2007 IMO Shortlist Problems/A1: Difference between revisions

Revision as of 20:26, 10 December 2019

(New Zealand) You are given a sequence $a_1,a_2,\dots ,a_n$ of numbers. For each $i$ ( $1\leq i\leq n$ ) define

$d_i=\max\{a_j:1\leq j\leq i\}-\min\{a_j:i\leq j\leq n\}$

and let

$d=\max\{d_i:1\leq i\leq n\}$ .

(a) Prove that for arbitrary real numbers $x_1\leq x_2\leq \dots \leq x_n$ ,

$\max\{|x_i-a_i|:1\leq i\leq n\}\geq \frac{d}{2}$ .

(b) Show that there exists a sequence $x_1\leq x_2\leq \dots \leq x_n$ of real numbers such that we have equality in (a).

This problem needs a solution. If you have a solution for it, please help us out by adding it.

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 == Problem ==
-(''New Zealand'') let's solve this problem bois @poco @john
+(''New Zealand'')
-You are given a sequence <math>a_1,a_2,\dots ,a_n</math> of numbers. For each <math>i</math> (<math>1\leq 1\leq n</math>) define
+You are given a sequence <math>a_1,a_2,\dots ,a_n</math> of numbers. For each <math>i</math> (<math>1\leq i\leq n</math>) define
 <center><math>d_i=\max\{a_j:1\leq j\leq i\}-\min\{a_j:i\leq j\leq n\}</math></center>