DE-BUNDESWETTBEWERB 2005 Problem 2

Let $a$ be such an integer, that $3a$ can be written in the form $x^2 + 2y^2$ , with integers $x$ and $y$ . Prove that the number $a$ can also be written in this form.

Additional problems:

a) Find a general (necessary and sufficent) criterion for an integer $n$ to be of that form.

b) In how many ways can the integer $n$ be represented in that way?