PAMO 2025

Let $\triangle{ABC}$ be a triangle. A circle passing through $B$ and $C$ intersects $AB$ and $AC$ at points $D$ and $E$ respectively. The orthogonal projections of points $B$ and $E$ onto $CD$ are noted respectively $B'$ and $E'$ . The orthogonal projections of points $D$ and $C$ onto $BE$ are noted respectively $D'$ and $C'$ . Show that points $B'$ , $C'$ , $D'$ and $E'$ lie on the same circle.

Let $x_1,x_2,\ldots,x_{20}$ and $y_1,y_2,\ldots,y_{20}$ be 2 sequences, so that each term of each sequence is either $0$ or $1$ . How many couples $((x_i),(y_i))$ where $i=1,2,\ldots,20$ of such sequences exist such that $$x_1y_1+x_2y_2+\ldots+x_{20}y_{20}$$ is an even number ?

A positive integer $N$ with $m$ digits is called $a$ $repetitive$ $number$ $with$ $m$ $digits$ if the number formed by the last $m$ digits of $N^k$ is always $N$ for all positive integer $k$ . For example $5$ is a $repetitive$ $number$ $with$ $1$ $digit$ . Find all $repetitive$ $numbers$ $with$ $6$ $digits$ .

Find all integers $n$ such that it exists an integer $x$ such that : $$x^5-nx-(n+1)=0$$

Find all functions : $f : \mathbb{Z} \to \mathbb{Z}$ such that: $f(1)=1$ and : $$f(x+y)+f(x-y)=\max\{f(2x);2f(y)\}$$ for all integers $x$ and $y$ .