59th Ukrainian National Mathematical Olympiad

Question

Consider the collection of lines of the form

y = (k + n) x + (k - n)

on a plane, where

k, n

are any integers. Is there a point with integer coordinates that doesn't belong to any of such lines?

Answer: yes, there is.

Accepted Answer

Let x=1, then the second coordinate of all the points that belong to the lines is y = (k+n) + (k-n) = 2k which is even. Therefore, the point (1, 1) doesn't belong to any of the lines.