Problem:
Score: 600 points

Problem Statement

E869120 is initially standing at the origin (0, 0) in a two-dimensional plane.

He has N engines, which can be used as follows:

When E869120 uses the i-th engine, his X- and Y-coordinate change by x_i and y_i, respectively. In other words, if E869120 uses the i-th engine from coordinates (X, Y), he will move to the coordinates (X + x_i, Y + y_i).

E869120 can use these engines in any order, but each engine can be used at most once. He may also choose not to use some of the engines.

He wants to go as far as possible from the origin.
Let (X, Y) be his final coordinates. Find the maximum possible value of \sqrt{X^2 + Y^2}, the distance from the origin.

Constraints

1 \leq N \leq 100

-1 \ 000 \ 000 \leq x_i \leq 1 \ 000 \ 000

-1 \ 000 \ 000 \leq y_i \leq 1 \ 000 \ 000

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
x_1 y_1
x_2 y_2
: :
x_N y_N

Output

Print the maximum possible final distance from the origin, as a real value.
Your output is considered correct when the relative or absolute error from the true answer is at most 10^{-10}.

Sample Input 1

3
0 10
5 -5
-5 -5

Sample Output 1

10.000000000000000000000000000000000000000000000000

The final distance from the origin can be 10 if we use the engines in one of the following three ways:

Use Engine 1 to move to (0, 10).

Use Engine 2 to move to (5, -5), and then use Engine 3 to move to (0, -10).

Use Engine 3 to move to (-5, -5), and then use Engine 2 to move to (0, -10).

The distance cannot be greater than 10, so the maximum possible distance is 10.

Sample Input 2

5
1 1
1 0
0 1
-1 0
0 -1

Sample Output 2

2.828427124746190097603377448419396157139343750753

The maximum possible final distance is 2 \sqrt{2} = 2.82842....
One of the ways to achieve it is:

Use Engine 1 to move to (1, 1), and then use Engine 2 to move to (2, 1), and finally use Engine 3 to move to (2, 2).

Sample Input 3

5
1 1
2 2
3 3
4 4
5 5

Sample Output 3

21.213203435596425732025330863145471178545078130654

If we use all the engines in the order 1 \rightarrow 2 \rightarrow 3 \rightarrow 4 \rightarrow 5, we will end up at (15, 15), with the distance 15 \sqrt{2} = 21.2132... from the origin.

Sample Input 4

3
0 0
0 1
1 0

Sample Output 4

1.414213562373095048801688724209698078569671875376

There can be useless engines with (x_i, y_i) = (0, 0).

Sample Input 5

1
90447 91000

Sample Output 5

128303.000000000000000000000000000000000000000000000000

Note that there can be only one engine.

Sample Input 6

2
96000 -72000
-72000 54000

Sample Output 6

120000.000000000000000000000000000000000000000000000000

There can be only two engines, too.

Sample Input 7

10
1 2
3 4
5 6
7 8
9 10
11 12
13 14
15 16
17 18
19 20

Sample Output 7

148.660687473185055226120082139313966514489855137208

Solution in Java (OpenJDK 11.0.6) (judge verdict: Accepted)

import java.util.Comparator;
import java.util.InputMismatchException;
import java.io.FileNotFoundException;
import java.util.Arrays;
import java.io.FileInputStream;
import java.io.PrintWriter;
import java.io.IOException;
import java.io.InputStream;

public class Main {
    InputStream is;

    int __t__ = 1;
    int __f__ = 0;
    int __FILE_DEBUG_FLAG__ = __f__;
    String __DEBUG_FILE_NAME__ = "src/B3";

    FastScanner in;
    PrintWriter out;

    class Point {
        final int x;
        final int y;

        public Point(int x, int y) {
            this.x = x;
            this.y = y;
        }
    }

    public void solve() {
        int n = in.nextInt();
        Point[] ps = new Point[n];
        for (int i = 0; i < n; i++) {
            int x = in.nextInt();
            int y = in.nextInt();
            ps[i] = new Point(x, y);
        }
        Arrays.sort(ps, Comparator.comparingDouble(p -> Math.atan2(p.x, p.y)));
        double res = 0;
        for (int i = 0; i < n; i++) {
            long curX = 0;
            long curY = 0;
            double cur = 0;
            for (int j = 0; j < n; j++) {
                curX += ps[(i+j)%n].x;
                curY += ps[(i+j)%n].y;
                res = Math.max(res, curX * curX + curY * curY);
            }
        }
        System.out.println(Math.sqrt(res));
    }

    public void run() {
        if (__FILE_DEBUG_FLAG__ == __t__) {
            try {
                is = new FileInputStream(__DEBUG_FILE_NAME__);
            } catch (FileNotFoundException e) {
                e.printStackTrace();
            }
            System.out.println("FILE_INPUT!");
        } else {
            is = System.in;
        }
        in = new FastScanner(is);
        out = new PrintWriter(System.out);

