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abc \geq 1 then \sum \frac 1{1+b+c} \leq 1

Source: GMB-IMAR 2005, Juniors, Problem 1

October 10, 2005
AMCUSAMOInequalitythree variable inequality

Problem Statement

Let a,b,ca,b,c be positive real numbers such that abc≥1abc\geq 1. Prove that 11+b+c+11+c+a+11+a+b≤1. \frac{1}{1+b+c}+\frac{1}{1+c+a}+\frac{1}{1+a+b}\leq 1. [hide="Remark"]This problem derives from the well known inequality given in [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=185470#p185470]USAMO 1997, Problem 5.
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