Hello everyone.

I would like to start 'marathon' of the exercises from different competition in the secondary school level. (ages 15-18)

let's start with inequality

Exercise 1.

Let $\displaystyle a,b$ be positive real numbers. Prove that

$\displaystyle \left(1+\frac{a}{b} \right)^m+\left(1+\frac{b}{a} \right)^m \ge 2^{m+1}$

where $\displaystyle m$ is natural.

Enjoy!