Problem:
Score : 700 points

Problem Statement

We will define the median of a sequence b of length M, as follows:

Let b' be the sequence obtained by sorting b in non-decreasing order. Then, the value of the (M / 2 + 1)-th element of b' is the median of b. Here, / is integer division, rounding down.

For example, the median of (10, 30, 20) is 20; the median of (10, 30, 20, 40) is 30; the median of (10, 10, 10, 20, 30) is 10.

Snuke comes up with the following problem.

You are given a sequence a of length N.
For each pair (l, r) (1 \leq l \leq r \leq N), let m_{l, r} be the median of the contiguous subsequence (a_l, a_{l + 1}, ..., a_r) of a.
We will list m_{l, r} for all pairs (l, r) to create a new sequence m.
Find the median of m.

Constraints

1 \leq N \leq 10^5

a_i is an integer.

1 \leq a_i \leq 10^9

Input

Input is given from Standard Input in the following format:

N
a_1 a_2 ... a_N

Output

Print the median of m.

Sample Input 1

3
10 30 20

Sample Output 1

30

The median of each contiguous subsequence of a is as follows:

The median of (10) is 10.

The median of (30) is 30.

The median of (20) is 20.

The median of (10, 30) is 30.

The median of (30, 20) is 30.

The median of (10, 30, 20) is 20.

Thus, m = (10, 30, 20, 30, 30, 20) and the median of m is 30.

Sample Input 2

1
10

Sample Output 2

10

Sample Input 3

10
5 9 5 9 8 9 3 5 4 3

Sample Output 3

8

Solution in PyPy3 (2.4.0) (judge verdict: Accepted)

#!/usr/bin/env python3
import sys
from math import *
from itertools import *
from collections import *
from functools import *
try:
    from math import gcd
except Exception:
    from fractions import gcd


class BinaryIndexTree:
    def __init__(self, n):
        self._size = n
        self._tree = [0] * (n+1)

    def sum(self, i):
        s = 0
        i += 1
        while 0 < i:
            s += self._tree[i-1]
            i -= i & -i
        return s

    def add(self, i, x):
        i += 1
        while i <= self._size:
            self._tree[i-1] += x
            i += i & -i


def solve(N: int, a: "List[int]"):
    c = N * (N + 1) // 2
    th = (c + 1) // 2

    xs = list(sorted(set(a)))
    lo = 0
    hi = len(xs)

    def _f(x):
        ss = [0] + list(accumulate([1 if x <= v else -1 for v in a]))
        n = max([abs(s) for s in ss])
        bit = BinaryIndexTree(2 * n + 1)

        ret = 0
        for s in ss:
            ret += bit.sum(s + n) 
            bit.add(s + n, 1)
        return ret

    while (lo + 1) < hi:
        mid = (lo + hi) // 2
        if th <= _f(xs[mid]):
            lo = mid
        else:
            hi = mid
    return xs[lo]


def main():
    def iterate_tokens():
        for line in sys.stdin:
            for word in line.split():
                yield word
    tokens = iterate_tokens()
    N = int(next(tokens))  # type: int
    a = [ int(next(tokens)) for _ in range(N) ]  # type: "List[int]"
    result = solve(N, a)
    print(result)

if __name__ == '__main__':
    main()