Context:
Compute the depth of the polynomial ring $k[x,y]$ with respect to the ideal $(x,y^2)$, where $k$ is a field.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A fair 20-sided die, numbered from 1 to 20, is rolled 20 times. The results of the rolls are summed. Determine the number below which 99% of the sums will fall.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $X=l^2(\mathbb{Z})$ be the Hilbert space of all real sequences $x= (x_{k})_{k}$ $(k\in \mathbb{Z})$ for which the norm $\|x\|= \left( \sum_{k=-\infty}^{\infty} |x_{k}|^{2}\right)^{\frac{1}{2}}$ is finite. Denote by ${e_{k}:k\in\mathbb{Z}}$ the canonical basis in $X$, and define $L: X\longrightarrow X$ by
$$L
\left(\sum_{k=-\infty}^{\infty}x_{k}e_{k}
\right)=\sum_{k=-\infty}^{\infty}(x_{k}-x_{k-1})e_{k}$$
Determine the norm $\|L\|$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the triple integral \( \iiint x\ \text{d}x \) over the region bounded by the surfaces \( z = x^2 + y^2 \) and \( 4x + z = -2 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of ring homomorphisms from the ring \( \mathbb{Z}[X] / (X^{2} + 1) \) to \( \mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of Q-field homomorphisms from the algebraic closure of the rational numbers to the complex numbers that keep the rational numbers fixed.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of zeros (counting multiplicities) of the polynomial \(f(z) = z^{100} + 8z^{10} - 3z^3 + z^2 + z + 1\) within the closed unit disc \(|z| < 1\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the class number of a field using the Minkowski bound of 6 and the ideal norms: 1, 3, 4, and 6.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Currently, Selena’s analog clock says $4{:}00$. Suddenly her clock breaks, so the hour hand moves $12$ times as fast as it normally does, but the minute hand stays the same speed. Find the degree measure of the smaller angle formed by the minute and the hour hand $2024$ minutes from now.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function for accumulated value over time, \( a(t) = xt^{2} + yt + z \), where \( 100a(2) = 152 \) and \( 200a(4) = 240 \), find the accumulated value at time 8 of an investment of $1600 made at time 6. Use the conditions to determine the constants \( x, y, \) and \( z \), and then calculate the accumulated value at time 8.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the derivative of the vector expression $A^Tx$ with respect to the matrix $A$, where $A \in \mathbb{R}^{n \times m}$ and $x \in \mathbb{R}^n$. Express your answer as $\frac{d(A^Tx)}{dA}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the Cumulative Distribution Function (CDF) of the absolute difference between two independent uniform random variables $X_1$ and $X_2$, both distributed on the interval [0, 1].
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit \( \lim_{n\to\infty}\frac{a_{12}(n)}{a_{22}(n)} \), where \( a_{ij}(n) = \begin{pmatrix}2 & 1\\ 1 & 0\end{pmatrix}^n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $S_n = \sum_{i=1}^n X_i$ be a symmetric random walk. Compute the conditional expectation $E(X_i \mid \mathcal{G}_{n+1})$ for $1 \leq i \leq n$, where $\mathcal{G}_{n} = \sigma(S_{n},S_{n+1},\dots)$. Assume $S_n$ is a martingale.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that \( n > 3 \), and \( a_1 = a_2 = \cdots = a_{n-1} = 1 \) and \( a_n = -2 \), find the value of the expression:
\[
f_n = \sum_{i=1}^{n}|a_{i}| - \sum_{1 \le i < j \le n}|a_{i} + a_{j}| + \sum_{1 \le i < j < k \le n}|a_{i} + a_{j} + a_{k}| - \cdots + (-1)^{n-1}|a_{1} + a_{2} + \cdots + a_{n}|
\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{n\to\infty}\left(\frac{1}{n!} \int_0^e \log^n x \ dx\right)^n$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of idempotents in the ring \( \mathbb{Z}[i] \otimes_{\mathbb{Z}} \mathbb{Z}[i] \), where \( \mathbb{Z}[i] = \{a+bi : a,b \in \mathbb{Z} \} \) is the ring of Gaussian integers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of polynomials of degree 3 contained in the ideal \( I = (x^5 + x^2 + x + 1, x^7 + x^5 + x^3 + x) \) in \( \mathbb{F}_2[x] \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the exterior derivative $d\beta$ of the two-form $\beta = z(y^2 - x^2)\,dx\,dy - x^2y\,dx\,dz - xy^2\,dy\,dz$ in three-dimensional real space and simplify the result if possible.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the figure, \(\triangle ABC\) is an equilateral triangle. Points \(D\) and \(E\) are on side \(BC\) such that \(BC = 3DE\). Construct an equilateral \(\triangle DEF\). Connect \(AF\), draw \(DG\) parallel to \(AF\) at point \(G\), and draw \(EH\) parallel to \(AF\) intersecting side \(AC\) at point \(H\). Let \(GI \perp AF\) and \(HJ \perp AF\). If the area of \(\triangle BDF\) is \(45\) and the area of \(\triangle DEF\) is \(30\), find the value of \(\frac{GI}{HJ}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Identify a group $G$ that contains subgroups isomorphic to the cyclic group $\mathbb{Z}_n$ for every positive integer $n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the norm of the function \( f(x) = \sum_k x_k a_k \), where \( x = (x_1, x_2, \dots) \in l^p \) and \( a = (a_1, a_2, \dots) \in l^q \). Assume that \( p \) and \( q \) are appropriate indices for the \( l^p \) and \( l^q \) spaces, respectively.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( I = \int_{0}^{1}\int_{0}^{1} \left\{\frac{x}{y}\right\}\mathrm{d}x\mathrm{d}y \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the winding number of the loop \( \alpha(z) = 3z^3 + 5z^2 + 2 \) around the origin \( 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $0<t<\frac{\pi}{2}$.

Evaluate

\[\lim_{t\rightarrow \frac{\pi}{2}} \int_0^t \tan \theta \sqrt{\cos \theta}\ln (\cos \theta)d\theta\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_0^\frac{\pi}{2}\cos ^2x\log(\tan x)dx. \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle \(ABC\), where \(AB = 8\), \(AC = 7\), and \(BC = 10\), let \(I\) be the incenter. If \(AI\) intersects \(BC\) at \(K\) and the circumcircle of \(\triangle ABC\) at \(D\), find \(\frac{DK}{DI}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( V = \left\lbrack  {\left( {x,y,z}\right) |0 \leq  x,y,z \leq  {2008}}\right\rbrack \) be a set of points in a 3-D space.
 If the distance between two points is either \( 1,\sqrt{2},2 \) , we color the two points differently.
 How many colors are needed to color all points in \( V \) ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $a, b, c$ be three distinct positive integers. Define $S(a, b, c)$ as the set of all rational roots of $px^2 + qx + r = 0$ for every permutation $(p, q, r)$ of $(a, b, c)$. Determine the maximum number of elements in $S(a, b, c)$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of \(\alpha\) such that the roots of the quintic equation \(x^5 + 5\lambda x^4 -x^3 + (\lambda \alpha -4)x^2 - (8\lambda +3)x + \lambda\alpha - 2 = 0\) are independent of the parameter \(\lambda\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\triangle{ABC}$ be an isoceles triangle with $\angle A=90^{\circ}$. There exists a point $P$ inside $\triangle{ABC}$ such that $\angle PAB=\angle PBC=\angle PCA$ and $AP=10$. Find the area of $\triangle{ABC}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( a,b,c \) be real nonzero numbers such that \( a + b + c = {12} \) and
 \( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{abc} = 1 \)
 Compute the largest possible value of \( {abc} - \left( {a + {2b} - {3c}}\right) \) .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a tight upper bound for the function \( f(n) \) defined by the recurrence relation \( f(n) = f(\sqrt{n}) + \frac{1}{n} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the asymptotic upper bound on the number of elementary matrices required to decompose a matrix in $GL(n,K)$ as $n$ approaches infinity, where $K$ is an infinite field.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{-\infty}^{\infty} \frac{1}{z^{4}-1}\,dz \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{-1}^1 f(x) \,dx \) for the function \( f(x) \) defined as follows:

\[f(x) = \begin{cases} \frac{|\sin(x)|}{x}, & \text{if } x \neq 0 \\
1, & \text{if } x = 0 \end{cases} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the conic hull of the set \( T = \left\{(x_1,x_2) : (x_1 + 1)^2 + x_2^2 = 4 \right\} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that \( \int^{\infty}_{-\infty}e^{tx}f(x)dx = \sin^{-1}\left(t-\sqrt{\frac{1}{2}}\right) \), find the value of \( \left(\int^{\infty}_{-\infty}xf(x)dx\right)^2 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the value of \( x \) such that the matrix \( M_x = \begin{pmatrix} 1 & x+1 & 0 & 0 \\ 0 & x & x-1 & 0 \\ 0 & x-1 & x & 0 \\ 1 & 0 & x+1 & x \end{pmatrix} \) is diagonalizable.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a sample $X_1, X_2, \ldots, X_{15}$ from a distribution with probability density $\exp(-(x-\theta))$ for $x \ge \theta$, we test $H_0: \theta=0$ against $H_1: \theta>0$. The test statistic is $T = \min\{x_1, x_2, \ldots, x_{15}\}$. If the observed value of $T$ is $t=0.1$, calculate the p-value of this test.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the line integral \(\oint_C \vec{F} \cdot d\vec{R}\) for the vector field \(\vec{F}=(ye^x, x^2+e^x, z^2e^z)\) and the curve \(C\) defined by \(\vec{r}(t)=(1+\cos t, 1+\sin t, 1-\cos t-\sin t)\), where \(0\leq t\leq 2\pi\). Given that the curl of \(\vec{F}\) is \((0, 0, 2x)\), apply Stokes' theorem to solve this problem.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the Lebesgue measure of the set of games that have a finite and odd number of Nash equilibria, given a fixed finite player set and finite action sets for these players.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of \(a\) in the interval \((2, \infty)\) such that \[ \int_{0}^{1}\frac{x^8-x^2}{x^{12}+ax^6+1}dx=-\frac{\pi}{24}. \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Start by writing the integers $1, 2, 4, 6$ on the blackboard. At each step, write the smallest positive integer $n$ that satisfies both of the following properties on the board.
$n$ is larger than any integer on the board currently.
$n$ cannot be written as the sum of $2$ distinct integers on the board.
Find the $100$-th integer that you write on the board. Recall that at the beginning, there are already $4$ integers on the board.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the functional equation \( f(x) = 3f(x+1) - 3f(x+2) \) and the condition \( f(3) = 3^{1000} \), find the value of \( f(2019) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A help center receives students according to a Poisson process with rate $r$. When there are $n \ge 1$ students in the center, the first one to leave does so at a random exponential time with rate $2r$. Calculate the expected time until there are two students in the center, starting from a state with no students.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_\Gamma z^{\frac{1}{2}} \, dz \) for the contour \( \Gamma \) in the right half-plane from \( z = -3i \) to \( z = 3i \), choosing appropriate branches of \( \log(z) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the convergence of the complex series \(\sum_{n=1}^\infty\dfrac{n!}{n^n}z^n\) using the Ratio test.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the differential equation model for population growth given by:

