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 an acute triangle with 
and 
. Let 
be the midpoint of the side 
. The circumcircle of triangle 
intersects the side 
again at 
and the circumcircle of triangle 
intersects the side 
again at 
. The point 
lies on the perpendicular bisector of the segment so that the points and 
lie on the same side of 
and 
. Prove that the circumcircles of triangles 
and 
are tangent.
![\[ p = \frac{2ab}{a + b} \]](http://latex.artofproblemsolving.com/0/f/4/0f4297ec59fe97c92fda75af4cc835be6b9a2507.png)
, ![\[ p, a, b \in \mathbb{C} \]](http://latex.artofproblemsolving.com/b/4/4/b4499970ac49d770a810d7345fedef7773cd03f5.png)
such that there exists a pair 
of positive integers, such that 
is not divisible by the cube of any prime, and 
has rational sides and equal angles. Show that it is regular.
such that the equation
be a right angled triangle with 
and 
its altitude. We draw parallel lines from 
to the vertical sides of the triangle and we call 
their points of intersection with and respectively. The parallel line from 
to 
intersects the line at the point 
. Let 
be the symmetric of with respect to the line and 
the projections of 
onto and respectively. If 
is the point of intersection of the lines 
and 
, prove that 
.
and 
that satisfy the equation 
Prove that
Prove that
Let 
Prove that
