limit \( \lim_{n \rightarrow \infty} \sqrt[n]{\binom{n^2k}{n}} \) without using Stirling's approximation.
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Context:
Express the function \( g(x) \) in terms of \( f(x) \) given the equation \( f(x) = g(x) \cdot g(-x) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the determinant of the matrix \( A \) defined as:
\[
A = \begin{pmatrix} 3 & 2 & 2 & . & . & . & 2 \\
2 & 3 & 2 & . & . & . & 2 \\
2 & 2 & 3 & . & . & . & 2 \\
. & . & . & . & . & . & . \\
. & . & . & . & . & . & . \\
. & . & . & . & . & . & . \\
2 & 2 & 2 & . & . & . & 3 \end{pmatrix}
\]
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Context:
Determine the range of values of $a$ for which the function \( f(x) = \frac{(x-1)}{x(x+a)} \) has no stationary points.
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Context:
Evaluate the integral \(\int_{-\infty}^{\infty}\frac{\sin(cx) - \sin(dx)}{x^2} \mathrm{d}x\) where \(c\) and \(d\) are positive, non-zero constants, using a suitable indented contour.
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Context:
Determine the dimension of the linear span of the powers of an $n \times n$ matrix $A$ up to its $k$-th power, given that $A$ has rank $n$ and satisfies the Cayley–Hamilton theorem.
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Context:
Determine the parameters of the binary restriction of the Hamming code $Ham(3,4)$, which is originally a $[21,18,3]$ code over $\mathbb{F}_4$. What are the parameters for $Ham(3,4)\mid_{F_2}=Ham(3,4)\cap F_2^{21}$?
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Context:
Evaluate the expression \( S = \frac{A^3-B^3}{B^3}+\frac{A^4+B^4}{B^4} \) where \( A=\int_0^1(1-x^{2023})^{\frac{1}{2022}}dx \) and \( B=\int_0^1(1-x^{2022})^{\frac{1}{2023}}dx \).
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Context:
Find the smallest connected graph that is regular but not vertex-transitive.
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Context:
Determine the number of distinct $7 \times 7$ symmetric matrices where each row contains all integers from $1$ to $7$ exactly once, and both the main and secondary diagonals also contain all integers from $1$ to $7$ exactly once.
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Context:
Evaluate the limit \(\lim_{x\to0^+}\int_x^1\frac{f(t)}{t^3}\,\mathrm{d}t\), where \(f:[0,1]\) is a continuous function.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the flux integral \(\int_S (\nabla \times \vec{F}) \cdot \vec{n} \, dS\) where \(\vec{F} = (y,z,x^2y^2)\) and \(S\) is the surface given by \(z = x^2 + y^2\) with \(0 \leq z \leq 4\), oriented so that \(\vec{n}\) points downwards.
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Context:
Given that a scalar random variable \( y = z + v \), where the probability density function (pdf) of \( v \) is \( p_{v}(t) = \frac{t}{4} \) on the interval \([0,4]\), and the pdf of \( z \) is \( p_{z}(t) = t \) on the interval \([0,2]\), both densities are zero elsewhere. If a single measurement value \( y = 3 \) is available, compute the maximum a posteriori (MAP) estimate of \( y \).
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Context:
Calculate the derivative of \( \frac{1}{2} \log(\det(X+X^T)) \) with respect to \( X \), given that the derivative of \( \log(\det(X)) \) with respect to \( X \) is \( X^{-1} \).
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Context:
设复数 $z=(\omega+2)^{3}(\omega-3)^{2}$ 。求当 $\omega$取遍所有模长为 1 的复数时, $z$ 的模长的最大值。原始答案为$m^{\frac{p}{q}}$的形式，其中$p,q$互质，请给出$m+p+q$的值。
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Context:
In triangle \( \triangle ABC \), the sides are given as \( AB = n \), \( AC = n-1 \), and \( BC = n+1 \), where \( n \) is a natural number. The angles opposite these sides are \( A \), \( 2A \), and \( \pi - 3A \) respectively. Determine the value of \( n \).
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Context:
It is known that $\cos 157^o = a$, where $a$ is given. Calculate $1^o$ in terms of $a$.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{
$1^{\circ} = \dfrac{1}{157} \cdot \arccos(a)$
}


Context:
Determine the cardinality of the set of functions that are either multiplicative or completely multiplicative, defined on the set of natural numbers and taking natural numbers as their values.
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Context:
Given the joint probability density function \(f(x,y,z) = kxyz^2\) for \(0 < x < 1\), \(0 < y < 1\), and \(0 < z < 2\), find the probability \(P(Z > X + Y)\).
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Context:
There are 1001 numbers arranged in a circle. For any two neighboring numbers \(x\) and \(y\), the inequalities \(|x-y| \geqslant 4\) and \(x+y \geqslant 6\) are satisfied. Determine the minimum possible sum of all the numbers.
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Context:
Find the limit of the sequence \( a_n = \frac{n!(0.5)^{n-1}}{\sqrt{n}\{[(n-1)/2]!\}^2} \) as \( n \to \infty \).
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Context:
Determine the number of sets of four primes \((p, q, r, s)\) such that their sum is a prime number, \(p^2 + qs\) and \(p^2 + qr\) are both perfect squares, and all conditions are met simultaneously.
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Context:
Task 2. Lёnya has cards with digits from 1 to 7. How many ways are there to glue them into two three-digit numbers (one card will not be used) so that their product is divisible by 81, and their sum is divisible by 9?
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Context:
For an ordered 10-tuple of nonnegative integers \( {a}_{1},{a}_{2},\ldots ,{a}_{10} \) ,we denote
 \( f\left( {{a}_{1},{a}_{2},\ldots ,{a}_{10}}\right)  = \left( {\mathop{\prod }\limits_{{i = 1}}^{{10}}\left( \begin{matrix} {20} - \left( {{a}_{1} + {a}_{2} + \cdots  + {a}_{i - 1}}\right) \\  {a}_{i} \end{matrix}\right) }\right)  \cdot  \left( {\mathop{\sum }\limits_{{i = 1}}^{{10}}\left( \begin{matrix} {18} + i \\  {19} \end{matrix}\right) {a}_{i}}\right) . \)
 When \( i = 1 \) ,we take \( {a}_{1} + {a}_{2} + \cdots  + {a}_{i - 1} \) to be 0 . Let \( N \) be the average of \( f\left( {{a}_{1},{a}_{2},\ldots ,{a}_{10}}\right) \) over all 10-tuples of
 nonnegative integers \( {a}_{1},{a}_{2},\ldots ,{a}_{10} \) satisfying
 \( {a}_{1} + {a}_{2} + \cdots  + {a}_{10} = {20}. \)
 Compute the number of positive integer divisors of \( N \) .
