Question 122
Find all functions $f : \mathbb{R}^{+} \times \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ that satisfy the following conditions for all positive real numbers $x, y, z$
$$
\begin{gathered} f(f(x, y), z) = x^2 y^2 f(x, z), \\ f(x, 1 + f(x, y)) \geq x^2 + xyf(x, x). \end{gathered}
$$
$$
\begin{gathered} f(f(x, y), z) = x^2 y^2 f(x, z), \\ f(x, 1 + f(x, y)) \geq x^2 + xyf(x, x). \end{gathered}
$$