$\left.\begin{array}{l}{\left[\begin{array}{l}\text { equations in integers }\end{array}\right]} \\ \text { [Completing the square. Sums of squares }]\end{array}\right]$

Solve the equation $x^{2}+y^{2}=x+y+2$ in integers.
Let's write the equation as $(2 x-1)^{2}+(2 y-1)^{2}=10$.

From this, it is clear that one term equals 9, and the other equals 1.

## Answer

$\{2,0\},\{2,1\},\{-1,0\},\{-1,1\}$.