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Context:
Note that there are exactly three ways to write the integer 4 as a sum of positive odd integers where the order of the summands matters:
$1 + 1 + 1 + 1 = 4,$
$1 + 3 = 4,$
$3 + 1 = 4.$
Let $f(n)$ be the number of ways to write a natural number $n$ as a sum of positive odd integers where the order of the summands matters. Find the remainder when $f(2008)$ is divided by 100.
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Context:
Evaluate the double integral \( \iint (x^2 \cdot y + y \cdot \sin(x^8)) \, dx \, dy \) over the region \( D = \{(x,y) : x^2 + y^2 \leq 2, y > 0\} \).
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Context:
Given that \(a, b, c > 0\) satisfy the equation \((8a - 7b - 3c)^2 = 6(a^2 - b^2 - c^2)\), find the ratio \(a:b:c\) in terms of constants \(b_1\) and \(c_1\) such that \(b = ab_1\) and \(c = ac_1\).
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Context:
Determine the number of integer solutions \((i, j)\) to the equation:
\[3i^2 + 2j^2 = 77 \cdot 6^{2012}.\]
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Context:
Evaluate the surface integral of the vector field $\mathbf{F}(x, y, z) = \left(\frac{1}{x}, \frac{1}{y}, \frac{1}{z}\right)$ over a sphere $S$ of radius $R$ with outward orientation.
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Context:
In $\triangle ABC$, $AB$, $BC$, and $CA$ have lengths $3$, $4$, and $5$, respectively. Let the incircle, circle $I$, of $\triangle ABC$ touch $AB$, $BC$, and $CA$ at $C'$, $A'$, and $B'$, respectively. Construct three circles, $A''$, $B''$, and $C''$, externally tangent to the other two and circles $A''$, $B''$, and $C''$ are internally tangent to the circle $I$ at $A'$, $B'$, and $C'$, respectively. Let circles $A''$, $B''$, $C''$, and $I$ have radii $a$, $b$, $c$, and $r$, respectively. If $\frac{r}{a}+\frac{r}{b}+\frac{r}{c}=\frac{m}{n}$ where $m$ and $n$ are positive integers, find $m+n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the order of convergence of the sequence \(\{y_n\}_{n=0}^{\infty }\) defined by \(y_{n+1}=3y_n-2 {y_n}^3\) with \(y_0=1/2\), given that the limit of the sequence is \(1\).
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Context:
Evaluate the limit \( \lim_{x\rightarrow 0} \frac{\sin x^2}{ \ln ( \cos x^2 \cos x + \sin x^2 \sin x)} \).
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Context:
Calculate the number of open subsets of the real line, denoted as \( o(\mathbb{R}) \).
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Context:
In a game, Ann and Bob alternately toss a fair coin. In each round, they flip the coin $2n$ times. Let $X_n$ be the number of heads in these $2n$ flips during the first $n$ rounds. Define $r_m$ as the probability that there exists an $n$ for which $X_n = m$. Using a renewal theorem, find $\lim\limits_{m\to\infty} r_m$. Provide your answer as a single value.
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Context:
Evaluate the integral \( \int_0^1 \frac{1+x^4}{1+x^6} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{\alpha\rightarrow 0} \int_{0}^1 \frac{(1 - x)^\alpha}{\ln x} dx$$
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Context:
Determine the upper bound of the function $g(x)$ that satisfies the differential equation $g'(x) = \frac{1}{x^{2} + (g(x))^{2}}$ with the initial condition $g(1) = 0$. Provide your answer as a single value.
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Context:
Determine the minimum value of \( k \in \mathbb{R}^{+} \) such that the inequality
\[
\frac{a^3}{(b-c)^2} + \frac{b^3}{(c-a)^2} + \frac{c^3}{(a-b)^2} \geq k (a+b+c)
\]
holds for all distinct positive real numbers \( a, b, c \) that do not form the sides of a triangle.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{n \to \infty} n \int_0^\pi (\sin x - \cos x)^n dx$$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of ordered pairs \((\alpha_1, \alpha_2)\) of permutations in the symmetric group \(S_n\) such that \(\alpha_1 \circ \alpha_2 = \alpha_2 \circ \alpha_1\), where \(\alpha_1, \alpha_2 \in S_n\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find all prime numbers $p$ such that $4p^4 + 1$ is also a prime number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the integral of the differential form \( \omega = \dfrac{x \, dy \wedge dz + y \, dz \wedge dx + z\, dx \wedge dy}{(x^2+y^2+z^2)^{3/2}} \) over the unit sphere \( S^2 \) using the parametrization \( f(u, v) = (\sin u \cos v, \sin u \sin v, \cos u) \), where \( (u, v) \in [0, \pi] \times [0, 2\pi] \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many distinct arrangements of the letters in the word "MATHEMATICS" exist if both 'T's appear before both 'A's, both 'A's appear before both 'M's, or both 'M's appear before the 'E'?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the value of \(\aleph_1^{\aleph_0}\) in set theory.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the largest square-free positive integer $n$ such that the ring of integers of the number field $\mathbb{Q}(\sqrt{-n})$ is a unique factorization domain.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Seven students count from $1$ to $1000$ as follows:
Alice says all the numbers, except she skips the middle number in each consecutive group of three numbers. That is, Alice says $1, 3, 4, 6, 7, 9, \cdots, 997, 999, 1000.$
Barbara says all of the numbers that Alice doesn't say, except she also skips the middle number in each consecutive group of three numbers.
Candice says all of the numbers that neither Alice nor Barbara says, except she also skips the middle number in each consecutive group of three numbers.
Debbie, Eliza, and Fatima say all of the numbers that none of the students with the first names beginning before theirs in the alphabet say, except each also skips the middle number in each of her consecutive groups of three numbers.
Finally, George says the only number that no one else says.
