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59th Ukrainian National Mathematical Olympiad

Ukraine geometry

Problem

Consider the family of parabolas of the form on a plane, where are any integer numbers. Is there a point with integer coordinates that doesn't belong to any parabola?
Solution
Let , then the second coordinate is , which is divisible by three. Therefore, doesn't belong to any parabola.
Final answer
(1, 1)

Techniques

Cartesian coordinatesQuadratic functionsModular Arithmetic