MathDB

2

Part of 2010 IberoAmerican

Problems(2)

Problem 2, Iberoamerican Olympiad 2010

Source:

9/24/2010
Determine if there are positive integers a,ba, b such that all terms of the sequence defined by x_{1}= 2010,x_{2}= 2011\\ x_{n+2}= x_{n}+ x_{n+1}+a\sqrt{x_{n}x_{n+1}+b}  (n\ge 1) are integers.
inductionnumber theory proposednumber theory
Problem 5, Iberoamerican Olympiad 2010

Source:

9/25/2010
Let ABCDABCD be a cyclic quadrilateral whose diagonals ACAC and BDBD are perpendicular. Let OO be the circumcenter of ABCDABCD, KK the intersection of the diagonals, L≠O L\neq O the intersection of the circles circumscribed to OACOAC and OBDOBD, and GG the intersection of the diagonals of the quadrilateral whose vertices are the midpoints of the sides of ABCDABCD. Prove that O,K,LO, K, L and GG are collinear
geometrycircumcircleparallelogramanalytic geometryrectanglesymmetrycyclic quadrilateral