International Mathematical Olympiad · 2014

Problems

Problem 1Let $a_0<a_1<a_2<\cdots$ be an infinite sequence of positive integers. Prove that there exists a unique integer $n\ge1$ such that $a_n<\frac{a_0+a_1+\cdots+a_n}{n}\le a_{n+1}.$ Solutions: 1
Problem 2Let $n\ge2$ 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.Solutions: 1
Problem 3Convex 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 $\angle CHS-\angle CSB=90^\circ$ , $\angle THC-\angle DTC=90^\circ$ . Prove that line $BD$ is tangent to the circumcircle of triangle $TSH$ .Solutions: 1
Problem 4Points $P$ and $Q$ lie on side $BC$ of an acute-angled triangle $ABC$ so 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 the intersection of lines $BM$ and $CN$ lies on the circumcircle of triangle $ABC$ .Solutions: 1
Problem 5For each positive integer $n$ , the Bank of Cape Town issues coins of denomination $\tfrac{1}{n}$ . Given a finite collection of such coins (of not necessarily different denominations) with total value at most $99+\tfrac12$ , prove that it is possible to split this collection into $100$ or fewer groups, such that each group has total value at most $1$ .Solutions: 1
Problem 6A 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; 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 colour at least $\sqrt{n}$ of the lines blue in such a way that none of its finite regions has a completely blue boundary.Solutions: 1