Problem
ALG-B2-M05-P011 Shifted reciprocals
#11
★★★★☆ Level 4 of 5
Let \(a,b,c\ge0\) and \(a+b+c=6\). Prove \[\frac{1}{1+a}+\frac{1}{1+b}+\frac{1}{1+c}\ge1.\]
Hint. The function \(f(x)=\frac{1}{1+x}\) is convex for \(x\ge0\).
By Jensen, \[\frac13\sum\frac{1}{1+a}\ge\frac{1}{1+(a+b+c)/3}=\frac{1}{3}.\] Multiplying by \(3\), we get the result.
A good problem on recognizing a shifted function.