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.
with orthocenter 
. Extend 
to meet 
at 
. The angle bisector of 
meets the midpoint of 
. If 
, compute the area of 
.
with side length of 
. The point 
is selected on the diagonal 
such that 
. Denote the circumcenters of 
as 
. Find the length of 
such that 
. Denote the midpoints of 
as 
respectively, compute the area of 
given the area of is 
.
with 
. Point 
lies inside the rectangle such that 
is equilateral. Given that 
and the midpoint of are on the same line, compute the length of .
be reals with 
and 
. Prove that
people seated around a circular table, and 
cookies
be one of these people. Find the least such that no matter cookies are distributed initially, there is a strategy to pass receives at least one cookie.
be a triangle with 
. Denote the orthocenter of triangle , 
the midpoint of segment ![$[AH]$](http://latex.artofproblemsolving.com/0/3/b/03b8986ebe750b377f987f87b41a1dbc4c128e17.png)
, 
the midpoint of segment ![$[BC]$](http://latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
and the intersection point of the angle bisector of 
with the segment ![$[MN]$](http://latex.artofproblemsolving.com/1/6/2/162e03b9cf481dcfb3d5bdf078be84feab5d2f6e.png)
. Prove that 
be a triangle with . Denote the orthocenter of triangle , the midpoint of segment , the midpoint of segment and the intersection point of the angle bisector of with the segment . Prove that 
be the circumcenter of . is the midpoint of , we want to prove that is on the Thales circle around which is equivalent to 
.
, is also the angular bisector of 
. Hence it suffices to prove that 
i.e. that 
and 
are parallel. But since clearly 
(both are perpendicular to ) this is equivalent to 
being a parallelogram i.e. it suffices to check that 
and 
have the same length i.e. that is half as long as .
be the point of reflection of by . It is well-known that lies on the circumcircle. Moreover, we have 
so 
so is opposite to on the circumcircle. But then 
are collinear so triangles 
and 
are similar whence finally 
.
altitudes of 
. Clearly 
is radical axis of the circles of diameters 
, thus 
. Since is perpendicular bisector of , we get its intersection with the bisector of lies onto the circle 
.
the reflection of in , 
the orthogonal projection of into the bisector of 
. We aim to show that 
are collinear which should do it.
passes through the circumcenter of thus
.
are collinear and we are done.
be non-negative reals such that 
.
