8122: USACO 2021 US Open, Silver Problem 2. Do You Know Your ABCs?
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Description
Farmer John's cows have been holding a daily online gathering on the "mooZ" video meeting platform. For fun, they have invented a simple number game to play during the meeting to keep themselves entertained.
Elsie has three positive integers A, B, and C (1≤A≤B≤C). These integers are supposed to be secret, so she will not directly reveal them to her sister Bessie. Instead, she tells Bessie N (4≤N≤7) distinct integers $x_1,x_2,…,x_N\ (1≤x_i≤10^9)$, claiming that each $x_i$ is one of A, B, C, A+B, B+C, C+A, or A+B+C. However, Elsie may be lying; the integers $x_i$ might not correspond to any valid triple (A,B,C).
This is too hard for Bessie to wrap her head around, so it is up to you to determine the number of triples (A,B,C) that are consistent with the numbers Elsie presented (possibly zero).
Each input file will contain T (1≤T≤100) test cases that should be solved independently.
This is too hard for Bessie to wrap her head around, so it is up to you to determine the number of triples (A,B,C) that are consistent with the numbers Elsie presented (possibly zero).
Each input file will contain T (1≤T≤100) test cases that should be solved independently.
Input
The first input line contains T.
Each test case starts with N, the number of integers Elsie gives to Bessie.
The second line of each test case contains N distinct integers $x_1,x_2,…,x_N$.
The second line of each test case contains N distinct integers $x_1,x_2,…,x_N$.
Output
For each test case, output the number of triples (A,B,C) that are consistent with the numbers Elsie presented.
Sample 1 Input
10
7
1 2 3 4 5 6 7
4
4 5 7 8
4
4 5 7 9
4
4 5 7 10
4
4 5 7 11
4
4 5 7 12
4
4 5 7 13
4
4 5 7 14
4
4 5 7 15
4
4 5 7 16
Sample 1 Output
1
3
5
1
4
3
0
0
0
1
For x={4,5,7,9}, the five possible triples are as follows:
(2,2,5),(2,3,4),(2,4,5),(3,4,5),(4,5,7).