With each , define Find all integers such that is an integer.

Find all functions such that for all positive real numbers and .

(Andrew Brahms, USA)

Consider a triangle and some point on . The perpendicular from to intersects the circumcircle of at and the perpendicular from to intersects the circumcircle of at . Show that the line does not depend on the choice of .

(Shining Sun, USA)

Let be given points in the plane, and let be a real number. Alice and Bob play the following game. Firstly, Alice constructs a connected graph with vertices at the points , i.e., she connects some of the points with edges so that from any point you can reach any other point by moving along the edges.Then, Alice assigns to each vertex a non-negative real number , for , such that . Bob then selects a sequence of distinct vertices such that and are connected by an edge for every . (Note that the length is not fixed and the first selected vertex always has to be .) Bob wins if
otherwise, Alice wins. Depending on , determine the largest possible value of for which Bobby has a winning strategy.

Let ,such that Prove that : Where is a positive real number solution of equation :

Let be an acute triangle. Points , and lie on a line in this order and satisfy . Let and be the midpoints of and , respectively. Suppose triangle is acute, and let be its orthocentre. Points and lie on lines and , respectively, such that and are concyclic and pairwise different, and and are concyclic and pairwise different. Prove that and are concyclic.

In each cell of a board, a nonnegative real number is written in such a way that the sum of the numbers in each row is equal to , and the sum of the numbers in each column is equal to . Define to be the largest value in row , and let . Similarly, define to be the largest value in column , and let .
What is the largest possible value of ?

Proposed by Paulius Aleknavičius, Lithuania

In how many ways can a deck of 52 cards be divided among 13 players, each with 4 cards, so that one player has all 4 suits and the others have one suit?

Let and . Find max (and prove):

Let be a positive integer and let be real numbers.
Prove that


Show that the equality holds if and only if is an arithmetic sequence.

Let be an integer. In a configuration of an board, each of the cells contains an arrow, either pointing up, down, left, or right. Given a starting configuration, Turbo the snail starts in one of the cells of the board and travels from cell to cell. In each move, Turbo moves one square unit in the direction indicated by the arrow in her cell (possibly leaving the board). After each move, the arrows in all of the cells rotate counterclockwise. We call a cell good if, starting from that cell, Turbo visits each cell of the board exactly once, without leaving the board, and returns to her initial cell at the end. Determine, in terms of , the maximum number of good cells over all possible starting configurations.

Proposed by Melek Güngör, Turkey

In convex quadrilateral . Let be a point on side , and be a point on the extension of such that . Let be the circumcenter of , and be a point on the side extension of satisfying . Line BP intersects AC at point Q. Prove that