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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.
be the feet of perpendiculars from 
to . Note that 
, so 
. Some length chasing![\[YE=DY+BD-BE\]](http://latex.artofproblemsolving.com/8/0/b/80b4244a52be298f324cdf123e5a738e406d4846.png)
![\[=\frac12(AD+DC-AC)+BD-\frac12(AB+BC-AC)\]](http://latex.artofproblemsolving.com/e/4/d/e4d8bc7c4ca0da86810416c719e478ce0d76f711.png)
![\[=\frac12(AD+BD-AB)=DX\]](http://latex.artofproblemsolving.com/6/e/1/6e155df05123841782f4b7f9b10a246aea2c2e78.png)
, so 
. Thus 
so 
is cyclic.
such that 
and 
are incenters of triangles 
,
to 
are cyclic.
