406654: GYM102471 I Moon

Let $$$S$$$ be a sphere with radius $$$1$$$ and center $$$(0, 0, 0)$$$. Let $$$a_0,a_1,\ldots,a_n$$$ be $$$n+1$$$ points on the surface of $$$S$$$. The positions of $$$a_1,\ldots,a_n$$$ are fixed while the position of $$$a_0$$$ is a uniform random point on the surface of $$$S$$$. Let $$$f$$$ be $$$1$$$ if there exists a hemisphere of $$$S$$$ that contains $$$a_0,\ldots,a_n$$$(possibly on the border) and $$$0$$$ otherwise. Calculate the expected value of $$$f$$$.

The first line contains an integer $$$n$$$ denoting the number of points ($$$0\le n\le 100000$$$).

The $$$i$$$-th line of the next $$$n$$$ lines contains three integers $$$x, y, z$$$ denoting the point $$$a_i=\left(\frac{x}{\sqrt{x^2+y^2+z^2}}, \frac{y}{\sqrt{x^2+y^2+z^2}}, \frac{z}{\sqrt{x^2+y^2+z^2}}\right)$$$ ($$$-1000000\le x, y, z\le 1000000, x^2+y^2+z^2\neq 0$$$).

It is guaranteed that $$$a_1,\ldots,a_n$$$ are distinct.

Output the answer.

The answer will be considered correct if its absolute or relative error doesn't exceed $$$10 ^{-6}$$$.

3
1 0 0
0 1 0
0 0 1

0.875000000000