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.
such that 
has a prime divisor greater than 
.
be an interior point of the convex quadrilateral 
. Construct triangles 
and 
on the outside of the quadrilateral such that the similarities 
and 
hold. Let 
and 
be the projections of on the lines 
and 
, respectively. Prove that if the quadrilateral 
is cyclic, then![\[EF\cdot CD=EG\cdot DA=EH\cdot AB=EI\cdot BC.\]](http://latex.artofproblemsolving.com/0/8/e/08e13ecbf07fb1e8950fa34e40cb7f08a8c5a7e1.png)
, let 
be all the positive integers smaller than that are coprime to . Find all 
such that
for all 
.
and 
are arranged so that 
lies on the segment 
and lies on the segment 
. Point 
is the midpoint of 
. Prove that 
.
be an acute triangle with incentre 
and 
. Let lines 
and 
intersect the circumcircle of at 
and 
, respectively. Consider points 
and such that 
and 
are parallelograms (with 
, and 
). Let 
be the point of intersection of lines 
and 
. Prove that points 
, and are concyclic.
Find the range of values that can be taken by :
are positive reals.
such that for all 
,
be a convex quadrilateral with side lengths 
, 
, 
. Let and be the midpoints of diagonal and respectively and 
. Find the area of the quadrilateral .
be reals such that 
. Prove that
and 
Prove that
specifies the greatest common divisor of 
and 
.
Prove that
and 
Prove that
Let 
Prove that
.
.