I would be very happy if the great Rabbi Shlita would answer us…

I am sending all our questions to Rabbi Shalit, and we would be very happy if he answered.

“And whoever touches the water of the rite will be unclean until evening” (19:21). Here is the opinion of Rabbi Akiva (Yoma 14), that just as the one who touches the water of the rite becomes the first to become unclean, it is the law that the one who sprinkles the water of the rite becomes the first to become unclean [and not according to the opinion of the Sages, see also Isaiah 14:11]. On the other hand, in Yoma (8:11) it is explained that the High Priest separates him seven days before Yom Kippur, and sprinkles the water of the rite upon him every day, because it is a high thing that he should be clean for the service of the Lord. And from that time on he worked every day, and they did not fear that he might be unclean and disqualified from work, because by the very nature of the law he is clean.

And they said (ibid. 14:1), This is the law of the Lord, that the one who is unclean is not unclean, because he is unclean with this water of sin. And my father said that I am unclean, because they would sprinkle water on him after work at the end of the day before sunset, and he would be immersed immediately and become pure in the evening sun.

And it is written in the verse "Ha-Dkei'l" (Haggiga 22, Shul-Yo'd 6:9) that if he put impure vessels into a large vessel and immersed them, he would be unclean; if the vessel was large and the mouth of the vessel was very narrow, the immersion would extend to the vessels inside it, for example, if immersion extended to the large vessel, immersion extended to the vessels inside it. But if the large vessel was pure, immersion did not arise for the vessels inside it, until the mouth of the large vessel was as wide as the mouth of the nod.

And according to this verse, one should perform this ritual near sunset and only after the service, they could perform it even before the service, as a simple advice, that they take an unclean vessel inside a large, pure vessel, since the mouth of the large vessel is not as wide as the mouth of the nod, and the law is that if the large vessel is pure, immersion is not permissible for the inside, but if the large vessel is impure, immersion is permissible for the whole of it. And now the high priest will touch the clean large vessel, and then they will immerse the vessel.

And he will be ashamed. What is your mind, on the side that was touched, the impurity of death, then he will purify the large vessel, and both vessels will be purified by immersion. And if it is pure, then the large vessel is not impure, and therefore the inner vessel remains impure. Now they will place the water of sin in the inner vessel, and so on. For example, if the Ḥaḍg is impure, and it is necessary to purify it and it is found, as stated, that the inner vessel was purified along with the large vessel by immersing them together, and therefore the water of sin did not become impure in it, and the Ḥaḍg is purified by purifying these water of sin. And if we truly consider that the Ḥaḍg is pure for the time of performing a religious act, and therefore he did not impure the large vessel by touching it, it is found that the inner vessel remains impure, and the water of sin that was placed in it in the inner vessel is invalidated, and therefore they do not impure at all by purifying it.

I don't know why this patent is preferable to a simple answer that was purified on the eve of the sun.
And I also thought that if the inner vessel is not pure, then the water itself is impure according to the law of impurity from drinking and not because it is water of sin. Again, the pure priest is impure.

"You shall not forget the memory of Amalek from under heaven." (25:19) He shall blot out the name of Amalek, from man to woman, from child to suckling, from ox to sheep, so that there may be no mention of Amalek, even on the beast, to say, "This beast was like Amalek." (Rashi) The Menach (Mitzvah Terad) quoted Rashi's words, and wrote: And this saying, that even the beasts must be killed, is not explained in the Rambam and in the teaching, and I do not know from where Rashi derived this, etc., and from the act of Deshaul [which was explained by Samuel to kill even the beasts (1 Samuel 15:3)] there is no evidence that God commanded him at that time through Samuel, but we have not heard that it should be a mitzvah for generations, therefore.

Indeed, we found a source for Rashi's words in Midrash Rabbah (Lamentations, end of chapter 3), and so on: Ramiah said, "Pursue them with fury and destroy them" (Lamentations 3:60) and Moses said, "I will surely blot out the remembrance of Amalek from under heaven" (Exodus 17:14) etc., from under heaven, so that they may not say, "This tree is Amalek's, this camel is Amalek's, this Rachel is Amalek's, etc." It is clear that there is a commandment to also blot out the beasts of Amalek, as Rashi says.

But the second, from what is written (1 Samuel 27:8-9): ‘And David and his men went up and raided the Geshurites, the Girzites, the Amalekites, and the rest’ and he took sheep and cattle, donkeys, camels, and clothing’.

It is clear in the text that David took the Amalekites' livestock, and did not kill them.

And even more, from what is written (1 Chronicles 18:11): ‘Them also King David dedicated to the Lord’ with the silver and gold that he carried off from all the nations, from Edom and Moab and the Ammonites and the Philistines and Amalek’, the king did not refrain from taking the silver and gold that he carried off from Amalek, and they were not lost from under heaven, and he dedicated them to the Most High to be placed in the Temple. (See the response to Ung Yot in the introduction, letter 6, in the Hajj).

Also on the same subject, I wondered, is it a custom of Israel to write the name of Amalek under the shoe, and to erase his name, and the origin of this custom, in the words of Chaim Melunil (Hebrew Purim letter 4), which brought what was customary in the provinces and in France, where the boys take pebbles and write Haman on them, and knock them one over the other, on which it is written, "And the name of the wicked shall rot," and it is written, "For I will erase the memory of Amalek," etc. [And so it is in the Book of Habits (Hebrew Megillah 6:18) and in the Book of Yesod Yosef (chapter 22), and so on. In the book Sichot Chafetz Chaim (Chapter 2, page 8a) he brought that there was also a custom of the Chafetz Chaim, ayah.

And apparently we are concerned with it, because of the mitzvah to erase the memory of Amalek [as in the words of Erachot Chaim and as stated in Rama (Ochat 33:17)].

And 22, does not the mitzvah to kill Amalek require him to be killed and to disappear from the world, and so how does one fulfill the mitzvah to kill his name alone by killing him?

And even more so in 22, which mitzvah is the one after he was created and brought into the world, since before the writing, his name was not mentioned at all, but that he wrote it, and to the extent that they wrote it, they erased it again to restore it to its former state. [Review carefully the language of the Rambam (Safah made a kaft) in the foundation of the commandment to remember Amalek].

It is possible that when they bought or dedicated the property of Amalek, it was no longer an Amalekite animal, but an Israelite animal. The assumption is that the obligation to destroy is not on the object itself, but that no Amalekite property will remain in the world (and this is what Rashi wrote, that the name of Amalek will be called upon it. Although according to Rashi, “the proverb of Amalek is” and not “the proverb of Amalek was”. And so is the version of the Midrash Icha Rabbah that you cited: “the proverb of Amalek” and nothing more).
The custom of wiping out the name of Amalek with shoes is not a fulfillment of the commandment to kill Amalek, of course, but rather some remembrance of a commandment that no longer exists (and look carefully at the Rambam, he made a mistake). Just as the reading of Zachor is not killing Amalek, but remembering him.