AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
such that the conditions
for all 
and 
for all 
where 
is the Euler's totient function.
and 
be positive integers. Find all pairs of non-negative integers 
and 
that always satisfy the following condition: white dots and black dots on a circle, there always exist a line cutting the circle into two arcs, one of which consists of exactly white dots and black dots.
divides 
for positive integers 
then
be a triangle with circumcircle 
and circumcenter 
. Let 
and 
represent the 
-excircle and -excenter, respectively. Denote by 
the circle tangent to 
, 
, and on the arc not containing , and similarly for 
. Let the tangency points of 
with line be 
, respectively. Let 
be the intersection point of 
and . Define 
as the point on segment 
such that 
. Suppose that 
intersects again at 
. Let 
be the touch point of the -mixtilinear incircle and , and let 
be the antipode of with respect to . Let 
be the intersection of 
and 
.
is the radical axis of and .
where 
and 
are positive integers that do not exceed 
?
denote the set of positive integers
such that for all 
it holds that 
intersecting forming 
. By isogonal conjugate and pedal circle we get that 
are concurrent at 
isogonal conjugate to 
wrt. . Following the Miquel structure for 
so 
lie on a circle. Similarly so 
lie on a circle. By simple angle chasing so 
lie on a circle.
such that 
is also prime.
be projection point from 
.
cuts circumcircle 
at 
.
be antipode of 
are cyclic.
are equidistant from the circumcenter of 
.