Determine all $4$-tuples $(a,b, c, d)$ of positive real numbers satisfying $a + b +c + d = 1$ and $\max (\frac{a^2}{b},\frac{b^2}{a}) \cdot \max (\frac{c^2}{d},\frac{d^2}{c}) = (\min (a + b, c + d))^4$
Source: Dutch IMO TST2 2019 p2
Tags: algebra, system of equations, max and min, max, min
Determine all $4$-tuples $(a,b, c, d)$ of positive real numbers satisfying $a + b +c + d = 1$ and $\max (\frac{a^2}{b},\frac{b^2}{a}) \cdot \max (\frac{c^2}{d},\frac{d^2}{c}) = (\min (a + b, c + d))^4$