Problem:
Score : 600 points

Problem Statement

Given are N points (x_i, y_i) in a two-dimensional plane.

Find the minimum radius of a circle such that all the points are inside or on it.

Constraints

2 \leq N \leq 50

0 \leq x_i \leq 1000

0 \leq y_i \leq 1000

The given N points are all different.

The values in input are all integers.

Input

Input is given from Standard Input in the following format:

N
x_1 y_1
:
x_N y_N

Output

Print the minimum radius of a circle such that all the N points are inside or on it.

Your output will be considered correct if the absolute or relative error from our answer is at most 10^{-6}.

Sample Input 1

2
0 0
1 0

Sample Output 1

0.500000000000000000

Both points are contained in the circle centered at (0.5,0) with a radius of 0.5.

Sample Input 2

3
0 0
0 1
1 0

Sample Output 2

0.707106781186497524

Sample Input 3

10
10 9
5 9
2 0
0 0
2 7
3 3
2 5
10 0
3 7
1 9

Sample Output 3

6.726812023536805158

If the absolute or relative error from our answer is at most 10^{-6}, the output will be considered correct.

Solution in PHP7 (7.0.15) (judge verdict: Time Limit Exceeded)

<?php

const MAX_X = 1000;
const MAX_Y = 1000;
const COMPUTATIONAL_ERROR = 1E-9;

list($n) = explode(' ', trim(fgets(STDIN)));
$points = [];

for ($i = 0; $i < $n; ++$i) {
    $points[] = explode(' ', trim(fgets(STDIN)));
}

$max_r = sqrt(MAX_X ** 2 + MAX_Y ** 2) / 2 + COMPUTATIONAL_ERROR;
$upper_r = $max_r;
$lower_r = 0;
$r_diff = COMPUTATIONAL_ERROR ** 2;
while ($upper_r - $lower_r > $r_diff) {
    $radius = ($upper_r + $lower_r) / 2;
    $intersections = getIntersections($points, $radius);
    $filtered_intersections = filterIntersection($intersections, $points, $radius);
    if (empty($filtered_intersections)) {
        $lower_r = $radius;
    } else {
        $upper_r = $radius;
    }
}

echo $radius;

function getIntersections(array $points, float $radius): array
{
    $point_num = count($points);
    $intersections = [];
    for ($i = 0; $i < $point_num; ++$i) {
        for ($j = $i + 1; $j < $point_num; ++$j) {
            $x_diff = $points[$i][0] - $points[$j][0];
            $y_diff = $points[$i][1] - $points[$j][1];
            $distance = sqrt($x_diff ** 2 + $y_diff ** 2);
            if ($distance > ($radius + COMPUTATIONAL_ERROR) * 2) {
                continue;
            }

            $median = [($points[$i][0] + $points[$j][0]) / 2, ($points[$i][1] + $points[$j][1]) / 2];
            $h = sqrt($radius ** 2 - ($distance / 2) ** 2);
            if ($x_diff * $y_diff > 0) {
                $intersections[] = [$median[0] - $h * $y_diff / $distance, $median[1] + $h * $x_diff / $distance];
                $intersections[] = [$median[0] + $h * $y_diff / $distance, $median[1] - $h * $x_diff / $distance];
            } else {
                $intersections[] = [$median[0] - $h * $y_diff / $distance, $median[1] - $h * $x_diff / $distance];
                $intersections[] = [$median[0] + $h * $y_diff / $distance, $median[1] + $h * $x_diff / $distance];
            }
        }
    }

    return $intersections;
}

function filterIntersection(array $intersections, array $points, float $radius): array
{
    return array_filter($intersections, function ($intersection) use ($points, $radius) {
        foreach ($points as $point) {
            $distance = sqrt(($intersection[0] - $point[0]) ** 2 + ($intersection[1] - $point[1]) ** 2);
            if ($distance > $radius + COMPUTATIONAL_ERROR) {
                return false;
            }
        }

        return true;
    });
}