(solved by Grégoire de Lambert, Luc Lehericy and Jean-François Martin). Find a polynomial in two variables $P$ such that the set of values taken by $P(x, y)$ when $x$ and $y$ range over the real numbers is exactly the strictly positive real numbers.
. It is easy to see that $P(x, y)=x^{2}+(x y+1)^{2}$ works.