KZ-MO 2023

$a,b,c$ are positive real numbers such that $\max\{\frac{a(b+c)}{a^2+bc},\frac{b(c+a)}{b^2+ca},\frac{c(a+b)}{c^2+ab}\}\le \frac{5}{2}$ . Prove inequality $$\frac{a(b+c)}{a^2+bc}+\frac{b(c+a)}{b^2+ca}+\frac{c(a+b)}{c^2+ab}\le 3$$

Given $x,y>0$ such that $x^2y^2+2x^3y=1$ . Find the minimum value of sum $x+y$