Data structure for finding max, inserting and deleting in O(1) and O(n) space
Suppose we have such data structure. We can find in $O(1)$ the max, delete the max in $O(1)$ and repeat it $n$ times. Hence, we can sort $n$ numbers in $O(n)$. Therefore, constructing such data structure must take $\Omega(n\log(n))$ (like sorting, in general, can't be done better than $n\log(n)$. Hence, you might have some constraint on data). 
Also, as deleting is in $O(1)$, means it must be found. Hence, Searching and Finding is in $O(1)$. 

Therefore, you must know the position of each value (something like counting sort, but with this constraint that $max &lt;= n$ to take the $O(n)$ space!). Hence, you can act like a counting sort by saving the max value in a variable besides the array, by accepting some constraint on data (not in general).