PMO 2019

Find all functions $f : R \to R$ such that $f(2xy) + f(f(x + y)) = xf(y) + yf(x) + f(x + y)$ for all real numbers $x$ and $y$ .

Twelve students participated in a theater festival consisting of $n$ different performances. Suppose there were six students in each performance, and each pair of performances had at most two students in common. Determine the largest possible value of $n$ .

Find all triples $(a, b, c)$ of positive integers such that $a^2 + b^2 = n\cdot lcm(a, b) + n^2$ $b^2 + c^2 = n \cdot lcm(b, c) + n^2$ $c^2 + a^2 = n \cdot lcm(c, a) + n^2$ for some positive integer $n$ .

In acute triangle $ABC$ with $\angle BAC > \angle BCA$ , let $P$ be the point on side $BC$ such that $\angle PAB = \angle BCA$ . The circumcircle of triangle $AP B$ meets side $AC$ again at $Q$ . Point $D$ lies on segment $AP$ such that $\angle QDC = \angle CAP$ . Point $E$ lies on line $BD$ such that $CE = CD$ . The circumcircle of triangle $CQE$ meets segment $CD$ again at $F$ , and line $QF$ meets side $BC$ at $G$ . Show that $B, D, F$ , and $G$ are concyclic