Difference between revisions of "2007 IMO Problems"
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Revision as of 17:14, 29 March 2010
Problem 1
Real numbers 
are given. For each 
(
) define
![\[d_i=\max\{a_j:1\le j\le i\}-\min\{a_j:i\le j\le n\}\]](http://latex.artofproblemsolving.com/6/6/1/661c06518e6a3ae8ca5c84d0eb4e63ef7945723e.png)
and let
![\[d=\max\{d_i:1\le i\le n\}\]](http://latex.artofproblemsolving.com/6/7/7/677a05f003aa103531ff367920be9f9dd13797f8.png)
.
(a) Prove that, for any real numbers 
,
![\[\max\{|x_i-a_i|:1\le i\le n\}\ge \dfrac{d}{2} (*)\]](http://latex.artofproblemsolving.com/a/7/a/a7ac71dcf18b9336f8b3d4cbb575615c1aefca54.png)
(b) Show that there are real numbers 
such that equality holds in (*)
