Let $n$ be a posititve integer. On a $n \times n$ grid there are $n^2$ unit squares and on these we color the sides with blue such that every unit square has exactly one side with blue. a) Find the maximun number of blue unit sides we can have on the $n \times n$ grid. b) Find the minimun number of blue unit sides we can have on the $n \times n$ grid.

Let $f: \mathbb Z^+ \to \mathbb Z$ be a function such that a) $f(p)=1$ for every prime $p$. b) $f(xy)=xf(y)+yf(x)$ for every pair of positive integers $x,y$ Find the least number $n \ge 2021$ such that $f(n)=n$

Let $p=ab+bc+ac$ be a prime number where $a,b,c$ are different two by two, show that $a^3,b^3,c^3$ gives different residues modulo $p$

On a isosceles triangle $\triangle ABC$ with $AB=BC$ let $K,M$ be the midpoints of $AB,AC$ respectivily. Let $(CKB)$ intersect $BM$ at $N \ne M$, the line through $N$ parallel to $AC$ intersects $(ABC)$ at $A_1,C_1$. Show that $\triangle A_1BC_1$ is equilateral.