9.2. Is there a point with integer coordinates on the coordinate plane, the distance from which to the origin is equal to $\sqrt{2 \cdot 2017^{2}+2 \cdot 2018^{2}}$?
Answer. There is. Solution. We will prove that for any natural number $n$, there exists a point with integer coordinates, the distance from which to the origin is $\sqrt{2 n^{2}+2(n+1)^{2}}$. To do this, we transform the last expression $2 n^{2}+2(n+1)^{2}=2 n^{2}+2 n^{2}+4 n+2=\left(4 n^{2}+4 n+1\right)+$ $1=(2 n+1)^{2}+1^{2}$. Therefore (by the Pythagorean theorem), the condition of the problem is satisfied by the point with coordinates $(2 n+1 ; 1)$.