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Beis Shamai maintains that we may bring Shelamim on Yom Tov without performing Semichah, and we may bring Olos. Beis Hillel maintains that we may bring Shelamim and Olos and perform Semichah on Yom Tov.
One who did not bring the Korbenos Chagigah and Re'iyah on Shavuos may bring them for six days after Shavuos. (1)
We derive from Pesach that an overnight stay in Yerushalayim is required for Sukos.
When Shavuos falls on Erev Shabbos, the Korbanos are brought on Sunday according to Beis Shamai, and Beis Hillel maintains that the Korbanos may be brought on Yom Tov. (2)
It is a Mitzvah to count both the days and the weeks of the Omer.
The verse says, "You shall designate," and, "You shall harvest." This teaches that the Korbanos of the Chag may be brought after the Chag at a time that it is permitted to do Melachah. (3)
A BIT MORE
1. That we have seven days to bring the Korbanos of Shavuos is derived from Pesach. It is not derived from Sukos, which has eight days to bring the Korbanos, because of the principle of "Tafasta Merubah Lo Tafasta, Tafasta Mu'at Tafasta" (only the minimum may be derived, not the maximum).
2. Even though when Shavuos falls on Erev Shabbos it is impossible to bring the Korbanos the next day because it is Shabbos, Beis Shamai still maintains that the Korbanos may not be brought on Yom Tov.
3. Although we derive from Pesach that the Korbanos of Shavuos may be brought for seven days, we need these additional verses to teach that Melachah is permitted during the six days after Shavuos, since we might have thought that Melachah is prohibited as it is during Pesach.
The verse says, "You shall designate," and, "You shall harvest." This teaches that the Korbanos of the Chag may be brought after the Chag. The Gemara says that if not for these verse we might have thought think that the six days after Shavuos are prohibited in Melachah just like Pesach. The Chasam Sofer asks, how do we know from the verse that says, "You shall harvest," that the six days after Shavuos are permitted in Melachah? Perhaps the verse refers to a situation in which the crops and roots are completely dried out, in which case the Torah permits one to harvest the crops on Shabbos and Yom Tov, and it is prohibited only mid'Rabanan. The Chasam Sofer answers that the verse refers to a person who harvests the entire field, because the verse says that when he harvests he should leave Pe'ah on the corners of the field. Harvesting an entire field is forbidden on Shabbos and Yom Tov, even if the crops are dried, since it involves a lot of strenuous labor, and strenuous labor, according to the Ramban, is forbidden by the Torah even if one is not technically doing a Melachah.
It is a Mitzvah for each person to count the Omer himself. It must be counted standing, and a blessing is recited. Both the days and the weeks must be counted. (Shulchan Aruch OC 489:1)
The Poskim disagree about the law in the case of a person who counted the days of the Omer but not the weeks. According to some opinions, it suffices b'Di'eved, while other opinions maintain that one has not fulfilled the Mitzvah at all. Therefore, one should count again without reciting another blessing. If one failed to count again, he may count the rest of the days with the recital of the blessing. If one counted the weeks and not the days, all Poskim agree that he did not fulfill the Mitzvah at all. (Mishnah Berurah)
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