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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

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