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Sequence of floors is arithmetic progression

Source: APMO 2013, Problem 3

May 3, 2013
floor functionarithmetic sequencealgebra

Problem Statement

For 2k2k real numbers a1,a2,...,aka_1, a_2, ..., a_k, b1,b2,...,bkb_1, b_2, ..., b_k define a sequence of numbers XnX_n by X_n = \sum_{i=1}^k [a_in + b_i]   (n=1,2,...). If the sequence XNX_N forms an arithmetic progression, show that ∑i=1kai\textstyle\sum_{i=1}^k a_i must be an integer. Here [r][r] denotes the greatest integer less than or equal to rr.
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