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 
.
is cut into one hundred rectangles of the same perimeter 
. What is the largest possible value of ?
