Olympiad Maths Prep

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Geometry Difficulty 5.2 AIME, harder Prove it Ukraine

Consider the collection of lines of the form y=(k+n)x+(k−n)y = (k+n)x + (k-n) on a plane, where k,nk, n are any integers. Is there a point with integer coordinates that doesn't belong to any of such lines?

Answer: yes, there is.

Solution

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

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