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, 
be the orthocenter of it and 
be any point on the side 
. The points 
are on the segments 
, respectively, such that the points 
and 
are cyclic. The segments 
and 
intersect at 
is a point on 
such that 
is tangent to the circumcircle of triangle 
at 
and 
intersect at 
. Prove that the points 
and lie on the same line.
be a finite set, and let 
be the set of all functions from to . Let 
be an element of , and let 
be the image of under . Suppose that 
for every 
in with 
. Show that 
.
and 
intersect each other at points 
and 
. Their external common tangent (closer to ) touches at 
and at 
. Let 
be the reflection of in line 
. Prove that 
.
satisfying :![\[n^{2p}=m^2+n^2+p+1\]](http://latex.artofproblemsolving.com/7/4/e/74ee592cb8b3f3a3996ea34f40b78fde9d3672fc.png)
and 
are integers and 
is a prime number.
are taken in the same plane and in another plane,will the same result hold??
,
and 
.
.
,
and 
is greater than 
.