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 Python (3.8.2) (judge verdict: Accepted)

import sys

import numba as nb

input = sys.stdin.readline


@nb.njit("i8(List(i8, reflected=True),i8)", cache=True)
def binary_search(a, key):
    """Meguru type binary search"""
    ok = -1
    ng = len(a)

    def is_ok(a, key, idx):
        if a[idx] < key:
            return True
        else:
            return False

    while (abs(ok - ng) > 1):
        mid = (ok + ng) // 2
        if is_ok(a, key, mid):
            ok = mid
        else:
            ng = mid

    return ok


@nb.njit("i8(i8,List(i8, reflected=True))", cache=True)
def solve(N, L):
    L.sort()
    ans = 0
    for a in range(N - 2):
        for b in range(a + 1, N - 1):
            x = binary_search(L, L[a] + L[b])
            ans += (x - b)
    return ans


def main():
    N = int(input())
    L = list(map(int, input().split()))

    ans = solve(N, L)

    print(ans)


if __name__ == "__main__":
    main()