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A Generalisation on an RMO Problem

\[\dfrac {a^k}{(a-b)(a-c)} + \dfrac {b^k}{(b-a)(b-c)} + \dfrac {c^k}{(c-a)(c-b)} > \dfrac {k(k-1)}{2}\]

If \(a\), \(b\) and \(c\) are distinct positive reals such that \(abc = 1\), and \(k \geq 3\) is a positive integer, show that the inequality above holds.

Note: The RMO problem had the case \(k=3\).

Note by Sharky Kesa
1 week, 2 days ago

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@Svatejas Shivakumar To be clear, your approach works for general \(n\) through a simple generalization. Calvin Lin Staff · 5 days, 3 hours ago

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@Svatejas Shivakumar The claim of "One can prove by induction" is not obvious, in part since you didn't use the induction hypothesis to prove \( k = 4 \).

Can you elaborate on the induction approach?

Note: The approach that I'm thinking of simply uses what you have, without any inductive step. It essentially boils down to finding that "4th factor" and showing that it is greater than k(k-1)/2. Calvin Lin Staff · 1 week ago

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