Let $a, b$ be positive integers. Prove that if $a^b + b^a \equiv 3 \pmod{4}$, then either $a+1$ or $b+1$ can't be written as the sum of two integer squares. (Proposed by Orestis Lignos, Greece)

Let $f: \mathbb{N} \to \mathbb{N}$ be an arbitrary function. Prove that there exist two positive integers $x$ and $y$ which satisfy $f(x+y) \le f(2x+f(y))$. (Proposed by David Anghel, Romania)

Prove that there are infinitely many integers $k\geqslant 2024$ for which there exists a set $\{a_1,\ldots,a_k\}$ with the following properties: $a_1{}$ is a positive integer and $a_{i+1}=a_i+1$ for all $1\leqslant i<k,$ and $2(a_1\cdots a_{k-2}-1)^2$ is divisible by $2(a_1+\cdots+a_k)+a_1-a_1^2.$ (Proposed by Prajit Adhikari, Nepal)

Vlad draws 100 rays in the Euclidean plane. David then draws a line $\ell$ and pays Vlad one pound for each ray that $\ell$ intersects. Naturally, David wants to pay as little as possible. What is the largest amount of money that Vlad can get from David? Proposed by Vlad Spătaru

Let $ABC$ be an acute triangle so that $2BC = AB + AC,$ with incenter $I{}.$ Let $AI{}$ meet $BC{}$ at point $A'.{}$ The perpendicular bisector of $AA'{}$ meets $BI{}$ and $CI{}$ at points $B'{}$ and $C'{}$ respectively. Let $AB'{}$ intersect $(ABC)$ at $X{}$ and let $XI{}$ intersect $AC'{}$ at $X'{}.$ Prove that $2\angle XX'A'=\angle ABC.{}$ (Proposed by Kang Taeyoung, South Korea)