409975: GYM103886 B Cereal Robber

Jesse has stolen a treasured cereal box from his principal, and needs to avoid getting caught!

Jesse's school can be modeled by a rectangle on a 2-D coordinate plane, with its lower left corner at $$$(0,0)$$$ and top right corner at $$$(H, K)$$$.

Jesse will use his escape-scanner 3000 to find a place to escape. It must be placed on integer coordinates, and will scan every point within a radius $$$R$$$.

However, Jesse's principal has hired $$$n$$$ hall monitors to catch him, standing at distinct integer coordinates in the school. If the scanner scans a point containing one of the monitors, scans on the border, or scans outside the school, it fails (and Jesse gets caught)!

What is the biggest radius $$$R$$$ he can achieve so that his escape-scanner does not fail?

Your answer is considered correct if its absolute or relative error does not exceed $$$10^{-6}$$$.

Formally, let your answer be $$$a$$$, and the jury's answer be $$$b$$$. Your answer is accepted if and only if $$$\frac{|a - b|}{\max{(1, |b|)}} \le 10^{-6}$$$.

The first line contains $$$H$$$, $$$K$$$, and $$$n$$$ ($$$1 \le H, K, n \le 200$$$).

The next $$$n$$$ lines, contain two integers each, $$$x_i$$$ and $$$y_i$$$ $$$(0 \le x_i \le H, 0 \le y_i \le K)$$$, signifying the location of a hall monitor.

It is guaranteed that all $$$(x_i, y_i)$$$ are distinct.

Print the maximum value of $$$R$$$ that will allow Jesse to use his escape-scanner 3000.

10 10 5
1 4
2 3
9 9
4 6
5 5

2.8284271

The maximal radius of scanning can be achieved when Jesse places his scanner on the point $$$(7, 3)$$$.

Problem Credits: Chongtian Ma