MN-MO 1999 Problem 2

Let $a,b,c$ be the real numbers with $a\ge\frac85b>0$ and $a\ge c>0$ . Prove the inequality $$\frac45\left(\frac1a+\frac1b\right)+\frac2c\ge\frac{27}2\cdot\frac1{a+b+c}.$$