2014 IMO Problems

Problem 3

Points $P$ and $Q$ lie on side $BC$ of 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 lines $BM$ and $CN$ intersect on the circumcircle of $\triangle{ABC}$ .

Solution

Problem 4

Points $P$ and $Q$ lie on side $BC$ of 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 lines $BM$ and $CN$ intersect on the circumcircle of $\triangle{ABC}$ .

Solution

Problem 5

For 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 $99+\tfrac{1}{2}$ , prove that it is possible to split this collection into $100$ or fewer groups, such that each group has total value at most $1$ .

Solution

Problem 6

A set of lines in the plane is in $\textit{general position}$ 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 are; we call these its $\textit{finite regions}$ . 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.

Solution

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2014 IMO Problems

Contents

Problem 3

Problem 4

Problem 5

Problem 6