Problem:
Score : 400 points

Problem Statement

Takahashi has N sticks that are distinguishable from each other. The length of the i-th stick is L_i.

He is going to form a triangle using three of these sticks. Let a, b, and c be the lengths of the three sticks used. Here, all of the following conditions must be satisfied:

a < b + c

b < c + a

c < a + b

How many different triangles can be formed? Two triangles are considered different when there is a stick used in only one of them.

Constraints

All values in input are integers.

3 \leq N \leq 2 \times 10^3

1 \leq L_i \leq 10^3

Input

Input is given from Standard Input in the following format:

N
L_1 L_2 ... L_N

Constraints

Print the number of different triangles that can be formed.

Sample Input 1

4
3 4 2 1

Sample Output 1

1

Only one triangle can be formed: the triangle formed by the first, second, and third sticks.

Sample Input 2

3
1 1000 1

Sample Output 2

0

No triangles can be formed.

Sample Input 3

7
218 786 704 233 645 728 389

Sample Output 3

23

Solution in OCaml (4.02.3) (judge verdict: Accepted)

let scan_int () = Scanf.scanf " %d" (fun x -> x)
let scan_array len read_func = Array.init len (fun _ -> read_func())
let print_int n = Printf.printf "%d\n" n

let rec binary_search left right array value =
  if right - left <= 1 then right
  else
    let mid = (left + right) / 2 in
    if array.(mid) < value then binary_search mid right array value
    else binary_search left mid array value
  

let () =
  let n = scan_int() in
  let l = scan_array n scan_int in
  let _ = Array.fast_sort compare l in
  let ans = ref 0 in
  let _ =
    for i = 0 to n-2 do
      for j = i+1 to n-1 do
        let sum = l.(i) + l.(j) in
        let k = binary_search j n l sum in
        ans := !ans + (k-j-1)
      done
    done in
  print_int !ans