What number does George say?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the limit of the following nested square root expression as \( n \) approaches infinity:

\[ \lim_{n \to \infty}\left( 1 + \sqrt{2 + \sqrt{3+ \ldots + \sqrt {n + \sqrt{n+1}}}}\right) \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit \(\lim_{n\rightarrow\infty} \int^{1}_{0}\frac{x^n}{\ln(1+x^n)}dx.\) Use the inequality \(\ln(1+t) \leq t\) to assist in your evaluation.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are 7 dwarfs, D1 through D7, and 7 tasks, T1 through T7, in a mine. Each dwarf can perform exactly one task. However, D1 cannot perform T1, D4 cannot perform T2, D5 cannot perform T6, and D6 cannot perform T3 or T7. Determine the number of ways to assign the tasks to the dwarfs under these constraints.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a Brownian motion $B_t$ starting at $0$. Compute the probability $P(T_1 < T_{-1} < T_2)$, where $T_a = \inf\{t \geq 0 \mid B_t = a\}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Solve the equation \( x = e^{\frac{1}{x}} \) for \( x \) without using the Lambert \( W \)-Function.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A stock trader has 100 shares of a company and uses a random strategy for buying or selling shares based on the outcome of a die roll at each trading session. The strategy is as follows:

- Outcome = 1: Do nothing
- Outcome = 2 or 4: Buy one share
- Outcome = 3 or 5: Sell one share
- Outcome = 6: Buy two shares

Calculate the approximate probability that after 18 independent trading sessions, the trader will have more than 110 shares of the company.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ l = \lim_{t \to 1^{-}} (1-t) \sum_{r=1}^\infty \frac{t^r}{1+t^r} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a sequence $X_1, X_2, \dots$ of independent exponential random variables, each with a mean of 1. Define $N$ as the minimum value of $n$ such that $\sum_{i=1}^n X_i > k$, where $k$ is a positive real number. Compute the expected value of $N$, $E[N]$, using Wald's equation $E[\sum_{i=1}^N X_i]=E[N]E[X]$. What is $E[N]$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the line integral \( \oint_{\partial S} \mathbf{F} \cdot d\mathbf{r} \) where \( S \) is the region of the plane bounded by the graph of \( x^2-y^2=4 \) and the lines \( y=2 \) and \( y=-2 \), and \( \mathbf{F}(x, y) = \left(-\frac{y}{x^2 + y^2}, \frac{x}{x^2 + y^2}\right) \). Use Green's theorem to simplify the integral.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( \psi \) be a 7th primitive root of unity. Determine the number of elements in the set \( \{a = a_1\psi + a_2\psi^2 + a_3\psi^3 + a_4\psi^4 + a_5\psi^5 + a_6\psi^6 : a_i \in (0,1)\} \) such that \( \mathbb{Q}(a) = \mathbb{Q}(\psi) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of elements of order 6 in the automorphism group of the cyclic group \( \mathbb{Z}_{720} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the exponential generating function for the number of different ways to seat \( n \) people around a round table, denoted as \( r(n) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Price of the book increased by \( {20}\% \) , and then decreased by \( {10}\% \) . How many percents should we decrease current price so we
 get a price which is \( {54}\% \) percent of an original one?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a floating-point number $x$ with $e$ bits for the exponent and $m$ bits for the mantissa, and an elementary function $f$ with domain and range in the interval $[1,2)$, consider an algorithm $\psi$ that approximates $f$ in the same floating-point system. Assuming that $f(x)$ is not a floating-point number, determine the minimum accuracy achievable with $\psi$. Express your answer in terms of $e$ and $m$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a sample $X$ of one hundred observations $X_i$ such that $\forall i: X_i \in \mathbb{N}[-101, 101]$, with unique minimum $A$ and unique maximum $B$, the mean of the sample is $M = \bar{X}$. After negating $A$ and $B$, the mean of the modified sample becomes $-M$. What is the maximum possible value of $M$ in the original sample?
Please reason step by step, and put your final answer within \boxed{}.


Context:
求 $a+b+c+d$ such 的所有可能值之和，这些 $(a, b, c, d)$ 值是满足  $a \cdot b \cdot c \cdot d = 4792$  的（不一定不同的）素数的四元组。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given two non-zero square matrices \( A \) and \( B \) such that \( A^2 B = BA \), find the value of \( k \) if \( (AB)^{10} = A^k B^{10} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the coefficient of \(x^{46}\) in the power series expansion of \(\frac{1}{1 - x^3 - x^4 - x^{20}}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \(S\) be the set of ordered triples \((x,y,z)\) of real numbers for which
\[ \log_{10} (x + y) = z \text{ and } \log_{10} (x^2 + y^2) = z + 1. \]
There are real numbers \(a\) and \(b\) such that for all ordered triples \((x,y,z)\) in \(S\) we have
\[ x^3 + y^3 = a \cdot 10^{3z} + b \cdot 10^{2z}. \]
The original answer is in \(\frac{m}{n}\) format. Please find the value of \(m + n\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $p=2017$ be a prime. Find the remainder when\[\left\lfloor\dfrac{1^p}p\right\rfloor + \left\lfloor\dfrac{2^p}p\right\rfloor+\left\lfloor\dfrac{3^p}p\right\rfloor+\cdots+\left\lfloor\dfrac{2015^p}p\right\rfloor \]is divided by $p$. Here $\lfloor\cdot\rfloor$ denotes the greatest integer function.
Please reason step by step, and put your final answer within \boxed{}.


Context:
$\text{Define an ordered quadruple(a,b,c,d)as interesting if }1\leq a<b<c<d\leq10,\text{and a+d-b+c. How many ordered quadruples are there?}$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( I = \int_{|z-a|=r} \frac{e^{2z}}{(z-a)^3} \, dz \) using Cauchy's Integral Formula, given \( r > 0 \) and \( a \in \mathbb{C} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the directional derivative of the function \( f(x, y) = \begin{cases} \frac{xy^2}{x^2+y^2} & \text{if $(x, y) \neq (0, 0)$} \\ 0 & \text{if $(x, y) = (0, 0)$} \end{cases} \) at the point \((0, 0)\) in the direction of the vector \(<2, 3>\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A polyomino $P$ occupies $n$ cells of an infinite grid of unit squares. In each move, we lift $P$ off the grid and then we place it back into a new position, possibly rotated and reflected, so that the preceding and the new position have $n-1$ cells in common. We say that $P$ is a caterpillar of area $n$ if, by means of a series of moves, we can free up all cells initially occupied by $P$.
