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Combinatorial inequality for n-tuples with sum 1

Source: 2nd Memorial Mathematical Competition "Aleksandar Blazhevski - Cane" - Problem 3

January 12, 2021
combinatoricsinequalities

Problem Statement

Given a positive integer n≥3n \geq 3, let CnC_{n} be the collection of all nn-tuples a=(a1,a2,...,an)a=(a_{1},a_{2},...,a_{n}) of nonnegative reals aia_i, i=1,...,ni=1,...,n, such that a1+a2+...+an=1a_{1}+a_{2}+...+a_{n}=1. For k∈{1,...,n−1}k \in \left \{ 1,...,n-1 \right \} and a∈Cna \in C_{n}, consider the sum set σk(a)={a1+...+ak,a2+...+ak+1,...,an−k+1+...+an}\sigma_{k}(a) = \left \{a_{1}+...+a_{k},a_{2}+...+a_{k+1},...,a_{n-k+1}+...+a_{n} \right \}. Show the following. (a) There exist mk=max⁡{min⁡σk(a):a∈Cn}m_k=\max\{\min\sigma_k(a):a\in\mathcal{C}_n\} and Mk=min⁡{max⁡σk(a):a∈Cn}M_k=\min\{\max\sigma_k(a):a\in\mathcal{C}_n\}. (b) It holds that 1≤∑k=1n−1(1Mk−1mk)≤n−2\displaystyle{1\leq\sum_{k=1}^{n-1}(\frac{1}{M_k}-\frac{1}{m_k})\leq n-2}. Moreover, on the left side, equality is attained only for finitely many values of nn, whereas on the right side, equality holds for infinitely values of nn.
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