News: IMO Paper 2 (14 July 2005)

Problem 4. Consider the sequence a₁, a₂, ... defined by

a_n = 2ⁿ + 3ⁿ + 6ⁿ - 1  (n = 1, 2, ...).

Determine all positive integers that are relatively prime to every term of the sequence.

Problem 5. Let ABCD be a given convex quadrilateral with sides BC and AD equal in length and not parallel. Let E and F be interior points of the sides BC and AD respectively such that BE = DF. The lines AC and BD meet at P, the lines BD and EF meet at Q, the lines EF and AC meet at R. Consider all the triangles PQR as E and F vary. Show that the circumcircles of these triangles have a common point other than P.

Problem 6. In a mathematical competition 6 problems were posed to the contestants. Each pair of problems was solved by more than 2/5 of the contestants. Nobody solved all the 6 problems. Show that there were at least 2 contestants who each solved exactly 5 problems.

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