Greatest integer function inequalities

Question

Prove: $$\Big{[}{n\over 2}\Big{]} +\Big{[}{n\over 3}\Big{]} + \Big{[}{n\over 4}\Big{]}>n$$ for every $n>a$, where $a$ is a natural number Find $a$. $[n]$ is greatest integer function

Answer 1

Answer to your Problem

Please find the answer to your problem here.

Answer 2

Here's a sketch:

Note that $\lfloor n/2 \rfloor = n/2$ or $(n-1)/2$.

And $\lfloor n/3 \rfloor = n/3$ or $(n-1)/3$ or $(n-2)/3.$

And $\lfloor n/4 \rfloor = n/4$ or $(n-1)/4$ or $(n-2)/4$ or $(n-3)/4.$

So the smallest case for the left side is

$$\frac{n-1}{2}+\frac{n-2}{3}+\frac{n-3}{4} >n.$$

This reduces to $n>27$, so now you know that $a\leq 27$. So you can work your way down by trying $26, 25, \ldots$ in the inequality until you find an exception.