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 is a multiple of 
and has exactly positive factors.
such that:
with all 
is real number
opposite the vertex 
be tangent to the side 
at the point 
. Define the points 
on 
and 
on 
analogously, using the excircles opposite 
and 
, respectively. Suppose that the circumcentre of triangle 
lies on the circumcircle of triangle . Prove that triangle is right-angled.
be the circumcircle of triangle , 
is the midpoint of and 
be the second intersection of 
and . Tangent line of at intersects at 
, let 
be a transversal of and 
be intersections of 
with . Let 
be a tangent line of . Prove that 
is cyclic
be positive integers such that 
and 
. Find the maximum possible value of 
.
, let the incircle touch sides 
at 
, respectively. Let lie on the line through 
perpendicular to . Let 
be the intersections of 
with 
, respectively. Let 
be the projections of 
onto line . Let 
be the second intersections of 
with the incircle . Let 
be the intersection of 
and 
. Prove that the line 
bisects segment 
.
be the incentre of 
such that the incircle touches the sides at the points respectively. The perpendicular to at and meets at and 
respectively. 
and 
meet the incircle at and 
respectively. 
and 
intersect at 
Prove that 
bisects 
![$P\in\mathbb{Z}[x]$](http://latex.artofproblemsolving.com/f/3/d/f3d763f4197792a2956273485eb1e94700a43044.png)
has degree having real roots in 
. Suppose 
is the leading coefficient of . Show that, 
, let 
be all positive integers smaller than that are coprime to . Find all 
such that 
for all 
is the largest positive integer that divides both and 
. Integers and are coprime if 
.
is taken inside parallelogram 
so that the midpoint of 
is equidistant from and , and the midpoint of 
is equidistant form and . Let be the midpoint of 
. Prove that the angles 
and 
are equal.
is the orthocentre of triangle 
. Do this by proving that 
(two times median theorem) and 
(again two times median theorem in triangles 
) You will need the fact that 
. Tell me if you want more details.
be the mid point of ,
be the mid point of 
,
be the mid point of 
and 
be the mid point of 
.Let 
be the origin of the argand plane.Observe that 
is the circumcenter of 
.So we can assume the unit circle is the circumcircle of .
is a parallelogram we have 
.Now from 
we get 
and from 
we get 
.So 
.But this means that is the orthocenter of 
.From this the result follows.
be a parallelogram and be a point inside it. be any point in the plane.
be 
, midpoint of 
be and midpoint of be , midpoint of be .
is obviously the orthocentre of 
and hence we are done.
be the midpoints of 
respectively and 
the intersections of 
and 
respectively.
, 
, 
and 
. Thus, 
so 
. But 
and 
as 
. Q.E.D. be the midpoints of respectively and the intersections of 
and respectively., 
, and . Thus, so . But and as 
. Q.E.D.
are positive real numbers, 
then
and 
Prove that
then prove 
