Problem
ALG-B2-M03-P004 Nesbitt
#4
★★★☆☆ Level 3 of 5
Prove for \(a,b,c>0\): \[\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\ge\frac32.\]
Hint 1. Write \(a=\frac{a^2}{a}\).
Hint 2. The denominator sum after Cauchy is \(2(ab+bc+ca)\).
\[\sum\frac{a}{b+c}=\sum\frac{a^2}{a(b+c)}\ge\frac{(a+b+c)^2}{a(b+c)+b(c+a)+c(a+b)}=\frac{(a+b+c)^2}{2(ab+bc+ca)}\ge\frac32,\] since \((a+b+c)^2\ge3(ab+bc+ca)\).
Cauchy-Schwarz module training problem. Method tags: cauchy, engel-form.