MEMO 2022

Let $\Omega$ be the circumcircle of a triangle $ABC$ with $\angle CAB = 90$ . The medians through $B$ and $C$ meet $\Omega$ again at $D$ and $E$ , respectively. The tangent to $\Omega$ at $D$ intersects the line $AC$ at $X$ and the tangent to $\Omega$ at $E$ intersects the line $AB$ at $Y$ . Prove that the line $XY$ is tangent to $\Omega$ .

Let $ABCD$ be a convex quadrilateral such that $AC = BD$ and the sides $AB$ and $CD$ are not parallel. Let $P$ be the intersection point of the diagonals $AC$ and $BD$ . Points $E$ and $F$ lie, respectively, on segments $BP$ and $AP$ such that $PC=PE$ and $PD=PF$ . Prove that the circumcircle of the triangle determined by the lines $AB, CD, EF$ is tangent to the circumcircle of the triangle $ABP$ .

Determine all functions $f : \mathbb {N} \rightarrow \mathbb {N}$ such that $f$ is increasing (not necessarily strictly) and the numbers $f(n)+n+1$ and $f(f(n))-f(n)$ are both perfect squares for every positive integer $n$ .

We call a positive integer $\textit{cheesy}$ if we can obtain the average of the digits in its decimal representation by putting a decimal separator after the leftmost digit. Prove that there are only finitely many $\textit{cheesy}$ numbers.