2014 IMO Problems
Let be an infinite sequence of positive integers, Prove that there exists a unique integer such that
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.
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
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 .
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 .
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.
Let be the set of integers. Determine all functions such that, for all integers and ,
In triangle , point lies on side and point lies on side . Let and be points on segments and , respectively, such that is parallel to . Let be a point on line , such that lies strictly between and , and . Similarly, let be the point on line , such that lies strictly between and , and .
Prove that points , and are concyclic.
A social network has users, some pairs of whom are friends. Whenever user is friends with user , user is also friends with user . Events of the following kind may happen repeatedly, one at a time: Three users , , and such that is friends with both and , but and are not friends, change their friendship statuses such that and are now friends, but is no longer friends with , and no longer friends with . All other friendship statuses are unchanged. Initially, users have friends each, and users have friends each. Prove that there exists a sequence of such events after which each user is friends with at most one other user.
Find all pairs of positive integers such that
The Bank of Bath issues coins with an on one side and a on the other. Harry has of these coins arranged in a line from left to right. He repeatedly performs the following operation:
If there are exactly coins showing , then he turns over the coin from the left; otherwise, all coins show and he stops. For example, if the process starting with the configuration would be , which stops after three operations.
(a) Show that, for each initial configuration, Harry stops after a finite number of operations.
(b) For each initial configuration , let be the number of operations before Harry stops. For example, and . Determine the average value of over all possible initial configurations .
Let be the incenter of acute triangle with . The incircle of is tangent to sides , , and at , , and , respectively. The line through perpendicular to meets ω again at . Line meets ω again at . The circumcircles of triangles and meet again at . Prove that lines and meet on the line through perpendicular to .
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