AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
be non-negative real numbers such that 
Prove that
be the circumcenter, symmedian point of 
. Show that the tangent of 
at 
,the tangent of 
at 
, the tangent of 
at 
are concurrent.
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.
be an acute triangle. Let 
be a point on side 
and 
be a point on side 
such that lines 
and 
are parallel. Let 
be an interior point of 
. Suppose rays 
and 
meet side at points 
and 
, respectively, such that both and lie between and . Suppose that the circumcircles of triangles 
and 
intersect at a point 
. Prove that the points 
, and 
are collinear.
. Prove that:
and 
, define 
as the greatest exponent 
such that 
divides 
. Prove that there are infinitely many such that 
and infinitely many 
such that 
.
with circumcircle 
is given. Let be the midpoint of , be the foot of altitude from to , and 
be the intersection point of and . Point 
lies on the arc of (the one that does not contain ) such that 
. Prove that 
.
and 
intersect at two distinct points 
and 
. A line through intersects and again at 
and 
, respectively. Suppose that 
. Show that the centre of lies on .
be a point on 
such that is the midpoint of the arc not containing . We have, ![\[\angle AOB = 180 - \angle ACB = 180 - (180 - 2\angle ADB) = 2\angle ADB.\]](http://latex.artofproblemsolving.com/1/4/6/1468d9e55c461c38a14c3486b725f228c37dd19c.png)
be a point on such that is the midpoint of the arc not containing . We have, as the mid-point only? Even if you consider to be any point (obviously not coinciding with either or ) on the arc 
not containing , your logic remains true..
and 
.
