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.
Prove that 
When does equality hold?
be a parallelogram such that 
Let 
and 
be the midpoints of 
and 
respectively. Assuming that 
is a cyclic quadrilateral, find 
be a polynomial with integer coefficients. Define 
and 
for 
.
such that 
, then prove that either 
or 
.
have 
distinct real roots, which sum up to 
. If 
, find the values of 
and their corresponding roots.
be a positive integer greater than . Find all functions 
satisfying:
for all integers 
be a triangle, and let 
be an arbitrary point on line 
, where 
is the circumcenter of . Define 
and 
as the orthocenters of triangles 
and 
. Prove that 
passes through a fixed point which is independent of the choice of .
be a triangle inscribed in a circle, where the tangents to the circle at points 
and 
intersect at the point . Let 
be a point on the arc 
(not containing ) such that 
and 
. Let the lines and 
intersect at point . Let 
be the reflection of with respect to the line . The lines 
and 
intersect at point 
, and intersects the circumcircle of again at point 
. lies on the circumcircle of 
.
be circumscribed about circle 
, and let 
be the orthocenter of . The circle touches line at 
. The tangent to the circle 
at meets at 
. Let 
be the midpoint of 
, and let the line 
meet again at 
. The tangent to parallel to meets the line 
at 
. Prove that 
is tangent to .
.
