return the nth element of a list in OCaml?

(In my opinion) A simpler solution without using a tuple can be:

let rec get_nth mylist index = match mylist with
    | [] -> raise (Failure "empty list")
    | first::rest -> 
        if index = 0 then first 
        else get_nth rest (index-1)
;;

Using tuples as parameters like this is not common in OCaml. Usually you would use currying and define your function like this:

let get_nth list n = ...

This would have the signature 'a list -> int -> 'a. Also note that you have a 'a paramter here, which means, there is no real reason to restrict your function to ints alone.

Now let's look at the problem. If you want to get the zeroth element, what would your function look like?

let get_nth list 0 = List.head list (* this is not actually valid in OCaml *)

now if you have a function to get the nth element from a list of m items (N.B. n > m), how could you use that function to build another function which get's the n+1st element from a list of m+1 elements? Let that function for n+1 elements be get_nth'

let get_nth' list n' = get_nth (List.tail list) (n'-1)

Now all you need to do is to combine the two and you are done. I'll leave that last part up to you.

If you follow this advice you will get something which is more complicated, than it has to be. However it is easier to understand what is happening this way.

You may not notice but List.nth function is already there in List module.

If you want to write it using recursion:

let rec get_nth = function
    | [], _ -> raise (Failure "get_nth")
    | _, n when n < 0 -> raise (Invalid_argument "get_nth")
    | x::_, 0 -> x
    | x::xs, n -> get_nth(xs, n-1)