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 $ .

N/A

N/A

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) $ .