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 Haskell (GHC 8.8.3) (judge verdict: Accepted)

import Data.Vector((!),fromList)
import Data.List
import qualified Data.ByteString.Char8 as BS
import Data.Maybe
getIntList = map (fst . fromJust . BS.readInteger) . BS.words <$> BS.getLine
binarySearch :: Integral a => (a -> Bool) -> a -> a -> a
binarySearch pred ok ng
  | abs (ok-ng) <= 1 = ok
  | pred mid = binarySearch pred mid ng
  | otherwise = binarySearch pred ok mid
    where mid = (ok+ng) `div` 2

main = do
    n <- readLn :: IO Int
    l <- fromList . sort <$> getIntList
    let list = [max 0 (right-left+1) |
           i <- [0 .. n - 3],
           let a = l ! i,
           j <- [i + 1 .. n - 2],
           let b = l ! j,
           -- condition :: l!k < a+b,abs (a-b) < l!k , j < k
           let left = binarySearch (\k -> abs (a-b) < l ! k) (n-1) j,
           let right = binarySearch (\k -> l ! k < a+b) 0 n
           ]
    print $ sum list