Problem
ALG-B2-M03-P014 Mixed denominators
#14
★★★★★ Level 5 of 5
Prove for \(a,b,c>0\): \[\frac{a^2}{a+b+c}+\frac{b^2}{a+2b+c}+\frac{c^2}{a+b+3c}\ge\frac{(a+b+c)^2}{3a+4b+5c}.\]
Hint 1. This is exactly Engel form.
Hint 2. Add the denominators.
By Cauchy, the left side is at least \[\frac{(a+b+c)^2}{(a+b+c)+(a+2b+c)+(a+b+3c)}=\frac{(a+b+c)^2}{3a+4b+5c}.\]
Cauchy-Schwarz module training problem. Method tags: cauchy, cyclic-sum.