Problem
ALG-B1-M06-P016 All variables are equal
#16
★★★☆☆ Level 3 of 5
Let \(a+b+c=3\) and \(a^2+b^2+c^2=3\). Prove that \(a=b=c=1\).
Consider the sum \((a-1)^2+(b-1)^2+(c-1)^2\).
We have \((a-1)^2+(b-1)^2+(c-1)^2=a^2+b^2+c^2-2(a+b+c)+3=3-6+3=0\).
A sum of squares is zero only when each square is zero. Hence \(a=b=c=1\).
The useful step is not merely expanding, but guessing the center of the squares.