MathLabs

Problem 3

For 2k real numbers a_1, ..., a_k, b_1, ..., b_k, define X_n=sum from i=1 to k of floor(a_i n+b_i), for n=1,2,... . If the sequence X_n is an arithmetic progression, prove that sum from i=1 to k of a_i is an integer. Here floor(r) denotes the greatest integer less than or equal to r.
Step 2 of 5: Squeeze X_n
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ain+bi−1<⌊ain+bi⌋≤ain+bi⟹An+B−k<Xn≤An+B.a_i n+b_i-1<\lfloor a_i n+b_i\rfloor\le a_i n+b_i\Longrightarrow An+B-k<X_n\le An+B.
Detailed analysis

Apply the defining floor inequality to every i and sum. The strict lower bound is enough for the later comparison.