Difference between revisions of "2002 OIM Problems/Problem 3"
(Created page with "== Problem == Pablo was copying the following problem: <cmath>\text{Consider all sequences of 2004 real numbers}(x_0,x_1,x_2,\cdots , x_{2003})\text{, such that}</cmath>\ <c...") |
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Pablo was copying the following problem: | Pablo was copying the following problem: | ||
| − | <cmath>\text{Consider all sequences of 2004 real numbers}(x_0,x_1,x_2,\cdots , x_{2003})\text{, such that}</cmath> | + | <cmath>\text{Consider all sequences of 2004 real numbers}(x_0,x_1,x_2,\cdots , x_{2003})\text{, such that}</cmath> |
| − | <cmath>x_0=1\text{,}</cmath> | + | <cmath>x_0=1\text{,}</cmath> |
<cmath>0\le x_1 \le 2x_0 \text{,}</cmath> | <cmath>0\le x_1 \le 2x_0 \text{,}</cmath> | ||
Revision as of 16:31, 13 December 2023
Problem
Pablo was copying the following problem:
![\[\text{Consider all sequences of 2004 real numbers}(x_0,x_1,x_2,\cdots , x_{2003})\text{, such that}\]](http://latex.artofproblemsolving.com/6/3/3/63312b554c40322152a5ccdc7d8ab282d4894343.png)
![\[x_0=1\text{,}\]](http://latex.artofproblemsolving.com/7/1/0/7105128b22cc531f7e576cfa1adc6450043414a5.png)
![\[0\le x_1 \le 2x_0 \text{,}\]](http://latex.artofproblemsolving.com/7/c/e/7ce41adc2095765e2245ca25606984ca92429b43.png)
![\[0\le x_2 \le 2x_1 \text{,}\]](http://latex.artofproblemsolving.com/0/2/2/022158d922f05ea61ff38d739667b0e1158463d8.png)
![\[\vdots\]](http://latex.artofproblemsolving.com/7/6/4/76403adfdd35fb846675aaeaac92daec680a4fba.png)
![\[0\le x_{2003} \le 2x_{2004} \text{.}\]](http://latex.artofproblemsolving.com/6/a/0/6a0dd99faf558b05a6ab6570b9bdbdd432731893.png)
![\[Among all these sequences, find the one for which the following expression takes its largest value: S = ....\]](http://latex.artofproblemsolving.com/0/f/3/0f38b5f54f9c8143b94b18a8f034e91aed342b10.png)
When Pablo was going to copy the expression for 
, they erased the blackboard. The only thing he could remember was that was of the form
![\[S=\pm x_1\pm x_2\pm \cdots\pm x_{2002}+x_{2003}\]](http://latex.artofproblemsolving.com/e/9/f/e9ff6369960142dea4bafc4a3b8ccf6c2d55413e.png)
where the last term, 
, 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
Solution
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