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A Shofar that was used for Idolatry: 
One who blew with a Shofar that was used for idolatry does not fulfill his obligation. However this only applies if the Shofar is owned by a Jew. If however the Shofar is owned by a gentile who worships it and one borrowed the Shofar from the gentile then although initially it is forbidden to use it for the Mitzvah, nevertheless Bedieved one fulfills his obligation.
If someone worshiped a cow with an attached horn, is that horn invalid?
It is forbidden to use such a Shofar whether the cow was worshiped by a Jew or gentile.
Is it permitted to purchase horns from countries that idolize animals?
This matter requires further analysis.
 586/6 [Parts of this Halacha are missing from our print of the Alter Rebbe’s Shulchan Aruch and are hence supplemented from other Halachas of Admur (11/13; 649/12) and the rulings of the Michaber and Rama on this Halacha.]
 Such as a Jew prostrated himself to the Shofar. [ibid] or a Jew prostrated himself to an animal which has a Shofar. [11/13] Or the animal was offered as a sacrifice to an idol. [Michaber 586/4] In the first case the Shofar is only forbidden if owned by a Jew. In the second case it is disputed whether it is forbidden Bedieved. [See M”A 586/6; M”B 11/35; from Admur 11/13 it is implied it is forbidden even Bedieved although from 649/12 it is implied that it is only initially forbidden. Vetzaruch Iyun.] In the third case the Shofar is completely forbidden even if owned by a gentile.
 The reason: As a Shofar is required to have a minimum of length of one Tefach and this Shofar, which is required to be burnt due to it being an item of idolatry, is considered as if it was already burnt and hence does not contain the minimum length. The reason for this is because all items that are required to be burnt are Halachicly considered as if they were already burnt. [ibid]
 Michaber 586/3; Admur 649/11
Whether the Jew is the one who prostrated to the Shofar or a gentile prostrated to the Shofar and then acquired it to the Jew without nullifying its state of idolatry. The reason for this is because once a Jew owns an idol, that idol can never be nullified and must be destroyed and this applies irrelevant of whether the Jew sanctified the item into an idol or the gentile. [Michaber ibid; Admur 649/13]
 Being that the item is repulsive in the eyes of Hashem. [Admur 649/12; See also 13/11]
 Michaber ibid; Admur ibid
The reason it is permitted Bedieved is because the Shofar is able to avoid being burnt if the Gentile nullifies its state of idolatry and hence it is not considered to lose its measurement of a Tefach’s length. [Admur ibid]
Other Opinions: The Rama 586/3 rules that even Bedieved the Shofar is forbidden as in actuality the gentile has not yet nullified its state of idolatry. Admur does not record this opinion in 649/12 and he motions to look in 586 in which he says that he explained that the Shofar is permitted. Practically majority of opinions are lenient like the Michaber and so is the final ruling. [M”B 586/18]