Inequalities with floor function

Question

How large should $n$ be in order for the following inequality to hold?

$$\left\lfloor \frac{n}{m} \right\rfloor \leq 2 \left\lfloor \frac{n}{2m} \right\rfloor$$

Thanks.

Answer 1

For $n,m > 0$ it is always the case that $$\left\lfloor n/m\right\rfloor \geq 2 \left\lfloor n/2m\right\rfloor $$ because any leftover fractional part that gets discarded is doubled on the right hand side.

We have $$n = km+q = 2\ell m + p$$ with $0 \leq q < m$ and $p = q$ or $p=q+m$

Then $$ (k-2\ell ) m = p-q$$ so $k > 2\ell$ if $k$ is odd.

Therefore, $$\left\lfloor n/m\right\rfloor \leq 2 \left\lfloor n/2m\right\rfloor $$ only if $\left\lfloor n/m\right\rfloor$ is even.