Let . Prove that

Solution

The LHS is equal to which by AM – GM is less or equal to

where . Since it follows from Bernoulli inequality that .

## Root inequality

Let be positive real numbers such that . Prove that

Solution

Well if we apply AM-GM to we obtain

(1)

and similarly if we apply AM – GM to we obtain

(2)

We have successively,

## Inequality with roots

Let be positive real numbers. Prove that

Solution

We apply the AM – GM inequality, thus:

Hence it suffices to prove that which holds because it is equivalent to .

## Inequality in acute triangle

Let be an acute triangle. Prove the following inequality

Solution

The solution can be found at cut -the – knot

## Inequality in a triangle

Given a triangle let denote the median points of the sides respectively. Prove that