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 Perl (v5.18.2) (judge verdict: Time Limit Exceeded)

#!/usr/bin/perl

use strict;
use warnings;

my ($n, $k);
{
    my $in = <STDIN>;
    ($n, $k) = split(" ", $in);
}
my @c;
{
    my $in = <STDIN>;
    @c = split(" ", $in);
}
my @d = grep {$_ > 0} @c;
my @e = grep {$_ < 0} @c;
my $d1 = @d*(@d-1)/2+@e*(@e-1)/2;
my $e1 = @d*@e;
my $f = $n*($n-1)/2-$d1-$e1;

my $ans = 0;
my @g;
if ($e1+$f<$k)
{
    $k -= $e1 + $f;
    for (my $i=0; $i<@d; $i++)
    {
        for (my $j=$i+1; $j<@d; $j++)
        {
            push @g, $d[$i]*$d[$j];
        }
    }
    for (my $i=0; $i<@e; $i++)
    {
        for (my $j=$i+1; $j<@e; $j++)
        {
            push @g, $e[$i]*$e[$j];
        }
    }
    @g = sort {$a <=> $b} @g;
    $ans = $g[$k-1];
}
elsif ($k<=$e1)
{
    for (my $i=0; $i<@d; $i++)
    {
        for (my $j=0; $j<@e; $j++)
        {
            push @g, $d[$i]*$e[$j];
        }
    }
    @g = sort {$a <=> $b} @g;
    $ans = $g[$k-1];
}
print "$ans\n";