HMIC 2023

Let $\mathbb{Q}^{+}$denote the set of positive rational numbers. Find, with proof, all functions $f: \mathbb{Q}^{+} \rightarrow \mathbb{Q}^{+}$ such that, for all positive rational numbers $x$ and $y$, we have

$$f(x)=f(x+y)+f\left(x+x^{2} f(y)\right)$$

A prime number $p$ is mundane if there exist positive integers $a$ and $b$ less than $\frac{p}{2}$ such that $\frac{a b-1}{p}$ is a positive integer. Find, with proof, all prime numbers that are not mundane.

Triangle $A B C$ has incircle $\omega$ and $A$-excircle $\omega_{A}$. Circle $\gamma_{B}$ passes through $B$ and is externally tangent to $\omega$ and $\omega_{A}$. Circle $\gamma_{C}$ passes through $C$ and is externally tangent to $\omega$ and $\omega_{A}$. If $\gamma_{B}$ intersects line $B C$ again at $D$, and $\gamma_{C}$ intersects line $B C$ again at $E$, prove that $B D=E C$.

Let $n>1$ be a positive integer. Claire writes $n$ distinct positive real numbers $x_{1}, x_{2}, \ldots, x_{n}$ in a row on a blackboard. In a move, William can erase a number $y$ and replace it with either $\frac{1}{y}$ or $y+1$ at the same location. His goal is to make a sequence of moves such that after he is done, the numbers are strictly increasing from left to right. (a) Prove that there exists a positive constant $A$, independent of $n$, such that William can always reach his goal in at most $A n \log n$ moves. (b) Prove that there exists a positive constant $B$, independent of $n$, such that Claire can choose the initial numbers such that William cannot attain his goal in less than $B n \log n$ moves.

Let $a_{1}, a_{2}, \ldots$ be an infinite sequence of positive integers such that, for all positive integers $m$ and $n$, we have that $a_{m+n}$ divides $a_{m} a_{n}-1$. Prove that there exists an integer $C$ such that, for all positive integers $k>C$, we have $a_{k}=1$.