Since everybody’s throwing out theories, here’s mine.
MC encounters an accident before taking his entrance exams. He person in the accident, but is so shaken up by the experience that he fails the exam. Person he saved turns out to be the FMC (Yankee girl/teacher.)
FMC wanted to thank MC for saving him but she finds out he failed the exam because of the accident. FMC now signed up as an instructor in the cram school to watch over MC and help him pass as a way to pay him back.