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Sequence of rational points P_i forming unit distances

Source: Japanese MO Finals 2000

February 10, 2011
analytic geometrynumber theory unsolvednumber theory

Problem Statement

Consider the points O(0,0)O(0,0) and A(0,1/2)A(0,1/ 2) on the coordinate plane. Prove that there is no finite sequence of rational points P1,P2,…,PnP_1,P_2,\ldots ,P_n in the plane such that OP1=P1P2=…=Pn−1Pn=PnA=1OP_1=P_1P_2=\ldots =P_{n-1}P_n=P_nA=1
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