IMO 2013 – Day 1

1. Prove that for any two positive integers k, n there exist positive integers m_1, m_2, \ldots, m_k such that 1+\dfrac{2^k-1}{n}=\left(1+\dfrac{1}{m_1}\right)\left(1+\dfrac{1}{m_2}\right)\dots\left(1+\dfrac{1}{m_k}\right).

2. Given 2013 red and 2014 blue points in the plane, no three of them on a line. We aim to split the plane by lines (not passing through these points) into regions such that there are no regions containing points of both the colors. What is the least number of lines that always suffice?

3. Let ABC be a triangle and let A_1, B_1, and C_1 be points of contact of the excircles with the sides BC, AC, and AB, respectively. Prove that if the circumcenter of \triangle A_1B_1C_1 lies on the circumcircle of \triangle ABC, then \triangle ABC is a right triangle.

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