Difference between revisions of "2014 IMO Problems"
m (Created page with "==Problem== 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 po...") |
Etmetalakret (talk | contribs) |
||
(9 intermediate revisions by 3 users not shown) | |||
Line 1: | Line 1: | ||
− | ==Problem== | + | ==Problem 1== |
+ | Let <math>a_0<a_1<a_2<\cdots \quad </math> be an infinite sequence of positive integers, Prove that there exists a unique integer <math>n\ge1</math> such that | ||
+ | <cmath>a_n<\frac{a_0+a_1+\cdots + a_n}{n}\le a_{n+1}.</cmath> | ||
+ | |||
+ | [[2014 IMO Problems/Problem 1|Solution]] | ||
+ | |||
+ | ==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== | ||
+ | Convex quadrilateral <math>ABCD</math> has <math>\angle{ABC}=\angle{CDA}=90^{\circ}</math>. Point <math>H</math> is the foot of the perpendicular from <math>A</math> to <math>BD</math>. Points <math>S</math> and <math>T</math> lie on sides <math>AB</math> and <math>AD</math>, respectively, such that <math>H</math> lies inside <math>\triangle{SCT}</math> and | ||
+ | <cmath>\angle{CHS}-\angle{CSB}=90^{\circ},\quad \angle{THC}-\angle{DTC} = 90^{\circ}.</cmath> | ||
+ | |||
+ | Prove that line <math>BD</math> is tangent to the circumcircle of <math>\triangle{TSH}.</math> | ||
+ | |||
+ | [[2014 IMO Problems/Problem 3|Solution]] | ||
+ | |||
+ | ==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>. | ||
+ | |||
+ | [[2014 IMO Problems/Problem 4|Solution]] | ||
+ | |||
+ | ==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>. | ||
+ | |||
+ | [[2014 IMO Problems/Problem 5|Solution]] | ||
+ | |||
+ | ==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]] | ||
+ | |||
+ | {{IMO box|year=2014|before=[[2013 IMO Problems]]|after=[[2015 IMO Problems]]}} |
Latest revision as of 18:35, 14 March 2021
Problem 1
Let be an infinite sequence of positive integers, Prove that there exists a unique integer such that
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
Convex quadrilateral has . Point is the foot of the perpendicular from to . Points and lie on sides and , respectively, such that lies inside and
Prove that line is tangent to 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.
2014 IMO (Problems) • Resources | ||
Preceded by 2013 IMO Problems |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 2015 IMO Problems |
All IMO Problems and Solutions |