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Operations on powers of two

Source: Kvant Magazine No. 8 2023 M2758

January 9, 2024
combinatoricsOperations

Problem Statement

The numbers 2,4,…,21002,4,\ldots,2^{100} are written on a board. At a move, one may erase the numbers a,ba,b from the board and replace them with ab/(a+b).ab/(a+b). Prove that the last numer on the board will be greater than 1.
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