2003 OIM Problems/Problem 3

Problem

Pablo was copying the following problem from a blackboard:

$\textit{Consider all sequences of 2004 real numbers }(x_0,x_1,x_2,\cdots , x_{2003})\textit{, such that}$

$x_0=1\text{,}$

$0\le x_1 \le 2x_0 \text{,}$

$0\le x_2 \le 2x_1 \text{,}$

$\vdots$

$0\le x_{2003} \le 2x_{2004} \text{.}$

$\textit{Among all these sequences, find the one for which the following expression takes its largest value:}$

$S = ...$

When Pablo was going to copy the expression for $S$ , they erased the blackboard. The only thing he could remember was that $S$ was of the form

$S=\pm x_1\pm x_2\pm \cdots\pm x_{2002}+x_{2003}$

where the last term, $x_{2003}$ , had a coefficient +1, and the previous ones had a coefficient +1 or -1. Show that Paul, despite not having the complete statement, can find with certainty the solution to the problem.

~translated into English by Tomas Diaz. ~orders@tomasdiaz.com

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2003 OIM Problems/Problem 3

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