Difference between revisions of "2014 IMO Problems"
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| + | ==Problem 4== | ||
| + | Points <math>P</math> and <math>Q</math> lie on side <math>BC</math> of acute-angled <math>\triangle{ABC}</math> so that <math>\angle{PAB}=\angle{BCA}</math> and <math>\angle{CAQ}=\angle{ABC}</math>. Points <math>M</math> and <math>N</math> lie on lines <math>AP</math> and <math>AQ</math>, respectively, such that <math>P</math> is the midpoint of <math>AM</math>, and <math>Q</math> is the midpoint of <math>AN</math>. Prove that lines <math>BM</math> and <math>CN</math> intersect on the circumcircle of <math>\triangle{ABC}</math>. | ||
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| + | [[2014 IMO Problems/Problem 4|Solution]] | ||
==Problem 5== | ==Problem 5== | ||
Revision as of 04:46, 9 October 2014
Problem 4
Points 
and 
lie on side 
of acute-angled 
so that 
and 
. Points 
and 
lie on lines 
and 
, respectively, such that is the midpoint of 
, and is the midpoint of 
. Prove that lines 
and 
intersect on the circumcircle of .
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.
