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Find M so one number exceeds the target

Source: Indonesian Mathematical Olympiad 2013 Problem 3

September 5, 2013
inequalitiesinequalities unsolved

Problem Statement

Determine all positive real MM such that for any positive reals a,b,ca,b,c, at least one of a+Mab,b+Mbc,c+Mcaa + \dfrac{M}{ab}, b + \dfrac{M}{bc}, c + \dfrac{M}{ca} is greater than or equal to 1+M1+M.