How many caterpillars of area $n=10^{6}+1$ are there?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the degree of the canonical divisor of the curve $C = \mathbb{P}^2$ using the differential form $\omega = dx \wedge dy$ on $\mathbb{C}^2$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Alice, Bob, Chad, and Denise decide to meet for a virtual group project between 1 and 3 PM, but they don't decide what time. Each of the four group members sign on to Zoom at a uniformly random time between 1 and 2 PM, and they stay for 1 hour. The group gets work done whenever at least three members are present. What is the expected number of minutes that the group gets work done?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the value of the following double summation: \[ \sum_{r=0}^{2019} \sum_{k=0}^{r} (-1)^k (k+1)(k+2) \binom{2021}{r-k} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find all $t\\in \\mathbb R$, such that $\\int_{0}^{\\frac{\\pi}{2}}\\mid \\sin x+t\\cos x\\mid dx=1$
Please reason step by step, and put your final answer within \boxed{}.


Context:
A trough is 3 feet long and 1 foot high, with a vertical cross-section parallel to an end shaped like the graph of $x^2$ from -1 to 1. The trough is full of water. Calculate the amount of work required to pump all the water over the top of the trough. Assume the weight of water is 62 pounds per cubic foot.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a regular triangular pyramid, the diameter of the circumscribed sphere is 9 times the height of the pyramid. Determine the angle between two of its lateral faces.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the linear transformation $T:V\to V$ defined by $T(X) = AX + XA$, where $A = \begin{pmatrix}1&1&0\\0&2&0\\0&0&-1 \end{pmatrix}$, find the determinant $\det(T)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Sally has five red cards numbered1through5and four blue
 cards numbered 3 through 6 . She stacks the cards so that the
 colors alternate and so that the number on each red card
 divides evenly into the number on each neighboring blue carc
 What is the sum of the numbers on the middle three cards?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit:
\[ \lim_{x\to 0}\frac{\int_0^{x^2}{\frac{1-\cos{t^2}+at^4}{t}}dt}{(1-\cos{(\frac{x}{3})})^4} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find all non negative integer pairs $(a,b)$ such that $3^a5^b-2024$ is a square of a positive integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit of the series: $$\lim_{n \to \infty} \sum_{k=0}^{\infty} \frac{(2\pi)^{2k}(-1)^k (n+1)!}{(n+2k+1)!}$$ where $n$ is an integer greater than or equal to 4.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For each permutation \( \left( {{a}_{1},{a}_{2},\ldots ,{a}_{11}}\right) \) of the numbers1,2,3,4,5,6,7,8,9,10,11,we can determine at least \( k \) of \( {a}_{i} \) s when
 we get \( \left( {{a}_{1} + {a}_{3},{a}_{2} + {a}_{4},{a}_{3} + {a}_{5},\ldots ,{a}_{8} + {a}_{10},{a}_{9} + {a}_{11}}\right) \) . \( k \) can be at most ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the Stieltjes integral: $$\int_{-\pi}^{\pi} (x+2) \, d(x \cdot \text{sign}(\sin x))$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( I = \int_{0}^{\infty} \frac{\sin\left(x\right)\sin\left(\frac{x}{3}\right)}{x\left(\frac{x}{3}\right)}\,dx \) using Laplace Transforms.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a circle of radius 2, there are many line segments of length 2 that are tangent to the circle at their midpoints. Find the area of the region consisting of all such line segments.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the contour integral \( \oint_{|z|=1} \sqrt{z} \, dz \) around the unit circle, where \( z = e^{2\pi it} \) for \( t \) from 0 to 1. Assume a branch cut along the positive real axis.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the improper integral \( \int_{0}^{\infty}{\frac{\cos 2x - \cos x}{x^2}dx} \) using the method of indented contour.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the volume bounded by the surface defined by the parametric equations: $$(a(t-\sin(t)),a(1-\cos(t))\cos(\theta),a(1-\cos(t))\sin(\theta))$$ where both $\theta$ and $t$ vary between $[0,2\pi]$. Express your answer in terms of $a$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral: \[ \int_0^\pi \cos(a \sin(x)) e^{a \cos(x)} \, dx \] where \( a \neq 0 \) and \( a \in \mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the contour integral \( \frac{1}{2\pi i}\oint_{|z|=1}\frac{e^{\frac{c(z-z^{-1})}{2}}}{z^{n+1}}\,dz \) using the unit circle \( z=e^{i\theta} \), and simplify the expression by exploiting the \( 2\pi \)-periodicity and even-odd symmetries of the integrand.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimal value of the expression: \[ \log_{\frac{3}{2}x_{1}}\left(\frac{1}{2}-\frac{1}{36x_{2}^{2}}\right)+\log_{\frac{3}{2}x_{2}}\left(\frac{1}{2}-\frac{1}{36x_{3}^{2}}\right)+\cdots+ \log_{\frac{3}{2}x_{n}}\left(\frac{1}{2}-\frac{1}{36x_{1}^{2}}\right) \] where \(x_{1}, x_{2}, \ldots, x_{n}\) are real numbers in the interval \(\left(\frac{1}{4},\frac{2}{3}\right)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a $9\times 9$ matrix $B$ that satisfies $3B^2-B^4=I$. If $B$ has three distinct eigenvalues, each with geometric multiplicity $2$, determine the number of non-zero entries in the Jordan Canonical Form of $B$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find all functions \( g: \mathbb{R} \rightarrow \mathbb{R} \) that satisfy the equation:

\[ g(x+g(y))+g(2y)=2x+6y \] for all real numbers \( x \) and \( y \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the definite integral: $$\int_{1}^{3} \frac{2x^3}{1+e^{-x}}\mathrm dx$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given in the polar coordinate system, point P moves on the curve $\rho^2\cos\theta-2\rho=0$, the minimum distance from point P to point $Q(1, \frac{\pi}{3})$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of the limit: \[ \lim_{n\to\infty} \frac{a_n}{b_n} \] where \( a_n = \sum_{m=1}^{\lfloor n/2 \rfloor} \frac {n!}{2^m(n-2m)!} \) is the number of elements of order 2 in the symmetric group \( S_n \), and \( b_n = \sum_{m=1}^{\lfloor n/4 \rfloor} \frac {n!}{2^{2m}(n-4m)!} \) is the number of elements of order 2 in the alternating group \( A_n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of invertible elements in the quotient ring \(\mathbb{F}_5 [x]/\langle x^4 + 2x^2 \rangle\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine all positive integers \( p \) so that \( \frac{{3p} + {25}}{{2p} - 5} \) is also a positive integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the remainder when $35!$ is divided by $1457$ using Wilson's theorem.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of integer pairs \((x, y)\) that satisfy the equation \(x^{2} + 6xy + 5y^{2} = 10^{100}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\mathbb{X} = C^1([0,1], \mathbb{R})$ with the norm $\|f\|_{\mathbb{X}} = \|f\|_{\infty} + \|f'\|_{\infty}$. Define the operator $A: \mathbb{X} \rightarrow \mathbb{R}$ by $Af = f'\left(\frac{1}{2}\right)$. Find the operator norm of $A$. \[ \|A\|_{op} = \inf\{c \geq 0: \|Af\| \leq c \|f\|_{\mathbb{X}} \} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $ABCDEF$ be a regular hexagon with side length $1$. Denote by $X$, $Y$, and $Z$ the midpoints of sides $\overline{AB},\overline{CD},\overline{EF}$, respectively. What is the area of the convex hexagon whose interior is the intersection of the interiors of $\triangle{ACE}$ and $\triangle{XYZ}$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the surface integral \( \iint (xz\cos\alpha + yz\cos\beta + \cos\gamma) \, dS \) over the spherical cap \( S_1 \) formed by the intersection of the sphere \( x^2 + y^2 + z^2 = 25 \) with the plane \( z = 3 \). The unit normal is denoted by \( n = (\cos\alpha, \cos\beta, \cos\gamma) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the surface integral of the curl of the vector field \( F = \langle 2y, xz, 1 \rangle \) over the surface defined by \( z = f(x,y) = 3 - 2x - \frac{3}{2}y \). Use a normal vector obtained from the cross product of two vectors on the surface.