NL-BXMO-TST 2020

For an integer $n \ge 3$ we consider a circle with $n$ points on it. We place a positive integer at each point, where the numbers are not necessary need to be different. Such placement of numbers is called stable as three numbers next to always have product $n$ each other. For how many values of $n$ with $3 \le n \le 2020$ is it possible to place numbers in a stable way?

In an acute-angled triangle $ABC, D$ is the foot of the altitude from $A$ . Let $D_1$ and $D_2$ be the symmetric points of $D$ wrt $AB$ and $AC$ , respectively. Let $E_1$ be the intersection of $BC$ and the line through $D_1$ parallel to $AB$ . Let $E_2$ be the intersection of $BC$ and the line through $D_2$ parallel to $AC$ . Prove that $D_1, D_2, E_1$ and $E_2$ on one circle whose center lies on the circumscribed circle of $\vartriangle ABC$ .

Find all functions $f: R \to R$ that satisfy $$f (x^2y) + 2f (y^2) =(x^2 + f (y)) \cdot f (y)$$ for all $x, y \in R$

Three different points $A,B$ and $C$ lie on a circle with center $M$ so that $| AB | = | BC |$ . Point $D$ is inside the circle in such a way that $\vartriangle BCD$ is equilateral. Let $F$ be the second intersection of $AD$ with the circle . Prove that $| F D | = | FM |$ .

A set S consisting of $2019$ (different) positive integers has the following property: the product of every 100 elements of $S$ is a divisor of the product of the remaining $1919$ elements. What is the maximum number of prime numbers that $S$ can contain?