IMO 2014 - Competition Day 2

Now that the second day of the competition is over, the job is over for the students, and it is now up to the team leaders and deputy leaders to debate and fight for marks in Coordination.

Here are all six problems from the two days:

And here are the questions from Day 2, if you want to give them a try:

Problem 4 (submitted by Georgia)

Let \(P\) and \(Q\) be on segment \(BC\) of an acute triangle \(ABC\) such that \(\angle PAB=\angle BCA\) and \(\angle CAQ=\angle ABC\). Let \(M\) and \(N\) be the points on \(AP\) and \(AQ\), respectively, such that \(P\) is the midpoint of \(AM\) and \(Q\) is the midpoint of \(AN\). Prove that the intersection of \(BM\) and \(CN\) is on the circumference of triangle \(ABC\).

Problem 5 (submitted by Luxembourg)

For every positive integer \(n\), Cape Town Bank issues some coins that have \(\dfrac{1}{n}\) value each. Consider a finite collection of coins (coins do not necessarily have different values) where the sum of their values is less than \(99+\dfrac{1}{2}\). Prove that we can divide the collection into at most \(100\) groups such that the sum of the coins' values in each group does not exceed \(1\).

Problem 6 (submitted by Austria/USA)

A set of lines in the plane is in general position if no two lines are parallel and no three lines 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 colour at least \(\sqrt{n}\) 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 \(c\).