Problem 2.
 Let n≥k≥3  be integers. Show that for every integer sequence 1≤a₁<a₂<…<a_k≤n one can choose non-negative integers b₁,b₂,…,b_k , satisfying the following conditions:

1. 0≤b_i≤n for each 1≤i≤k,
2. all the positive b_i are distinct,
3. the sums a_i+b_i, 1≤i≤k, form a permutation of the first k terms of a non-constant arithmetic progression.