Problem 1. Let Γ be the circumcircle of ∆ABC. Let D be a point on the side BC. The tangent to Γ at A intersects the parallel line to BA through D at point E. The segment CE intersects Γ again at F. Suppose B, D, F, E are concyclic. Prove that AC, BF, DE are concurrent.

Problem 2. Show that r = 2 is the largest real number r which satisfies the following condition: If a sequence a_{1}, a_{2}, … of positive integers fulfills the inequalities

for every positive integer n, then there exists a positive integer M such that a_{n}+2 = a_{n }for every n ≥ M.

Problem 3. Determine all positive integers k for which there exist a positive integer m and a set S of positive integers such that any integer n > m can be written as a sum of distinct elements of S in exactly k ways.

**Problem 4. **Let Z denote the set of all integers. Find all polynomials *P*(*x*) with integer coefficients that satisfy the following property:

For any infinite sequence *a*1*, a*2*, … *of integers in which each integer in Z appears exactly once, there exist indices *i < j *and an integer *k *such that *a _{i} *+

*a*

_{i}_{+1}+

*· · ·*+

*a*=

_{j }*P*(

*k*).

**Problem 5. **Let *n ≥ *3 be a fixed integer. The number 1 is written *n *times on a blackboard. Below the blackboard, there are two buckets that are initially empty. A move consists of erasing two of the numbers *a *and *b*, replacing them with the numbers 1 and *a *+ *b*, then adding one stone to the first bucket and gcd(*a, b*) stones to the second bucket. After some finite number of moves, there are *s *stones in the first bucket and *t *stones in the second bucket, where *s *and *t *are positive integers. Find all possible values of the ratio *t/s.*