55rd Ukrainian National Mathematical Olympiad - Third Round

If $a,b$ have the same parity, define $x,y$ as: $$x=\frac{a+b}{2},y=\frac{a-b}{2}.$$ Then they are obviously integers, and by substituting them we verify that the equation does hold.

If $a,b$ don't have the same parity, the right-hand side of the equation is an odd integer. For example, if $a$ is even and $b$ is odd, then $a^4+6a^2{b}^2$ is even and $b^4$ is odd, therefore, the equality cannot hold.