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DP - Problem: Knockout Tournament

This problem K from the NWERC 2016.

Knockout Tournament

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 the figure below.


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?


  • One line with an integer \(n\), the total number of players.
  • \(n\) lines, each with an integer \(r\), the rating of a player. The first rating given is Dale's rating.


  • \(2 \leq n \leq 4096\)
  • \(1 \leq r \leq 10^5\)


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 Test Cases

Sample input 1

Sample output 1

Sample input 2

Sample output 2

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Allowed extensions: .java, .cpp, .py