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maximum of {i,j,k}

Source: China high school competition 2011

October 17, 2011
floor functioncombinatorics proposedcombinatorics

Problem Statement

Given n≥4n\ge 4 real numbers an>...>a1>0a_{n}>...>a_{1} > 0. For r>0r > 0, let fn(r)f_{n}(r) be the number of triples (i,j,k)(i,j,k) with 1≤i<j<k≤n1\leq i<j<k\leq n such that aj−aiak−aj=r\frac{a_{j}-a_{i}}{a_{k}-a_{j}}=r. Prove that fn(r)<n24{f_{n}(r)}<\frac{n^{2}}{4}.
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