Is it obligatory to make up ten days or a full month of fasting, knowing that I missed ten days three years ago and was an adult at the time? And is it permissible to delay making up these days until after exams due to my preoccupation with studying?

If you deliberately broke your fast in Ramadan (definition) without a legitimate Shar'i excuse while you were an adult, you have committed a grave sin. You must repent to Allah, increase your seeking of forgiveness, and perform righteous deeds. You are only required to make up for the days you missed.

If you delayed making up for these days until the next Ramadan without an excuse, you must pay an expiation for the delay, which is one mudd of the staple food of the people of the land, given to a needy person for each day you missed. However, if you were unaware of the impermissibility of this delay, then there is no expiation for you. This expiation is not repeated for the number of years you delayed making up the fast. As for delaying making up the fast until after exams, it is permissible according to those who hold that making up fasts can always be done at leisure, and impermissible according to those who hold that it must be done immediately if the original fast was abandoned without excuse.