AoPS Academy
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.
with real coefficient. It is known that 
and 
. Prove that 
can't have 
distinct real roots.
satisfy the condition 
. Inside this triangle, the point 
is chosen on the bisector of the angle 
such that 
. Prove that 
.
, a point 
onto its perpendicular bisector; reflection of in 
will cut 
at 
; extend 
to 
, and construction is done. See that 
or 
.
so that 
. Finally reflection 
of 
in 
is incenter of 
, because 
, thus 
and 
.
