Problem
ALG-B2-M12-P019 Set 5. Shifted denominators
#19
★★★★★ Level 5 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 numerators \(1,1,1\).
By Cauchy, \[\sum\frac1{a+2}\ge\frac9{a+b+c+6}=\frac9{9}=1.\]
Mock set 5: the problem is intended for independent method selection without an explicit cue in the statement.