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be an acute scalene triangle , 
be the orthogonal projection of 
on 
, 
and 
are the midpoints of 
and 
respectively. Let 
and 
are points on the minor arcs 
and 
of the circumcircle of triangle respectively such that 
. Show that the circumcircles of triangles 
and 
are tangent if and only if lies on 
.
, 
, 
, ... be an infinite sequence of real numbers satisfying the equation 
for all 
, where and are two different positive reals., , , ... be bounded?
, the altitude from meets the circumcircle again at 
. Let 
be the circumcenter. The lines 
and 
intersect the side at and , respectively. Prove that![\[\frac{S_{AQC}}{S_{CMT}} = \biggl( \frac{ \sin B}{\cos C} \biggr)^2 .\]](http://latex.artofproblemsolving.com/d/1/7/d173437034a42cd5ab1fc664a5d5e87fc86d1ae0.png)
and 
it follows that 
Thus
such that:
