Find all positive integers $n$ for which there exist positive integers $a, b,$ and $c$ that satisfy the following three conditions: $\bullet$ $a+b+c=n$ $\bullet$ $a$ is a divisor of $b$ and $b$ is a divisor of $c$ $\bullet$ $a < b < c$

The bisector of the diagonal $BD$ of a rectangle $ABCD$ (with $AB < BC$) intersects the lines $BC$ and $BA$ at points $E$ and $F$, respectively. The line passing through point $F$ and parallel to segment $AC$ intersects line $CD$ at point $G$. Prove that lines $EG$ and $AC$ are perpendicular

Find all the quaterns $(x,y,z,w)$ of real numbers (not necessarily distinct) that solve the following system of equations: $$x+y=z^2+w^2+6zw$$$$x+z=y^2+w^2+6yw$$$$x+w=y^2+z^2+6yz$$$$y+z=x^2+w^2+6xw$$$$y+w=x^2+z^2+6xz$$$$z+w=x^2+y^2+6xy$$

A whole number is written on each square of a board of $2019\times 2021$ squares. If the number written in each square is equal to the arithmetic mean of the numbers written in two of its neighboring squares, how many different numbers written on the blackboard can there be at most? Note: Two squares on the board are neighbors when they have a common side.

Prove that there are infinitely many positive integers $a, b$ and $c$ such that their greatest common divisor is $1$ (ie: $gcd(a, b, c) = 1$) and satisfy that: $$a^2=b^2+c^2+bc$$

Find all functions $f:\mathbb{R}\to \mathbb{R}$ such that for any real numbers $x$ and $y$ the following is true: $$x^2+y^2+2f(xy)=f(x+y)(f(x)+f(y))$$

In a country there are $2021$ cities. Each pair of cities is either linked by a single road or not linked at all. It is known that for any subset of $2019$ cities, the total number of roads between them is the same. If the total number of roads in that country is $A$, find all possible values of $A$.