Let be the set of integers. Determine all functions such that, for all integers and ,
In triangle , point lies on side and point lies on side . Let and be points on segments and , respectively, such that is parallel to . Let be a point on line , such that lies strictly between and , and . Similarly, let be the point on line , such that lies strictly between and , and .
Prove that points , and are concyclic.
A social network has users, some pairs of whom are friends. Whenever user is friends with user , user is also friends with user . Events of the following kind may happen repeatedly, one at a time: Three users , , and such that is friends with both and , but and are not friends, change their friendship statuses such that and are now friends, but is no longer friends with , and no longer friends with . All other friendship statuses are unchanged. Initially, users have friends each, and users have friends each. Prove that there exists a sequence of such events after which each user is friends with at most one other user.
Find all pairs of positive integers such that
The Bank of Bath issues coins with an on one side and a on the other. Harry has of these coins arranged in a line from left to right. He repeatedly performs the following operation:
If there are exactly coins showing , then he turns over the coin from the left; otherwise, all coins show and he stops. For example, if the process starting with the configuration would be , which stops after three operations.
(a) Show that, for each initial configuration, Harry stops after a finite number of operations.
(b) For each initial configuration , let be the number of operations before Harry stops. For example, and . Determine the average value of over all possible initial configurations .
Let be the incenter of acute triangle with . The incircle of is tangent to sides , , and at , , and , respectively. The line through perpendicular to meets ω again at . Line meets ω again at . The circumcircles of triangles and meet again at . Prove that lines and meet on the line through perpendicular to .