Maths Olympiad Prep

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Geometry Difficulty 5.2 AIME, harder Prove it Ukraine

Consider the family of parabolas of the form y=kx2+(k−n)x+(k+n)y = kx^2 + (k-n)x + (k+n) on a plane, where k,nk, n are any integer numbers. Is there a point with integer coordinates that doesn't belong to any parabola?

Solution

Let x=1x=1, then the second coordinate is y=k+(k−n)+(k+n)=3ky = k + (k-n) + (k+n) = 3k, which is divisible by three. Therefore, (1;1)(1; 1) doesn't belong to any parabola.

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