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be the circumcenter, symmedian point of 
. Show that the tangent of 
at 
,the tangent of 
at 
, the tangent of 
at 
are concurrent.
be distinct natural numbers each of which has exactly 
positive integer factors. For instance, 
has exactly positive integer factors 
. Assume that no prime larger than 
divides any of the 
's. Show that there must be some perfect cube among the 's.
be an integer. Prove that 
can be written as a polynomial with integer coefficients in the variables 
, 
and 
.
, where 
are real numbers such that 
is an integral multiple of 
. Using (a) or otherwise show that if 
, then 
for all positive integers 
.
be the orthocenter of an acute-angled triangle 
. The circle 
centered at the midpoint of 
and passing through intersects the sideline at points 
and 
. Similarly, define the points 
, 
, 
and 
., , , , and are concyclic.
be the midpoints of 
.
to 
, Perpendicular from 
to 
and the Perpendicular from 
to 
. (Let them pass through 
the center of circle)
and 
we have![\[ -AB+2AR+BD-CD+AC-2AS=0 \]](http://latex.artofproblemsolving.com/2/5/3/253d6aabc1201c8495b22ac3095f1b67b7c7323f.png)
be an equilateral triangle and 
and 
,
and 
.
.