Algebra · Symmetric expressions · Algebraic identities · Quadratic forms
Let \(p\), \(q\), \(r\) be real numbers such that
\[ \frac{1}{p} + \frac{1}{q} + \frac{1}{r} = 0 \qquad \text{and} \qquad p + q + r = 1 . \]
Prove that for all real numbers \(a\), \(b\), \(c\),
\[ a^{2} + b^{2} + c^{2} = (pa + qb + rc)^{2} + (qa + rb + pc)^{2} + (ra + pb + qc)^{2} . \]
Sign in to check answers, open hints, read the full solution, and track your progress. Statements are always free.
Serbian Regional Competition 2009, high school grade I, category A, problem 2. Organized by the Mathematical Society of Serbia (DMS). Source