Count pairs

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Problem

88% Success 5427 Attempts 30 Points 1.5s Time Limit 256MB Memory 1024 KB Max Code

You are given an array $$A$$ consisting of $$N$$ non-negative integers. You are also given 2 integers $$p$$(a prime number) and $$k$$. You are required to count number of pairs $$(i, j)$$ where, $$1 \leq i < j \leq N$$ and satisying: $$ (A_i^2 + A_j^2 + A_i*A_j) \text{ mod }p = k$$ where \(a \text{ mod } p = b\) means that \(b\) is the remainder when \(a\) is divided by \(p\). In particular, \(0 \leq b < p\). You are given $$T$$ test cases. Input format The first line contains a single integer $$T$$ denoting the number of test cases. For each test case: The first line contains three space-separated integers $$N$$, $$k$$, and $$p$$ denoting the length of the array, the required remainder/modulo, and the prime number respectively. The second line contains $$N$$ space-separated integers denoting the integer array $$A$$. Output format For each test case, print the number of pairs satisfying the equation in a separate line. Constraints $$ 1 \le T \le 1000 \\ 1 \le N \le 5 \times 10^5 \\ 1 \le p \le 10^9 \\ \text{p is prime} \\ 0 \le k < p \\ 0 \le A_i \le 10^9 \\ \text{Sum of N over all test cases does not exceed } 10^6 $$

Sample Input

2
4 1 2
3 2 1 0
8 0 3
9 2 1 8 4 5 9 10

Sample Output

5
7

Explanation

In the first sample, $$N = 4, k = 1, p = 2, A = [3, 2, 1, 0]$$. Caculations for each pair $$(i, j)$$  are shown below: $$(1, 2)$$ : $$A_1^2 + A_2^2 + A_1*A_2 \text{ mod } p = (9 + 4 + 6) \text{ mod } 2 = 1$$. $$(1, 3)$$ : $$A_1^2 + A_3^2 + A_1*A_3 \text{ mod } p = (9 + 1 + 3) \text{ mod } 2 = 1$$. $$(1, 4)$$ : $$A_1^2 + A_4^2 + A_1*A_4 \text{ mod } p = (9 + 0 + 0) \text{ mod } 2 = 1$$. $$(2, 3)$$ : $$A_2^2 + A_3^2 + A_2*A_3 \text{ mod } p = (4 + 1 + 4) \text{ mod } 2 = 1$$. $$(2, 4)$$ : $$A_2^2 + A_4^2 + A_2*A_4 \text{ mod } p = (4 + 0 + 0) \text{ mod } 2 = 0$$. $$(3, 4)$$ : $$A_3^2 + A_4^2 + A_3*A_4 \text{ mod } p = (1 + 0 + 0) \text{ mod } 2 = 1$$. Therefore, the number of pairs satisfying the equation are 5. In the second sample, $$N = 8, k = 0, p = 3, A = [9, 2, 1, 8, 4, 5, 9, 10]$$. Valid pairs $$(i, j)$$ are: $$(1, 7), (2, 4), (2, 6), (3, 5), (3, 8), (4, 6), (5, 8)$$.

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