I think it is a shonen manga. I haven't read it in years. But I remember details of the story.

It is about a boy who is not enrolled in a martial arts school but is seeking to be partners with a student who has already a partner who is a loner with a stoic personality. There are students partnered with each other. Two partners.

Students participate in deadly games In the first round, some students trapped in a cave die when the teacher detonates explosives causing the cave to collapse and the boy seeks to join the dead student's partner. In the second round they are facing a dilemma where they try to find an antidote to the poison that they were injected with.

There is another major character who is a girl with a female partner

Please help me remember Name of a manga?

You may be looking for the manga "Battle Royale"

No I am afraid that is not the title I am looking for.

Anyone else?