AoPS Academy
with another positive integer 
, as long as 
is a perfect square. For example, if the number on the blackboard is 
, Carmela can replace it with 
, because 
, then replace it with 
, because 
. If the number on the blackboard is initially 
, determine all integers that Carmela can write on the blackboard after finitely many operations.
with circumcenter 
and orthocenter 
, let 
be an arbitrary point on the circumcircle of triangle such that does not lie on line 
and that line 
is not parallel to line 
. Let 
be the point on the circumcircle of triangle such that 
is perpendicular to , and let 
be the point on line 
such that 
. Let 
and 
be the points on the circumcircle of triangle such that 
is a diameter, and 
and 
are parallel. Let 
be the midpoint of 
.
and 
are perpendicular.
and 
are perpendicular.
with 
.Let 
be the circumcircle of 
and be centeriod of triangle .
is altitude of .
intersect with at 
.prove that circumcircle of triangle 
is tangent to 
.
be the orthocenter and the circumcenter . And let 
intersect at point 
. and let
intersect the arc at point 
. By a homegeneity with center and relation 
the nine pointe circle goes to the circumcircle of so we can find that 
so 
and angles
.we know that 
so we find that 
so the circumcircle of 
is tangant to the line 
such that 
and 
