AoPS Academy
be triangle with circumcenter 
and orthocenter 
. 
intersect again at 
respectively. Lines through parallel to 
intersects 
at 
respectively. Point 
such that 
is a parallelogram. Prove that lines 
and 
are concurrent at a point on 
.
be a triangle with incenter 
, and circumcircle 
. Let the circle with diameter 
intersects at 
and 
. Let 
be the midpoint arc 
and let 
intersect again at a point 
. Let the circumcircle of 
intersect the circumcircle of 
at a point 
. Let the tangent at to intersect 
at a point 
. Prove that 
and 
intersect on .
satisfy,
Where {
} is a permutation of . Prove that 
be a non-constant monic polynomial with integer coefficients and let 
be an infinite sequence. Prove that there are infinitely many primes, each of which divides at least one term of the sequence 
.
for some positive integers 
. Show that there must exist two among them one of which is entirely contained in the other.
for some positive integers . Show that there must exist two among them one of which is entirely contained in the other.
with the vertex 
. If 2 rectangles has the same 
or 
value, then one must be contained in another. Then there are at most 
other rectangles whose or value is less than 
or 
respectively. All the other rectangles will have 
value more than 
, so they all contain .
