59th Ukrainian National Mathematical Olympiad

Consider the following expression: $$x^4+y^4+(x+y{)}^4=2x^4+4x^{3}y+6x^2{y}^2+4x{y}^3+2y^4,$$ thus $$2(x^4+y^4+(x+y{)}^4)=4(x^4+2x^{3}y+3x^2{y}^2+2x{y}^3+y^4)=(2(x^2+xy+y^2){)}^2.$$ Hence for $x=20$, $y=19$ the following holds:

$$n^2=(2(x^2+xy+y^2){)}^2\text{ or }n=2(20^2+20\cdot 19+19^2)=2282.$$