$$ \begin{cases} \dfrac{\mathrm{d}N}{\mathrm{d}t} = rN(t) - r\dfrac{N^2(t)}{kt + b} \\ N(0) = N_0, \end{cases} $$

where $k>0$ and $b<0$. Determine the value of:

$$ \lim_{{t \to \frac{b}{k}}}{N'(t)} $$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a differentiable function that serves as a lower bound for \( \min(a,b) \) given \( a, b \geq 0 \) and \( a + b = c \), where \( c \) is a constant.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two chess players, $P_1$ and $P_2$, are competing in the final of a championship. The series consists of matches, and the winner is the one who has a two-game lead over the other and wins at least 6 games. If $P_2$ wins the first 4 matches, what is the probability that $P_1$ will win the championship? Express your answer as a fraction.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the maximum number of acute angles possible in a non-convex polygon with $n$ sides.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of $s$ for which the integral \[ \int_0^\infty \frac{\mathrm{d} x}{x \cdot \ln x \cdot \ln^{(2)} x \cdot \ln^{(3)} x \cdots (\ln^{(k)} x)^s } \] converges, where $\ln^{(k)} x$ denotes the $k$-fold iterated logarithm of $x$. Provide a condition or range for $s$ that ensures convergence.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the nature of the singularity at the point \( z = \pi/2 \) for the function \( f(z) = \tan^2 z \). Specifically, identify if it is a regular point, an essential singularity, or a pole, and if it is a pole, state its degree.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim_{n \rightarrow \infty} n \sin (2\pi e n!). \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimum order of an ordinary differential equation whose fundamental set of solutions includes the functions $e^x$, $xe^x$, and $\cos 2x$. \( \boxed{\text{Order}} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the triple integral \( \iiint_S {z{\sqrt{x^2+y^2}}}\,dx\,dy\,dz \) over the solid region \( S \) bounded by the cylinder \( x^2+y^2=1 \), the plane \( z=0 \), and the plane \( z=x \) in the first octant (\( z\ge0 \)).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the injective dimension of \( \mathbb{Z}_n \) as a \( \mathbb{Z} \)-module.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the fundamental group of a Klein bottle with two points removed.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\mathbb{Z} * \mathbb{Z} * \mathbb{Z}}


Context:
Determine $d(A)$ for the set $A = \{f_n:[0,1] \to \mathbb{R} \mid f_n(x) = x^n, n \in \mathbb{N} \}$, where $E = ([0, 1], \mathbb{R})$ is the set of bounded functions equipped with the sup norm and the metric it creates.
Please reason step by step, and put your final answer within \boxed{}.


Context:
All the positive integers that are coprime to $2021$ are grouped in increasing order, such that the nth group contains $2n - 1$ numbers. Hence the first three groups are $\{1\}$, $\{2, 3, 4\}$, $\{5, 6, 7, 8, 9\}$. Suppose that $2022$ belongs to the $k$th group. Find $k$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $a_n$ be a sequence of real numbers such that the series $\sum |a_n|^2$ is convergent. Determine the range of values of $p$ for which the series $\sum |a_n|^p$ is also convergent.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are 2017 turtles in a room. Every second, two turtles are chosen uniformly at random and combined to form one super-turtle. After 2015 seconds, only two turtles remain. What is the probability that there is a turtle that has never been combined with another turtle? Express this probability as a fraction \( \frac{p}{q} \) in simplest form, where \( p \) and \( q \) are relatively prime positive integers. Find \( p + q \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A palindrome is a whole number that reads the same forwards and backwards. If one neglects the colon, certain times displayed on a digital watch are palindromes. Three examples are: $ \boxed{1:01} $, $ \boxed{12:21} $.How many times during a 12-hour period will be palindromes?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the area enclosed by the region defined by the equation: \[ \left\lfloor\frac{\left|3x + 4y\right|}{5}\right\rfloor + \left\lfloor\frac{\left|4x - 3y\right|}{5}\right\rfloor = 3 \] where \( \lfloor\cdot\rfloor \) denotes the greatest-integer function.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Use Stokes' Theorem to calculate \( \iint_S \text{curl} \mathbf{F} \cdot d\mathbf{S} \) for the surface \( S = \{(x,y,z): x^2+y^2+z^2=2; -1\leq z\leq 1\} \) and the vector field \( \mathbf{F}(x,y,z) = (-x-y+xz^2 , x+z , z+e^{x+y}) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate \( \iint_R (x^2+y^2) \, dx \, dy \), where \( R \) is the region in the first quadrant bounded by the curves \( xy=2, \, xy=4, \, x^2-y^2=3, \, x^2-y^2=5. \) Use the transformation \( u=xy \) and \( v=x^2-y^2 \) to simplify the integral. Express \( x^2+y^2 \) in terms of \( u \) and \( v \), and evaluate the integral over the transformed region. Provide the final numerical result.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are 2 boys for every 3 girls in Ms. Johnson's math class. If there are 30 students in her class, what percent of them are boys?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of values of \(x\) in the interval \((0, 20)\) such that \(\{x\} + \{x^2\} = 1\), where \(\{\}\) denotes the fractional part of a number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the surface integral \( \iint_Y F \cdot N\, dS \) using Gauss's theorem, where \( Y = (x-z)^2 + (y-z)^2 = 1 + z^2 \) for \( 0 \leq z \leq 1 \) and \( N \) is the outward-pointing normal vector. The vector field is given by \( F = (y, x, 1 + x^2z) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the sum of all possible surface area of a cube two of whose vertices are $(1,2,0)$ and $(3,3,2)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given \(\mathbf{m}=(\cos \theta) \mathbf{i}+(\sin \theta) \mathbf{j}\) and \(\mathbf{n}=(\sqrt{2}-\sin \theta) \mathbf{i}+(\cos \theta) \mathbf{j}\), where \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors along the \(x\)-axis and \(y\)-axis respectively, and \(\theta \in (\pi, 2\pi)\). If the magnitude of the vector \(\mathbf{m} + \mathbf{n}\) is \(|\mathbf{m} + \mathbf{n}| = \frac{8 \sqrt{2}}{5}\), find the value of \(5 \cos \left(\frac{\theta}{2} + \frac{\pi}{8}\right) + 5\).
Please reason step by step, and put your final answer within \boxed{}.


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


Context:
Determine the greatest value of \( b \) such that any real-valued function \( f \) satisfying the following properties must also satisfy \( f(1) < 5 \):

1) \( f \) is infinitely differentiable on the real numbers.
2) \( f(0) = 1, f'(0) = 1, \) and \( f''(0) = 2 \).
3) \( |f'''(x)| < b \) for all \( x \) in \([0,1]\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the function \( f(x) \) that satisfies the functional equation \( f(x+1) = x f(x) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A group of farmers produced fewer than 1000 apples. Each farmer, in turn, took either exactly one-half or exactly one-third of the apples remaining in the collection, without cutting any apples. After all farmers took their shares, the remaining apples were given to charity. What is the greatest number of farmers that could have participated in this apple sharing process?
Please reason step by step, and put your final answer within \boxed{}.


Context:
An unbiased die is cast seven independent times. Calculate the conditional probability that each side appears at least once, given that side 1 appears exactly twice.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Jacob rolls two fair six-sided dice. If the outcomes of these dice rolls are the same, he rolls a third fair six-sided die. Compute the probability that the sum of the outcomes of all the dice he rolls is even.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the value of \( n \) in the following limit without using L'Hopital's rule:

\[\lim_{x \to 0} \frac{(e^x+e^{-x})\cos(nx)}{e^x-1}\]

given that \(\lim_{x \to 0} f(x) = 2 - n\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest positive integer \( c \) such that for all non-negative integers \( n \), the inequality \( f(n) \leq cn^2 \) holds, where \( f(n) \) is defined recursively by:

- Base case: \( f(0) = 4 \).
- Recursive case: \( f(n+1) = 7f\left(\lfloor \frac{n+1}{3} \rfloor \right) + 5(n+1)^2 \) for \( n > 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $Y_1 < Y_2 < \ldots < Y_8$ are the order statistics of $n$ independent observations from a continuous distribution with a 70th percentile $\pi_{0.7} = 27.3$, find the probability $P(Y_7 < 27.3)$. Assume no specific distribution is given.
Please reason step by step, and put your final answer within \boxed{}.


