Problem
ALG-B2-M10-P002 Three quadratic denominators
#2
★★★★★ Level 5 of 5
Let \(a,b,c\ge0\), \(a+b+c=3\). Prove \[\frac1{a^2+a+1}+\frac1{b^2+b+1}+\frac1{c^2+c+1}\ge1.\]
Hint. Check the convexity of \(f(x)=1/(x^2+x+1)\) on \(x\ge0\).
For \(x\ge0\), the function \(f(x)=1/(x^2+x+1)\) is convex. Jensen gives \[\frac13\sum f(a)\ge f\left(\frac{a+b+c}{3}\right)=f(1)=\frac13.\] Multiplying by \(3\), we get the result.
Teaching goal: choose a combination of methods and always check the equality case.