Školjka

%V0 Real numbers $a_{1}$, $a_{2}$, $\ldots$, $a_{n}$ are given. For each $i$, $(1 \leq i \leq n )$, define $$d_{i} = \max \{ a_{j}\mid 1 \leq j \leq i \} - \min \{ a_{j}\mid i \leq j \leq n \}$$ and let $d = \max \{d_{i}\mid 1 \leq i \leq n \}$. (a) Prove that, for any real numbers $x_{1}\leq x_{2}\leq \cdots \leq x_{n}$, $$\max \{ |x_{i} - a_{i}| \mid 1 \leq i \leq n \}\geq \frac {d}{2}. \quad \quad (*)$$ (b) Show that there are real numbers $x_{1}\leq x_{2}\leq \cdots \leq x_{n}$ such that the equality holds in (*). Author: Michael Albert, New Zealand