IR-TST 2017 Problem 15

Find all functions $f: \mathbb {R}^+ \times \mathbb {R}^+ \to \mathbb {R}^+$ that satisfy the following conditions for all positive real numbers $x,y,z:$ $$f\left ( f(x,y),z \right )=x^2y^2f(x,z)$$ $$f\left ( x,1+f(x,y) \right ) \ge x^2 + xyf(x,x)$$ Proposed by Mojtaba Zare, Ali Daei Nabi