2007 IMO Problems

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 (*)

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