## A factorization

Let be a right triangle at . Factor .

**Solution**

Since we have successively:

## Equality

Prove that in any triangle it holds that

**Solution**

We have successively:

## On a Lambert W integral

Prove that

## A Fibonacci series

Let denote the – th Fibonacci number. Prove that

**Solution**

Let denote the given sum. Then,

## A floor series

Let . Prove that

- .
- .
- where denotes the sign function.

**Solution**

- We have successively:
However,

and the result follows.

- Let . Then, and . It follows that . Hence,
since is integer for all . The result follows from the previous question as well as the fact that if-f is odd; .

- It follows from the previous question since .