Chagigah, Chapter 2, Mishnah 2. This mishnah deals with a dispute that continued for many generations, and it is described as the first significant dispute in the history of the Jewish people. The mishnah opens by teaching that already in the early periods of the Mishnah, shortly after the Men of the Great Assembly, the Sages disagreed about semichah.
What is semichah?
One who brings a private offering must press his hands down with all his strength on the head of the animal. The question is whether this may be done on Yom Tov: even though a person may become obligated to bring an offering on Yom Tov, may he place his full weight on the head of the animal? For by doing so he is leaning on an animal, and a person may not lean on an animal on Shabbos or Yom Tov.
The root of the dispute:
It depends on whether the semichah must be performed specifically right before the offering is brought. If one is obligated to perform semichah immediately before the offering is sacrificed, then it must necessarily be done on Yom Tov, since the Torah requires it. But if the semichah need not be done immediately before the offering, it can be performed a day earlier, on Erev Yom Tov.
The five pairs and their dispute:
Yosei ben Yoezer says "shelo lismoch" - one may not perform semichah on Yom Tov. Yosei ben Yochanan, his contemporary, says "lismoch" - one may perform semichah on Yom Tov, and in fact it must be done then, since the semichah must be performed immediately before the offering is brought.
Yehoshua ben Perachyah says "shelo lismoch", while Nittai, who was with him in his generation, says "lismoch".
In the next generation, Yehudah ben Tabbai says "shelo lismoch", while Shimon ben Shatach says "lismoch".
In the generation after that, Shemayah says "lismoch", while Avtalyon says "shelo lismoch".
In Hillel's generation there was no dispute: the two Torah scholars who stood at the head were Hillel and Menachem, and they did not disagree. "Yatza Menachem" - Menachem left, and regarding his departure there are two opinions: some say that he went to serve in the government, and others say that, unfortunately, he strayed from the path of Torah. Either way, he no longer headed the Sanhedrin and left his position.
"Nichnas Shammai" - Shammai entered in place of Menachem, and from then on Hillel and Shammai stood at the head. Shammai says "shelo lismoch", and Hillel says "lismoch".
"Harishonim hayu nesi'im, ushenayim lahem avos beis din":
In each and every pair, the sage mentioned first was the Nasi of the Sanhedrin, and the sage mentioned second was the Av Beis Din - the secondary position in the leadership of the Sanhedrin. It emerges that in every generation the Nasi and the Av Beis Din disagreed about whether semichah should be performed on Yom Tov on the private offerings that a person brings on Yom Tov.