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A very strange binary number is rational

Source: KoMaL A. 708

May 21, 2023
algebra

Problem Statement

Let SS be a finite set of rational numbers. For each positive integer kk, let bk=0b_k=0 if we can select kk (not necessarily distinct) numbers in SS whose sum is 00, and bk=1b_k=1 otherwise. Prove that the binary number 0.b1b2b3…0.b_1b_2b_3… is a rational number. Would this statement remain true if we allowed SS to be infinite?
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