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 Ruby (2.3.3) (judge verdict: Time Limit Exceeded)

n = gets.chomp.to_i
l = gets.chomp.split.map(&:to_i)
l.sort!

ans = 0

def ok?(a,b,c)
  # p 'called'
  if a < b+c && b<c+a && c<a+b
    true
  else
    false
  end
end
def ok1?(a,b,c)
  # p 'called'
  # p ['ok1',a,b,c, a < b+c && b<c+a]
  if a < b+c && b<c+a
    true
  else
    false
  end
end
def ok2?(a,b,c)
  # p 'called'
  # p ['ok2',a,b,c,c<a+b]
  if c<a+b
    true
  else
    false
  end
end


0.upto(n-3).each do |i|
  (i+1).upto(n-2).each do |j|
    left = 0
    lmin = j+1
    lmax = n-1
    while true
      lmid = (lmin + lmax)/2
      # p ['l',lmin, lmid, lmax]
      if ok1?(l[i],l[j],l[lmid])
        lmax = lmid
      else
        lmin = lmid
      end
      if lmax - lmin <= 1
        left = ok1?(l[i],l[j],l[lmin]) ? lmin : lmax
        break
      end
    end

    right = 0
    rmin = j+1
    rmax = n-1
    while true
      rmid = (rmin + rmax)/2
      # p ['r',rmin, rmid, rmax]
      if ok2?(l[i],l[j],l[rmid])
        rmin = rmid
      else
        rmax = rmid
      end
      if rmax - rmin <= 1
        right = ok2?(l[i],l[j],l[rmax]) ? rmax : lmin
        break
      end
    end
    # p ['---------', i, j, '', left, right]
    # p ['---------', l[i], l[j], '', l[left], l[right]]
    if ok1?(l[i],l[j],l[left]) && ok2?(l[i],l[j],l[right])
      ans += right - left + 1
    end
  end
end

puts ans