Problem
ALG-B2-M08-P020 Choose the substitution yourself
#20
★★★★★ Level 5 of 5
Let \(a,b,c\ge0\), \(a+b+c=3\). Prove \[\sqrt{a}+\sqrt{b}+\sqrt{c}\le3+\frac{(a-1)^2+(b-1)^2+(c-1)^2}{2}.\]
Hint. Use the tangent \(\sqrt{t}\le 1+\frac{t-1}{2}\), then notice that a nonnegative reserve has been added.
For \(t\ge0\), concavity of the square root gives \(\sqrt{t}\le1+\frac{t-1}{2}\). Summing for \(a,b,c\), we get \(\sum\sqrt{a}\le3+\frac{a+b+c-3}{2}=3\). The right side in the problem is \(3+\frac12\sum(a-1)^2\ge3\), so the result follows. Equality occurs at \(a=b=c=1\).
Teaching goal: recognize the appropriate substitution and check the domain.