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 4 of 5: Compare slopes
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A(n+1)+B−k<nd+X1<A(n+1)+B⟹∣A−d∣<kn.A(n+1)+B-k<nd+X_1<A(n+1)+B\Longrightarrow |A-d|<\frac{k}{n}.
Detailed analysis

Combine the two bounds and the bound for X_1. After subtracting the common terms, the distance between A and d is less than k/n.