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim_{n\to +\infty}\frac{1}{n^2}\sum_{i=1}^{n}\log\binom{n}{i}. \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the kernel of the ring homomorphism \( g: k[x_1,x_2] \to k[u,v] \) defined by \( x_1 \mapsto u^2+v^2 \) and \( x_2 \mapsto uv \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A line passes through the point (1, 2). The abscissa of the intersection of the tangent to this line with the x-axis is half the abscissa of the intersection of the tangent with the line itself. Determine the abscissa of the intersection point of the tangent with the x-axis.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of group homomorphisms from $\Bbb Z/2\Bbb Z \times \Bbb Z/2\Bbb Z$ to $S_{3}$, where $S_{3}$ is the symmetric group of permutations of three elements.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{0}^{\infty} \frac{\cos x - e^{-x^2}}{x} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a primitive root of 71.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a doctor's clinic, patients arrive at an average rate of 10 per hour. The consultation time per patient is exponentially distributed with an average of 6 minutes per patient. The doctor does not admit any patient if at any time 10 patients are waiting. Calculate the expected number of patients waiting in the steady state of this M/M/1/R queue.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of roots of the equation \(2\cos^{2} \theta = 1 + 3^{\sin \theta} + 3^{\csc \theta}\) within the interval \([0, 2\pi]\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many labeled trees with 10 vertices have exactly one vertex of degree 4?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a Markov chain on the state space \( S = \{1, 2, \ldots, 21\} \) with the given transition matrix. The initial distribution is \( P(X(0) = 11) = 1 \). State 20 is in a closed recurrent class, while state 11 is in a transient class. Determine \( \lim_{n \to \infty} P(X(n) = 20) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $A = \left\{a_{1}, a_{2}, \cdots, a_{n}\right\}$ be a set of numbers, and let the arithmetic mean of all elements in $A$ be denoted by $P(A)\left(P(A)=\frac{a_{1}+a_{2}+\cdots+a_{n}}{n}\right)$. If $B$ is a non-empty subset of $A$ such that $P(B) = P(A)$, then $B$ is called a "balance subset" of $A$. Find the number of "balance subsets" of the set $M = \{1,2,3,4,5,6,7,8,9\}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For a standard Brownian motion $W_t$, compute the probability $P(T_{1} < T_{-3} < T_3)$, where $T_a$ denotes the hitting time of level $a$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of elements in the conjugacy class of the permutation \( \tau = (56789)(3456)(234)(12) \) in the symmetric group \( S_9 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \( \displaystyle\lim_{z\to3i}\frac{-3z}{z^2-9-6iz} \). Determine whether the limit is \( \infty \) or does not exist.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Johnny buys a pack of gum each day, aiming to collect all 6 different prizes, with each pack having an equal probability of containing any of the prizes. Calculate the standard deviation of the number of days Johnny needs to buy gum packs to collect all the different prizes.
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Context:
Evaluate the limit: $$\lim_{x\to 0}\frac{\sin x^4-x^4\cos x^4+x^{20}}{x^4(e^{2x^4}-1-2x^4)}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a set of linearly independent vectors $\{v_1, v_2, \ldots, v_{2014}\}$, find the scalar $\lambda \in \mathbb{R}$ such that the vectors $\{v_1+v_2, v_2+v_3, \ldots, v_{2014}+\lambda v_1\}$ are also linearly independent.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For two independent Brownian motions \( W_t \) and \( \tilde{W}_t \), and a constant \( \rho \) such that \( -1 \leq \rho \leq 1 \), define \( X_t = \rho W_t + \sqrt{1-\rho^2} \tilde{W}_t \). Find the conditions on \( \rho \) for which \( X_t \) is a martingale.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the norm formula for the ring of Kleinian integers, \( \mathbb{Z}[\frac{1+\sqrt{-7}}{2}] \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find an equivalent expression for the remainder of the series \( \sum_{n=1}^{\infty} \frac{1}{n^3} \) given by \( R_n = \sum_{k=n+1}^{\infty} \frac{1}{k^3} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim_{x\to1^{-}} (1-x)\left(\sum_{i=0}^{\infty} x^{i^2}\right)^{2} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim _{x\to \infty }\left(x\sqrt{x^2-x}-x^2\cos\left(\frac{1}{\sqrt{x}}\right)\right) \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $n$ be a positive integer. Consider a random ordered triple $(a, b, c)$ of nonnegative integers such that $a + b + c = n$, chosen uniformly at random from among all such triples. Let $M_n$ be the expected value of the largest of $a$, $b$, and $c$. Determine the value that $\frac{M_n}{n}$ approaches as $n$ approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the line integral \( \int_{C} F \cdot \, dr \) where \( C \) is the curve consisting of line segments from \((4,1)\) to \((4,3)\) to \((1,3)\) to \((1,1)\), and \( F(x,y) = (x+y)i + (y-1)^3 e^{\sin(y)}j \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of \( r \) for which a non-trivial fixed point appears in the equation \( x = r\sin\pi x \), given that \( 0 \leq x \leq 1 \) and \( 0 \leq r \leq 1 \). The trivial fixed point is \( x = 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim_{x\to 0} \left( \frac{1+x\cdot2^x}{1+x\cdot3^x} \right)^{1/x^2} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a sequence of independent and identically distributed (i.i.d.) random variables $\{X_i\}$ with mean $0$ and variance $\sigma^2 > 0$, find the limit of the expected value of the squared fraction of the sample mean's absolute value as the number of variables approaches infinity: $$\lim_{n \to \infty} E\left(\left|\frac{\overline X_n}{\overline X_n + \frac{1}{n}}\right|^2\right)$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a set of points in the plane such that only two distinct Manhattan distances occur between all pairs of distinct points. What is the maximal number of points in such a set?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of ring homomorphisms from the polynomial ring \(\mathbb{Z}[X]\) to the integers \(\mathbb{Z}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of the fourth derivative of the inverse function \((f^{-1}(y))^{(4)}\) evaluated at \(y=1\), where \(f(x) = \sin{x} + x^2 + 1\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
For a zebra group of order $2^8p^b$, where $p$ is a prime, determine the maximum number of abelian 2-group stripes that can appear in its chief series.
