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.
, on line 
it is given point 
such that 
(point 
is between points 
and ), and on line 
it is given point 
(
) such that 
. If 
, prove that 
, on line it is given point such that (point is between points and ), and on line it is given point () such that . If , prove that 
is the centroid of triangle 
. Letting 
be the midpoint of 
, 
= 
= and so is the centre of circle 
. So 
is a right angle which implies the desired result.
.By using Menelaus we get 
is centroid. Since 
and 
. So 
and this implies the desired result
