# 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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