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: Accepted)

# ABC 143 D - Triangles
# https://atcoder.jp/contests/abc143/tasks/abc143_d

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

class Stick
  def initialize(length, index)
    @length = length
    @index = index
  end

  attr_reader :length, :index
end

sticks = l.map.with_index(0) { |length, i| Stick.new(length, i) }

# puts sticks.inspect
# puts ""

count = 0
0.upto(n - 3) do |i|
  a = sticks[i]
  (i + 1).upto(n - 2) do |j|
    b = sticks[j]
    candidates = sticks[(j + 1)..(n - 1)]
    t = a.length - b.length
    c_over = candidates.bsearch { |stick| stick.length <= t }
    if c_over.nil?
      c_last_index = n - 1
    else
      c_last_index = c_over.index - 1
    end
    # puts "a: #{a.length}, b: #{b.length} -> #{c_last_index - j}"
    # puts ""
    count += c_last_index - j
  end
end

puts count