How to find collisions in a secret-prefix Merkle–Damgård given an adversary that can choose the IV?
Algorithm C would give you the answer directly, except it needs an IV to start off with, and you don't initially know that.  Here's some hints:


The IV you give algorithm C needn't be the secret IV $K$; instead, it can be the intermediate state of some message
So, how could you use your access to function $F_K$ to find the intermediate state in some message?