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How many countries in the world are entirely surrounded by just one other country?

Three

Lesotho sits inside South Africa, and San Marino and the Vatican both sit inside Italy. Every other landlocked country touches at least two neighbours.

Which capital city entirely contains another country's capital?

Rome, which contains Vatican City

Vatican City is a sovereign state of 0.49 km2 sitting inside Rome, so Rome contains a foreign capital within its city limits. San Marino and Andorra are separate territories entirely.

Which sovereign country receives the fewest tourists each year?

Tuvalu

Tuvalu receives on the order of 2,000 visitors a year. There are only a couple of flights a week and the runway doubles as a public space between them.

How many countries are entirely surrounded by a single other country?

Three

Lesotho sits inside South Africa; San Marino and the Vatican both sit inside Italy. There are no others.

Which national capital has the smallest population?

Ngerulmud, Palau

Ngerulmud has a few hundred residents and was purpose-built in 2006, largely to house government buildings. Vatican City is smaller still but is a whole state, not a capital of one.