NMO Selection Tests for the Junior Balkan Mathematical Olympiad

Set $a=x_1^2+y_1^2$. If $a$ is odd, the numbers $x_i$ and $y_i$ do not have the same parity, so $x_i+y_i$ is odd. Since $\sum (x_i+y_i)=0$, it follows that $n$ is even.

Suppose $a=4k+2$. Then $x_i$ and $y_i$ are both odd. The equality $x_1+x_2+\cdots +x_n=0$ implies $n$ is even.

Finally, if $a=4k$, then $x_i$ and $y_i$ are even. The numbers $a_i=\frac{x_i}{2}$ and $b_i=\frac{y_i}{2}$ satisfy the initial conditions. Furthermore, $a_1^2+b_1^2=\frac{a}{4}$. Repeating the argument for $\frac{a}{4}$ instead of $a$, after a finite number of steps we end up in a previous case.