Alice is drawing a shape on a piece of paper. She starts by placing her pencil at the origin, and then draws line segments of length one, alternating between vertical and horizontal segments. Eventually, her pencil returns to the origin, forming a closed, non-self-intersecting shape. Show that the area of this shape is even if and only if its perimeter is a multiple of eight.

Show that $n!=a^{n-1}+b^{n-1}+c^{n-1}$ has only finitely many solutions in positive integers. Proposed by Dorlir Ahmeti, Albania

Let $ABC$ be a triangle with circumcircle $\omega$ and $D$ be any point on $\omega.$ Suppose that $P$ is the midpoint of chord $AD$ and points $X, Y$ are chosen on lines $AC, AB$ such that reflections of $B, C$ with respect to $AD$ lie on $XP, YP,$ respectively. If the circumcircle of triangle $AXY$ intersects $\omega$ at $I$ for the second time, prove that $\angle PID$ equals the angle formed by lines $AD$ and $BC.$ Proposed by tenplusten.

Let $ABCD$ be a parallelogram with $AC=BC.$ A point $P$ is chosen on the extension of ray $AB$ past $B.$ The circumcircle of $ACD$ meets the segment $PD$ again at $Q.$ The circumcircle of triangle $APQ$ meets the segment $PC$ at $R.$ Prove that lines $CD,AQ,BR$ are concurrent.

For each integer $n\ge 1,$ compute the smallest possible value of \[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\]over all permutations $(a_1,\dots,a_n)$ of $\{1,\dots,n\}.$ Proposed by Shahjalal Shohag, Bangladesh

Determine all integers $n\geqslant 2$ with the following property: every $n$ pairwise distinct integers whose sum is not divisible by $n$ can be arranged in some order $a_1,a_2,\ldots, a_n$ so that $n$ divides $1\cdot a_1+2\cdot a_2+\cdots+n\cdot a_n.$ Arsenii Nikolaiev, Anton Trygub, Oleksii Masalitin, and Fedir Yudin