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 Rust (1.15.1) (judge verdict: Accepted)

#[allow(unused_imports)]
use std::cmp::{Ordering};

#[allow(unused_imports)]
use std::collections::{BTreeMap, BTreeSet, BinaryHeap, HashMap, HashSet, VecDeque};

#[allow(unused_imports)]
use std::io::{stdin, stdout, BufWriter, Write};

#[allow(unused_imports)]
use std::iter::FromIterator;

#[allow(unused_macros)]
macro_rules! min {
    ($x: expr) => ($x);
    ($x: expr, $($z: expr),+) => {{
        std::cmp::min($x, min!($($z),*))
    }}
}

#[allow(unused_macros)]
macro_rules! max {
    ($x: expr) => ($x);
    ($x: expr, $($z: expr),+) => {{
        std::cmp::max($x, max!($($z),*))
    }}
}

#[allow(unused_macros)]
macro_rules! get { 
    ([$t: ty]) => { 
        { 
            let mut line = String::new(); 
            stdin().read_line(&mut line).unwrap(); 
            line.split_whitespace().map(|t|t.parse::<$t>().unwrap()).collect::<Vec<_>>()
        }
    };
    ([$t: ty];$n: expr) => {
        (0..$n).map(|_|get!([$t])).collect::<Vec<_>>()
    };
    ($t: ty) => {
        {
            let mut line = String::new();
            stdin().read_line(&mut line).unwrap();
            line.trim().parse::<$t>().unwrap()
        }
    };
    ($($t: ty),*) => {
        { 
            let mut line = String::new();
            stdin().read_line(&mut line).unwrap();
            let mut iter = line.split_whitespace();
            ($(iter.next().unwrap().parse::<$t>().unwrap(),)*)
        }
    };
    ($t: ty; $n: expr) => {
        (0..$n).map(|_|get!($t)).collect::<Vec<_>>()
    };
    ($($t: ty),*; $n: expr) => {
        (0..$n).map(|_|get!($($t),*)).collect::<Vec<_>>()
    };
}

#[allow(unused_macros)]
macro_rules! debug {
    ($($a:expr),*) => {
        #[cfg(debug_assertions)]
        writeln!(&mut std::io::stderr(), concat!("[DEBUG] ", $(stringify!($a), "={:?} "),*), $($a),*).unwrap();
    }
}

const BIG_STACK_SIZE: bool = true;

#[allow(dead_code)]
fn main() {
    use std::thread;
    if BIG_STACK_SIZE {
        thread::Builder::new()
            .stack_size(32 * 1024 * 1024)
            .name("solve".into())
            .spawn(solve)
            .unwrap()
            .join()
            .unwrap();
    } else {
        solve();
    }
}

fn lower_bound<F>(l: isize, r: isize, criterion: F) -> isize
where
    F: Fn(isize) -> bool,
{
    let mut l = l;
    let mut r = r;
    while r - l > 1 {
        let mid = l + (r - l) / 2;
        if criterion(mid) {
            r = mid;
        } else {
            l = mid;
        }
    }
    l
}

fn solve() {
    let (n,k) = get!(usize, usize);
    let mut a = get!([isize]);
    a.sort();

    let mut minus = Vec::new();
    let mut zeros = Vec::new();
    let mut plus = Vec::new();
    for &e in &a {
        if e < 0 {
            minus.push(e);
        } else if e == 0 {
            zeros.push(e);
        } else {
            plus.push(e);
        }
    }

    let pminus = minus.len()*plus.len();
    let pplus = plus.len()*max!(0, plus.len() as isize -1) as usize /2 + minus.len()*max!(0, minus.len() as isize-1) as usize/2;

    if pminus >= k {
        let crit = |val: isize| {
            let mut count = 0;
            for i in 0..plus.len() {
                let crit = |index: isize| {
                    let index = index as usize;
                    minus[index]*plus[plus.len()-i-1] > val
                };
                let index = lower_bound(-1, minus.len() as isize, crit);
                count += (index+1) as usize;
            }
            count > k-1
        };
        let val = lower_bound(-1000_000_000_000_000_000, 0, crit)+1;
        println!("{}", val);
    } else if n*(n-1)/2-pplus+1 <= k {
        let base = n*(n-1)/2-pplus+1;
        let k = k-base;
        let crit = |val: isize| {
            let mut count = 0;
            for i in 0..plus.len() {
                let pcrit = |index: isize| {
                    let index = index as usize;
                    plus[i]*plus[index] > val
                };
                let index = lower_bound(0, plus.len() as isize, pcrit);
                count += max!(0, index-i as isize) as usize;
            }
            for i in 0..minus.len() {
                let mcrit = |index: isize| {
                    let index = index as usize;
                    minus[minus.len()-i-1]*minus[minus.len()-index-1] > val
                };
                let index = lower_bound(0, minus.len() as isize, mcrit);
                count += max!(0, index-i as isize) as usize;
            }
            count > k
        };
        let val = lower_bound(0, 1000_000_000_000_000_001, crit)+1;
        println!("{}", val);
    } else {
        println!("0");
    }
}