If a boy or girl become Bar Mitzva or Bas Mitzva on the the 10th day of Av which turns out to be Tisha B'Av because the 9th of Av was on Shabbos, are they obligated to fast?
The Piskei Tshuvos (550:4) brings many opinions on the matter. The Avnei Nezer paskens that they need not fast. The fact that we pasken that on Shabbos Tisha B'Av we are noheg Aveilus b'Tzina proves that Sunday's fast is only to make up for the fast we didn't do on Shabbos. Since the katan had no obligation to fast on Shabbos he need not fast on Sunday.
Many poskim including the Divrei Malkiel, Shevet HaLevi, Lehoros Nosson, and Tzafnas Panei'ach all say that if you become a gadol on Sunday you need to fast. They base this on a Tshuvas Rashba that says we are not noheg any Aveilus on Shabbos Tisha B'Av which indicates that Tisha B'Av is totally transplanted to Sunday and not Tashlumim of Shabbos therefore the Chiyuv depends on today and not yesterday.
Rav Yisroel Yaakov Fisher in Even Yisroel says that they need to fast but for a different reason. While technically they are not obligated to fast because it is Tashlumim of Shabbos, since all Klal Yisroel is holding a day of mourning, its obligations apply to them as well, as they are now full fledged adult members of Klal Yisroel.
Important Note: We bring this tshuvah as a starting point for discussion and not to convey any halacha. We try to convey the Tshuva to the best of our ability. We admit that our understanding may not be accurate. One should learn the tshuva to verify the accuracy of our interpretation. Please understand that this Tshuva may not be the final word on this topic. One should consult a Rav before drawing any conclusions.