Difference between revisions of "1997 IMO Problems/Problem 3"

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==Solution==
 
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==See Also==
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{{IMO box|year=1997|num-b=2|num-a=4}}
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[[Category:Olympiad Geometry Problems]]
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[[Category:3D Geometry Problems]]

Latest revision as of 01:01, 17 November 2023

Problem

Let $x_{1}$, $x_{2}$,...,$x_{n}$ be real numbers satisfying the conditions

$|x_{1}+x_{2}+...+x_{n}|=1$

and

$|x_{i}| \le \frac{n+1}{2}$, for $i=1,2,...,n$

Show that there exists a permutation $y_{1}$, $y_{2}$,...,$y_{n}$ of $x_{1}$, $x_{2}$,...,$x_{n}$ such that

$|y_{1}+2y_{2}+...+ny_{n}|\le \frac{n+1}{2}$


Solution

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See Also

1997 IMO (Problems) • Resources
Preceded by
Problem 2
1 2 3 4 5 6 Followed by
Problem 4
All IMO Problems and Solutions