Problem:
Score: 400 points

Problem Statement

We have N integers A_1, A_2, ..., A_N.

There are \frac{N(N-1)}{2} ways to choose two of them and form a pair. If we compute the product of each of those pairs and sort the results in ascending order, what will be the K-th number in that list?

Constraints

All values in input are integers.

2 \leq N \leq 2 \times 10^5

1 \leq K \leq \frac{N(N-1)}{2}

-10^9 \leq A_i \leq 10^9\ (1 \leq i \leq N)

Input

Input is given from Standard Input in the following format:

N K
A_1 A_2 \dots A_N

Output

Print the answer.

Sample Input 1

4 3
3 3 -4 -2

Sample Output 1

-6

There are six ways to form a pair. The products of those pairs are 9, -12, -6, -12, -6, 8.

Sorting those numbers in ascending order, we have -12, -12, -6, -6, 8, 9. The third number in this list is -6.

Sample Input 2

10 40
5 4 3 2 -1 0 0 0 0 0

Sample Output 2

6

Sample Input 3

30 413
-170202098 -268409015 537203564 983211703 21608710 -443999067 -937727165 -97596546 -372334013 398994917 -972141167 798607104 -949068442 -959948616 37909651 0 886627544 -20098238 0 -948955241 0 -214720580 277222296 -18897162 834475626 0 -425610555 110117526 663621752 0

Sample Output 3

448283280358331064

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

import qualified Data.Text as T
import qualified Data.Text.IO as T
import qualified Data.Text.Read as T
import Data.List (sort, tails)

main :: IO ()
main = do
  (n : k : _) <- map unsafeTextToInt . T.words <$> T.getLine :: IO [Int]
  as <- sort . map unsafeTextToInt . T.words <$> T.getLine :: IO [Int]
  let
    ps = mergeAll $ zipWith f as (tail . tails $ as)
  print $ ps !! pred k

merge :: [Int] -> [Int] -> [Int]
merge xs [] = xs
merge [] ys = ys
merge xs@(x : xs') ys@(y : ys') | y < x     = y : merge xs ys'
                                | otherwise = x : merge xs' ys

merges :: [[Int]] -> [[Int]]
merges (xs : xs' : xss) = merge xs xs' : merges xss
merges xss = xss

mergeAll :: [[Int]] -> [Int]
mergeAll [] = []
mergeAll (xs : []) = xs
mergeAll xss = mergeAll . merges $ xss

f :: Int -> [Int] -> [Int]
f a as | a < 0     = map (a *) . reverse $ as
       | otherwise = map (a *) as

{-
f :: [Int] -> Int
f = f' (- 10 ^ 18 - 1) (10 ^ 18 + 1)
  where
    f' x y | succ x == y = y
           | p m         = f' x m
           | otherwise   = f' m y
      where
        m = (x + y) `div` 2
    p = (k <=) . g
    
-}

unsafeTextToInt :: T.Text -> Int
unsafeTextToInt s = case T.signed T.decimal s of
  Right (n, _) -> n