,
be positive integers. Julia and Florian play a game on a 
board. Julia
is called 
,
is equal to the number of distinct elements in the set 
.
of positive real numbers that satisfy the system:![\[
\begin{aligned}
11bc - 36b - 15c &= abc \\
12ca - 10c - 28a &= abc \\
13ab - 21a - 6b &= abc.
\end{aligned}
\]](http://latex.artofproblemsolving.com/1/7/0/170e1cfaee90f5c37428fa75192ac889261a8f83.png)
be a positive integer. Prove that there are infinitely many primes, each of which divides at least one term of the sequence 
,
.
Y by Adventure10, Mango247
A triangle 
is given. A circle 
, passing through 
, is tangent to side 
at point 
and intersects sides 
and 
at 
and 
respectively. Prove that the smaller arcs 
and 
of are equal iff is tangent to the circumcircle of 
at .
is given. A circle 
, passing through 
, is tangent to side 
at point 
and intersects sides 
and 
at 
and 
respectively. Prove that the smaller arcs 
and 
of are equal iff is tangent to the circumcircle of 
at .
.
.
and 
.
and 
.
be the center of 
(asume that 
)
(where 
is circumcentre). Hence 
)
be the second intersection of line 
with 
which implies that 
or 
.
