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n points with integer distances

Source: Turkey National Olympiad Second Round 2013 P6

November 28, 2013
geometryrectanglegeometry proposed

Problem Statement

Let nn be a positive integer and P1,P2,…,PnP_1, P_2, \ldots, P_n be different points on the plane such that distances between them are all integers. Furthermore, we know that the distances PiP1,PiP2,…,PiPnP_iP_1, P_iP_2, \ldots, P_iP_n forms the same sequence for all i=1,2,…,ni=1,2, \ldots, n when these numbers are arranged in a non-decreasing order. Find all possible values of nn.
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