50th Mathematical Olympiad in Ukraine, Third Round (January 23, 2010)

Obviously $x\ne 0$ and $y\ne 0$. Now we have to consider three cases:

1st case. $x>0$, $y>0$. Hence $2y=2$, and we have the first ray of the answer.

2nd case. $x<0$, $y<0$, analogous to the first case. It gives us the second ray of the answer.

3rd case. $xy<0$. It follows that the equality $2y=0$ must be held, but this is impossible.



Fig.05