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China South East Mathematical Olympiad 2013 problem 8

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August 10, 2013
algebra unsolvedalgebra

Problem Statement

n≥3n\geq 3 is a integer. α,β,γ∈(0,1)\alpha,\beta,\gamma \in (0,1). For every ak,bk,ck≥0(k=1,2,…,n)a_k,b_k,c_k\geq0(k=1,2,\dotsc,n) with ∑k=1n(k+α)ak≤α,∑k=1n(k+β)bk≤β,∑k=1n(k+γ)ck≤γ\sum\limits_{k=1}^n(k+\alpha)a_k\leq \alpha, \sum\limits_{k=1}^n(k+\beta)b_k\leq \beta, \sum\limits_{k=1}^n(k+\gamma)c_k\leq \gamma, we always have ∑k=1n(k+λ)akbkck≤λ\sum\limits_{k=1}^n(k+\lambda)a_kb_kc_k\leq \lambda. Find the minimum of λ\lambda
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