Problem
ALG-B2-M03-P015 Sum of roots
#15
★★★★★ Level 5 of 5
Let \(u,v,w>0\) and \(u+v+w=6\). Prove \[\sqrt{u+v}+\sqrt{v+w}+\sqrt{w+u}\le6.\]
Hint 1. Use Cauchy as \((\sum r_i)^2\le n\sum r_i^2\).
Hint 2. The sum of the squares of the roots is \(2(u+v+w)\).
By Cauchy, \[(\sqrt{u+v}+\sqrt{v+w}+\sqrt{w+u})^2\le3((u+v)+(v+w)+(w+u))=6(u+v+w)=36.\] Hence the sum of roots is at most \(6\).
Cauchy-Schwarz module training problem. Method tags: cauchy, rms-am.