Let $x_0=2024^{2024}$ and $x_{n+1}=|x_n-\pi|$ for $n \ge 0$. Show that there exists a value of $n$ such that $x_{n+2}=x_n$.

We are given a unit square in the plane. A point $M$ in the plane is called median if there exists points $P$ and $Q$ on the boundary of the square such that $PQ$ has length one and $M$ is the midpoint of $PQ$. Determine the geometric locus of all median points.

A positive integer $n$ is called egyptian if there exists a strictly increasing sequence $0<a_1<a_2<\dots<a_k=n$ of integers with last term $n$ such that \[\frac{1}{a_1}+\frac{1}{a_2}+\dots+\frac{1}{a_k}=1.\](a) Determine if $n=72$ is egyptian. (b) Determine if $n=71$ is egyptian. (c) Determine if $n=72^{71}$ is egyptian.

Let $ABCD$ be a rectangle with $AB<BC$ and circumcircle $\Gamma$. Let $P$ be a point on the arc $BC$ (not containing $A$) and let $Q$ be a point on the arc $CD$ (not containing $A$) such that $BP=CQ$. The circle with diameter $AQ$ intersects $AP$ again in $S$. The perpendicular to $AQ$ through $B$ intersects $AP$ in $X$. (a) Show that $XS=PS$. (b) Show that $AX=DQ$.

A fortress is a finite collection of cells in an infinite square grid with the property that one can pass from any cell of the fortress to any other by a sequence of moves to a cell with a common boundary line (but it can have "holes"). The walls of a fortress are the unit segments between cells belonging to the fortress and cells not belonging to the fortress. The area $A$ of a fortress is the number of cells it consists of. The perimeter $P$ is the total length of its walls. Each cell of the fortress can contain a guard which can oversee the cells to the top, the bottom, the right and the left of this cell, up until the next wall (it also oversees its own cell). (a) Determine the smallest integer $k$ such that $k$ guards suffice to oversee all cells of any fortress of perimeter $P \le 2024$. (b) Determine the smallest integer $k$ such that $k$ guards suffice to oversee all cells of any fortress of area $A \le 2024$.

For each integer $n$, determine the smallest real number $M_n$ such that \[\frac{1}{a_1}+\frac{a_1}{a_2}+\frac{a_2}{a_3}+\dots+\frac{a_{n-1}}{a_n} \le M_n\]for any $n$-tuple $(a_1,a_2,\dots,a_n)$ of integers such that $1<a_1<a_2<\dots<a_n$.