is written on the blackboard. Every time, Carmela can erase the number 
on the black board and replace it with a new number 
,
is a perfect square. Prove or disprove that all positive integers 
can be written exactly once on the blackboard.
and 
,
be some point lying on 
such that 
lie on 
be some circle passing through 
be its center. Let 
be some point lying on the circumcircle of 
.
is tangent to 
is tangent to 
be an acute triangle, and let 
to 
and 
,
meet the circumcircle of 
and 
.
,
be the midpoint of 
.
and 
exist on 
and 
are tangent to 
,
,
,
.
at the orthocenter of 
be its orthocenter, and let 
,
be a point lying on 
such that 
.
.
be the midpoint of minor arc 
on the circumcircle of 
at points 
,
the intersection of 
and 
.
and 
denote the 
,
,
and 
,
be the intersection of 
and 
.
and 
are tangent to each other.
and 
be circles tangent to 
meets 
.
similarly.
and 
.
and 
.
and 
touch externally at 
and the radius of 
is any point on 
and 
are points on 
and 
are tangent to 
,
intersect 
,
respectively. 
is the intersection of the tangent at 
.
?
Y by
Let 
and 
be points on the circle 
such that 
is a convex quadrilateral. Suppose that 
and 
intersect at a point 
such that 
is between 
and and that 
and 
intersect at a point 
. Let 
be the point on such that 
and 
are parallel. Let 
be the reflection of through and suppose that is inside the circle .
Show that
, and are collinear.
and 
be points on the circle 
such that 
is a convex quadrilateral. Suppose that 
and 
intersect at a point 
such that 
is between 
and and that 
and 
intersect at a point 
. Let 
be the point on such that 
and 
are parallel. Let 
be the reflection of through and suppose that is inside the circle .Show that
, and are collinear.
![[asy]
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[/asy]](http://latex.artofproblemsolving.com/a/1/4/a141a66de0a3d713500f6c07bd3c953585431298.png)
is a cyclic quad.
therefore proving that 
and 
and let 
meet 
at 
and 
,
.
and 
bisects 
so if 
then 
is cyclic so we'll prove it in order to solve the Problem. 
so 
