I only remember a few details. Superpowers were a known thing in the setting, reasonably common. The kids are living together a suite or small building. There was a boy with super-strength who was prone to berserk rates and a duplicator. All students were required to go into single combat against one of their classmates to begin training. The combat was in an artificial environment (a physical arena with prop buildings, not holograms) and was violent, sometimes deadly. In one combat, a girl who could teleport things close to her won by teleporting away the muscles of her opponent's legs. The berserk guy fought the duplicator and reduced most of the dupes to a bloody pulp after he lost himself. I'm pretty sure they also had a suspected shapeshifter fight.
I read this on a website in 2016 and it already has several chapters available. There might have been a physical book available for purchase too. I probably found the link off of TvTropes.org.