MathDB

In the Cartesian plane is given a set of points with integer coordinate

T=\{ (x;y)\mid x,y\in\mathbb{Z} ; \ |x|,|y|\leq 20 ; \ (x;y)\ne (0;0)\}

We colour some points of

T

such that for each point

(x;y)\in T

then either

(x;y)

(-x;-y)

is coloured. Denote

N

to be the number of couples

{(x_1;y_1),(x_2;y_2)}

such that both

(x_1;y_1)

and

(x_2;y_2)

are coloured and

x_1\equiv 2x_2 \pmod {41}, y_1\equiv 2y_2 \pmod {41}

. Find the all possible values of

N

Problem Statement