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


Context:
Determine the smallest number of terms, \( n \), required to approximate the series \( \sum_{k=1}^{\infty} 28 k e^{-0.45 k^2} \) with an accuracy of \( 10^{-7} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many odd $3$-digit numbers have exactly four $1$’s in their binary (base $2$) representation? For example, $225_{10} = 11100001_2$ would be valid.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{n\to\infty}\left[\frac{\left(\frac{3n}{e}\right)^{3n}\cdot \sqrt{6\pi n}}{n^n\cdot \left(\frac{n}{e}\right)^{n}\cdot \sqrt{3\pi n}}\right]^{\frac{1}{n}}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the fractional change in the wavelength of the Lyman-alpha line (transition from n = 2 to n = 1) for a hydrogen atom when considering the reduced mass of the electron.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many ways are there to distribute 20 different presents to 4 different children such that no child receives exactly 6 presents?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( A = k[x,y] \) where \( k \) is an algebraically closed field, and let \( M = A/(xy) \) be an \( A \)-module. Calculate the support of \( M \), denoted as \( \text{Supp}(M) = \{ P \in \text{Spec}(A) : M_P \neq 0 \} \), where \( M_P = S^{-1}M \) and \( S = A \setminus P \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Identify a function that grows faster than $n^{kn}$ but slower than $2^{n^2}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the expression: \(\cos\left(\frac{\pi}{11}\right)\cos\left(\frac{2\pi}{11}\right)\cos\left(\frac{3\pi}{11}\right)\cdots\cos\left(\frac{10\pi}{11}\right) = -2^n \). Find the value of \(n\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the codimension of the variety defined by the zero set of the determinants of all $m \times m$ submatrices of a symbolic $m \times n$ matrix, where $m \leq n$. Express your answer in terms of $m$ and $n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a square matrix $A = [a_{ij}] \in \mathbb{R}^{n \times n}$ with entries $\pm 1$ and balanced columns (i.e., $\sum_{i=1}^n a_{ij}=0$ for each column $j$), determine the maximum possible spectral norm of $A$. Provide your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find all functions \( f: \mathbb{R} \to \mathbb{R} \) that satisfy the functional equation \((x-y)(f(x)+f(y))=(x+y)f(x-y)\), given that \(f(0)=0\) and \(f\) is an odd function.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the maximum value of \( C \) such that for any continuously differentiable function \( f \) on the interval \([0,1]\) with \( f(0) = f(1) = 0 \), the inequality \( \max_{x \in [0,1]} |f'(x)| \geq C \int_0^1 |f(x)| \, dx \) holds.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $A \in \mathbb{R}^{n \times p}$ and $B \in \mathbb{R}^{n \times d}$ be matrices such that both have a spectral norm (largest singular value) equal to 1. If we concatenate the columns of $A$ and $B$ to construct $M = [A|B]$, determine the spectral norm of $M$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of distinct partitions of the set \(\{1,2,3, \ldots , 100\}\) that satisfy the following conditions: (a) The partition consists of exactly three non-empty parts, and (b) the elements 1, 2, and 3 are each in different parts.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a real number \( \lambda \) such that \((A+B)^{-1} \leq (A+\lambda I)^{-1}\), where \(A\) and \(B\) are real symmetric non-singular matrices and \(A+B\) is non-singular.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the splitting field of the polynomial \(x^3 + x^2 + 1\) over the real numbers \(\Bbb R\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the covariance of the Gaussian random variables $x[n]$ and $y[n]$ in the AR(1) model, given that $w[n]$ and $z[n]$ are independent random variables.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of the expression \( \cot^{-1}(\tan 2x) + \cot^{-1}(-\tan 3x) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \(\mathbb{F}_9\) be the field with nine elements. Determine the number of triples \((a, b, c)\) of elements in \(\mathbb{F}_9\) such that \(a^2 + b^2 = c^2\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine all values of the parameter \(a\) for which the inequality \(3 - |x-a| > x^2\) is satisfied by at least one negative \(x\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the spectral radius of the complete tripartite graph $K_{4,4,12}$. The spectral radius is the largest eigenvalue of the graph's adjacency matrix.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the cardinality of the quotient ring \( \mathbb{Z}[x]/(x^2-3,2x+4) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two water particles fall freely in succession from a $300 \mathrm{~m}$ high cliff. The first one has already fallen $\frac{1}{1000} \mathrm{~mm}$ when the second one starts to fall.

How far apart will the two particles be at the moment when the first particle reaches the base of the cliff? (The result should be calculated to the nearest $\frac{1}{10} \mathrm{~mm}$. Air resistance, etc., are not to be considered.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider four positive integers \(a_1, a_2, a_3, a_4\), each less than 45. Determine the number of different permutations \((a_1, a_2, a_3, a_4)\) such that \(a_1 + a_2 + a_3 + a_4 < 90\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a matrix $A \in M_{2\times 2}(\mathbb{R})$ that satisfies the equation $A + A^{-1} = I_2$, calculate the sum $I_2 + A + A^2 + \cdots + A^{2019}$. Provide your answer as a single matrix expression.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the cardinality of the set $X$, where $X$ is the collection of all relations $R$ over the set of natural numbers $\mathbb{N}$ such that the transitive closure of $R$, denoted by $R^{*}$, equals $\mathbb{N\times\mathbb{N}}$. Provide your answer as a cardinal number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In an experiment where a fair coin is tossed an infinite number of times, determine the probability that the sequence of heads and tails will never repeat itself.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A company has a manager and $n \geq 3$ employees. The manager flips a coin $2n$ times: each time he flips heads, he gives one task to one of his employees, chosen randomly and uniformly. Let $X_1$ and $X_2$ be the number of tasks given to employee 1 and employee 2, respectively. Find the covariance $\text{Cov}(X_1, X_2)$. Provide your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the quadrilateral pyramid \(PABCD\), the lateral face \(PAD\) is an equilateral triangle and is perpendicular to the base \(ABCD\). The base \(ABCD\) is a rhombus with side length 2, and \(\angle BAD = 60^\circ\). Let \(N\) be the midpoint of \(PB\), and the cross-section \(DAN\) intersects \(PC\) at \(M\). Find the dihedral angle between the planes \(P-AD-M\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the orthogonal trajectories of the family of curves given by the equation \( y^2 = x^2(1 - cx) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the degree of the field extension \( \mathbb{Q}(\xi+\xi^5+\xi^8) / \mathbb{Q} \), where \( \xi = \cos \left(\frac{2\pi}{13}\right) + i\sin \left(\frac{2\pi}{13}\right) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{\epsilon\to 0^+} \int_{\Lambda/s}^\Lambda \int_{-\infty}^\infty \frac{i}{y^2-x^2+i\epsilon} \, dy \, dx$$ where \(s\) and \(\Lambda\) are positive constants.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of ways to tile a $3 \times n$ rectangle using $2 \times 1$ rectangles and $1 \times 1$ squares, given that no two $2 \times 1$ rectangles share a common point and each $2 \times 1$ rectangle is placed with its longer side parallel to the shorter side of the $3 \times n$ rectangle. Assume $n > 3$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a non-zero function $f(x)$ such that $\frac{d}{dx}f(x) = f(x-1)$. Provide the function $f(x)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the maximum possible order of an element in the matrix group \({\rm GL}_m(\mathbb{F}_p)\), where \(m\) and \(p\) are positive integers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the Euler characteristic of a surface formed by joining two projective planes, each with two punctures, along the punctures.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of zeros of the equation \( ze^{3-z} - 5^2 = 0 \) inside the unit disk.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the improper integral: $$\int_{0}^\infty \int_{-\infty}^{\infty} \mathrm{exp}[-((y^2+z^2)/2)]\,dy\,dz$$ using the transformation $y=r \cos \theta$ and $z=r \sin \theta$. What is the value of the integral?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In parallelogram \( ABCD \), point \( F \) is on side \( AB \) and point \( H \) is on the extension of side \( BC \) such that \( \frac{AB}{BF} = \frac{BC}{BH} = 5 \). Point \( G \) is chosen so that \( BFGH \) forms a parallelogram. If line \( GD \) intersects \( AC \) at point \( X \) and \( AC = 100 \), find the length of \( AX \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A teacher has \( n \) students and divides them into groups, assigning one student as president and another as vice president in each group. Determine the exponential generating function for the number of ways \( t_n \) this can be done.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the denominator of the coefficient of the integral of the polynomial \((x-\alpha)^5(x-\beta)^4\) with respect to \(x\), given that \(\beta > \alpha\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the double integral \(\displaystyle \iint_R (x^2+y^2) \, dx \, dy\), where \(R\) is the region in the first quadrant bounded by the curves \(xy=2\), \(xy=4\), \(x^2-y^2=3\), and \(x^2-y^2=5\). Use the transformation \(u=xy\) and \(v=x^2-y^2\) to express \(x^2+y^2\) in terms of \(u\) and \(v\), and evaluate the integral over the region \(1 \le u \le 3\), \(1 \le v \le 5\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A trapezoid has one angle measuring $60^{\circ}$. If a circle can be inscribed in the trapezoid and another circle can be circumscribed around it, find the ratio of the lengths of its bases.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of $n$ for which the parametric equations \( \begin{pmatrix} X \\ Y \end{pmatrix} = \begin{pmatrix} \sum_{i=1}^N \frac{\cos (it)}{i^n} \\ \sum_{i=1}^N \frac{\sin (it)}{i^n} \end{pmatrix} \) are divergent.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the base $b$ such that for any $\epsilon > 0$, the following asymptotic behavior holds for the function \( f_n(2,2) = \sum_{k=0}^n \binom{2n}{k} 2^k \):
\[ \frac{f_n(2,2)}{(b+\epsilon)^n} \to 0 \quad \text{and} \quad \frac{f_n(2,2)}{(b-\epsilon)^n} \to \infty \quad \text{as} \quad n \to \infty. \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the function \(f:\mathbb{R^3} \to \mathbb{R}\) defined by \(f(x, y, z)=\left(x^{2}+y^{2}+z^{2}+R^{2}-r^{2}\right)^2-4 R^{2}\left(x^{2}+y^{2}\right)\). Let \(Z=\{ (x,y,z) \in \mathbb{R^3} \mid f(x,y,z)=0\}\). For \(R=3\) and \(r=2\), find the surface area of \(Z\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimum value of the sum of the lengths of line segments connecting the midpoints of the hypotenuses of $n$ right triangles to a common point, given that the sum of the areas of the triangles is a constant.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $ P(x)$ be a nonzero polynomial such that, for all real numbers $ x$, $ P(x^2 - 1) = P(x)P(-x)$. Determine the maximum possible number of real roots of $ P(x)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the value of the supremum $A = \sup \limits_{f} \{m^*(f[0, 1])\}$, where $f$ are all functions satisfying the following conditions: $f: [0, 1] \rightarrow [0, 1]$; for all $x$, $f(x) - x \in \mathbb{Q}$; and for all $x, y$, if $x - y \in \mathbb{Q}$, then $f(x) = f(y)$. Find the value of $A$, given that $0 < A \leq 1$. \(m^*\) denotes the outer measure.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the system of differential equations \( \frac{dx}{dt} = -x - y \) and \( \frac{dy}{dt} = 2x - y \), suppose there is an error of magnitude \(10^{-4}\) in measuring the initial conditions \(x(0)\) and \(y(0)\). Determine the largest error that can occur when evaluating \(x(t)\) and \(y(t)\) for \(0 \leq t < \infty\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that \(X\) and \(Y\) are jointly continuous random variables with conditional density \(f_{X|Y}(x|y) = I_{(x,x+1)}(y)\) and marginal density \(f_{X}(x) = I_{(0,1)}(x)\), find the probability \(P[X + Y < 1]\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the value of the exterior $2$-form
$$\omega = (x_1 + x_2)e_1^* \wedge e_2^* + (x_2 + x_3)e_2^* \wedge e_3^* + \cdots + (x_{n-1} + x_n)e_{n-1}^* \wedge e_n^*$$
onumber
$$\text{on the pair of vectors } (x_1e_1 + \cdots + x_ne_n, e_1 + \cdots + e_n),$$
where $x_1, \ldots, x_n$ are variables.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Seventeen people are seated at the vertices of a large circular table, with a constant distance between each pair of neighbors. After an hour, they are rearranged randomly. Choose any two people. Let \( X \) be the positive difference between their distances before and after the rearrangement. Calculate the expected value of \( X \). Note that the maximum distance is 8.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a circle, there are \(a+b+c\) ducks, each choosing one of rock, paper, or scissors. Specifically, \(a\) ducks choose rock, \(b\) ducks choose paper, and \(c\) ducks choose scissors. A move occurs when:
- A rock duck is behind a scissors duck, and they switch places.
- A paper duck is behind a rock duck, and they switch places.
- A scissors duck is behind a paper duck, and they switch places.

