Non-empty set $M$, that consists of positive integer numbers, has the following property: if for some(not necessarily distinct) positive integers $a_1,\ldots,a_{2024}$ the number $a_1\ldots a_{2024}$ is in $M$, then the number $a_1+a_2+\ldots+a_{2024}$ is also in $M$ Prove that all positive integer numbers, starting from $2049$, are in the $M$ M. Zorka
Problem
Source: Belarusian national olympiad 2024
Tags: combinatorics
RagvaloD
25.07.2024 12:48
Am I missing something, or set $M=\{2024n\}$ has the property, but don't contain all positive integer numbers from $2049$ ?
NO_SQUARES
25.07.2024 12:59
RagvaloD wrote: Am I missing something, or set $M=\{2024n\}$ has the property, but don't contain all positive integer numbers from $2049$ ? I think you missed that condition is follow: If $a_1 \cdot a_2 \cdot ... \cdot a_n \in M$, then $a_1+... +a_n \in M$.
$1^{2023} \cdot 2024n \in M$ and thus $2024n+2023 \in M$.
RagvaloD
25.07.2024 13:11
NO_SQUARES wrote: RagvaloD wrote: Am I missing something, or set $M=\{2024n\}$ has the property, but don't contain all positive integer numbers from $2049$ ? I think you missed that condition is follow: If $a_1 \cdot a_2 \cdot ... \cdot a_n \in M$, then $a_1+... +a_n \in M$.
$1^{2023} \cdot 2024n \in M$ and thus $2024n+2023 \in M$.
Ok, I understood condition incorrectly first, because I was thinking, that $a_1,...,a_n$ should be also in $M$, which is wrong