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.
is the midpoint of the base 
of trapezoid 
. On base 
, point 
is selected. Line 
intersects line 
at point 
, and the perpendicular from on the bases intersects line 
at point 
. Prove that 
.
is the midpoint of the base of trapezoid . On base , point is selected. Line intersects line at point , and the perpendicular from on the bases intersects line at point . Prove that .
cuts at 
. So by Thales's theorem,
. But 
, so 
. We also have that 
, hence 
is isoceles at . So 
intersects 
.Enough to show: 
or 
.notice that 
and 
are homothetic and 
are collinear so