Algebra · Symmetric expressions · Algebraic identities · Quadratic forms
Problem 2, 2009
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 \(a\), \(b\), \(c\),
\[ a^{2} + b^{2} + c^{2} = (pa + qb + rc)^{2} + (qa + rb + pc)^{2} + (ra + pb + qc)^{2} . \]
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Serbian Regional Competition 2009, high school grade I, category A, problem 2. Organized by the Mathematical Society of Serbia (DMS). Source