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ab in terms of inverse and sum / differences of a,b

Source: 2004 Moldova JBMO TST p8

February 20, 2021
algebraProduct

Problem Statement

The positive real numbers aa and bb (a>ba> b) are written on the board. At every step, with numbers written on the board, one of the following operations can be performed: a) choose one of the numbers and write its square or its inverse. b) choose two numbers written on the board ¸and write their sum or their positive difference. Show how the product aba \cdot b can be obtained with the help of the defined operations.