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 Haskell (GHC 7.10.3) (judge verdict: Time Limit Exceeded)

{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE TupleSections #-}
{-# LANGUAGE MultiWayIf #-}
{-# LANGUAGE TypeFamilies #-}

import Control.Monad
import Control.Monad.ST
import Control.Applicative
import Control.Monad.Primitive
import Data.List
import Data.Maybe
import Data.Bits
import qualified Data.ByteString.Char8 as BS
import qualified Data.Vector as V
import qualified Data.Vector.Mutable as VM
import qualified Data.Vector.Unboxed as VU
import qualified Data.Vector.Unboxed.Mutable as VUM
import qualified Data.Vector.Generic as VG
import qualified Data.Vector.Generic.Mutable as VGM
import qualified Data.IntMap as IM


readIntBS = fst . fromJust . BS.readInt

main :: IO ()
main = do
  n <- readLn :: IO Int
  as <- map readIntBS . BS.words <$> BS.getLine :: IO [Int]

  fen <- newFenwickTree (2*n+1) 0 (+) :: IO (IOUFenwickTree Int)

  let
    m = n*(n+1)`div`2
    solve l r
      | l == r = return l
      | l+1 == r = aux r >>= \a -> return $ if a then r else l 
      | otherwise = do
          a <- aux $ (l+r)`div`2
          if a then
            solve ((l+r)`div`2) r
            else
            solve l ((l+r)`div`2)
        where
          aux x = do
            let
              as' = map (+(n+1)) . scanl (+) 0 $ map (\a -> if a >= x then 1 else -1) as :: [Int]

            (\(tree,_,_,_) -> forM_ [1..2*n+1] $ \i -> VUM.unsafeWrite tree i 0) fen
            y <- sum <$> (forM as' $ \a -> queryFenwickTree fen a >>= \q -> modifyFenwickTree fen a (+1) >> return q)
            return $ y >= m`div`2
        

  print =<< solve (minimum as) (maximum as)


type MFenwickTree v s a = (v s a, a, (a -> a -> a), Int) -- (フェニック木, 単位元, 二項演算)
type STFenwickTree s a = MFenwickTree VM.MVector s a
type IOFenwickTree a = STFenwickTree RealWorld a
type STUFenwickTree s a = MFenwickTree VUM.MVector s a
type IOUFenwickTree a = STUFenwickTree RealWorld a

newFenwickTree :: (Applicative m, PrimMonad m, VGM.MVector v a) => Int -> a -> (a -> a -> a) -> m (MFenwickTree v (PrimState m) a)
newFenwickTree n e f = (,e,f,n) <$> VGM.replicate (n+1) e

modifyFenwickTree :: (Applicative m, PrimMonad m, VGM.MVector v a) => MFenwickTree v (PrimState m) a -> Int -> (a -> a) -> m ()
modifyFenwickTree (tree,e,f,size) i g = do
  aux i
  where
    aux j = do
      when (j <= size) $ do
        VGM.unsafeWrite tree j =<< g <$> VGM.unsafeRead tree j
        aux (j + (j .&. (-j)))

queryFenwickTree :: (Applicative m, PrimMonad m, VGM.MVector v a) => MFenwickTree v (PrimState m) a -> Int -> m a
queryFenwickTree (tree,e,f,size) i = do
  aux i
  where
    aux j
      | j <= 0 = return e
      | otherwise = f <$> VGM.unsafeRead tree j <*> aux (j - (j .&. (-j)))

lowerBoundFenwickTree :: (Num a, Ord a, Eq a, Applicative m, PrimMonad m, VGM.MVector v a) => MFenwickTree v (PrimState m) a -> a -> m Int
lowerBoundFenwickTree (tree,e,f,size) x = do
  aux k 0 x
  where
    k = (2 ^ (floor $ logBase 2 (fromIntegral size)))
    aux i j w
      | i == 0 = return (j+1)
      | otherwise = do
          if i+j <= size then do
            a <- VGM.unsafeRead tree (i+j)
            if a < w then aux (i `div` 2) (i+j) (w-a) else aux (i `div` 2) j w
            else
            aux (i `div` 2) j w