For every positive $a, b,c, d$ such that $a + c \le ac$ and $b + d \le bd$ prove that $\frac{ab}{a + b} +\frac{bc}{b + c} +\frac{cd}{c + d} +\frac{da}{d + a} \ge 4$
Source: Kyiv mathematical festival 2013
Tags: 4-variable inequality, inequalities
For every positive $a, b,c, d$ such that $a + c \le ac$ and $b + d \le bd$ prove that $\frac{ab}{a + b} +\frac{bc}{b + c} +\frac{cd}{c + d} +\frac{da}{d + a} \ge 4$