#P293D. Ksusha and Square
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ID: 2657
Type: RemoteJudge
4000ms
256MiB
Tried: 0
Accepted: 0
Difficulty: (None)
Uploaded By:
Hydro
Tags> Ksusha and Square
No submission language available for this problem.
Description
Ksusha is a vigorous mathematician. She is keen on absolutely incredible mathematical riddles.
Today Ksusha came across a convex polygon of non-zero area. She is now wondering: if she chooses a pair of distinct points uniformly among all integer points (points with integer coordinates) inside or on the border of the polygon and then draws a square with two opposite vertices lying in the chosen points, what will the expectation of this square's area be?
A pair of distinct points is chosen uniformly among all pairs of distinct points, located inside or on the border of the polygon. Pairs of points p, q (p ≠ q) and q, p are considered the same.
Help Ksusha! Count the required expectation.
The first line contains integer n (3 ≤ n ≤ 105) — the number of vertices of Ksusha's convex polygon. Next n lines contain the coordinates of the polygon vertices in clockwise or counterclockwise order. The i-th line contains integers xi, yi (|xi|, |yi| ≤ 106) — the coordinates of the vertex that goes i-th in that order.
Print a single real number — the required expected area.
The answer will be considered correct if its absolute and relative error doesn't exceed 10 - 6.
Input
The first line contains integer n (3 ≤ n ≤ 105) — the number of vertices of Ksusha's convex polygon. Next n lines contain the coordinates of the polygon vertices in clockwise or counterclockwise order. The i-th line contains integers xi, yi (|xi|, |yi| ≤ 106) — the coordinates of the vertex that goes i-th in that order.
Output
Print a single real number — the required expected area.
The answer will be considered correct if its absolute and relative error doesn't exceed 10 - 6.
Samples
3
0 0
5 5
5 0
4.6666666667
4
-1 3
4 5
6 2
3 -5
8.1583333333
3
17 136
859 937
16 641
66811.3704155169