Laura 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 1.
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 $10^{-6}$.
Sample Input 1 | Sample Output 1 |
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4 3 1 2 4 |
0.364285714 |
Sample Input 2 | Sample Output 2 |
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5 1 1 3 3 3 |
0.125 |