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)

{-
  This code was written after reading the editorial.
-}

import qualified Data.Vector.Unboxed as VU
import qualified Data.Vector.Unboxed.Mutable as VUM
import qualified Data.ByteString.Lazy.Char8 as B
import Control.Monad.Primitive
import Data.Function
import Data.List
import Data.Maybe
import Control.Monad
import Control.Monad.ST
import Data.Bits
import Debug.Trace

solve :: Int -> [Int] -> Int
solve n as = binsch 1 (10^9 + 1)
  where
    binsch xMin xMax | xMin + 1 == xMax = xMin
                     | try mid n as     = binsch mid xMax
                     | otherwise        = binsch xMin mid
      where mid = (xMax + xMin) `div` 2

try :: Int -> Int -> [Int] -> Bool
try x n as =
  t >= ((n * (n+1) `div` 2) + 1) `div` 2
  where
    t :: Int
    t = numIncr n ss
    ss :: [Int]
    ss = scanl' (+) 0 $ map (\a -> if a >= x then 1 else -1) $ as

numIncr :: Int -> [Int] -> Int
numIncr n ss = runST $ do
  v <- VUM.replicate (2*n+2) 0
  liftM sum $ forM ss $ \s -> do
    let i = s + (n+1)
    bit_add v (2*n + 1) i 1
    (+ (-1)) <$> bit_intv v i

{--
  bit_add v n i x ... a_i += x
--}
bit_add :: VUM.MVector s Int -> Int -> Int -> Int -> ST s ()
bit_add v n i x = forM_ (unfoldr f i) $ VUM.modify v (+ x)
  where f j | j > n = Nothing
            | otherwise = Just (j, j + (j .&. (-j)))

{--
  bit_intv v i    ... \Sum_{j=1}^{i} a_i
--}
bit_intv :: VUM.MVector s Int -> Int -> ST s Int
bit_intv v i = sum <$> (forM (unfoldr f i) $ VUM.read v)
  where f j | j == 0    = Nothing
            | otherwise = Just (j, j - (j .&. (-j)))


bsReadInts :: B.ByteString -> [Int]
bsReadInts = map (fst . fromJust . B.readInt) . B.words

tmain :: B.ByteString -> Int
tmain cont =
  let n : as = bsReadInts cont
  in solve n as

main = putStrLn . show . tmain =<< B.getContents