COMC 2013 C Problem 4

For each real number x, let [x] be the largest integer less than or equal to x. For example, [5]=5,[7.9]=7 and [-2.4]=-3. An arithmetic progression of length k is a sequence a_{1}, a_{2}, \ldots, a_{k} with the property that there exists a real number b such that a_{i+1}-a_{i}=b for each 1 \leq i \leq k-1.

Let \alpha>2 be a given irrational number. Then S=\{[n \cdot \alpha]: n \in \mathbb{Z}\}, is the set of all integers that are equal to [n \cdot \alpha] for some integer n.

(a) Prove that for any integer m \geq 3, there exist m distinct numbers contained in S which form an arithmetic progression of length m.

(b) Prove that there exist no infinite arithmetic progressions contained in S.