Find the maximum number of moves that can occur, expressed in terms of \(a\), \(b\), and \(c\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{n\to\infty} 1+\sqrt{2+(3+(4+\dots+(n)^{\frac{1}{n}})^{\frac{1}{4}})^{\frac{1}{3}}}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the integral \( \iint_R \sin\left(\frac{y-x}{y+x}\right)\, dy\, dx \) using the substitution \( x = u+v, \ y = u-v \). The region \( R \) is defined by \( \{(x,y): x+y \leq \pi, \ y \geq 0, \ x \geq 0\} \). Determine the new bounds for \( u \) and \( v \) after the substitution.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the surface integral \( \int_{S}\nabla\times\vec{F}\cdot{\rm d}\vec{S} \) using Stokes' Theorem, where \( \vec{F}(x,y,z) = y^2\vec{i} + 2xy\vec{j} + xz^2\vec{k} \) and \( S \) is the surface of the paraboloid \( z = x^2 + y^2 \) bounded by the planes \( x=0, y=0 \), and \( z=1 \) in the first quadrant, with the surface pointing upward.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the equation \(\frac{d}{114}=\left(1-\frac{1}{2^{2}}\right)\left(1-\frac{1}{3^{2}}\right) \cdots\left(1-\frac{1}{c^{2}}\right)\), find the value of \(d\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose $a$ is a 28-digit palindromic number. Given that $a$ is a multiple of $13$ and all the digits except the 13th, 14th, 15th, and 16th are $1$. Let $A, B, C, D$ be the 13th, 14th, 15th, and 16th digits respectively. Find the minimum value of $A+B+C+D$. \( \boxed{} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose that the bivariate random variable \((X,Y)\) has the joint density function:
\[
f_{X,Y}(x,y)=\left\{
 \begin{array}{ll}
  1  & \mbox{when } y > 0 , x+y<1 \text{ and } y<x+1 \\
  0 & \mbox{otherwise }
 \end{array}
\right.
\]
Find \(\operatorname{Cov}(X,Y)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of ways to rearrange the sequence "1234123567" such that no two identical numbers are adjacent to each other.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The area of trapezoid \(ABCD\) is \(S\), and the ratio of the bases \(\frac{AD}{BC} = 3\). On a line intersecting the extension of the base \(AD\) beyond point \(D\), there is a segment \(EF\), such that \(AE \parallel DF\), \(BE \parallel CF\), and \(\frac{AE}{DF} = \frac{CF}{BE} = 2\). Find the area of triangle \(EFD\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the size of the set generated by an unrestricted number of operations on the elements of the set \( F = \{c, x, \exp(x), \ln(x) \} \), where \( c \) is a constant term. The operations allowed are addition \( f+g \), multiplication \( f\cdot g \), division \( f/g \), and composition \( f\circ g \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the cumulative distribution function (CDF) of the standard normal distribution at 0.5365, i.e., find \( N(0.5365) \), where \( N(x) \) denotes the CDF of the standard normal random variable.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the limit: \[ \lim_{n \rightarrow \infty} \int_0^{1} \min\left( \frac{e^{-nx^{2}}}{\sqrt{x}}, n \right) \, d\lambda \] where \( \lambda \) is the Lebesgue measure on the interval \([0,1]\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( B(t) \) be a standard Brownian motion. Evaluate the conditional probability \( P(B(3) \leq 3 \mid B(1) = 1) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \(\alpha =2^{1/5} \) and \(\zeta=e^{2i\pi/5} \). Let \(K=\mathbb {Q}(\alpha\zeta) \). How many field automorphisms does \(K\) have?
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the largest number of solid $2\text{ in } \times 2\text{ in } \times 1\text{ in}$ blocks that can fit in a $3\text{ in x } 2\text{ in x } 3\text{ in}$ box?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^n \left(e^{(1+\frac{k}{n})^2} - \frac{3e^{(1 + \frac{3k}{n})}}{2\sqrt{1 + \frac{3k}{n}}}\right). \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the cohomology groups $H^q(G, M)$ for a free group $G$ with $n$ free generators and a trivial $G$-module $M$, where $q \geq 3$. Provide your answer in terms of $n$ and $M$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of different ways to color a 5×5 chessboard using 5 distinct colors such that each row contains all 5 colors and no two adjacent squares have the same color.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral using contour integration: $$\int_0^{2\pi}e^{\cos\theta} \sin(n\theta-\sin\theta)~d\theta$$ where $n\in\mathbb{Z}$. Provide your answer in terms of $n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the number of unique colorings of a regular hexagon where each vertex is either Black or White and each edge is either Red, Green, or Blue. The hexagon is acted upon by the dihedral group $D_6$. Use Burnside's Lemma to determine the number of distinct colorings.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $f(x)$ and $g(x)$ be periodic functions with periods 5 and 10, respectively, such that \( \lim_{x\to0}\frac{f(x)}{x}=\lim_{x\to0}\frac{g(x)}{x}=k \) where \( k>0 \). Determine the value of:
\[ \lim_{n\to\infty}\frac{f(5(4+\sqrt{15})^n)}{g(10(4+\sqrt{14})^n)}. \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the paraboloid $P$ defined by $P = \{(x, y, z) \in \mathbb{R}^3: z = 4 - x^2 - y^2, x^2 + y^2 < 4\}$. Find the unit normal field of $P$ and compute the integral of the vector field $F(x, y, z) = (x, y, xy + z)$ over $(P, \nu)$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{n\to\infty}n^2\left(\sin(2\pi en!)-\frac{2\pi}{n}\right)$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Points \(A_1, B_1, C_1,\) and \(D_1\) are taken on the planes of the faces of the tetrahedron \(ABCD\) such that the lines \(AA_1, BB_1, CC_1,\) and \(DD_1\) are parallel. Determine the ratio of the volumes of tetrahedrons \(ABCD\) and \(A_1B_1C_1D_1\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the volume of the set \(\Omega = \bigcup_{t \in [0,1]} T(t)\), where \(T(t)\) is the triangle formed by the points \(A(t) = (t,t^3,t)\), \(B(t) = (t,t,t)\), and \(C(t) = (0,2t,t)\). Assume \(T(t)\) is empty if the points are collinear.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A supermarket has four entrances. The arrivals at each entrance follow Poisson processes with rates \(\lambda_1 = 120, \lambda_2 = 100, \lambda_3 = 150,\) and \(\lambda_4 = 180\) customers per hour. 40% of all customers are children. The probability that a child customer makes a purchase is 0.2, while the probability that an adult customer makes a purchase is 0.4. Calculate the probability that the fifth child customer to make a purchase arrives within the first 20 minutes.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle ABC, segments BD, DF, and FE divide the triangle into four equal-area triangles. Points E and D are on side CA, and point F is on side BC. Given that BF = 2DE, find the ratio of AC to BC.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of $x$ for which the series \( \sum_{n=1}^{\infty}\frac{\sin (nx)}{n} \) converges pointwise.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral: $$\int_{0}^{1} \left((1-x^7)^{\frac{1}{3}}-(1-x^3)^{\frac{1}{7}}\right)\,dx$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the input signal $x(t) = \cos(t)$ and the impulse response $h(t) = u(t)$, find the output $y(t)$ using the Fourier Transform.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are 2011 positive numbers such that both their sum and the sum of their reciprocals equal 2012. Let \(x\) be one of these numbers. Determine the maximum value of \(x + \frac{1}{x}.\)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the sum \( \sum\limits_{i,j,k=0}^n{n \choose i+j}{n \choose i+k}{n \choose k+j} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In $\triangle BAC$, $\angle BAC = 40^\circ$, $AB = 10$, and $AC = 6$. Points $D$ and $E$ lie on $\overline{AB}$ and $\overline{AC}$ respectively. What is the minimum possible value of $BE + DE + CD$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the value of \( c \in \mathbb{R}^+ \) for which the Banach fixed-point theorem is satisfied for the function \( g(x_1, x_2) = \frac{1}{6} \begin{pmatrix} x_1x_2+cx_2-1 \\ x_1^2-x_2+1 \end{pmatrix} \) on the set \( M = [-1,1]\times [-1,1] \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two boys start moving from the same point \( \mathrm{A} \) on a circular track but in opposite directions. Their speeds are \( 5\mathrm{{ft}} \) ,per second
 and 9 ft. per second. If they start at the same time and finish when they first me at the point A again, then the number of time
 they meet, excluding the start and finish, is
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( f \) be a quadratic function that satisfies the following condition: For two distinct real numbers \( a \) and \( b \), if \( f(a) = f(b) \), then \( f(a^2 - 6b - 1) = f(b^2 + 8) \). Find the value of \( \frac{f(8) - f(2)}{f(2) - f(1)} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the range of values for $m$ such that the equation $|x^2-4|+x^2+mx=0$ has exactly 2 distinct roots in the interval $(1,3)$. Provide your answer as an interval for $m$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A number is additive if it has three digits, all of them are different and the
 sum of two of the digits is equal to the remaining one. (For example,
 \( {123}\left( {1 + 2 = 3}\right) ,{945}\left( {4 + 5 = 9}\right) ) \) . Find the sum of all additive number
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the vector spaces in \(\mathbb{R}^2\) with the basis \(B=\{ (2,1), (5,3)\}\) and \(C=\{ (1,1), (2,3)\}\). Let the linear transformation \(f: \mathbb{R}^2 \to \mathbb{R}^2\) be defined by \(f(x,y)=(x+y,-2x+y)\). The matrix representation of \(f\) with respect to the bases \(B\) and \(C\) is \(Cf_B=\begin{pmatrix} \alpha_{11} & \alpha_{12}\\ \alpha_{21} & \alpha_{22} \end{pmatrix}\). Compute the value of \(\alpha_{12} + 2\alpha_{22}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A system consists of 7 components with independent, identically distributed lifetimes. The system will operate until any one of the components fails. If the lifetime \( X \) of each component has the probability density function:
\[
f(x) =
\begin{cases}
\frac{3}{x^4},  & \text{for } x > 1 \\
0, & \text{otherwise}
\end{cases}
\]
what is the expected lifetime until failure of the system?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the asymptotic complexity of the recurrence relation \(B(n) = 5\cdot B(n/3) + cn^2\) with the initial condition \(B(1) = c\). Express your answer in the form \(O(n^d)\) for some value of \(d\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Estimate the upper bound of the number of solutions \(N_{\sqrt{2}}(1)\) for the inequality:
\[ \left \lvert \sqrt{2} - \frac{p}{q} \right \rvert < \frac{1}{q^{3}} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the matrix \(A = \begin{bmatrix} 1 & 3 & 4\\ 3 & 6 & 9\\ 1 & 6 & 4 \end{bmatrix},\) where \(B\) is an invertible \(3\times 3\) matrix that satisfies \(A \cdot A^{T} \cdot A +3B^{-1} =0,\) determine the value of \(\det( \operatorname{adj} (A^{-1}(B^{-1}){2B^{T}})).\)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine all values of \(N\) such that for any arrangement of \(N\) cubes (using an infinite supply of white, blue, and red cubes), the final remaining cube's color is independent of the robot's starting position. We call such \(N\) lucky. Find all lucky \(N\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of $n$ such that $$(1+2i)^{4n}=(1-2i)^{4n}.$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
$ABCD$ is a regular tetrahedron with side length $1$. Find the area of the cross section of $ABCD$ cut by the plane that passes through the midpoints of $AB$, $AC$, and $CD$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a closed formula for the determinant of an \( n \times n \) matrix where the diagonal elements are 0, the last row is \([0, 0, \ldots, 1]\), the first row is \([1, 0, \ldots, 0]\), and all other elements are 0. For example, when \( n=2 \), the matrix is \( \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \), and when \( n=3 \), the matrix is \( \begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the maximum degree of a Lagrange polynomial required to approximate the function \( f(x) = \cos(x) \) on the interval \([-\frac{\pi}{2}, \frac{\pi}{2}]\) with an error of less than or equal to 0.01 using the Lagrange error bound formula.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the disc of convergence for the series \( \sum_{n=0}^\infty z^{n^3} \).
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\{ z \in \mathbb{C} \mid |z| < 1 \}}


