CF1188B.Count Pairs

You are given a prime number $p$ , $n$ integers $a_1, a_2, \ldots, a_n$ , and an integer $k$ .

Find the number of pairs of indexes $(i, j)$ ( $1 \le i < j \le n$ ) for which $(a_i + a_j)(a_i^2 + a_j^2) \equiv k \bmod p$ .

The first line contains integers $n, p, k$ ( $2 \le n \le 3 \cdot 10^5$ , $2 \le p \le 10^9$ , $0 \le k \le p-1$ ). $p$ is guaranteed to be prime.

The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ( $0 \le a_i \le p-1$ ). It is guaranteed that all elements are different.

Output a single integer — answer to the problem.

In the first example:

$(0+1)(0^2 + 1^2) = 1 \equiv 1 \bmod 3$ .

$(0+2)(0^2 + 2^2) = 8 \equiv 2 \bmod 3$ .

$(1+2)(1^2 + 2^2) = 15 \equiv 0 \bmod 3$ .

So only $1$ pair satisfies the condition.

In the second example, there are $3$ such pairs: $(1, 5)$ , $(2, 3)$ , $(4, 6)$ .