Customising the Track (Codeforces 1543B)
Tags: Math, combinatorics

Highway 201 is the most busy street in Rockport. Traffic cars cause a lot of hindrances to races, especially when there are a lot of them. The track which passes through this highway can be divided into $$$n$$$ sub-tracks. You are given an array $$$a$$$ where $$$a_i$$$ represents the number of traffic cars in the $$$i$$$-th sub-track. You define the inconvenience of the track as $$$\sum\limits_{i=1}^{n} \sum\limits_{j=i+1}^{n} \lvert a_i-a_j\rvert$$$, where $$$|x|$$$ is the absolute value of $$$x$$$. You can perform the following operation any (possibly zero) number of times: choose a traffic car and move it from its current sub-track to any other sub-track.Find the minimum inconvenience you can achieve.

Input: The first line of input contains a single integer $$$t$$$ ($$$1\leq t\leq 10\,000$$$) — the number of test cases.The first line of each test case contains a single integer $$$n$$$ ($$$1\leq n\leq 2\cdot 10^5$$$).The second line of each test case contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$0\leq a_i\leq 10^9$$$).It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$2\cdot 10^5$$$.

Output: For each test case, print a single line containing a single integer: the minimum inconvenience you can achieve by applying the given operation any (possibly zero) number of times.

Note: For the first test case, you can move a car from the $$$3$$$-rd sub-track to the $$$1$$$-st sub-track to obtain $$$0$$$ inconvenience.For the second test case, moving any car won't decrease the inconvenience of the track.

Sample Inputs:
3
3
1 2 3
4
0 1 1 0
10
8 3 6 11 5 2 1 7 10 4

Sample Outputs:
0
4
21

Solution (Python):
for s in [*open(0)][2::2]:
    n = len((a := s.split()))
    r = sum(map(int, a)) % n
    print(r * (n - r))

Solution (Python):
def main():
    cont = int(input())
    for _ in range(cont):
        n = int(input())
        a = list(map(int, input().split()))
        s = sum(a) % n
        res = s * (n - s)
        print(res)
if __name__ == '__main__':
    main()

Solution (Python):
def solve():
    n = int(input())
    a = list(map(int, input().split()))
    extra = sum(a) % n
    print(extra * (n - extra))
for _ in range(int(input())):
    solve()