Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

2007 IMO Shortlist Problems/A1

Revision as of 20:26, 10 December 2019 by Pocofrosty12 (talk | contribs) (→‎Problem)

Problem

(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).

Solution

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

Resources

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2007_IMO_Shortlist_Problems/A1&oldid=112745"