Let there be the triangle ABC and the points E ∈ (AC), F ∈ (AB), such that BE and CF are concurrent in O.
If {L} = AO ∩ EF and K ∈ BC, such that LK ⊥ BC, show that EKL = FKL.

is acute triangle. is orthocenter, is the midpoint of , is the midpoint of smaller arc . Point is on such that, is perpendicular to . Prove that: is paralelogram(Synthetic sol needed).

Let be the circumcircle of . Let be a point on the side . The tangent to at intersects the parallel line to through at point . The segment intersects again at . Suppose , , , are concyclic. Prove that , , are concurrent.

(Note I couldn't find a post w/ this from AoPS search so I'm posting, please do tell if there exists a post.)

Solve over positive real numbers the functional equation

For any nonzero integer, define as the largest integer such that . Find all integers that are not powers of such that the sequence is unbounded.

Evaluate

For which positive integers do there exist infinitely many positive integers such that divides ?

Evan Chen and Ankan Bhattacharya

........

Determine all pairs of positive integers for which there exist positive integers and such that
holds for all integers (Note that denotes the greatest common divisor of integers and )

Proposed by Valentio Iverson, Indonesia

Dear Mathlinkers,

1. ABC an acuatangle triangle
2. H the orthcenter of ABC
3. DEF the orthic triangle of ABC
4. A* the midpoint of AH
5. X the point of intersection of AH and EF.

Prove : X is the orthocenter of A*BC.

Sincerely
Jean-Louis