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Indeed, after the people of Israel became coalesced as a nation in its land, appointed a king in Shaul, and his kingdom stabilized, the prophet Shmuel approached him, saying:
I am the one the Lord sent to anoint you king over His people Israel. Therefore, listen to the Lord’s command! "Thus said the Lord of Hosts: I am exacting the penalty for what Amalek did to Israel, for the assault he made upon them on the road, on their way up from Egypt." Now go, attack Amalek, and proscribe all that belongs to him. Spare no one, but kill alike men and women, infants and sucklings, oxen and sheep, camels and asses!" (1 Shmuel 15:1-3)
However, King Shaul did not fulfill the mitzva properly. He had pity on King Agag of Amalek and on the best of the sheep and cattle. As a result, God took the kingdom away from Shaul and gave it to David. Nevertheless, the damage that was already done was devastating. Because of Shaul’s weakness and compassion, many Amalekites survived and continued harassing Israel. A few years later, a band of Amalekites attacked Tziklag, where the families of David and his men lived, burning down the city and taking all the women and children captive. With God’s help, David and his men managed to rescue the captives and vanquish the marauders. But since David was not yet king and did not have the army of Israel at his disposal, he was unable to eradicate them. According to 1 Samuel 30:17, "four hundred young men…mounted camels and got away." Apparently, other groups of Amalekites survived elsewhere as well. Even after David became king and continued to fight against the Amalekites, he was unable to destroy them all, because they were scattered throughout the land. The Sages also relate that because Shaul delayed killing Agag, Agag’s line was preserved, eventually resulting in the birth of Haman the Agagite, who attempted to wipe out the Jewish people (Megilla 13a).
Even though the mitzva to eradicate Amalek is primarily incumbent upon the Jewish people as a whole, every individual Jew is commanded to fulfill it, as well. Therefore, if a Jew meets an Amalekite, and has the ability to kill him, but refrains from doing so, he has neglected this mitzva (Sefer Ha-ĥinukh §604). The descendants of Amalek are currently unknown, but if one would ascertain that a particular person is an Amalekite, and that person follows their ways, it would be a mitzva to kill him. 5
^ 5.Responsa Kol Mevaser 2:42 derives from Rambam, Ramban (Shemot 17:16), and Sefer Ha-ĥinukh §604 that the mitzva of eradicating Amalek is primarily incumbent upon the king and the Jewish people as a whole, but that an individual who is able to kill an Amalekite must do so.
I asserted that some Amalekites remained after Shaul neglected to kill Agag. This is clear from bb 21a-b, which relates that David’s general Yoav erred in killing only the men of Amalek but not the women, because his school teacher had taught him an incorrect interpretation of the relevant verses. This incident took place during the war mentioned in 1 Melakhim 11:15-16, which was fought mainly against Edom, i.e., the descendants of Esav. Since the Amalekites represented only a small part of the descendants of Esav, this implies that there were groups of Amalekites within the Edomites, and it was concerning those Amalekites that Yoav erred. This explanation is cited in Responsa Kol Mevaser 2:42.

Rabbi Eliezer Melamed
Rosh Yeshiva of Har Bracha and rabbi of the settlement.

Peninei Halakha 18. Torah Verses in the Prayers and in the Berakha of She-asa Li Kol Tzorki
Chapter 10: The Laws of Tisha Be-Av
Lessons
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Kuzari -Rabbi Ari Shvat Kuzari class 9 - "Seeing is Believing" (parag. 21-30)
These paragraphs elaborate on the theme that seeing and knowing is better than any attempt to prove logically, and begins explaining the difference between Israel and gentiles.

Ein Aya Various Universal Stages of the Geula Process
Rav Kook examines the various stages of redemption, explaining how (in addition to the obvious oft-mentioned stages of ingathering the exiles, reviving the Hebrew language, army, state etc.) the messianic dream of world prosperity, the State of Israel and world unity can and are realistically and logically gradually coming true.

Kuzari -Rabbi Ari Shvat Kuzari class 8- "Answering Questions on the Kuzari's Proof from Mass Revelation
How do we know that the "claim" of mass revelation to 2,000,000 witnesses at Mt. Sinai is really true? This important class answers all of the questions skeptics ask about this claim of the Kuzari.

Ein Aya Armies Still Necessary for Balance & the War Against Wars
Rav Kook explains why the world was originally divided into the various seemingly contradicting ideologies and cultures, in order to develop each one respectively. Swords or armies symbolize how each respective ideology defends themselves, as well as deters their opposing ideologies and cultures. On the other hand, the messianic era will be one of peace, and Rav Kook explains the transition to that stage, which mankind is already undergoing.

The Land of Israel LGBT'S IN ISRAEL
The question was asked, how can one make Aliyah with the LGBT parades?

Kuzari -Rabbi Ari Shvat Kuzari class 7 - Five Accumulative Proofs of G-d
As a preparation for the Kuzari's classic proof of G-d from the mass-revelation at Sinai, we start here with 5 other directions to strengthen our belief which also contribute to what the Kuzari will present as well.

Ein Aya Muscle & Meaning: The Dual Nature of Gevurah (Physical Strength)
Is physical strength and fitness a necessity or an ideal? Although it if often totally overlooked among topics of Judaism, Rav Kook writes that it clearly is also a necessity to deter the many enemies of Israel, but even in Y'mot HaMashiach, in the Messianic era, to a certain extent, it's ideal continues even after our enemies will have been finished off.









