Problem
ALG-B2-M08-P009 Sum of pairwise roots
#9
★★★★☆ Level 4 of 5
Prove for \(a,b,c\ge0\): \[\sqrt{ab}+\sqrt{bc}+\sqrt{ca}\le a+b+c.\]
Hint. Set \(a=x^2\), \(b=y^2\), \(c=z^2\).
After substitution, we need \(xy+yz+zx\le x^2+y^2+z^2\). This is equivalent to \(\frac12((x-y)^2+(y-z)^2+(z-x)^2)\ge0\).
Teaching goal: recognize the appropriate substitution and check the domain.