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PrintNMO Selection Tests for the Junior Balkan Mathematical Olympiad
Romania geometry
Problem
In the Cartesian plane are given points of integer coordinates, with pairwise integer distances. Prove that more than of the distances are even integers.
Solution
Among the given points there cannot exist simultaneously a point with odd coordinates and a point with even coordinates. Indeed, that be the case, let and be such points. The square of the distance between them is , which is not the square of an integer.
By the same argument as above, if an (even, odd)-coordinate point existed, there cannot exist an (odd, even)-coordinate point.
Let now be the number of points with coordinates of the same parity, then notice that all distances among them are even integers. So there exist points with coordinates of different parity, again all distances among them being even integers.
Therefore, at least distances are even numbers (by definition if ). Since there are distances in all, one has
By the same argument as above, if an (even, odd)-coordinate point existed, there cannot exist an (odd, even)-coordinate point.
Let now be the number of points with coordinates of the same parity, then notice that all distances among them are even integers. So there exist points with coordinates of different parity, again all distances among them being even integers.
Therefore, at least distances are even numbers (by definition if ). Since there are distances in all, one has
Techniques
Cartesian coordinatesIntegersLinear and quadratic inequalities