Context:
Solve the differential equation \( y'=\frac{\sin x + e^x (\sin x + \cos x) + x\cos x - y(1 + \cos x + e^x)}{e^x + \sin x + x - 2y} \) and find the curve that passes through the point (0,0).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of ring homomorphisms from \(\mathbb{Z}_{12}\) to \(\mathbb{Z}_{28}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the longest geometric progression that can be formed using prime numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of different ring homomorphisms from \(\mathbb{Z}[i]\) to \(\mathbb{Z}/(85)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the dimension of the algebraic set $X = V(y-x^2, y^2-y+z^2)$ over an algebraically closed field $F$. The dimension of an algebraic set is defined as the transcendence degree of its rational function field over $F$ if the set is irreducible, or the maximum dimension of its irreducible components if the set is reducible.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given two sequences encoded as follows: a=1, b=2, c=4, d=8, ..., y=2^24, z=2^25, and a '|' symbol indicating the end of a message, determine the time at which the sequences synchronize. For example, the sequence "*ea|babab" has an offset of 17 and a length of 25, while "*d|abaca" has an offset of 8 and a length of 17. Calculate the synchronization time for these sequences.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Minimize the value of the function \( P = a^2 + b^2 + c^2 + 2\sqrt{3abc} \) given that \( a, b, c \) are positive real numbers such that \( a + b + c = 1 \). Find the minimum value of \( P \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $z$ and $w$ be differentiable functions of $x$ and $y$ satisfying the following equations:

$$ xw^3+yz^2+z^3=-1$$

$$ zw^3-xz^3+y^2w=1$$

Find $\frac{\partial z}{\partial x}$ and evaluate it at the point $(x,y,z,w)=(1,-1,-1,1)$. Provide your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the dimension of the subspace $W$ of $\mathcal{L}(\mathbb{R}[x])$, where $W$ is generated by linear operators on $\mathbb{R}[x]$ that have no adjoint. The vector space $\mathbb{R}[x]$ consists of all polynomials with real coefficients, and the inner product is defined by $\langle f,g\rangle=\int_{0}^{1}f(t)g(t)dt$. What is the dimension of $W$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the contour integral \( \oint_{|z| = 1} \! dz \log\left(\frac{z - a}{z - b}\right) \) given \(|a| < 1\) and \(|b| < 1\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the limit \( \lim_{a\to \infty}\frac{1}{a}\int_1^a a^{\frac{1}{x}} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( \omega \) be a circle with center O and radius 10, and let H be a point such that \( OH = 6 \). A point P is called snug if, for all triangles ABC with circumcircle \( \omega \) and orthocenter \( H \), P lies on \( \triangle ABC \) or in the interior of \( \triangle ABC \). Find the area of the region consisting of all snug points.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the derivative \( \frac{d}{dx}\int_{0}^{x} \sin(1/t) \, dt \) at \( x = 0 \). Determine if the derivative exists at this point, and if so, find its value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the coefficient of $x^{60}$ in the expansion of the product $\prod_{m=1}^{11} (x^m - m)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $a$ and $b$ are roots of the equation $x^2 - 3cx - 8d = 0$, and $c$ and $d$ are roots of the equation $x^2 - 3ax - 8b = 0$, find the value of $a + b + c + d$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of $p + q$ if the following two lines intersect at an angle of $60^\circ$, given that $a = -(p/q)$, where $p$ and $q$ are coprime positive integers:

