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The given description provides several clues to identify the border district of India. Firstly, it is situated at a height of 10,000 feet, offering scenic beauty. Secondly, it is surrounded by two nations in the north and south-west. Thirdly, the district has the Nyamjang-chu river running through it. Additionally, the main crops grown in the district are potatoes, maize, and millet.
Taking these clues into consideration, we can conclude that the correct answer is option 2: Tawang.
Lohit (option 1) is a district in Arunachal Pradesh, but it does not fulfill the criteria mentioned in the description. Kangra (option 3) is a district in Himachal Pradesh and West Sikkim (option 4) is a district in Sikkim, neither of which match the given criteria.
Tawang is a district in Arunachal Pradesh that fits all the provided details. It is situated at a high altitude, has international borders in the north and south-west, is crossed by the Nyamjang-chu river, and is known for its scenic beauty. Tawang is also famous for its monasteries and tourist attractions like Sella Pass, P. Tso.