        solve();
    }

    public static void main(final String[] args) {
        new Main().run();
    }
}


class FastScanner {
    private InputStream stream;
    private byte[] buf = new byte[1024];
    private int curChar;
    private int numChars;

    public FastScanner(InputStream stream) {
        this.stream = stream;
        // stream = new FileInputStream(new File("dec.in"));
    }

    int read() {
        if (numChars == -1)
            throw new InputMismatchException();
        if (curChar >= numChars) {
            curChar = 0;
            try {
                numChars = stream.read(buf);
            } catch (IOException e) {
                throw new InputMismatchException();
            }
            if (numChars <= 0)
                return -1;
        }
        return buf[curChar++];
    }

    public boolean isSpaceChar(int c) {
        return c == ' ' || c == '\n' || c == '\r' || c == '\t' || c == -1;
    }

    public boolean isEndline(int c) {
        return c == '\n' || c == '\r' || c == -1;
    }

    public int nextInt() {
        return Integer.parseInt(next());
    }

    public int[] nextIntArray(int n) {
        int[] array = new int[n];
        for (int i = 0; i < n; i++)
            array[i] = nextInt();

        return array;
    }

    public int[][] nextIntMap(int n, int m) {
        int[][] map = new int[n][m];
        for (int i = 0; i < n; i++) {
            map[i] = nextIntArray(m);
        }
        return map;
    }

    public long nextLong() {
        return Long.parseLong(next());
    }

    public long[] nextLongArray(int n) {
        long[] array = new long[n];
        for (int i = 0; i < n; i++)
            array[i] = nextLong();

        return array;
    }

    public long[][] nextLongMap(int n, int m) {
        long[][] map = new long[n][m];
        for (int i = 0; i < n; i++) {
            map[i] = nextLongArray(m);
        }
        return map;
    }

    public double nextDouble() {
        return Double.parseDouble(next());
    }

    public double[] nextDoubleArray(int n) {
        double[] array = new double[n];
        for (int i = 0; i < n; i++)
            array[i] = nextDouble();

        return array;
    }

    public double[][] nextDoubleMap(int n, int m) {
        double[][] map = new double[n][m];
        for (int i = 0; i < n; i++) {
            map[i] = nextDoubleArray(m);
        }
        return map;
    }

    public String next() {
        int c = read();
        while (isSpaceChar(c))
            c = read();
        StringBuilder res = new StringBuilder();
        do {
            res.appendCodePoint(c);
            c = read();
        } while (!isSpaceChar(c));
        return res.toString();
    }

    public String[] nextStringArray(int n) {
        String[] array = new String[n];
        for (int i = 0; i < n; i++)
            array[i] = next();

        return array;
    }

    public String nextLine() {
        int c = read();
        while (isEndline(c))
            c = read();
        StringBuilder res = new StringBuilder();
        do {
            res.appendCodePoint(c);
            c = read();
        } while (!isEndline(c));
        return res.toString();
    }

    public int[][] nextPackedIntArrays(int packN, int size) {
        int[][] res = new int[packN][size];
        for (int i = 0; i < size; i++) {
            for (int j = 0; j < packN; j++) {
                res[j][i] = nextInt();
            }
        }
        return res;
    }
}

class ModCombination {
    private final ModLib modLib;

    private final long[] fact;
    private final long[] invfact;

    private ModCombination(ModLib modLib, long[] fact, long[] invfact) {
        this.modLib = modLib;
        this.fact = fact;
        this.invfact = invfact;
    }

    public static ModCombination create(int size, ModLib modLib) {
        long[] fact = new long[size];
        fact[0] = 1;
        for (int i = 0; i < size - 1; i++) {
            fact[i+1] = modLib.mult(fact[i], i + 1);
        }
        long[] invfact = new long[size];
        invfact[size-1] = modLib.div(1, fact[size-1]);
        for (int i = size - 2; i >= 0; i--) {
            invfact[i] = modLib.mult(invfact[i+1], i + 1);
        }
        return new ModCombination(modLib, fact, invfact);
    }

    public long fact(int x) {
        return fact[x];
    }

    public long invfact(int x) {
        return invfact[x];
    }

    public long comb(int a, int b) {
        if (b > a) {
            return 0;
        }
        return modLib.mult(fact[a], modLib.mult(invfact[b], invfact[a-b]));
    }

}

class ModLib {
    final long mod;

    public ModLib(long mod) {
        this.mod = mod;
    }

    public long add(long a, long b) {
        return (a + b) % mod;
    }

    public long sub(long a, long b) {
        return (a - b + mod) % mod;
    }

    public long mult(long a, long b) {
        return ((a % mod) * (b % mod)) % mod;
    }

    public long pow(long a, long x) {
        long res = 1;
        for ( ; x > 0; x >>= 1) {
            if (x % 2 == 1) res = mult(res, a);
            a = mult(a, a);
        }

        return res;
    }

    public long div(long a, long b) {
        return mult(a, pow(b, mod - 2));
    }
}