[/quote]
,
its circumcentre. A line tangent to the circumcircle of the triangle 
intersects sides 
at 
at 
.
be the image of 
under 
.
is tangent to the circumcircle of triangle 
be a semicircle with diameter 
.
touches 
,
in a point 
.
touches 
and 
and 
are perpendicular.
with no parallel sides is inscribed in a circle 
.
are inscribed in triangles 
,
and 
,
,
,
.
and 
Prove that
,
,
,
are integers such that 
.
and 
of 
such that for each integer 
with 
,![\[ n\mid a_i - b_i - c_i
\]](http://latex.artofproblemsolving.com/1/a/e/1ae8d54f068b646ac7a1e43d75e7f26510cfdc95.png)
be an odd prime and 
be integers so that the integers 
are divisible by 
Prove that 
be real numbers such that 
Prove that 
Where 
.
and 
respectively denote the midpoints of the arcs 
denote the midpoint of arc 
(the arc 
be the circle that is tangent to 
be the circle that is tangent to 
,
friends, 
people are seated around a circular table in a fixed order. Ali places 
.
lie on 
,
is tangent to circumcircle of 
.
is tangent to circumcircle of triangle 
Y by Adventure10, Mango247
Let 
be a triangle with 
, 
its incenter and 
the intersection point of line 
with side 
. Let 
and 
be points on sides 
and 
, respectively, such that 
and 
. The circumcircle of triangle 
intersects again line at 
. Prove that quadrilateral 
is cyclic.
be a triangle with 
, 
its incenter and 
the intersection point of line 
with side 
. Let 
and 
be points on sides 
and 
, respectively, such that 
and 
. The circumcircle of triangle 
intersects again line at 
. Prove that quadrilateral 
is cyclic.
is parallel to 
and 
is cyclic.
