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be positive real integer such that 
Prove that
such that 
and 
that verify the following equation for all 
:
is a prime number
be an acute scalene triangle with circumcenter 
. Let 
be the feet of the altitudes from 
to 
respectively. Let 
be the midpoint of 
. Let 
be the circle with diameter 
. Let 
be the intersection of 
and 
. Let 
be the orthocenter of . Let 
be the intersection of 
and . Let 
be the lines through 
tangent to 
respectively. Let 
be the intersection of 
and 
. Let 
be the intersection of 
and 
. Let 
be the line through parallel to 
and let 
be the reflection of across . Prove that 
is tangent to 
.
, in which 
. Point lies on the side is such that 
. Prove that the line passing through the midpoints of the segments 
and 
is parallel to the bisector of the angle .
, in which . Point lies on the side is such that . Prove that the line passing through the midpoints of the segments and is parallel to the bisector of the angle .
