## Integral and inequality

Let be a continuous function such that

(1)

Prove that

**Solution**

We note that

On the other hand by Cauchy – Schwartz we have that

The result follows.

## Determinant of block matrices

Given a real matrix show that .

**Solution**

We simply note that

The rest is … history!

## Limit of a sum

Let be a real sequence such that , , and . Prove that

**Solution**

**Lemma: ** Let be a bounded sequence of, say, complex numbers and let be another complex number.

where the latter means “convergence on a set of density ” i.e. there exists a subsequence such that and is dense i.e.

*Proof: The proof is omitted because it is too technical.*

Hence,

All we need is that the logarithm is continuous and that as well as are bounded sequences.

**Note:** We can simplify the conditions to and .

*The exercise along with the solution may be found on AoPS.com . The proof of the claim may also be found there.*

## Rational number

Prove that there exist infinite many positive real numbers such that the sum

is rational.

**Solution**

It suffices to prove that is continuous. Indeed,

Thus is continuous. Since is non constant , it follows that is an interval. The result follows.