$$ \frac{x+2}{a} = \frac{y-1}{a-2} = \frac{z+3}{a+2} \\
 x = 0 \; \& \; y = z. $$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the ideals $I = (2, x)$ and $J = (3, x)$, determine whether the product $I \cdot J$ is equal to the ideal $(6, x)$ or if $(6, x)$ is contained in $I \cdot J$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $F$ be a field of characteristic $p > 0$ and let $f(t) = t^p - t - a \in F[t]$. Suppose also that $a \neq b^p - b$ for any $b \in F$. Determine the Galois group $Gal(K/F)$, where $K$ is a splitting field of $f$. \( \text{Provide your answer in terms of group structure.} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the value of $k \in \mathbb{R}$ such that the stochastic process \( Y_t = (W_t + t)e^{-W_t + kt}, \quad t \geq 0 \) has zero drift, where \( W_t \) is a Wiener process.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $A$ be an abelian group of order $p^5$, where $p$ is a prime number. Given that there exists a subgroup $N$ such that $A/N \simeq \mathbb{Z}/p^2\mathbb{Z}$, determine the number of possible groups $A$ that satisfy this condition.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A stick of unit length is cut into three pieces of lengths $X$, $Y$, and $Z$, with $X \le Y \le Z$. First, the stick is cut into two pieces at a randomly chosen point uniformly distributed along its length. Then, the smaller of these two pieces is cut again at a randomly chosen point uniformly distributed along its length. These two cuts are conducted independently. Find the covariance $\text{Cov}(X,Y)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many ways are there to place $31$ knights in the cells of an $8 \times  8$ unit grid so that no two attack one another?
(A knight attacks another knight if the distance between the centers of their cells is exactly $\sqrt5$.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Express the function $f(x) = g(x) \prod_{n=1}^x{n^k}$ in an explicit form without using the product operator, such that the resulting function $h(x) = \frac{f(x)}{g(x)}$ satisfies the conditions $h(0) = 0$, $h(1) = 1$, and for all natural numbers $n$, $h(n) = n^k \cdot h(n-1)$. What is the explicit form of $h(x)$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a 1-dimensional sufficient statistic for \( \theta \) given that \( X_1, X_2, \ldots, X_n \) is a sample from a normal distribution with probability density function:
\[ f(x\mid\theta)=\frac{1}{\sqrt{2\pi}}\exp{\left(-\frac{1}{2}(x-\theta)^2\right)} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a signal $s(t)$ sampled at $1000$ observations, where $h(k) = s(k\Delta t + t_0)$ for $k = 0, 1, \dots, 999$, with $\Delta t = 1/200$ and $t_0 = -2$ seconds, the discrete Fourier transform $|\hat{h}(m)|$ has its largest values at $m = 193$ and $m = 807$. What is the corresponding positive frequency in hertz for the signal $s$, rounded up to the nearest integer?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the partial derivative \( u_{xy} \) of the function \( u(x,y) = \frac{xy(x^2-y^2)}{x^2+y^2} \) at the point \((0,0)\), given that \( u(0,0) = 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $A$ be the set of nonnegative sequences $(a_n)$ such that $\sum_{n=1}^{\infty}a_n=1.$ Determine the range of the map $P:A\to \mathbb{R}$ defined by \[P((a_n))= \prod_{n=1}^\infty(1+a_n).\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the norm of the linear functional $L: \mathcal{H} \rightarrow \mathbb{F}$ defined by $L(\{\alpha_n\}) = \sum_{n=0}^\infty \alpha_n\lambda^n$, where $|\lambda|>1$ and $\mathcal{H} = l^2(\mathbb{N}\cup \{0\})$. \( \mathbb{F} \) denotes the field of scalars, which can be either the real numbers \( \mathbb{R} \) or the complex numbers \( \mathbb{C} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the volume of the region in \( \mathbb{R}^n \) defined by the inequalities \( x_1 + \cdots + x_n \geq a \) and \( x_i > 0 \) for all \( i \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a $5 \times 5$ matrix $B$ over the field $\mathbb{R}$, compute the dimension of the vector space of $5 \times 5$ matrices that commute with $B$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find all non-zero polynomials \( P(x) \) such that \( P\left(x^{2}-2 x\right) \equiv [P(x-2)]^{2} \) for all \( x \in \mathbf{R} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $b_1, b_2, b_3, \ldots$ be the sequence of positive integers $b$ such that $b$ and $a$ are relatively prime, and when $a$ is divided by $b$, the remainder is $4$ and the quotient is $7$. Find $b_{2014}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_0^\pi \sin(\sin(x))\sin(x)\, dx \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the residue of the function \( f(z) = \frac{1}{e^{e^{\frac{1}{z}}}} \) at the point \( z = 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The minimum value of the function \( y = |\cos x| + |\cos 2x| \) (for \( x \in \mathbf{R} \)) is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the range of values for \(a\) such that the function \(f(x) = (a+2)x^3 - 3ax^2 + 9ax - 1\) is decreasing for all real values of \(x\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Arnaldo and Bernaldo train for a marathon along a circular track, which has in its center a mast with a flag raised. Arnaldo runs faster than Bernaldo, so that every $30$ minutes of running, while Arnaldo gives $15$ laps on the track, Bernaldo can only give $10$ complete laps. Arnaldo and Bernaldo left at the same moment of the line and ran with constant velocities, both in the same direction. Between minute $1$ and minute $61$ of the race, how many times did Arnaldo, Bernaldo and the mast become collinear?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the expression \( \displaystyle \underset{(i \ne j \ne k)}{\sum_{i=0}^\infty \sum_{j=0}^\infty \sum_{k=0}^\infty \frac{1}{3^i 3^j 3^k}} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the expected first-passage time \( \mathbf{E}[T_{a,b}] \) for a standard Brownian motion \((B_t)_t\) with \(B_0=0\), where \(T_{a,b} = \inf\{t>0 : B_t \not\in(a,b)\}\) for \(a<0<b\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the first derivative of the function \( f(X) = \log \det(XX^TA^{-1}) \) with respect to the matrix \( X \), where \( X \in \mathbb{R}^{n \times r} \) and \( A \in \mathbb{R}^{n \times n} \) with \( A^{-1} \) being the inverse of \( A \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of Polish group topologies that can be placed on \( \text{PSL}_2(\mathbb{C}) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a linear transformation $T$ on a finite vector space $V$ such that $Ker(T) = Im(T)$, determine the minimal polynomial of $T$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given 11 sets \( M_{1}, M_{2}, \cdots, M_{11} \), each with exactly 5 elements, and for all \( i, j \) such that \( 1 \leq i < j \leq 11 \), \( M_{i} \cap M_{j} \neq \varnothing \), determine the minimum possible value of the maximum number of sets that have a non-empty intersection.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the initial value problem given by the differential equation \( x' = x + \mu(t + x^2) \) with the initial condition \( x(0) = 1 \). Find the sensitivity of the solution \( x(t, \mu) \) with respect to the parameter \( \mu \) by calculating \( \frac{\partial x}{\partial \mu} \bigg|_{\mu=0} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit of the sequence given by:
\[ \lim_{n \rightarrow \infty} \left( \frac{n \pi}{4} - \left( \frac{n^2}{n^2+1^2} + \frac{n^2}{n^2+2^2} + \cdots + \frac{n^2}{n^2+n^2} \right) \right). \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine strictly positive real numbers \( {a}_{1},{a}_{2},\ldots ,{a}_{n} \) if for any \( n \in  {N}^{ * } \) takes
 place equality: \( {a}_{1}^{2} + {a}_{2}^{2} + \ldots  + {a}_{n}^{2} = {a}_{1} + {a}_{2} + \ldots  + {a}_{n} + \frac{n\left( {{n}^{2} + {6n} + {11}}\right) }{3} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Segment $BD$ and $AE$ intersect at $C$, as shown, $AB=BC=CD=CE$, and $\angle A = \frac 52 \angle B$. What is the degree measure of $\angle D$?
[asy] unitsize(2cm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=4;  pair C=(0,0), Ep=dir(35), D=dir(-35), B=dir(145); pair A=intersectionpoints(Circle(B,1),C--(-1*Ep))[0]; pair[] ds={A,B,C,D,Ep};  dot(ds); draw(A--Ep--D--B--cycle);  label("$A$",A,SW); label("$B$",B,NW); label("$C$",C,N); label("$E$",Ep,E); label("$D$",D,E); [/asy]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of distinct Boolean functions \( f(x_1, x_2, \dots, x_n) \) that can be constructed using only AND and OR operators, without NOT, where the input is from \( \{0,1\}^n \) and the output is in \( \{0,1\} \). The function must output \(1\) for all inputs with all bits set to \(1\) and \(0\) for all inputs with all bits set to \(0\). Constant functions \(f \equiv 1\) and \(f \equiv 0\) are not valid. What is the number of such functions?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A puzzle club meets in a schoolroom every Friday after school. Last Friday, a member said, "I’ve hidden a list of numbers in this envelope that add up to the number of this room." A girl asked, "If you told us the number of numbers in the envelope and their product, would that be enough to determine the room number?" He replied, "No." She then said, "Well, at least I’ve worked out their product." What is the number of the school room they meet in?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \(a\) and \(b\) be elements of a group such that \(|a| = 2\), \(b \neq e\), and \(aba = b^2\). Determine the order of \(b\), denoted as \(|b|\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The area of a convex quadrilateral $ABCD$ is $18$. If $|AB|+|BD|+|DC|=12$, then what is $|AC|$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_0^{2\pi}\cos(\sin(e^{i \theta}))d\theta \) using complex analysis.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the vector field \(\vec{F} (x, y, z) = xy\hat{i} + (4x - yz)\hat{j} + (xy - z^{1/2}) \hat{k}\) and a circle \(C\) of radius \(R\) lying in the plane \(x + y + z = 5\), oriented counterclockwise when viewed from above, find the radius \(R\) if the line integral \(\int_C \vec{F} \cdot d\vec{r}\) equals \(\pi \sqrt{3}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Vasya, Petya, and Kolya live in the same house. Vasya and Petya go fishing at a lake, while Kolya stays home. Vasya starts walking home, and Kolya sets out on his bicycle to meet him. When Kolya meets Vasya, he gives him a ride home. Petya starts walking home at the same speed as Vasya when Kolya meets Vasya. Kolya then goes back to meet Petya and gives him a ride home. Petya's travel time from the lake to home is $\frac{4}{3}$ of Vasya's travel time. Calculate how many times slower Vasya's journey would be if he walked the entire way home.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\lbrace N(t)\rbrace_{t\geq 0}$ be a Poisson process with intensity $\lambda = 3$. Compute the probability $P\left[N(6) = 2 \,|\, N(8) = 4, N(3) = 1\right].$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $U$ be the vector subspace of $\mathbb{R}^5$ generated by $\{(1,3,-3,-1,-4),(1,4,-1,-2,-2),(2,9,0,-5,-2)\}$, and let $V$ be the vector subspace of $\mathbb{R}^5$ generated by $\{(1,6,2,-2,3),(2,8,-1,-6,-5),(1,3,-1,-5,-6)\}$. Determine the dimension of the vector space $U \cap V$. \( \text{Provide your answer as a single integer.} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of ordered pairs \((x, y)\) that satisfy the following system of equations: \((\cos^{-1} x)^2 + \sin^{-1}(y) = 1\) and \(\cos^{-1}(x) + (\sin^{-1}y)^2 = 1\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the eigenvalues of the linear map $g: V_3 \rightarrow V_3$ defined by $p(X) \mapsto p'(X)$, where $V_3$ is the vector space of polynomials with degree at most 3.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the smallest positive odd integer \( n \) such that the product of
 \( {2}^{\frac{1}{7}},{2}^{\frac{3}{7}},\ldots ,{2}^{\frac{2 - n + 1}{7}} \)
 is greater than 1000 ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the integral \( \int_{|z|=2} z^n(1 - z)^m\ dz \) for negative integers \( n \) and \( m \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the order of the distribution $S$ defined by $S(\psi)=\psi^{(n)}(y_0)$ for $\psi \in C_c^\infty (\mathbb R)$ and $n\in \mathbb N$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{0}^{2\pi } \frac{\sin^{2} (x) }{5+4\cos(x)}\,\mathrm{d}x \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function $f(x) = x^2 - 2ax + a(a+1)$ with the domain $f:[a, \infty) \to [a, \infty)$, one solution to the equation $f(x) = f^{-1}(x)$ is $5049$. Determine the other possible solution.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $2X^T = A^{-1}BA$, where $A$, $B$, and $X$ are $4 \times 4$ matrices, $\det(A) = 4$, and $\det(B^{-1}) = 3$, find $\det(X)$. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute \(d\alpha(V, W)\) at the point \((0,1)\) on \(\mathbb{R}^{2}\), where \(V = x \frac{\partial}{\partial y} - y \frac{\partial}{\partial x}\), \(W = y \frac{\partial}{\partial y}\), and the differential 1-form \(\alpha\) satisfies \(\alpha(V) = 2x - y^{2}\) and \(\alpha(W) = 2y\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of ways to choose $k$ numbers from the first $n$ natural numbers such that the longest consecutive sequence among the chosen numbers is exactly $m$ numbers long. For example, calculate the number of combinations when $k = 7$, $n = 14$, and $m = 3$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the diagram, a circle \(\omega\) with center \(P\) is tangent to line segment \(AB\) at point \(T\), and \(\angle APB = 90^\circ\). Point \(K\) is on circle \(\omega\), and a semicircle with diameter \(BK\) intersects \(PB\) at point \(Q\). Given that the radius \(R\) of the semicircle satisfies \(4R^2 - AT \cdot TB = 10\), and \(PQ = \sqrt{2}\), find the length of \(BQ\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let
 \( f\left( t\right)  = \frac{\left( {{10} + {9i}}\right) t - {10} + {9i}}{t + i}, \)
 where \( i = \sqrt{-1} \) . Let \( P = f\left( 0\right) ,Q = f\left( {2023}\right) \) ,and
 \( R = f\left( 1\right) \) . Determine \( {\sin }^{2}\left( {m\angle {PQR}}\right) \) .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Emma's calculator has ten buttons: one for each digit $1, 2, \ldots, 9$, and one marked "clear". When Emma presses a digit button, that digit is appended to the right of the display. When she presses the "clear" button, the display is erased. Starting with an empty display, Emma presses five buttons at random, with each button having an equal chance of being chosen. The expected value of the number displayed is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $100m + n$. (An empty display represents the number 0.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a set $\Omega$, let $A_1, A_2, \ldots, A_n$ be distinct subsets of $\Omega$. Determine the number of sets in the $\sigma$-field generated by $\{A_1, A_2, \ldots, A_n\}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a game involving two players, the following system of equations is given:

\[
\begin{aligned}
& a_{1} x_{1}+\ldots+a_{n} x_{n}=c \\
& b_{1} x_{1}+\ldots+b_{n} x_{n}=d
\end{aligned}
\]