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Context:
Evaluate the integral \( \int f(x) \, \mathrm{d}x \) where \( f(x) = \lim_{n \to \infty} \left[ 2x + 4x^3 + \cdots + 2nx^{2n-1} \right] \) for \( 0 < x < 1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the locus of the duals of lines that pass through the point (1, 1) and have variable slopes.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the coefficient of the term $x^{60}$ in the power series expansion of $C(x) = \frac{1}{1-x} \cdot \frac{1}{1-x^2} \cdot \frac{1}{1-x^5}$. This coefficient represents the cardinality of the set $A = \{(N, D, Q) \mid 0.05N + 0.1D + 0.25Q = 3 \text{ and } N, D, Q \in \mathbb{N}\}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( \phi \) be an automorphism of the quaternion group \( \mathcal{Q}_8 \) such that \( \phi(i) = -i \) and \( \phi(j) = -k \). Determine the order of \( \phi \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the lines $l_1: ax+2y-1=0$ and $l_2: 8x+ay+2-a=0$, if $l_1 \parallel l_2$, find the value of the real number $a$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( q(x) = 1 - (1-x) + (1-x)^2 - \cdots + (1-x)^{10} \). The roots of \( q(x) \) are \( \beta_i, i \in \{1, 2, \ldots, 10\} \). Find the value of \( \sum_{i=1}^{10} \frac{1}{\beta_i} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of singular values of a linear map $T: V \to W$ given that $\dim V = n$ and $\dim W = m$. Provide the correct number of singular values for the linear map $T$. \( \text{(Assume $T$ is represented by a matrix in the standard basis.)} \)
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Context:
Evaluate the limit: $$\lim_{x\to \infty} \int_1^x x\,e^{t^2-x^2} \,dt$$
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Context:
Let $n\geq 2$ be integer. On a plane there are $n+2$ points $O,\ P_{0},\ P_{1},\ \cdots P_{n}$ which satisfy the following conditions as follows.

[1] $\angle{P_{k-1}OP_{k}}=\frac{\pi}{n}\ (1\leq k\leq n),\ \angle{OP_{k-1}P_{k}}=\angle{OP_{0}P_{1}}\ (2\leq k\leq n).$
[2] $\overline{OP_{0}}=1,\ \overline{OP_{1}}=1+\frac{1}{n}.$
Find $\lim_{n\to\infty}\sum_{k=1}^{n}\overline{P_{k-1}P_{k}}.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim_{x \to \infty} \frac{((2x)!)^4}{(4x)! ((x+5)!)^2 ((x-5)!)^2}. \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the range of \( \alpha \in \mathbb{R} \) for which the following statement holds: If a sequence \( \{ x_n \}_{n=1}^{\infty} \) of real numbers satisfies \( \lim_{n \to \infty}{ n^\alpha \,\, |x_n - x_{n+1}|}=0 \), then it is a convergent sequence.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many distinct ways can the numbers $1, 2, 3, 4, 5, 6, 7$ be inserted into an empty binary search tree such that the resulting tree has a height of $6$? Note that a tree with a single node has a height of $0$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the Morley rank of the type $x = x$ in the theory $ACF_{0}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the flow of the vector field \( F(x,y,z) = (xz, yz, -z^2) \) out of the surface \( S = \{(x,y,z) \in \mathbb{R}^3 : z^2 = 1 + x^2 + y^2, 2 \leq z \leq 3\} \) using the definition of surface integral.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of distinct ways to color a regular hexagon such that each vertex is either Black or White, and each edge is one of the three colors Red, Green, or Blue, considering the symmetries of the dihedral group $D_6$. The group $D_6$ includes rotations by multiples of $\frac{\pi}{3}$ and reflections across axes through opposite vertices or midpoints of opposite edges.
