1857F · Sum and Product

Problem: You have an array $a$ of length $n$.

Your task is to answer $q$ queries: given $x,y$, find the number of pairs $i$ and $j$ ($1 \le i < j \le n$) that both $a_i + a_j = x$ and $a_i \cdot a_j = y$.

That is, for the array $[1,3,2]$ and asking for $x=3,y=2$ the answer is $1$:

• $i=1$ and $j=2$ fail because $1 + 3 = 4$ and not $3,$ also $1 \cdot 3=3$ and not $2$;
• $i=1$ and $j=3$ satisfies both conditions;
• $i=2$ and $j=3$ fail because $3 + 2 = 5$ and not $3,$ also $3 \cdot 2=6$ and not $2$;

Input Format: The first line contains one integer $t$ ($1\le t\le 10^4$) — the number of test cases.

The second line of each test case contains one integer $n$ ($1 \le n \le 2\cdot 10^5$) — the length of the array $a$.

The third line of each test case contains $n$ integers $a_1,a_2,\dots,a_n$ ($1 \le |a_i| \le 10^9$) — array $a$.

The fourth line of each test case contains the integer $q$ ($1 \le q \le 2\cdot 10^5$) — the number of requests.

The next $q$ lines contain two numbers each $x$ and $y$ ($1 \le |x|\le 2\cdot 10^9,1\le |y|\le 10^{18}$) — request.

It is guaranteed that the sum of $n$ over all test cases does not exceed $2\cdot 10^5$. This is also guaranteed for the sum of $q$ values.

Output Format: For each test case print a line with $q$ numbers — the answers to the queries.

Note: For the first test case, let's analyze each pair of numbers separately:

• pair $(a_1,a_2)$: $a_1 + a_2 = 4$, $a_1 \cdot a_2 = 3$
• pair $(a_1,a_3)$: $a_1 + a_3 = 3$, $a_1 \cdot a_3 = 2$
• pair $(a_2,a_3)$: $a_2 + a_3 = 5$, $a_2 \cdot a_3 = 6$

In the second test case, all combinations of pairs are suitable.