AoPS Academy
be an acute-angled triangle, with circumcenter 
, circumradius 
and orthocenter 
. Let 
be a point on 
such that 
. Define 
and 
similarly.
, prove that is equilateral.
be a positive integer. There are two numbers written on a blackboard, one red and one blue. Initially, both are 0. We define the following procedure: at each step, we choose a nonnegative integer 
(not necessarily distinct from the previously chosen ones), and, if the red and blue numbers are 
and 
respectively, we replace them with 
and 
, which we color blue and red (in this order). We keep doing this procedure until the blue number is at least 
.
lies in the interior of the triangle 
. The lines 
, and 
intersect 
, and 
at points 
, and 
, respectively. Prove that if two of the quadrilaterals 
, and 
are concyclic, then all six are concyclic.
lies in the interior of the triangle . The lines , and intersect , and at points , and , respectively. Prove that if two of the quadrilaterals , and are concyclic, then all six are concyclic.
and 
.
