Let be a cyclic quadrilateral with circumcenter , such that is not a diameter of its circumcircle. The lines and intersect at point , so that lies between and , and lies between and . Suppose triangle is acute and let be its orthocenter. The points and on the lines and , respectively, are such that and . The line through , perpendicular to , intersects at , and the line through , perpendicular to , intersects at . Prove that the points , , are collinear.

Amir Parsa Hosseini Nayeri, Iran

Let be a primes , be positive integers such that and .
Prove that

Show that in a non-obtuse triangle the perimeter of the triangle is always greater than two times the diameter of the circumcircle.

For positive real numbers satisfying , find the biggest value the following equation could acquire:

Find the area enclosed by the curves

Let be the positive number satisfied Find the minimum of

Adrian and Bertrand take turns as follows: Adrian starts with a pile of () stones. On their turn, each player must divide a pile. The player who can make all piles have at most 2 stones wins. Depending on n, determine which player has a winning strategy.

find f: R+ -> R such that:
f(x)/x + f(y)/y = (1/x + 1/y).f(sqrt(xy))

Let be positive integers such that no number divides some other number. If , prove that .

Let be real numbers such that . Prove that

Determine all functions with the property that holds for all .