Question 130
For a fixed positive integer $m$, let $A$ be a subset of $\{0,1,2, \ldots, 5^{m}\}$ consisting of $4m+2$ elements.
Prove that $A$ always contains three numbers $a, b, c$ with $a \lt b \lt c$ and $c + 2a \gt 3b$.
Prove that $A$ always contains three numbers $a, b, c$ with $a \lt b \lt c$ and $c + 2a \gt 3b$.