In a class, there are students and a teacher with marbles. The teacher then play a Marble distribution according to the following rules. At the start, the teacher distributed all her marbles to students, so that each student receives at least marbles from the teacher. Then, the teacher chooses a student , who has never been chosen before, such that the number of marbles that he owns in a multiple of . That chosen student then equally distribute half of his marbles to other students. The same goes on until the teacher is not able to choose anymore student.

Find all integer , such that for some initial numbers of marbles that the students receive, the teacher can choose all the student(according to the rule above), so that each student receiving equal amount of marbles at the end.

Let . Prove that

For a positive integer , define and , where and denote the product and sum of the digits of , respectively. Find all solutions to

Determine the largest integer for which the following statement holds: there exists at least one triple of integers such that
and all triples of real numbers, satisfying the equations, are such that are integers.

Marek Maruin, Slovakia

Let be an acute triangle such that with circumcircle and circumcentre . Let and be the tangents to at and respectively, which meet at . Let be the foot of the perpendicular from onto the line segment . The line through parallel to line meets at . Prove that the line passes through the midpoint of the line segment .

Proposed by Dominic Yeo, United Kingdom

If positive reals are such that , find the minimum value of expression