Difference between revisions of "2014 IMO Problems"
m |
m |
||
| Line 1: | Line 1: | ||
| + | ==Problem 2== | ||
| + | Let <math>n\ge2</math> be an integer. Consider an <math>n\times n</math> chessboard consisting of <math>n^2</math> unit squares. A configuration of <math>n</math> rooks on this board is <math>peaceful</math> if every row and every column contains exactly one rook. Find the greatest positive integer <math>k</math> such that, for each peaceful configuration of <math>n</math> rooks, there is a <math>k\times k</math> square which does not contain a rook on any of its <math>k^2</math> squares. | ||
| + | |||
| + | [[2014 IMO Problems/Problem 2|Solution]] | ||
| + | |||
==Problem 3== | ==Problem 3== | ||
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>. | 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>. | ||
Revision as of 04:47, 9 October 2014
Problem 2
Let 
be an integer. Consider an 
chessboard consisting of 
unit squares. A configuration of 
rooks on this board is 
if every row and every column contains exactly one rook. Find the greatest positive integer 
such that, for each peaceful configuration of rooks, there is a 
square which does not contain a rook on any of its 
squares.
Problem 3
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 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.
