The students that stood around Amon, although they came from poor clans, they were the best those clans had to offer. Different from those rich clans who sent their young generations who are talented yet the clans didn't wish to nurture directly. These poor clans didn't have enough resources to properly nurture any of their talented descendants. Hence, they sent their best in the hope that the royal academy could help them grow. Even if later they ended up leaving the clans and joining the royal kingdom, it was still better than letting them rot in their poor clans.
Hence, all these students even though having a feeling of inferiority against the big clans, they also possessed the pride of being the best from where they came from. All of them had the thought that if only they were provided the same resources as those big clans' descendants, they won't lose.
Most of all, they were all hot-headed teenagers. After being challenged in such a way by Amon, how could they back off?