Problem 1. Let n be a positive integer and let a_1, a_2, ..., a_k (k>=2)
be distinct integer in the set {1, 2, ..., n} such that n divides
a_i(a_{i+1}-1) for i=1, 2, ..., k-1. Prove that n does not divide
a_k(a_1-1)
Problem 2. Let ABC be a triangle with circumcenter O. The points P and Q
are interior points of the sides CA and AB respectively. Let K, L and M
be the midpoints of the segments BP, CQ, and PQ, respectively, and letΓ
be the circle passing through K, L, and M. Suppose that the line PQ is
tangent to the circleΓ. Prove that OP=OQ.
Problem 3. Suppose that s_1, s_2, s_3, ... is a strictly increasing sequence
of positive integers such that the subsequence s_{s_1}, s_{s_2}, s_{s_3}, ...
and s_{s_1+1}, s_{s_2+1}, s_{s_3+1}, ... are both arithmetic progressions.
Prove that the sequence s_1, s_2, s_3, ... is itself an arithmetic
progression.