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Are you allowed to blow shofar after chatzos. Do we say "davar shemitzvusoi byom kasher kol hayom(Megilla 20b)"?Or do we say its like tefilla and tefilla is assur after chatzos?

I find it hard to believe that you don't already know the answer to this one.

From a plain halacha perspective, the answer is clear. Tekias shofar is all day, as is clearly mentioned in שלחן ערוך. However, the gemara says that originally תקיעת שופר was during שחרית, so to do it בברכותיה it would have been before חצות. It was changed when the oppressors didn't forbade tekias shofar, and were on guard during shacharis.