Difference between revisions of "2007 IMO Shortlist Problems/A1"
(New page: == Problem == (''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 <center><math>d_i=\max\{a_...) |
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== Problem == | == Problem == | ||
− | (''New Zealand'') | + | (''New Zealand'') let's solve this problem bois @poco @john |
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 1\leq n</math>) define | ||
Revision as of 21:24, 10 December 2019
Problem
(New Zealand) let's solve this problem bois @poco @john
You are given a sequence of numbers. For each
(
) define
![$d_i=\max\{a_j:1\leq j\leq i\}-\min\{a_j:i\leq j\leq n\}$](http://latex.artofproblemsolving.com/1/8/9/189de4c31e3bbc18f2a728546428ea8bb222bebe.png)
and let
![$d=\max\{d_i:1\leq i\leq n\}$](http://latex.artofproblemsolving.com/7/f/d/7fdc1b9fb65f8df2b3b292342c6e1b2d77c44e21.png)
(a) Prove that for arbitrary real numbers ,
![$\max\{|x_i-a_i|:1\leq i\leq n\}\geq \frac{d}{2}$](http://latex.artofproblemsolving.com/e/d/8/ed80966e31623d83b93ce2db9fdac05d04549dac.png)
(b) Show that there exists a sequence of real numbers such that we have equality in (a).
Solution
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