## Homogeneity of inequality

Let . Prove that:

**Solution**

Due to homogeneity we may assume . Thus there exist positive such that

Hence,

## Galois theory … of the Euler’s totient function

Let and let be an -th primitive root of unity. Prove that

where denotes the Euler’s totient function.

**Solution**

We have

where since is the minimal polynomial of over .

The result follows.

## Complex inequality

Prove that forall and it holds that

**Solution**

Well,

Done!

## Bound of derivative

Let . Let denote the -th derivative. Prove that

**Solution**

We note that

Hence,

## Searching for the …function

Find all functions such that and

**Solution**

First of all we note that

Thus,

Since the integrand is positive it only remains that

Setting and at the last equation we have that:

Since is positive we conclude that

which satisfies the given conditions.