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Find points with sames integer distances as given

Source: Belarus TST 2024

July 17, 2024
combinatoricsnumber theory

Problem Statement

Points A1,…AnA_1, \ldots A_n with rational coordinates lie on a plane. It turned out that the distance between every pair of points is an integer. Prove that there exist points B1,…,BnB_1, \ldots ,B_n with integer coordinates such that AiAj=BiBjA_iA_j=B_iB_j for every pair 1≤i≤j≤n1 \leq i \leq j \leq n N. Sheshko, D. Zmiaikou
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