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!