2007 IMO Problems

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Problem 1


Real numbers $a_1, a_2, \dots , a_n$ are given. For each $i$ ($1\le i\le n$) define

\[d_i=\max\{a_j:1\le j\le i\}-\min\{a_j:i\le j\le n\}\]

and let

\[d=\max\{d_i:1\le i\le n\}\].

(a) Prove that, for any real numbers $x_1\le x_2\le \cdots\le x_n$,

\[\max\{|x_i-a_i|:1\le i\le n\}\ge \dfrac{d}{2}   (*)\]

(b) Show that there are real numbers $x_1\le x_2\le x_n$ such that equality holds in (*)