DAY 2: IMO 2005 México
4. Determine all positive integers relatively prime to all the terms of the infinite sequence \(a_n=2^n+3^n+6^n-1, n\ge1\).
5. Let ABCD be a fixed convex quadrilateral with BC=DA and BC not parallel with DA. Let two variable points E and F lie of the sides BC and DA, respectively and satisfy 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.
Prove that the circumcircles of the triangles PQR, as E and F vary, have a common point other than P.
6. In a mathematical competition in which 6 problems were posed to participants, every two of these problems were solved by more than 2/5 of the contestants. Moreover, no contestant solved all the 6 problems. Show that there are at least 2 contestants who solved exactly 5 problems each.
Reference:
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