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non-constant arithmetic progression, existence of a sequence

Source: BMO 2024 Problem 2

April 29, 2024
arithmetic sequencecombinatorics

Problem Statement

Let n≥k≥3n \ge k \ge 3 be integers. Show that for every integer sequence 1≤a1<a2<...<ak≤n1 \le a_1 < a_2 < . . . < a_k \le n one can choose non-negative integers b1,b2,...,bkb_1, b_2, . . . , b_k, satisfying the following conditions:
[*] 0≤bi≤n0 \le b_i \le n for each 1≤i≤k1 \le i \le k, [*] all the positive bib_i are distinct, [*] the sums ai+bia_i + b_i, 1≤i≤k1 \le i \le k, form a permutation of the first kk terms of a non-constant arithmetic progression.
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