Prove that for all positive real numbers $a,b,c,d$, $$\frac{2}{(a+b)(c+d)+(b+c)(a+d)} \leq \frac{1}{(a+c)(b+d)+4ac}+\frac{1}{(a+c)(b+d)+4bd}$$ and determine when equality occurs.
Source: JBMO Shortlist 2023, A1
Tags: inequalities, JBMO, JBMO Shortlist, algebra
Prove that for all positive real numbers $a,b,c,d$, $$\frac{2}{(a+b)(c+d)+(b+c)(a+d)} \leq \frac{1}{(a+c)(b+d)+4ac}+\frac{1}{(a+c)(b+d)+4bd}$$ and determine when equality occurs.