Algebra · Algebraic identities · Rational expressions · Factoring
Let \(a\) and \(b\) be real numbers for which
\[ \frac{a}{1+a} + \frac{b}{1+b} = 1 . \]
Prove that
\[ \frac{a}{1+b^{2}} - \frac{b}{1+a^{2}} = a - b . \]
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Slovenian High School Mathematics Competition for Vega Awards (MaSSA), drzavno (national) round 2006, 1. letnik, category A, problem 1. Organized by DMFA Slovenije (Society of Mathematicians, Physicists and Astronomers of Slovenia). Source