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# Tag Archives: Inequality

## Root inequality

Let be three positive real numbers such that . Prove that

**Solution**

By AM – GM we have,

However,

Hence and the exercise is complete.

## Nested radical inequality

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**