Expectation of minimum growth

Question

Suppose $\mathbb E[X] \geq n$, is it true that: $$\mathbb E[\min\{n, X\}] \geq n \cdot a$$ for some constant $a>0$?

Answer 1

Pick $X$ such that with probability $\frac1{n+2}$, $X=(n+1)(n+2) +1$ and $X=-1$ otherwise.

This is a counterexample when $n>0$.