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IMO Team Selection Contest I

Combinatorics Difficulty 8.8 Shortlist Prove it Estonia

Find the smallest real constant $C$ such that for any positive real numbers $a_1, a_2, a_3, a_4$ and $a_5$ (not necessarily distinct), one can always choose distinct subscripts $i, j, k$ and $l$ such that $\left|\frac{a_i}{a_j} - \frac{a_k}{a_l}\right| \le C$ .

Solution

See IMO 2016 shortlist, problem A2.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.