Problem
ALG-B1-M07-P003 Three squares
#3
★☆☆☆☆ Level 1 of 5
Prove that for all real \(a,b,c\), \(a^2+b^2+c^2\ge ab+bc+ca\).
Multiply the difference by \(2\) and group squares.
\(2(a^2+b^2+c^2-ab-bc-ca)=(a-b)^2+(b-c)^2+(c-a)^2\ge0\). Therefore the inequality holds. Equality occurs when \(a=b=c\).
A classical symmetric sum-of-squares form.