AoPS Academy
. Let 
and 
be the feet of the altitudes from 
to 
and from 
to 
respectively.
and 
as the reflections of across lines 
and , respectively. Let 
be the circumcircle of 
. Denote by 
the second intersection of line 
with , and by 
the intersection of ray 
with .
is the circumcenter of , prove that , , and are collinear if and only if quadrilateral 
can be inscribed within a circle.
+1 w
of positive integers satisfying the following two conditions:
, if 
is in 
, then so are 
and 
; and contains an integer 
such that 
is not divisible by 
.
+1 w
and 
of a tetrahedron 
are tangent to the edge in the same point. Prove that the points of tangency of these incircles to the edges 
are concyclic.
and of a tetrahedron are tangent to the edge in the same point. Prove that the points of tangency of these incircles to the edges are concyclic.
at 
respectively. Then the lines 
and 
pass through the point on the line for which 
. Thus 
and the points 
are concyclic.
such that 
.
?
