Algebra · Algebraic identities · Triangle inequality · Antisymmetric expressions · Fractions
Let \(a\), \(b\) and \(c\) be the side lengths of a triangle, and set
\[ p = \frac{a}{b} + \frac{b}{c} + \frac{c}{a}, \qquad q = \frac{a}{c} + \frac{c}{b} + \frac{b}{a}. \]
Prove that \(|p - q| < 1\).
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Serbian Municipal Competition 2003, high school grade I, category A, problem 3. Organized by the Mathematical Society of Serbia (DMS). Source