Ezzat and Two Subsequences (Codeforces 1557A)
Tags: Math, sortings

Ezzat has an array of $$$n$$$ integers (maybe negative). He wants to split it into two non-empty subsequences $$$a$$$ and $$$b$$$, such that every element from the array belongs to exactly one subsequence, and the value of $$$f(a) + f(b)$$$ is the maximum possible value, where $$$f(x)$$$ is the average of the subsequence $$$x$$$. A sequence $$$x$$$ is a subsequence of a sequence $$$y$$$ if $$$x$$$ can be obtained from $$$y$$$ by deletion of several (possibly, zero or all) elements.The average of a subsequence is the sum of the numbers of this subsequence divided by the size of the subsequence.For example, the average of $$$[1,5,6]$$$ is $$$(1+5+6)/3 = 12/3 = 4$$$, so $$$f([1,5,6]) = 4$$$.

Input: The first line contains a single integer $$$t$$$ ($$$1 \le t \le 10^3$$$)— the number of test cases. Each test case consists of two lines.The first line contains a single integer $$$n$$$ ($$$2 \le n \le 10^5$$$).The second line contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$-10^9 \le a_i \le 10^9$$$).It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$3\cdot10^5$$$.

Output: For each test case, print a single value — the maximum value that Ezzat can achieve.Your answer is considered correct if its absolute or relative error does not exceed $$$10^{-6}$$$.Formally, let your answer be $$$a$$$, and the jury's answer be $$$b$$$. Your answer is accepted if and only if $$$\frac{|a - b|}{\max{(1, |b|)}} \le 10^{-6}$$$.

Note: In the first test case, the array is $$$[3, 1, 2]$$$. These are all the possible ways to split this array:   $$$a = [3]$$$, $$$b = [1,2]$$$, so the value of $$$f(a) + f(b) = 3 + 1.5 = 4.5$$$.  $$$a = [3,1]$$$, $$$b = [2]$$$, so the value of $$$f(a) + f(b) = 2 + 2 = 4$$$.  $$$a = [3,2]$$$, $$$b = [1]$$$, so the value of $$$f(a) + f(b) = 2.5 + 1 = 3.5$$$.  Therefore, the maximum possible value $$$4.5$$$.In the second test case, the array is $$$[-7, -6, -6]$$$. These are all the possible ways to split this array:   $$$a = [-7]$$$, $$$b = [-6,-6]$$$, so the value of $$$f(a) + f(b) = (-7) + (-6) = -13$$$.  $$$a = [-7,-6]$$$, $$$b = [-6]$$$, so the value of $$$f(a) + f(b) = (-6.5) + (-6) = -12.5$$$.  Therefore, the maximum possible value $$$-12.5$$$.

Sample Inputs:
4
3
3 1 2
3
-7 -6 -6
3
2 2 2
4
17 3 5 -3

Sample Outputs:
4.500000000
-12.500000000
4.000000000
18.666666667

Solution (Python):
for _ in range(int(input())):
    n = int(input())
    s = [*map(int, input().split())]
    a = max(s)
    s.remove(a)
    print(float(sum(s) / (n - 1)) + a)

Solution (Python):
from math import *
for _ in range(int(input())):
    n = int(input())
    a = list(map(int, input().split()))
    print(max(a) + (sum(a) - max(a)) / (n - 1))

Solution (Python):
def solve(a, n):
    avg = sum(a)
    k = max(a)
    avg -= k
    avg = avg / (n - 1)
    return '{:.9f}'.format(avg + k)

def main():
    ans = []
    t = int(input())
    for _ in range(t):
        n = int(input())
        a = list(map(int, input().split()))
        ans.append(solve(a, n))
    for i in ans:
        print(i)
if __name__ == '__main__':
    main()