XXI SILK ROAD MATHEMATICAL COMPETITION

Without loss of generality $A>B$. Choose an arbitrary prime $p>2$ and let's find $x_1$ and $x_2$ so that $$x_1^2+A(2p{)}^2=x_2^2+B(2p{)}^2,$$ Hence, $x_2^2-x_1^2=4C{p}^2$ where $C=A-B$. Set $x_1=C{p}^2-1$ and $x_2=C{p}^2+1$. If $x_1$ and $x_2$ are both odd, then they are both coprime with $y=2p$, and we have $x_1^2+A{y}^2=x_2^2+B{y}^2$. If they are both even, then $\frac{x_1}{2}$ and $\frac{x_2}{2}$ are both coprime with $y=p$, and we have $(\frac{x_1}{2}{)}^2+A{y}^2=(\frac{x_2}{2}{)}^2+B{y}^2$. The number to which we have found two such forms will be not less than $p^2$, thus proving there are infinitely many such numbers.