Is it permissible to enter the mosque for an examination in a Quran memorization competition during menstruation?

A menstruating woman is not permitted to sit in the mosque. Scholars have based this on the Prophet's (peace be upon him) command for menstruating women to avoid the Eid prayer ground, and because the Eid prayer ground holds the same حكم (ruling/status) as a mosque, her entry into the mosque is prohibited.

However, there is no harm in a menstruating woman passing through the mosque if there is a need for it and she is certain that she will not contaminate the mosque, based on the Almighty's saying: ﴿وَلَا جُنُبًا إِلَّا عَابِرِي سَبِيلٍ حَتَّى تَغْتَسِلُوا﴾ (nor one who is in a state of major ritual impurity, except when passing through, until you have bathed). A menstruating woman is in the same category as one who is in a state of major ritual impurity (junub).

She is also permitted to sit in the unsheltered courtyard of the mosque or at its entrance to listen to a lesson, but she is not allowed to enter [the mosque itself].

A menstruating woman is permitted to recite the Qur'an without touching the Mus'haf (physical copy of the Qur'an).