Question 4
Let $a,b,c$ be positive real numbers satisfying
$$16(a+b+c)\ge\frac1a+\frac1b+\frac1c.$$
Prove that
$$\frac1{(a+b+\sqrt{2(a+c)})^3}+\frac1{(b+c+\sqrt{2(b+a)})^3}+\frac1{(c+a+\sqrt{2(c+b)})^3}\le\frac89.$$
Determine when equality holds.
$$16(a+b+c)\ge\frac1a+\frac1b+\frac1c.$$
Prove that
$$\frac1{(a+b+\sqrt{2(a+c)})^3}+\frac1{(b+c+\sqrt{2(b+a)})^3}+\frac1{(c+a+\sqrt{2(c+b)})^3}\le\frac89.$$
Determine when equality holds.