PMO 2023

Find all ordered pairs $(a, b)$ of positive integers such that $a^2 + b^2 + 25 = 15ab$ and $a^2 + ab + b^2$ is prime.

Find all primes $p$ such that $\dfrac{2^{p+1}-4}{p}$ is a perfect square.

In $\triangle ABC$ , $AB > AC$ . Point $P$ is on line $BC$ such that $AP$ is tangent to its circumcircle. Let $M$ be the midpoint of $AB$ , and suppose the circumcircle of $\triangle PMA$ meets line $AC$ again at $N$ . Point $Q$ is the reflection of $P$ with respect to the midpoint of segment $BC$ . The line through $B$ parallel to $QN$ meets $PN$ at $D$ , and the line through $P$ parallel to $DM$ meets the circumcircle of $\triangle PMB$ again at $E$ . Show that the lines $PM$ , $BE$ , and $AC$ are concurrent.

In chess, a knight placed on a chess board can move by jumping to an adjacent square in one direction (up, down, left, or right) then jumping to the next two squares in a perpendicular direction. We then say that a square in a chess board can be attacked by a knight if the knight can end up on that square after a move. Thus, depending on where a knight is placed, it can attack as many as eight squares, or maybe even less.

In a $10 \times 10$ chess board, what is the maximum number of knights that can be placed such that each square on the board can be attacked by at most one knight?

Silverio is very happy for the 25th year of the PMO. In his jubilation, he ends up writing a finite sequence of As and Gs on a nearby blackboard. He then performs the following operation: if he finds at least one occurrence of the string "AG", he chooses one at random and replaces it with "GAAA". He performs this operation repeatedly until there is no more "AG" string on the blackboard. Show that for any initial sequence of As and Gs, Silverio will eventually be unable to continue doing the operation.

Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(2f(x)) = f(x - f(y)) + f(x) + y$$ for all $x, y \in \mathbb{R}$ .

A set of positive integers is said to be pilak if it can be partitioned into 2 disjoint subsets $F$ and $T$ , each with at least $2$ elements, such that the elements of $F$ are consecutive Fibonacci numbers, and the elements of $T$ are consecutive triangular numbers. Find all positive integers $n$ such that the set containing all the positive divisors of $n$ except $n$ itself is pilak.

Let $\mathcal{S}$ be the set of all points in the plane. Find all functions $f : \mathcal{S} \rightarrow \mathbb{R}$ such that for all nondegenerate triangles $ABC$ with orthocenter $H$ , if $f(A) \leq f(B) \leq f(C)$ , then $$f(A) + f(C) = f(B) + f(H).$$