Algebra · Symmetric expressions · Algebraic identities · Conditional identities · Factoring
Let \(a\), \(b\), \(c\) be nonzero real numbers such that
\[ a(b + c) + b(c + a) + c(a + b) = ab + bc + ca . \]
Prove that the value of
\[ \frac{a^{2}(b + c) + b^{2}(a + c) + c^{2}(a + b)}{abc} \]
is an integer.
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Slovenian High School Mathematics Competition for Vega Awards (MaSSA), drzavno (national) round 2008, 1. letnik, category A, problem 2. Organized by DMFA Slovenije (Society of Mathematicians, Physicists and Astronomers of Slovenia). Source