AoPS Academy
, define 
to be the set of all triplets 
of natural numbers, with 
odd and 
, such that 
and divides 
. Find all values of 
for which is finite.
be positive real numbers . Prove that
and 
in the plane, draws the perpendicular bisector of 
. Show that if three lines forming a triangle are drawn, Janabel can mark the orthocenter of the triangle using this device, a pencil, and no other tools.
be an acute, non-isosceles triangle with circumcenter 
, incenter 
and 
tangent to 
, 
, 
at 
respectively. Suppose that 
cuts 
at 
. Prove that 
bisects segment .
be an acute, non-isosceles triangle with circumcenter , incenter and tangent to , , at respectively. Suppose that cuts at . Prove that bisects segment .
and 
be midpoint of . Since 
and 
we get 
is cyclic, which we wanted to show.
