In this paper, we find collisions of MD5 in the Matyas-Meyer-Oseas mode and Miyaguchi-Preneel mode with a complexity of 239 operations, which runs contrary to the cryptographer's belief that these modes are stronger against collision attacks than the Davies-Meyer mode due to the impossibility of the message modification. We then show that, our collision attack for the Matyas-Meyer-Oseas mode can give impact to some collision properties of the Davies-Meyer mode, which we call "free-start given-message collisions" and "NMAC colliding keys". These indicate that collisions of MMO-MD5 give some impacts on the original MD5. The attack is implemented on a PC and we present generated collisions of MMO-MD5.