Problem
ALG-B2-M09-P019 Fractions with a constraint
#19
★★★★★ Level 5 of 5
Let \(a,b,c\ge0\), \(a+b+c=3\). Prove \[\frac{a^2}{1+a}+\frac{b^2}{1+b}+\frac{c^2}{1+c}\ge\frac32.\]
Hint. Apply Cauchy in Engel form.
By Cauchy, \[\sum\frac{a^2}{1+a}\ge\frac{(a+b+c)^2}{(1+a)+(1+b)+(1+c)}=\frac{9}{6}=\frac32.\] Equality holds at \(a=b=c=1\).
The key is seeing that the denominator sum is also fixed.