Why did unresolved disputes begin specifically at this time? Why were there no unresolved disputes before?
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the reason is that until then questions were brought to the sanhedrin, which followed "acharei rabim lehatos", therefore any opinion that didnt win the majority vote was discarded.
however King Herod nullified the sanhedrin. therefore when hillel and shamai had different opinions, even though they followed majority in their own school, they couldn't come together for a global ruling.
Tosafos ד"ה יוסי in Chagiga Daf ט"ז says that the Machlokes between Beit Shammai and Beit Hillel was caused by students who didn't do enough שימוש by their teachers.
Essentially the students didn't spend enough time studying, possibly out of laziness, as we see Hillel accusing בני בתירא of doing in פסחים on Daf ס"ו.
Hillel and Shammai themselves only argued about 4 issues, as enumerated here.
Before that there was only one unsolved argument that went on for 5 generations; whether one can do Shmicha on a Korban on Yom Tov, as mentioned in Chagiga Daf ט"ז.