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Context:
Let $f$ and $g$ be functions of $x$ and $y$. Given that $f(1, 2) = 2$ and $g(1, 2) = 1$. Also at $(1, 2)$, $df = -2dx + 5dy$, $dg = 4dx - 3dy$. $z$ is a function of $(x, y)$ defined implicitly by $z^2f + zg + x - y = 0$ with $z(1, 2) = -1$. Find $\frac{\partial z}{\partial x}(1, 2)$. \( \boxed{?} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a triangle $ABC$, given that $r_1 + r_3 + r = r_2$, find the value of $\sec^2 A + \csc^2 B - \cot^2 C$, where $r_1 = \frac{\Delta}{s-a}$, $r_2 = \frac{\Delta}{s-b}$, $r_3 = \frac{\Delta}{s-c}$, and $r = \frac{\Delta}{s}$. Here, $\Delta$ is the area of the triangle and $s$ is the semi-perimeter.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A die is rolled repeatedly. Let $X$ be the random variable representing the number of rolls needed to get a 4, and $Y$ be the random variable representing the number of rolls needed to get a 1. Calculate the expected value $E[X|Y=7]$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( P(x) = 5x^{2015} + x^2 + x + 1 \) be a polynomial with roots \( x_1, x_2, x_3, \ldots, x_{2015} \). Find the value of the expression:
\[ \left(\prod_{i=1}^{2015} (1+x_i)\right) \left( \sum_{i=1}^{2015} \frac{1}{1-x_i} \right). \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of distinct equivalence classes for $2 \times 2$ matrices over the field of complex numbers, $\mathbb{C}$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Identify the smallest group that is not a complex reflection group.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $X$ be a set. Define mappings $F, G : \text{Map}(X, X)\times\text{Map}(X, X)\to\text{Map}(X, X)$ such that $F(f, g) = f\circ g$ and $G(f, g) = g\circ f$. What is the necessary and sufficient condition on $X$ that ensures $F=G$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A wire is shaped as a circle defined by the equation \(x^2 + y^2 = a^2\). The density at any point \((x, y)\) on the wire is given by \(|x| + |y|\). Calculate the moment of inertia of the wire about a diameter.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of real values of \(x\) that satisfy the equation:
\[ \left| 3 - x \right|^{\log_7(x^2) - 7\log_x(49)} = (3 - x)^3 \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine all positive integers \( n \) such that for any sequence \( a_1, a_2, \ldots, a_n \) with each \( a_i \in (0, 1] \), the inequality \( \sum_{i=1}^{n} \frac{\sqrt{1-a_{i}}}{a_{i}} \leq \frac{1}{\prod_{i=1}^{n} a_{i}} \) holds.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{0}^{\infty} \frac{\sin{x}}{x^2+1} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \(x\) be a nonnegative real number. The function \(f_n\) is defined by: \[ f_n(x) = 3x^n e^{-x^2} - 1. \] Let \(v_n\) be a real number for all \(n > 1\) such that \(v_n > 1\) and \(f_n(v_n) = 0\). Calculate the limit of \(v_n\) as \(n\) approaches infinity, i.e., find \(\ell\) such that \(\lim\limits_{n \to \infty} v_n = \ell.\)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit of the expression obtained by substituting \( n=2 \) in the equation \( r^{n-1}w'+\frac{1}{2}r^nw=a \), as \( r \) approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the smallest degree $n$ of a Taylor polynomial centered at $0$ for the function $f(x) = (1-x)^{-1}$, such that the error in the interval $[0, 0.5]$ is no greater than $10^{-8}$. Use the remainder formula:
\[ \frac{|f^{(n+1)}(c)|}{(n+1)!} \cdot |(x-\bar{x})^{(n+1)}| \le 10^{-8} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of possible solutions for the equation \( \int_{0}^{x} (t^2-8t+13)\,\mathrm{d}t = x\sin\left(\frac{a}{x}\right) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the order of magnitude of \( \log h(n) \) as \( n \to \infty \), given that \( \lim_{n \to \infty} n^n h(n) = \int_{\mathbb{R}} e^{-e^x} g(x) \, dx \) for some function \( g(x) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the interval for the value of $k \in (0, 2\pi]$ such that for the function $f(x) = \tan\left(\frac{3x - \pi}{2}\right)$, $f(x) = f(x+k)$ for all $x$ in the domain of $f$. What is the interval over which $k$ can be found?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a circle with center $A$, point $E$ is the tangent point, and $[BA] \perp [AC]$. Given that $\angle ABD = \angle DCB$ and $\angle DBC = \angle DCA$, with $D$ on the circle $\omega$, and $|EC| = x$, find the area of triangle $BCD$ in terms of the radius $r$ of the circle and $x$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the boundary of the manifold $D^2 \times S^1$, where $D^2$ is the closed unit disk and $S^1$ is the circle.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{-\infty}^{\infty} \frac{x \sin (\Omega x)}{x^2(x^2+1)} \, dx \) for \( \Omega \gg 1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
For which values of \(a \in \mathbb{R}\) does the second-order matrix differential equation \(\frac{d^2}{dt^2}x = \begin{pmatrix}1 & 1 \\ 0 & a\end{pmatrix}x\) have periodic solutions?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $W_t$ be a standard Brownian motion and define the process $$X_t = X_0 e^{t(1-\sigma^2/2) + \sigma W_t}$$ where $\sigma$ has an exponential distribution given by $$ P(\sigma \leq x) = 1-e^{-x}$$ for $x \geq 0$. Determine the probability $$P\left(\lim_{t \to \infty}X_0 e^{t(1-\sigma^2/2) + \sigma W_t}=0\right).$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the spectrum of the operator \( T \) defined on the space of double-sided sequences \( \ell_2(\mathbb{Z}) = \{ \{ a_n \}_{n = -\infty}^{\infty} ; \sum_{n= -\infty} ^{\infty} | a_n |^2 < \infty \} \) by \( Tx = y \) where \( y_n = -x_{n-1} + x_n - x_{n+1} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the number of ways the numbers from $1$ to $14$ can be split into $7$ pairs such that for each pair, the greater number is at least $2$ times the smaller number?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the radius of convergence of the series \( \sum_{n=0}^{\infty} 2^{n} z^{n!} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of elements of order 5 in a group of order 100.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A point mass is attracted towards a center by a force given by \( \frac{m k^2}{r^3} \), where \( r \) is the distance from the center. If the point starts at a distance \( r = a \) and is initially at rest, determine the time required for the point to reach the center. Note that the mass \( m \) can be eliminated from the equation \( m r'' = -\frac{m k^2}{r^3} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the expectation of the random variable \( T \), where \( T = \min\{n \geq 1 \mid X_1 + X_2 + \ldots + X_n > 1\} \) and \( X_1, X_2, \ldots \) are i.i.d. standard uniform random variables on \((0, 1)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
On an island, there are 12 blue chameleons, 15 green chameleons, and 7 red chameleons. When two chameleons of different colors meet, they both change into the third color. What is the minimum number of meetings required to convert all the chameleons into the same color?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the smallest number of vertices in a graph that is vertex-transitive but neither edge-transitive nor edge-flip-invariant.
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Context:
For which real number $a$ does the series \( \sum_{n=0}^\infty \frac{1}{u_n} \) converge, where the sequence \( \{u_n\} \) is defined recursively by \( u_0 = 2 \) and \( \int_{u_n}^{u_{n+1}} (\ln u)^a \, du = 1 \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Simplify the polynomial \( p_n(x) = a_0 \binom{n}{0} (1-x)^n + a_1 \binom{n}{1} x(1-x)^{n-1} + a_2 \binom{n}{2} x^2 (1-x)^{n-2} + \cdots + a_n \binom{n}{n} x^n \), where \( \{a_n\} \) is an arithmetic sequence with common difference \( d \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a convex $n$-gon with distinct letters at each vertex, determine a formula in terms of $n$ for the maximum number of distinct $n$-letter words that can be read by an observer standing outside the polygon. Each word must include all $n$ letters, with no letter hidden behind another from the observer's viewpoint.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the percentage of prime numbers among all numbers with 100 decimal digits.
