Speaking of Great Buddha statue in Japan, most people imagine the statue in Nara or Kamakura. It is not well-known that the seated Great Buddha statue was in Ueno Park, and the only face is left now.
The seated Great Buddha statue was built in 1631. The face fell from the body due to the earthquake in 1647, and a bronze Great Buddha was newly built. Again, the face fell from the body due to the fire in 1841 and 1855, and the statue was repaired.
The face fell from the body due to earthquake in 1923. This was 4th fall. There was a plan to repair the statue, but, it was not executed.
When you visit Ueno Park, you can see the only face of Great Buddha statue. Since the face is strongly fixed, it will not fall again. This wins the hearts and minds of student preparing for an exam. They do not want to fail an exam. They hope to receive a power from Great Buddha.
I think that the last paragraph seems to be a little difficult to understand for non-Japanese readers.
We use the same verb "ochiru" in Japanese when expressing "face fall from the body" and "student fails an exam". That is, we say "face ochiru from the body" and "student ochiru an exam".
The face of Great Buddha statue will not ochiru. Thus, students believe that if they pray to the Buddha face then they will not ochiru an exam.