The 55th International Mathematical Olympiad: Problems and Solutions

Day 1 (July 8th, 2014)

Problem 1
 

Let \( a_0< a_1< a_2< \cdots \) be an infinite sequence of positive integers. Prove that there exists a unique integer \( n\geq 1 \) such that \[ a_n< \frac{a_0+a_1+\cdots+a_n}n\leq a_{n+1}.\]

Problem 2
 

Let \( n\geq 2 \) be an integer. Consider an \( n\times n \) chessboard consisting of \( n^2 \) unit squares. A configuration of \( n \) rooks on this board is peaceful if every row and every column contains exactly one rook. Find the greatest positive integer \( k \) such that, for each peaceful configuration of \( n \) rooks, there is a \( k\times k \) square which does not contain a rook on any of its \( k^2 \) unit squares.

Problem 3
 

Convex quadrilateral \( ABCD \) has \( \angle ABC=\angle CDA=90^{\circ} \). Point \( H \) is the foot of the perpendicular from \( A \) to \( BD \). Points \( S \) and \( T \) lie on sides \( AB \) and \( AD \), respectively, such that \( H \) lies inside triangle \( SCT \) and \[ \angle CHS-\angle CSB=90^{\circ}, \quad\quad \angle THC-\angle DTC=90^{\circ}.\] Prove that line \( BD \) is tangent to the circumcircle of triangle \( TSH \).

Day 2 (July 9th, 2014)

Problem 4
 

Points \( P \) and \( Q \) lie on side \( BC \) of acute-angled triangle \( ABC \) such that \( \angle PAB=\angle BCA \) and \( \angle CAQ=\angle ABC \). Points \( M \) and \( N \) lie on lines \( AP \) and \( AQ \), respectively, such that \( P \) is the midpoint of \( AM \), and \( Q \) is the midpoint of \( AN \). Prove that lines \( BM \) and \( CN \) intersect on the circumcircle of triangle \( ABC \).

Problem 5
 

For each positive integer \( n \), the Bank of Cape Town issues coins of denominations \( \frac1n \). Given a finite collection of such coins (of not necessarily different denominations) with total value at most \( 99+\frac12 \), prove that it is possible to split this collection into \( 100 \) or fewer groups, such that each group has total value at most \( 1 \).

Problem 6
 

A set of lines in the plane is in general position 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 area; we call these its finite regions. Prove that for all sufficiently large \( n \), in any set of \( n \) lines in general position it is possible to color at least \( \sqrt n \) of the lines blue in such a way that none of its finite regions has a completely blue boundary.

Note: Results with \( \sqrt n \) replaced by \( c\sqrt n \) will be awarded points depending on the value of the constant \( c \).




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