Shekalim 8:2-3

Shekalim 8:2

If a utensil is found in Jerusalem on a path leading to where it can be immersed, it is assumed to be ritually impure; on a path leading away from a place to immerse it, ritually pure. The same path was not used going to and from the place of immersion (so that the clean and unclean utensils shouldn’t mix). This is the opinion of Rabbi Meir; Rabbi Yosi says that all utensils are assumed to be ritually clean except for funereal tools.

Shekalim 8:3

If a knife was found in Jerusalem on 14 Nisan (the day of the Passover offering), it is presumed clean and one may use it to offer his sacrifice right away. If it was found on 13 Nisan, it must be immersed. A meat cleaver found on either day needed to be immersed. If 14 Nisan fell on Shabbos, one may use the cleaver immediately; if it was found on 15 Nisan (the first day of Passover), one may use the cleaver immediately. (Since we may not immerse utensil on Shabbos and yom tov, we assume the owner did so beforehand.) If a cleaver was found tied to a knife, the cleaver has whatever presumption of status the knife has (so if found on 14 Nisan, it may be used immediately).