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Ukrainian National Mathematical Olympiad

Geometry Difficulty 5.2 AIME, harder Prove it Ukraine

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

Solution

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

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