IMO 2014 – Day 2


4. 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.

5. For every positive integer n, Cape Town Bank issues some coins that has \dfrac{1}{n} value. Let a collection of such finite coins (coins does not neccesarily have different values) which sum of their value is less than 99+\dfrac{1}{2}. Prove that we can divide the collection into at most 100 groups such that sum of all coins’ value does not exceed 1.

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 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

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