2022 Fermat Problem 24

There are functions f(x) with the following properties:

- f(x) = ax^2 + bx + c for some integers a, b and c with a > 0, and - f(p) = f(q) = 17 and f(p+q) = 47 for some prime numbers p and q with p < q.

For each such function, the value of f(pq) is calculated. The sum of all possible values of f(pq) is S. What are the rightmost two digits of S?

If there are no answer choices shown, enter a numerical answer.

Full credit to this problem is given to the CEMC, you may view all Fermat contests here.