Problem
ALG-B2-M05-P012 Concave fraction
#12
★★★★☆ Level 4 of 5
Let \(a,b,c\ge0\) and \(a+b+c=3\). Prove \[\frac{a}{1+a}+\frac{b}{1+b}+\frac{c}{1+c}\le\frac32.\]
Hint. Consider \(f(x)=\frac{x}{1+x}\).
The function \(f(x)=\frac{x}{1+x}\) is concave for \(x\ge0\). Jensen gives \[\frac13\sum\frac{a}{1+a}\le \frac{1}{1+1}=\frac12.\] Hence the sum is at most \(\frac32\).
Here the important part is identifying concavity correctly; otherwise the direction is easy to reverse.