IMO 1970 LL SWE53

A square ABCD is divided into (n −1)2 congruent squares,

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IMO 1970 LL SWE53

Origin: SWE

Problem

A square ABCD is divided into (n −1)2 congruent squares, with sides parallel to the sides of the given square. Consider the grid of all n2 corners obtained in this manner. Determine all integers n for which it is possible to construct a nondegenerate parabola with its axis parallel to one side of the square and that passes through exactly n points of the grid.