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Context:
Evaluate the integral: \[ \int_{-\infty}^\infty e^{-x^{-2}} \, dx \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
A casino patron will continue to make $5 bets on red in roulette until she has won 4 of these bets. Calculate her expected winnings when she stops.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find \( \lim_{t \to \infty} P (X>t+ \frac{x}{t} | X > t) \) where \( X \sim N(0,1) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a single generator for the kernel of the homomorphism \( \varphi : \mathbb{Z}[i] \to \mathbb{F}_{13} \) defined by \( \varphi(a+bi) = a + 5b \pmod{13} \). Given that \( \ker(\varphi) \) is the ideal generated by \( 13 \) and \( i-5 \), determine a single generator for \( \ker(\varphi) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the asymptotic bound for the expression \( \sum_{i=0}^{\log(n)} 2^{i} \sum_{k=0}^{\frac{n}{2^{i}}} (k+2)^{2} + \theta(n) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of homomorphisms from the dihedral group $D_{21}$ to the dihedral group $D_{15}$. Assume that $\phi(ab) = \phi(a)\phi(b)$, where $s$ and $r$ are the standard generators of $D_n$ with $s$ having order 2 and $r$ having order $n$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \(X_1, X_2, \ldots, X_n\) be a random sample from a Poisson distribution with parameter \(\lambda = 1\). Define the statistic \(T = \bar{X}(n - \bar{X})\), where \(\bar{X}\) is the sample mean. Find an upper bound for the probability \(P(T = 0)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose that \( {ABCDE} \) is a convex pentagon with
 \( \angle A = {90}^{o},\angle B = {105}^{o},\angle C = {90}^{o} \) and \( {AB} = 2,{BC} = {CD} = {DE} = \sqrt{2} \) . If the
 length of \( {AE} \) is \( \sqrt{a} - b \) where \( a,b \) are integers,what is the value of \( a + b \) ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \(\beta\) be an element in \(\mathbb{F}_{16}\) with a minimal polynomial over \(\mathbb{F}_{2}\) given by \(x^4+x+1\). Determine the minimal polynomial of \(\beta^7\) over \(\mathbb{F}_{2}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of the limit \( L = \lim\limits_{n\to\infty} \dfrac{2^{n^{1.1}}}{n!} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the double integral \( \iint_R xe^y \, dx \, dy \) where \( R \) is the region inside \( x^2+y^2 = 1 \) but outside \( x^2+y^2 = 2y \) with \( x \ge 0 \) and \( y \ge 0 \). Use the change of variables \( u = x^2 + y^2 \) and \( v = x^2 + y^2 - 2y \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{C}{xy^{2}dy-yx^{2}dx} \) where \( C \) is the circle \( x^{2}+(y-1)^{2}=1 \) using Green's theorem.
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Context:
A fly moves through the point \((\sqrt{3}, 1, 2)\) at time \(t = 0\) along the curve of intersection of the surfaces \(z = x^2 - y^2\) and \(z^2 = x^2 + y^2\). The temperature in the room is given by \(T(x,y,z) = \frac{1}{2}(4x^2 + y^2 + z^2)\). If the fly's vertical velocity at this time is 5, what is the rate of change of temperature \(\frac{dT}{dt}\) experienced by the fly at \(t = 0\)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Solve the recurrence relation given by:
\[ T(n) = \frac{1}{n}(T(0) + T(1) + \ldots + T(n-1)) + 5n \]
with the initial condition \( T(0) = 0 \). Find a closed-form expression for \( T(n) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \frac{1}{2\pi i}\int_{\gamma} \frac{1}{z^2(e^{z}-1)}dz \), where \( \gamma=\{z: z=\frac{1}{2}e^{2\pi i t},0\leq t\leq1 \} \). Note that \( z=0 \) is a pole singularity of order 3.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit using Taylor series expansion: $$\lim\limits_{h \to 0^-}{\int\limits_{0}^2{\frac{h}{h^2+(x-1)^2}}g(x)\:dx}$$ where $g(x)$ is a continuous function.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function \( f(x,y) = \frac{xy \sin x}{x^2+y^2} \) for \((x,y) \neq (0,0)\) and \( f(0,0) = 0 \), find the directional derivative of \( f \) at the origin in the direction of the unit vector \( \phi = (\phi_1, \phi_2) \). Calculate \( \frac{\partial f(0,0)}{\partial \phi} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the asymptotic expression for the average number of prime factors of a number as the number of digits goes to infinity. More formally, determine the asymptotic behavior as $N \to \infty$ of
\[ \frac{\sum_{1\le k\le N} M(k)}{N} \]
where
\[ M(p_1^{d_1}p_2^{d_2}\cdots p_k^{d_k}) = d_1+d_2+\cdots+d_k \]
For example, $M(24) = M(2^3\cdot3) = 4$. Provide your answer in terms of $N$. \( \boxed{} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a $2009 \times 2009$ grid, each small square contains a real number with an absolute value not exceeding 1. Additionally, for every $2 \times 2$ square, the sum of the 4 real numbers is 0. Determine the maximum possible sum of all the numbers in the grid.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a group \((G, \cdot)\) with order \(n\), where \(p\) is a prime divisor of \(n\), consider the set \(M\) defined as:
\[M=\left\{(g_1,g_2,\dots,g_p)\in G^p:g_1\cdot g_2\cdots g_p=e_G\right\}\]
Determine the cardinality of the set \(M\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle $ABC$, $F$ is on segment $AB$ such that $CF$ bisects $\\angle ACB$. Points $D$ and $E$ are on line $CF$ such that lines $AD,BE$ are perpendicular to $CF$. $M$ is the midpoint of $AB$. If $ME=13$, $AD=15$, and $BE=25$, find $AC+CB$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose $g$ is a function from the positive integers to the positive integers that satisfies $g(n+1) > g(n)$ and $g(g(n)) = 4n$ for all $n$. Find $g(10)$.

Provide your answer in the form \( \boxed{} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Vertices \( A, B, C \) of an equilateral triangle with side length 1 are on the surface of a sphere with radius 1 and center \( O \). Let \( D \) be the orthogonal projection of \( A \) onto the plane \( \alpha \) determined by points \( B, C, O \). Let \( N \) be one of the intersections of the line perpendicular to \( \alpha \) passing through \( O \) with the sphere. Find the angle \( \angle DNO \).
Please reason step by step, and put your final answer within \boxed{}.