Please define "[n]"? It is not the same has floor function. It is the fractional part?

Thank you,it is updated

add the space between "of" and "$20z$"... sorry for being so critical

$z=1.5$ so the answer is $30$. This is the largest since anything bigger would make $[6/z]<4$ which is absurd.

A bit too easy for an olympiad... We see that this function is just the floor function... Since $\lfloor{\frac{5}{z}}\rfloor$ and $\lfloor{\frac{6}{z}}\rfloor$ must be integers, we see that: $$\lfloor{\frac{5}{z}}\rfloor = 3$$$$\lfloor{\frac{6}{z}}\rfloor = 4$$ This implies that $z$ could be ranging from $\frac{5}{4}$ to $\frac{6}{4}$ (inclusive). Since we want the maximum, $z = \frac{6}{4}$, which implies that $20z = \boxed{30}$.