see the point (a,b) in the cordinates plane, and the circle w, with center in (a,b) and ray 1. with easy computations we see that is impossible have 2 points of form (x+2y, 2x-y) inside the circle.

"ray"-radius

$x=\frac{c+2d+(a+2b)}{5}$ $y=\frac{2c-d+(a-2b)}{5}$ and $c+2d\equiv-(a+2b)[5]$ $2c-d\equiv 2b-a[5]$ (it have unique solution $mod\ 5$) so $c$ and $d$ are unique because with $\{c,d\}\in \{-1,0,1\}$ then $(x,y)$ is unique