Problem
ALG-B2-M09-P001 Minimum of squares
#1
★★☆☆☆ Level 2 of 5
Let \(a+b+c=6\). Find the minimum value of \(a^2+b^2+c^2\).
Hint. Use \((a+b+c)^2\le3(a^2+b^2+c^2)\).
By Cauchy, \(36=(a+b+c)^2\le3(a^2+b^2+c^2)\). Therefore \(\sum a^2\ge12\). Equality occurs at \(a=b=c=2\), so the minimum is \(12\).
Basic problem: fixed sum and a convex quantity.