RIOPLATENSE 2019

Let $ABCDEF$ be a regular hexagon, in the sides $AB$ , $CD$ , $DE$ and $FA$ we choose four points $P,Q,R$ and $S$ respectively, such that $PQRS$ is a square. Prove that $PQ$ and $BC$ are parallel.

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $f(f(x)^2+f(y^2))=(x-y)f(x-f(y))$

In the dog dictionary the words are any sequence of letters $A$ and $U$ for example $AA$ , $UAU$ and $AUAU$ . For each word, your "profundity" will be the quantity of subwords we can obtain by the removal of some letters. For each positive integer $n$ , determine the largest "profundity" of word, in dog dictionary, can have with $n$ letters. Note: The word $AAUUA$ has "profundity" $14$ because your subwords are $A, U, AU, AA, UU, UA, AUU, UUA, AAU, AUA, AAA, AAUU, AAUA, AUUA$ .

Prove that there are infinite triples $(a,b,c)$ of positive integers $a,b,c>1$ , $gcd(a,b)=gcd(b,c)=gcd(c,a)=1$ such that $a+b+c$ divides $a^b+b^c+c^a$ .

Let $ABC$ be a triangle with $AB<AC$ and circuncircle $\omega$ . Let $M$ and $N$ be the midpoints of $AC$ and $AB$ respectively and $G$ is the centroid of $ABC$ . Let $P$ be the foot of perpendicular of $A$ to the line $BC$ , and the point $Q$ is the intersection of $GP$ and $\omega$ ( $Q,P,G$ are collinears in this order). The line $QM$ cuts $\omega$ in $M_1$ and the line $QN$ cuts $\omega$ in $N_1$ . If $K$ is the intersection of $BM_1$ and $CN_1$ prove that $P$ , $G$ and $K$ are collinears.

Let $\alpha>1$ be a real number such that the sequence $a_n=\alpha\lfloor \alpha^n\rfloor- \lfloor \alpha^{n+1}\rfloor$ , with $n\geq 1$ , is periodic, that is, there is a positive integer $p$ such that $a_{n+p}=a_n$ for all $n$ . Prove that $\alpha$ is an integer.