with no parallel sides is inscribed in a circle 
.
are inscribed in triangles 
,
and 
,
,
,
.
by 
and the number of distinct prime divisors of 
.
be a positive integer. Prove that there exist infinitely many positive integers 
such that 
and 
does not divide 
for any positive integers 
satisfying 
.
such that for any 
:
![\[
\begin{cases}
x - y z = 1,\\[2pt]
y - z x = 2,\\[2pt]
z - x y = 4.
\end{cases}
\]](http://latex.artofproblemsolving.com/b/2/d/b2d94e49491a71f86109103f985fe3f238cf4d95.png)
such that 
for all 
be a triangle with circumcircle 
and let 
be the 
-
and 
be the intersection points of 
intersects the tangent line at 
at a point 
.
.
and 
: Prove that :
board and made 
moves so that it visited all squares and returned to the starting square. At every moment, the distance from the center of the square the king was on to the center of the board was calculated. A move is called 
if this distance becomes smaller after the move. Find the maximum possible number of pleasant moves. (The chess king moves to a square adjacent either by side or by corner.)
,
,
,
such that 
and 
for 
.
,
Y by
In a rectangle 
two not intersecting circles 
and 
are drawn such that is tangent to 
and 
at points 
and 
respectively, and is tangent to 
and 
at 
and 
respectively. It is known that 
.
Find the distance between centers of circles and .
two not intersecting circles 
and 
are drawn such that is tangent to 
and 
at points 
and 
respectively, and is tangent to 
and 
at 
and 
respectively. It is known that 
.Find the distance between centers of circles
and .
