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.
and two fixed points 
on it. 
is another point on , and 
is the midpoint of 
. 
is the foot of the perpendicular from to 
passes through a fixed point (as varies);.
and two fixed points on it. is another point on , and is the midpoint of . is the foot of the perpendicular from to passes through a fixed point (as varies);.
be the point diametrically opposite to 
and let 
be the centre of . Let 
and 
. Observe that
Thus 
. Therefore 
is our desired fixed point.
, the locus of is precisely 
.
