Problem
ALG-B2-M08-P017 Deviations and squares
#17
★★★★★ Level 5 of 5
Let \(a,b,c\ge0\), \(a+b+c=3\). Prove \[(a-1)^2+(b-1)^2+(c-1)^2= a^2+b^2+c^2-3.\]
Hint. Set \(a=1+x\), \(b=1+y\), \(c=1+z\), \(x+y+z=0\).
After substitution, the left side is \(x^2+y^2+z^2\). The right side is \(\sum(1+x)^2-3=3+2(x+y+z)+x^2+y^2+z^2-3=x^2+y^2+z^2\). The identity is proved.
Teaching goal: recognize the appropriate substitution and check the domain.