▪ Switzerland Final Round 2011

1. At a party, there are people with a glass of fruit juice each sitting around a circular table. Once a second, they clink glasses obeying the following two rules:
(a) They do not clink glasses crosswise.
(b) At each point of time, everyone can clink glasses with at most one other person.
How many seconds pass at least until everyone clinked glasses with everybody else?
2. Let be an acute-angled triangle and let , , be points on , , , respectively, such thatProve that , and are the feet of the perpendiculars through , and on , and , respectively.

                          
3. For positive integers and , find the smalles possible value of .
4. Find all functions such that for any real numbers satisfying ,holds true.

5. Let be a triangle with circumcircle . The tangentlines to through and intersect at . The circle through , and intersects and again at and , respectively; and intersect at . Suppose is the midpoint of . Calculate the ratio .
6. Let be positive real numbers satisfying . Show that
7. For a given rational number , find all integers such that
8. Let be a parallelogram and the Orthocentre of . The line parallel to through intersects at and at while the line parallel to through intersects at and at . Show that , , and are concyclic.
9. For any positive integer let be the number of divisors of ending with or in base and let be the number of divisors of ending with digit or in base . Prove that for all nonnegative integers .
10. On each square of an -chessboard, there are two bugs. In a move, each bug moves to a (vertically of horizontally) adjacent square. Bugs from the same square always move to different squares. Determine the maximal number of free squares that can occur after one move.