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numbers in [0,1) with some j

Source: Iran TST 2023 ; Exam 1 Problem 1

March 15, 2023
alalgebra

Problem Statement

Suppose that n≥3n\ge3 is a natural number. Find the maximum value kk such that there are real numbers a1,a2,...,an∈[0,1)a_1,a_2,...,a_n \in [0,1) (not necessarily distinct) that for every natural number like j≤kj \le k , sum of some aia_i-s is jj.
Proposed by Navid Safaei
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