BALKAN 2024 Problem 2

Let $n \geq k \geq 3$ be integers. Show that for every integer sequence $1 \leq a_{1}<a_{2}<\ldots<$ $a_{k} \leq n$ one can choose non-negative integers $b_{1}, b_{2}, \ldots, b_{k}$, satisfying the following conditions: (i) $0 \leq b_{i} \leq n$ for each $1 \leq i \leq k$, (ii) all the positive $b_{i}$ are distinct, (iii) the sums $a_{i}+b_{i}, 1 \leq i \leq k$, form a permutation of the first $k$ terms of a non-constant arithmetic progression.