GR-HMS 2023

Find all quadruplets (x, y, z, w) of positive real numbers that satisfy the following system: $\begin{cases} \frac{xyz+1}{x+1}= \frac{yzw+1}{y+1}= \frac{zwx+1}{z+1}= \frac{wxy+1}{w+1} x+y+z+w= 48 \end{cases}$

Find all positive integers $N$ that are perfect squares and their decimal representation consists of $n$ digits equal to 2 and one digit equal to 5, where $n$ takes positive integer values.

A triangle $ABC$ with $AB>AC$ is given, $AD$ is the A-angle bisector with point $D$ on $BC$ and point $I$ is the incenter of triangle $ABC$ . Point M is the midpoint of segment $AD$ and point $F$ is the second intersection of $MB$ with the circumcirle of triangle $BIC$ . Prove that $AF\bot FC$ .

A class consists of 26 students with two students sitting on each desk. Suddenly, the students decide to change seats, such that every two students that were previously sitting together are now apart. Find the maximum value of positive integer $N$ such that, regardless of the students' sitting positions, at the end there is a set $S$ consisting of $N$ students satisfying the following property: every two of them have never been sitting together.