is written on the board. A move consists of adding either 
or 
to the number written on the board, but only if the chosen number is coprime with the current number (for example, if the current number is 
,
of positive integers is called central if for every positive integer 
, the arithmetic mean of the first 
terms of the sequence is equal to 
of positive integers such that for every central sequence 
there are infinitely many positive integers 
.
be a triangle. Let 
be a circle passing through 
intersecting 
at 
and 
at 
.
be the intersection of 
and 
and 
be the intersections of 
and 
with the tangent to 
at 
.
be the second intersection of 
and 
and 
Y by Adventure10, Mango247
Let 
be a parallelogram. Construct squares 
on the outer side of the parallelogram. Construct a square having 
as one of its sides and it is on the outer side of 
and call its center 
. Similarly do it for 
to obtain 
. Prove that 
.
be a parallelogram. Construct squares 
on the outer side of the parallelogram. Construct a square having 
as one of its sides and it is on the outer side of 
and call its center 
. Similarly do it for 
to obtain 
. Prove that 
.
the centers of the squares 
respectively. Since 
and 
it follows that 
by SAS 
and 
i.e 
is isosceles right at 
Likewise, 
is isosceles right at 
thus quadrilateral 
is a square with side length 
we deduce that 
and 
are congruent and perpendicular 
Similarly, 
equal 
