TSTST 2019 Problem 1

Find all binary operations $\diamond: \mathbb{R}_{>0} \times \mathbb{R}_{>0} \rightarrow \mathbb{R}_{>0}$ (meaning $\diamond$ takes pairs of positive real numbers to positive real numbers) such that for any real numbers $a, b, c>0$, - the equation $a \diamond(b \diamond c)=(a \diamond b) \cdot c$ holds; and - if $a \geq 1$ then $a \diamond a \geq 1$.