AoPS Academy
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 
.
are positiv real number, such that 
prove inequality 
is given with 
. Point 
is the orthogonal projection of point 
on the bisector of angle 
. Point 
is the midpoint of 
. Knowing that 
, 
, 
, calculate the length of the segment 
.
is given with . Point is the orthogonal projection of point on the bisector of angle . Point is the midpoint of . Knowing that , , , calculate the length of the segment .
and 
.