Players take turns assigning arbitrary real numbers to the variables \(a_{i}, b_{i}, x_{i}\) (where \(i=1,2,\ldots,n\)) and the constants \(c, d\). In each turn, a player can choose any remaining variable to assign a value, but they can only assign 0 to a variable if it is the last unassigned variable in its equation.

Determine the value of \(n\) for which the first player can always ensure that both equations are satisfied, regardless of the second player's choices.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the operator norms \( ||S||, ||T||, ||ST||, \) and \( ||TS|| \) for the operators \( S, T: C[0,1] \rightarrow C[0,1] \) defined by \( S(x)(z) = z \int_{0}^{1} x(t) \, dt \) and \( T(x)(z) = z \cdot x(z) \). It is known that \( ||T|| = ||S|| = ||TS|| = 1 \). Determine \( ||ST|| \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The variables \( a,b,c,d, \) traverse, independently from each other, the set of
 positive real values. What are the values which the expression
 \( S = \frac{a}{a + b + d} + \frac{b}{a + b + c} + \frac{c}{b + c + d} + \frac{d}{a + c + d} \)
 takes?
Please reason step by step, and put your final answer within \boxed{}.

\boxed{
The range of \( S \) is \( (1, 2) \).

}


Context:
Determine the rank of the linear transformation $A$ defined by left multiplication by a matrix $P$ of rank $m$ on the space of all linear transformations on an $n$-dimensional vector space $V$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the limit: \[ \lim_{n \to \infty} \sqrt[n]{|\sin n|} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Assuming \(g(x) = \int \frac{x^2 - 12}{(x^2 - 6x + k)^2}dx, \quad k \in \mathbb{N},\) where \(g(x)\) is a rational function, find the sum of all possible values of \(k\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit \( \lim_{n \to \infty} \left( \sqrt{|B_{2n}|} - \sqrt{|B_{2n+2}|} \right) \), where \( B_{2n} \) and \( B_{2n+2} \) are consecutive even Bernoulli numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the limit of the sequence \(b_n = n\left(\sqrt[n]{eb}-\sqrt[n]{b}\right)\), where \(b>0\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the surface integral \( \int_{\tau} z \, d\tau \) over the surface \( S: x^2+y^2+z^2=a^2 \) with \( z \geq 0 \) using Gauss's Divergence Theorem.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \frac{2}{\pi i}\int_C \left(ze^{1/z}+\tan\left(\frac{z}{2}\right)+\frac{1}{(z-1)(z-3)^2}\right)dz \), where \( C \) is the positively oriented circle of radius 2 centered at the origin in the complex plane.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the contour integral \( \oint_C (4z - 1) \, dz \) along the quarter-circle \(|z| = 1\) from the point \(z_1 = 0\) to \(z_2 = 1\) in the first quadrant.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the circle \(x^2 + y^2 + 2gx + 2fy + c = 0\) intersects the circle \(x^2 + y^2 - 5 = 0\) at the extremities of its diameter, and the centers of the circles \(x^2 + y^2 - 8x - 6y + 10 = 0\) and \(x^2 + y^2 - 4x + 2y - 2 = 0\) lie on the first circle, find the value of \(c + g + f\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the smallest value of the constant $p$ such that the series \( \sum_{n=1}^{\infty}\frac{2^{np}(n!)^2}{(2n)!n^{n}} \) is convergent.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the solutions to $z^3 = -8.$  Enter the solutions, separated by commas.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \iint_{D} \frac{\mathrm{d}x\,\mathrm{d}y}{\sqrt{1 - x^{2} - y^{2}}} \), where \( D = \{(x,y) \mid x^{2} + 4y^{2} \leq 1\} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
If \( 1 < {k}_{1} < {k}_{2} < \ldots  < {k}_{n} \) and \( {a}_{1},{a}_{2},\ldots ,{a}_{n} \) are integers such that for every integer \( I \)
 \( {k}_{i} \mid  N - {a}_{i} \) for some \( 1 \leq  i \leq  n \) ,find the smallest possible value of \( n \) .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the Krull dimension of the localization $S^{-1}A$, where $A$ is a commutative reduced ring and $S$ is the set of all non-zerodivisors in $A$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of distinct parametrizations for the surface defined by the equation \(|x|+|y|+|z|=1\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Assume we have a problem $R \in \mathit{NP}$ that can be solved with $M(x,y)$ in $O(n^3)$ time, where $y$ is additional information no longer than $3\log n$ bits. For what function $f(n)$ can we say that $R \in \mathit{TIME}(f(n))$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a game, Sheldon picks three non-zero real numbers, and Leonard arranges them as the coefficients of a quadratic equation \( ax^2 + bx + c = 0 \). Sheldon wins if the resulting equation has two distinct rational solutions; otherwise, Leonard wins. What is the maximum probability that Sheldon wins the game?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{-\tfrac\pi2}^{\tfrac\pi2} \ln(\cos \tau) \cos \tau \, d\tau \) using the beta function.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the norm of the linear functional \( l(x) = x(0) + \int_0^1 (2t-1)x(t) \, dt \) on the space of continuous functions \( C[0,1] \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the constant $c$ to which the sequence $\frac{1}{n} \sum_{i=1}^n X_i$ converges almost surely as $n$ approaches infinity, given a sequence of independent random variables $\{X_i, i\geq1\}$ with probabilities $\Bbb P(X_i=i^2 -1)=\frac{1}{i^2}$ and $\Bbb P(X_i=-1)=1-\frac{1}{i^2}$. \(X_i\) are independent random variables.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the present value of a 10-year deferred life annuity-due of one dollar per annum at Kimberly's issue age, given an interest rate of 3%. Use the formula for a deferred annuity and provide the final present value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( P(x) \) be a polynomial in \( \mathbb{R}[x] \) with a leading coefficient of 1. Determine the leading coefficient of the polynomial \( P(2x+1) - 2P(x) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Equilateral \( \bigtriangleup  {ABC} \) has side length600. Points \( P \) and \( Q \) lie outside of the plane of \( \bigtriangleup  {ABC} \) and are on the opposite sides o
 the plane. Furthermore, \( {PA} = {PB} = {PC} \) , and \( {QA} = {QB} = {QC} \) , and the planes of \( \bigtriangleup  {PAB} \) and \( \bigtriangleup  {QAB} \) form a \( {120}^{ \circ  } \)
 dihedral angle (The angle between the two planes). There is a point \( O \) whose distance from each of \( A,B,C,P \) and \( Q \) is \( d \) .
 Find \( d \) .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_0^{+\infty}\cos 2x\prod_{n=1}^{\infty}\cos\frac{x}{n}dx \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In an acute angle triangle $ABC,$ let $D$ be the foot of the altitude from $A,$ and $E$ be the midpoint of $BC.$ Let $F$ be the midpoint of $AC.$ Suppose $\angle{BAE}=40^o. $ If $\angle{DAE}=\angle{DFE},$ What is the magnitude of $\angle{ADF}$ in degrees $?$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the dimension of the complex vector space generated by the set \( E_n = \{ e^{2 \pi i k x} \mid -n \leq k \leq n \} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
For positive integers \( m, n \geq 2 \), and a real number \( x \) such that \( 0 \leqslant x < n \), define the function:

\[ f(n, x) = (1 - \{x\}) \cdot \binom{n}{\lfloor x \rfloor} + \{x\} \cdot \binom{n}{\lfloor x \rfloor + 1} \]

where \( \lfloor x \rfloor \) is the greatest integer less than or equal to \( x \), and \( \{x\} = x - \lfloor x \rfloor \). Given that:

\[ f\left(m, \frac{1}{n}\right) + f\left(m, \frac{2}{n}\right) + \cdots + f\left(m, \frac{mn - 1}{n}\right) = 123, \]

find the value of:

