Difference between revisions of "2014 IMO Problems"
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| + | ==Problem 5== | ||
| + | For each positive integer <math>n</math>, the Bank of Cape Town issues coins of denomination <math>\tfrac{1}{n}</math>. Given a finite collection of such coins (of not necessarily different denominations) with total value at most <math>99+\tfrac{1}{2}</math>, prove that it is possible to split this collection into <math>100</math> or fewer groups, such that each group has total value at most <math>1</math>. | ||
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| + | [[2014 IMO Problems/Problem 5|Solution]] | ||
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| + | ==Problem 6== | ||
A set of lines in the plane is in <math>\textit{general position}</math> if no two are parallel and no three pass through the same point. A set of lines in general position cuts the plane into regions, some of which have finite are; we call these its <math>\textit{finite regions}</math>. Prove that for all sufficiently large <math>n</math>, in any set of <math>n</math> lines in general position it is possible to colour at least <math>\sqrt{n}</math> of the lines blue in such a way that none of its finite regions has a completely blue boundary. | A set of lines in the plane is in <math>\textit{general position}</math> if no two are parallel and no three pass through the same point. A set of lines in general position cuts the plane into regions, some of which have finite are; we call these its <math>\textit{finite regions}</math>. Prove that for all sufficiently large <math>n</math>, in any set of <math>n</math> lines in general position it is possible to colour at least <math>\sqrt{n}</math> of the lines blue in such a way that none of its finite regions has a completely blue boundary. | ||
[[2014 IMO Problems/Problem 6|Solution]] | [[2014 IMO Problems/Problem 6|Solution]] | ||
Revision as of 05:45, 9 October 2014
Problem 5
For each positive integer 
, the Bank of Cape Town issues coins of denomination 
. Given a finite collection of such coins (of not necessarily different denominations) with total value at most 
, prove that it is possible to split this collection into 
or fewer groups, such that each group has total value at most 
.
Problem 6
A set of lines in the plane is in 
if no two are parallel and no three pass through the same point. A set of lines in general position cuts the plane into regions, some of which have finite are; we call these its 
. Prove that for all sufficiently large , in any set of lines in general position it is possible to colour at least 
of the lines blue in such a way that none of its finite regions has a completely blue boundary.
