레이팅 미적용Laura is organising a knockout tournament, in which her friend Dale takes part. Laura would like to maximise the probability of Dale winning the tournament by arranging the games in a favourable way. She does not know how to do it, so she asked you for help. Naturally, you refuse to cooperate with such a deplorable act—but then you realise that it is a very nice puzzle!
When the number of players is a power of two, the tournament setup can be described recursively as follows: the players are divided into two equal groups that each play their own knockout tournament, after which the winners of both tournaments play each other. Once a player loses, they are out of the tournament.
When the number of players is not a power of two, some of the last players in the starting line-up advance from the first round automatically so that in the second round the number of players left is a power of two, as shown in Figure .
Every player has a rating indicating their strength. A player with rating \(a\) wins a game against a player with rating \(b\) with probability \(\frac{a}{a+b}\) (independently of any previous matches played).
Laura as the organiser can order the starting line-up of players in any way she likes. What is the maximum probability of Dale winning the tournament?
The input consists of:
- One line with an integer \(n\) (\(2 \le n \le 4096\)), the total number of players.
- \(n\) lines, each with an integer \(r\) (\(1 \le r \le 10^5\)), the rating of a player. The first rating given is Dale's rating.
Output the maximum probability with which Dale can win the tournament given a favourable setup. Your answer should have an absolute or relative error of at most \(10^{-6}\).
4
3
1
2
4
0.364285714
5
1
1
3
3
3
0.125