AoPS Academy
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.
, the angle - bisector 
(
) and the median 
(
) intersect at point 
. Lines 
and 
intesect at point 
. The parallel through to 
intersects 
at point 
. Prove that 
, the angle - bisector () and the median () intersect at point . Lines and intesect at point . The parallel through to intersects at point . Prove that
is median, 
, thus 
. From angle bisector theorem, 
and 
. From 
we get 
. 
. From 
we get 
; with 
we are done.
