Possible values of z? Let $f:[-2,2]\to \mathbb {R} $ where $$f(x)=x^3+2(\sin x)^5+3(\tan x)^7+\left\lfloor\frac{x^2+1}{z}\right\rfloor $$ is an odd function then what are possible values of $z$?
$\lfloor\cdot\rfloor $ is the floor function.

There are some troubles with the domain ($\pi/2 \in [-2,2]$ but $\tan(\pi/2)$ is not defined). But I will ignore these.
Now, $f$ is odd if and only if for all $x$
$$f(x)+f(-x)=0$$
Now, you have
$$f(x)+f(-x) = \dots = 2 \left[\frac{x^2+1}{z} \right]$$
This can be identically zero when $0 \le \frac{x^2+1}{z} < 1$ for all $x$. From this you have $z >0$. For such $z$, you have that $\frac{x^2+1}{z}$ has a maximum at $x=2$, so you need 
$$\frac{2^2+1}{z}<1$$
i.e. $z>5$.
