This was an excellent entry and a blast to play. I like that you found a way to incorporate the requirement in couple different ways and the reward for taking damage made it worth taking the hit, but forced me to play that much more diligently.
I loved the art, music and sound design. I do think the bosses should have a health bar so I can feel my progress through the run. I also noticed two small things. On the first boss, when the four mouths open and shoot bullets out, it's almost like origin point was off, so it didn't really look like it was coming out of the bosses mouths, so it broke the immersion a little bit.
The power ups felt very rewarding to use, but there was one run between the first and second boss, where the powerups didn't display, and then the select text stayed up during my entire fight:
Overall, a great entry. Three awesome boss fights and a clever way to use the requirement in multiple ways. I really liked this one. Good job, team!