MathDB

On the Cartesian plane consider the set

V

of all vectors with integer coordinates. Determine all functions

f : V \rightarrow \mathbb{R}

satisfying the conditions: (i)

f(v) = 1

for each of the four vectors

v \in V

of unit length. (ii)

f(v+w) = f(v)+f(w)

for every two perpendicular vectors

v, w \in V

(Zero vector is considered to be perpendicular to every vector).

Problem Statement