Problem
ALG-B2-M08-P012 Sum equals one
#12
★★★★☆ Level 4 of 5
Let \(a+b+c=1\). Set \(a=\frac13+x\), \(b=\frac13+y\), \(c=\frac13+z\). Prove \(ab+bc+ca\le\frac13\).
Hint. After substitution, \(x+y+z=0\).
Since \(x+y+z=0\), \(ab+bc+ca=\frac13+xy+yz+zx\). But \(xy+yz+zx=-\frac12(x^2+y^2+z^2)\le0\). Hence \(ab+bc+ca\le1/3\).
Teaching goal: recognize the appropriate substitution and check the domain.