IR-TST 2023 Problem 1

Suppose that $n\ge3$ is a natural number. Find the maximum value $k$ such that there are real numbers $a_1,a_2,...,a_n \in [0,1)$ (not necessarily distinct) that for every natural number like $j \le k$, sum of some $a_i$ - s is $j$.

Proposed by Navid Safaei