Problem
ALG-B2-M11-P009 Three shifted denominators
#9
★★★★☆ Level 4 of 5
Let \(a,b,c\ge0\), \(a+b+c=3\). Prove \[\frac1{a+2}+\frac1{b+2}+\frac1{c+2}\ge1.\]
Hint. Apply Cauchy to the numerators \(1,1,1\).
By Cauchy, \[\sum\frac1{a+2}\ge\frac{(1+1+1)^2}{a+b+c+6}=\frac9{9}=1.\]
Teaching goal: the student should first recognise the type of estimate, then choose the tool.