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 5 of 5: Let n grow
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∣A−d∣<kn for every n≥1⟹A=d∈Z.|A-d|<\frac{k}{n}\text{ for every }n\ge1\Longrightarrow A=d\in\mathbb Z.
Detailed analysis

The same inequality holds for every positive integer n, so its left side must be zero. Hence A=d, and d is an integer; therefore the required sum of the a_i is an integer.