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Found problems: 2

2023 Iran Team Selection Test, 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 $j$. [i]Proposed by Navid Safaei [/i]
al algebra inequalities mathematical induction

2003 Portugal MO, 6
Given six irrational numbers, will it be possible to choose three such that the sum of any two of these three is irrational?
irrational number al natural number number theory algebra