Lifting the Stone
Time Limit: 2 Seconds
Memory Limit: 65536 KB
There are many secret openings in the floor which are covered by a big heavy stone.
When the stone is lifted up, a special mechanism detects this and activates poisoned
arrows that are shot near the opening. The only possibility is to lift the stone
very slowly and carefully. The ACM team must connect a rope to the stone and then
lift it using a pulley. Moreover, the stone must be lifted all at once; no side
can rise before another. So it is very important to find the centre of gravity
and connect the rope exactly to that point. The stone has a polygonal shape and
its height is the same throughout the whole polygonal area. Your task is to find
the centre of gravity for the given polygon.
The input consists of T test cases. The number of them (T) is given on the first
line of the input. Each test case begins with a line containing a single integer
N (3 <= N <= 1000000) indicating the number of points that form the polygon.
This is followed by N lines, each containing two integers Xi and Yi (|Xi|, |Yi|
<= 20000). These numbers are the coordinates of the i-th point. When we connect
the points in the given order, we get a polygon. You may assume that the edges
never touch each other (except the neighboring ones) and that they never cross.
The area of the polygon is never zero, i.e. it cannot collapse into a single
Print exactly one line for each test case. The line should contain exactly two
numbers separated by one space. These numbers are the coordinates of the centre
of gravity. Round the coordinates to the nearest number with exactly two digits
after the decimal point (0.005 rounds up to 0.01). Note that the centre of gravity
may be outside the polygon, if its shape is not convex. If there is such a case
in the input data, print the centre anyway.
Source: Central Europe 1999