Difference between revisions of "2014 IMO Problems/Problem 6"
m (→Problem) |
|||
| Line 4: | Line 4: | ||
==Solution== | ==Solution== | ||
| + | Hi, | ||
| + | I was wondering if the problem seems a bit ambiguous. | ||
| + | As there are n set of lines in consideration, with the constraint that no two are parallel and none of them are concurrent (constraints 1 and 2). | ||
| + | Picking any three set of lines from set of n lines (nC3) will give a triangle formed by these lines because of the above constraints. This means if we color any more than 2 set of lines, at least 1 triangle is going to have all the three lines as blue. This sets the upper limit on the number of lines to be colored blue as 2 and not <math>\sqrt{n}</math>. Is this right? | ||
| + | |||
| + | Thanks and regards | ||
| + | ---- | ||
{{alternate solutions}} | {{alternate solutions}} | ||
Revision as of 12:15, 9 September 2015
Problem
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.
Solution
Hi,
I was wondering if the problem seems a bit ambiguous.
As there are n set of lines in consideration, with the constraint that no two are parallel and none of them are concurrent (constraints 1 and 2).
Picking any three set of lines from set of n lines (nC3) will give a triangle formed by these lines because of the above constraints. This means if we color any more than 2 set of lines, at least 1 triangle is going to have all the three lines as blue. This sets the upper limit on the number of lines to be colored blue as 2 and not . Is this right?
Thanks and regards
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
See Also
| 2014 IMO (Problems) • Resources | ||
| Preceded by Problem 5 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 6 |
| All IMO Problems and Solutions | ||
