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such that 
divides 
for all positive integers 
and 
with 
.
be a triangle with circumcenter 
and internal bisector 
. Let cuts 
again at 
and 
cuts again at 
. Point 
lie on such that 
. The line pass through and perpendicular to intersects 
at 
respectively. Let circumcircle of 
cuts at 
. Prove that 
.
such that the number
is the product of two or more consecutive numbers.
be a triangle. The incircle of touches the sides 
and 
at the points 
and 
, respectively. Let 
be the point where the lines 
and 
meet, and let 
and 
be points such that the two quadrilaterals 
and 
are parallelogram.
.
be the orthocenter of an acute-angled triangle and 
a point on the circumcenter of triangle . Prove that the Simson line of bisects the segment ![$[P H]$](http://latex.artofproblemsolving.com/5/8/7/587225c92de7ccb1a3837835001032884cf125b8.png)
.
