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 1 of 5: Name the total slope
Detailed analysis
Collect the linear coefficients and constants appearing in the k floor terms.