USACO 2021 US Open, Silver

Problem 2. Do You Know Your ABCs?


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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 $C$ ($1\le A\le B\le C$). These integers are supposed to be secret, so she will not directly reveal them to her sister Bessie. Instead, she tells Bessie $N$ ($4\le N\le 7$) distinct integers $x_1,x_2,\ldots,x_N$ ($1\le x_i\le 10^9$), 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 $T$ ($1\le T\le 100$) test cases that should be solved independently.

INPUT FORMAT (input arrives from the terminal / stdin):

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 $x_1,x_2,\ldots,x_N$.

OUTPUT FORMAT (print output to the terminal / stdout):

For each test case, output the number of triples $(A,B,C)$ that are consistent with the numbers Elsie presented.

SAMPLE 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 OUTPUT:

1
3
5
1
4
3
0
0
0
1

For $x=\{4,5,7,9\}$, the five possible triples are as follows:

$$(2, 2, 5), (2, 3, 4), (2, 4, 5), (3, 4, 5), (4, 5, 7).$$

SCORING:

  • In test cases 1-4, all $x_i$ are at most $50$.
  • Test cases 5-6 satisfy $N=7$.
  • Test cases 7-15 satisfy no additional constraints.

Problem credits: Benjamin Qi

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