Maths Olympiad Prep

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Combinatorics Difficulty 8.8 Shortlist Prove it Estonia

Find the smallest real constant CC such that for any positive real numbers a1,a2,a3,a4a_1, a_2, a_3, a_4 and a5a_5 (not necessarily distinct), one can always choose distinct subscripts i,j,ki, j, k and ll such that aiajakalC\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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