\[ f\left(n, \frac{1}{m}\right) + f\left(n, \frac{2}{m}\right) + \cdots + f\left(n, \frac{mn - 1}{m}\right). \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two boats, Boat 1 and Boat 2, start simultaneously from docks A and B on a river, moving towards each other. Boat 1's speed in still water is twice that of Boat 2. Boat 1 leaves a floating cargo box on the water upon departure. After 20 minutes, the boats meet, and Boat 1 leaves another identical cargo box. Later, Boat 1 realizes it is missing cargo and turns around to search for it. When Boat 1 finds the second cargo box, Boat 2 encounters the first cargo box. How many minutes have passed since Boat 1 departed when it realizes its cargo is missing? Assume the time for turning around is negligible.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A manufacturer of car radios ships them to retailers in cartons of \( n \) radios. The profit per radio is \$59.50,\) less a shipping cost of \$25\) per carton, resulting in a profit of \( 59.5n - 25 \) per carton. To promote sales by assuring high quality, the manufacturer promises to pay the retailer \( 200X^2 \) if \( X \) radios in the carton are defective. Radios are produced independently, with a \( 5\% \) defect rate. Determine the number of radios that should be packed per carton to maximize the expected net profit per carton.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a triangle \(\triangle PQR\) with sides 16, 30, and 34, let the incircle touch the sides of \(\triangle PQR\) at points \(X, Y,\) and \(Z\). The ratio of the area of triangle \(\triangle XYZ\) to the area of triangle \(\triangle PQR\) can be expressed as \(\frac{m}{n}\), where \(m\) and \(n\) are relatively prime positive integers. Find \(m+n\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the improper integral \( \int_{0}^{1} \ln(e^{1/x}+1)\, dx \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the value of \( \limsup_{n \to \infty}(n!|a_n|)^{1/n} \) for an entire function \( f(z) \) that satisfies \( |f(z)| \le e^{(2\pi M+\epsilon)|z|} \) for any \( \epsilon > 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine $w^2+x^2+y^2+z^2$ if
\[ \begin{array}{l} \displaystyle \frac{x^2}{2^2-1}+\frac{y^2}{2^2-3^2}+\frac{z^2}{2^2-5^2}+\frac{w^2}{2^2-7^2}=1 \\ \displaystyle \frac{x^2}{4^2-1}+\frac{y^2}{4^2-3^2}+\frac{z^2}{4^2-5^2}+\frac{w^2}{4^2-7^2}=1 \\ \displaystyle \frac{x^2}{6^2-1}+\frac{y^2}{6^2-3^2}+\frac{z^2}{6^2-5^2}+\frac{w^2}{6^2-7^2}=1 \\ \displaystyle \frac{x^2}{8^2-1}+\frac{y^2}{8^2-3^2}+\frac{z^2}{8^2-5^2}+\frac{w^2}{8^2-7^2}=1 \\ \end{array}  \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Solve the equation: \(5x = 0.01^x\)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the set of positive numbers \( b \) such that \( b^x \geq 1+x \) for all real numbers \( x \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given $m,n\geq 2$.Paint each cell of a $m\times n$ board $S$ red or blue so that:for any two red cells in a row,one of the two columns they belong to is all red,and the other column has at least one blue cell in it.Find the number of ways to paint $S$ like this.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the expected value \( E\left(\frac{X_1}{X_2+1}\right) \) where \( X_1 \) and \( X_2 \) are independent random variables with the density function \( P[X_1=k]=P[X_2=k]=pq^k \) for \( k=0,1,2,\ldots \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two players are playing a game. One player thinks of a sequence of integers \((x_{1}, x_{2}, \ldots, x_{n})\), which are single-digit numbers, both positive and negative. The other player can ask for the value of the sum \(a_{1} x_{1}+\ldots+a_{n} x_{n}\), where \((a_{1}, \ldots, a_{n})\) is any chosen set of coefficients. What is the minimum number of questions needed for the guesser to determine the thought-of sequence?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of different quadratic functions of the form \( x \mapsto ax^2 + bx + c \pmod{12} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Frank alternates between flipping two weighted coins. The first coin has a \(\frac{2}{3}\) chance of landing heads and a \(\frac{1}{3}\) chance of landing tails. The second coin has a \(\frac{1}{4}\) chance of landing heads and a \(\frac{3}{4}\) chance of landing tails. Starting with the first coin, what is the probability that Frank sees two heads in a row before he sees two tails in a row?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider n balls, each labeled with a number from 1 to 365, placed in a row. A ball is selected at random. Let P be the probability that the number on the selected ball is the same as the number on any one of the balls placed to its left, with all balls to the left having distinct numbers. Determine the position K, counting from left to right, that maximizes P.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A consumer is searching for 1 unit of a product across stores with prices 1, 2, 3, ..., each occurring with probabilities 1/2, 1/4, 1/8, ..., 1/2^n respectively. The consumer incurs a search cost of 1 for each store visited. Determine the price at which the consumer should stop searching.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the possible range of values for \( a \) such that the area under the curve \( y = e^x + \cos(x) \ln(x + \sqrt{1+x^2}) \) from \( x = -a \) to \( x = a \) is less than \( \frac{3}{2} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$ \lim_{n\to\infty}\frac{3^{3n} \binom{2n}{n}^2}{\binom{3n}{n} \binom{6n}{3n}} $$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of orders on the rational numbers \( \mathbb{Q} \) that are compatible with the standard field operations. Is this number finite, infinite, or uncountably infinite?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the equations:
\[ 11\sin x + 7\sin y = 3(\sin z + 3\sin w) \]
\[ 11\cos x + 7\cos y = 3(\cos z + 3\cos w) \]
find the value of \( \frac{\cos(x-z)}{\cos(w-y)} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{r\rightarrow \infty}r\cdot \bigg(\frac{2}{\pi}\bigg)^{r+1}\cdot \int^{\frac{\pi}{2}}_{0}e^{r\log(x)}\sin(x)\,dx$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Apply one step of Richardson's Extrapolation to compute the improved value for $M$ given $N(3h) = 2.7172$ and $N(h) = 2.7182$. Assume $M = N(h) + Ah^{4/3} + Bh^{7/3}$ and use $h = 0.0001$ if needed.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine whether there exists a function \( f \) such that \( f(f(r)\sin \theta) = f(r)\cos \theta \) for all \( \theta \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the sum of the squares of the real solutions to the equation \( \sqrt[3]{5x - 2} = \frac{1}{5}(x^3 + 2) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a number \( x \) such that for all integers \( a \) in the set \( \{0, 1, 2, \ldots, 1926\} \), the congruence \( a^x \equiv a \pmod{1926} \) holds. Exclude \( x = 1 \) from consideration.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the isomorphism class of the quotient group $\mathbb{R}^+/U$, where $\mathbb{R}^+$ is the group of positive real numbers under multiplication and $U$ is the unit circle.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the contour integral \( \oint_{|z|=2} \frac{f'(z)}{f(z)} \, dz \), where \( f(z)=\frac{2 z^{4}-7 z^{3}-2 z-i}{(z-i)^{2}(3 z^{2}-z+9)^{2}} \). The function \( f(z) \) has no zeros or poles on the contour \( |z|=2 \), but may have poles inside the contour. Use the argument principle to determine the value of the integral.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int _0^\infty \dfrac{\ln x}{(x^2+1)^2}dx. \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the dimensions $n$ for which the existence of an $n \times n$ complex matrix $A$ with $\det(I-A^*A)<0$ implies that the operator norm of $A$ is greater than 1.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the Galois group \( \operatorname{Gal}(\mathbb{Q}(\sqrt{2},\sqrt{3})/\mathbb{Q}(\sqrt{2})) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
For how many distinct triangles $\triangle ABC$, with $AB = 2011$, are both $\cos(2\angle A + 3\angle C)$ and $\sin(2\angle B + \angle C)$ integers?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the Rao-Cramer lower bound for the parameter \( \theta \) of the distribution with the probability density function given by \[ f(x;\theta)=\frac{1}{\theta}x^{\frac{1-\theta}{\theta}} \] where \( 0<x<1 \) and \( 0<\theta<\infty \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A knight starts at position \((0, 0)\) on an infinite chessboard. In each move, the knight can move to one of the 8 possible positions: \((a \pm 1, b \pm 2)\) or \((a \pm 2, b \pm 1)\), each with equal probability. Calculate the probability that after 2023 moves, the knight is on a square \((a, b)\) such that \(a \equiv 4 \pmod{8}\) and \(b \equiv 5 \pmod{8}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The sequence $\{a_i\}_{i \ge 1}$ is defined by $a_1 = 1$ and\[ a_n = \lfloor a_{n-1} + \sqrt{a_{n-1}} \rfloor \]for all $n \ge 2$. Compute the eighth perfect square in the sequence.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a function \( g(x) \) that satisfies the following conditions: \( g''(0) > 0 \), \( g(x) \to -\infty \) as \( x \to \infty \) and \( x \to -\infty \), and \( g(x) = -g(-x) \) for all \( x \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a permutation \( \sigma \) such that \( \sigma(1,2)(3,4)\sigma^{-1} = (5,6)(3,1) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_C e^{\pi(x+iy)} \, dz \), where \( C \) is the square with vertices \( 0, 1, i, 1+i \) traversed once in the counterclockwise direction.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A beacon light is located 4 miles offshore, perpendicular to a straight shoreline, and rotates at 3 revolutions per minute. Determine the speed at which the spot of light moves along the shoreline when the beam forms a 60° angle with the shoreline.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the remainder when the smallest positive multiple of 15, consisting of exactly 3k digits with k 0's, k 3's, and k 8's, is divided by 11.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine all complex numbers $z \in \mathbb{C}$ for which the series \( \sum_{n=1}^\infty \frac{1}{n^3}(z+i)^{n^2} \) converges.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{ \{ z \in \mathbb{C} \mid |z + i| \leq 1 \} }


Context:
In the Cartesian coordinate system, there are 50 different parabolas of the form \( y = ax^2 + bx + c \) and another 50 different parabolas of the form \( x = a_1 y^2 + b_1 y + c_1 \). Determine the maximum number of regions these 100 parabolas can divide the coordinate plane into.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are 2022 distinct integer points on the plane. Let \( I \) be the number of pairs among these points with exactly1unit apart.
 Find the maximum possible value of \( I \) .
 (Note. An integer point is a point with integer coordinates.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the area of the region enclosed by the curve parameterized by \( g(t) = (t^4 - t^2, t^6 - t^2) \), where \( 0 \leq t \leq 1 \), using Green's Theorem.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $x=3^{a_1}5^{a_2}7^{a_3}$ and $y=3^{b_1}5^{b_2}7^{b_3}$ where $a_i,b_i$ are nonnegative integers such that $a_k + b_k \le 5$ for all $k$. Find the sum $$\sum \gcd(x,y),$$ where the sum is over all possible $x,y$. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \((x_n)\) be a sequence such that \(\lim\limits_{n\to\infty} x_n = \infty\) and \(\lim\limits_{n\to\infty} (x_{n+1} - x_n) = 0\). Define \(v_n = x_n - [x_n]\), where \([x_n]\) is the integer part of \(x_n\). Determine the set of adherence values of the sequence \((v_n)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the sum of all integer values of \(a\) in the interval \([-10, 10]\) such that for each \(a\), the inequality \(5 \leq x \leq 10\) implies \(a x + 3a^{2} - 12a + 12 > a^{2} \sqrt{x-1}\).
Please reason step by step, and put your final answer within \boxed{}.