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2020-05-18 03:10 AKDT

AAPS20 Session 06

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2020-05-18 06:30 AKDT
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Problem B
Dart Scoring

You have invented a new form of the game of darts, and it is based on the idea that the tightest cluster of darts wins — but they don’t necessarily have to be close to a target. In fact, the game is easy to play because it doesn’t require a target; players can just throw darts against a wall (preferably one that you don’t care about). When a player is done throwing their $n$ darts, they wrap a string tightly around the outside of all the outermost darts, so that the string encompasses all of the thrown darts. If $s$ is the length of the string needed to enclose the darts, then the score for that player’s darts is $100 \cdot n / (1 + s)$.

Now you need a program to determine the score of different dart configurations.

Input

Input consists of up to $200$ player turns, one turn per line. Each player’s turn has a list of $1$ to $100$ pairs of real numbers with at most $3$ digits past the decimal, which are the $(x, y)$ locations where the dart has landed. The bounds are $-30 \le x, y \le 30$. Input ends at the end of file.

Output

For each turn, print the score obtained, with at most $0.0001$ error.

Sample Input 1 Sample Output 1
10.8 -13.7
0.278 2.555 2.815 3.800 3.920 1.510
2.358 1.731 0.663 3.485 4.276 6.242 3.858 0.089 0.409 1.460 2.578 4.539
0.117 5.881 4.655 2.766 0.941 0.213 6.180 6.550 4.215 6.723 5.822 4.367 5.464 1.001
100.0000000000
29.5344250128
34.3557840779
30.3425374145