2021 IMO Shortlist A1

Let $n$ be an integer, and let $A$ be a subset of $\left\{0,1,2,3, \ldots, 5^{n}\right\}$ consisting of $4 n+2$ numbers. Prove that there exist $a, b, c \in A$ such that $a<b<c$ and $c+2 a>3 b$.

设$n$为整数，$A$为由$4 n+2$个数字组成的$\left\{0,1,2,3,\ldots, 5^{n}\right\}$的子集。证明 A$中存在$a, b, c \，使得 $a<b<c$ 且 $c+2 a>3 b$。