A person eating a seudah on Hoshanah Rabbah wanted to continue eating his seudah when Shmini Etzeres already started. He was "pores mappah", made kiddush and continued to eat. Now, after a meal eaten wholly on Hoshanah Rabbah, one says "Sukkos" in Yaleh V'Yavo in Bircas HaMazon; after a meal eaten wholly on Shmini Etzeres, one says "Shmini Etzeres". So what about this felow who started on Hoshanah Rabbah and ended on Shmini Etzeres: When he recites Bircas HaMazon what day should be mentioned in Yaleh V'Yavo, "Sukkos" or "Shmini Etzeres"? What's the reason why and please cite all sources.
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The Pri Megadim in Ashel Avraham Siman 188 Sif-Katan 18 discusses this exact shailo. He writes that it is similar to the shailo of when Rosh Chodesh comes out after Shabbos and one eats after nightfall. According to the Taz one mentions both Shabbos and Rosh Chodesh. He said so to in this case one would mention both Sukkos and Shmini Atzeres (which is quite surprising since it's not 2 seperate things.) However (if I understand correctly) he says that according to the Magen Avraham one "wouldn't say". What I believe he means is that one wouldn't say Sukkos and would only say Shmini Atzeres that is because in the case of Rosh Chodesh coming out Motzei Shabbos the M"A holds that if one eats when it's already night they only mention Rosh Chodesh and not Shabbos.