Problem
ALG-B2-M03-P007 Fractions with ones
#7
★★★★☆ Level 4 of 5
Let \(x_1,\ldots,x_n>0\). Prove \[\frac1{1+x_1}+\cdots+\frac1{1+x_n}\ge\frac{n^2}{n+x_1+\cdots+x_n}.\]
Hint 1. Apply Cauchy to numerators \(1,\ldots,1\).
Hint 2. The denominator sum is \(n+\sum x_i\).
By Cauchy, \[\sum_{i=1}^{n}\frac{1^2}{1+x_i}\ge\frac{(1+\cdots+1)^2}{(1+x_1)+\cdots+(1+x_n)}=\frac{n^2}{n+\sum x_i}.\]
Cauchy-Schwarz module training problem. Method tags: cauchy, fractions.