Holiday Contest 2025 - Individual Round - Problem 10

For positive integer a < b, denote f(a,b) = 2\ln\left(\prod_{k = a}^{b-1}\left|\dfrac{1+k+k^2+i}{1+k^2}\right|\right). For a prime p \equiv 3 \pmod{4}, let \ln(g(p)) = \sum_{1 \le a < b < p}f(a,b). Find the sum of g(p) \pmod{p} across all primes p \le 100 and p \equiv 3 \pmod 4 (where g(p) \pmod{p} gives the integer k s.t. k \equiv g(p) \pmod{p} and 0 \le k < p).