Problem

Source: ChInese TST 2009 P2

Tags: induction, ratio, LaTeX, inequalities, blogs, inequalities proposed



Given an integer $ n\ge 2$, find the maximal constant $ \lambda (n)$ having the following property: if a sequence of real numbers $ a_{0},a_{1},a_{2},\cdots,a_{n}$ satisfies $ 0 = a_{0}\le a_{1}\le a_{2}\le \cdots\le a_{n},$ and $ a_{i}\ge\frac {1}{2}(a_{i + 1} + a_{i - 1}),i = 1,2,\cdots,n - 1,$ then $ (\sum_{i = 1}^n{ia_{i}})^2\ge \lambda (n)\sum_{i = 1}^n{a_{i}^2}.$