Problem
ALG-B1-M03-P017 A Sum of Squares from a System
#17
★★★☆☆ Level 3 of 5
Real numbers \(x,y,z\) satisfy \(x+y+z=3\), \(x^2+y^2+z^2=3\). Prove that \(x=y=z=1\).
Consider \((x-1)^2+(y-1)^2+(z-1)^2\).
\((x-1)^2+(y-1)^2+(z-1)^2=x^2+y^2+z^2-2(x+y+z)+3=3-6+3=0\).
A sum of squares is zero only when all squares are zero. Hence \(x=y=z=1\).
The algebraic system becomes non-negativity.