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Which countries are landlocked by other landlocked countries, so you must cross two borders to reach the sea?

Liechtenstein and Uzbekistan

Liechtenstein is ringed by Switzerland and Austria, both landlocked. Uzbekistan's neighbours are Kazakhstan, Kyrgyzstan, Tajikistan, Afghanistan and Turkmenistan. None has ocean access, since the Caspian is a lake.

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 two countries share the shortest international land border?

Vatican City and Italy

The Vatican's entire perimeter is its border with Italy, roughly 3.4 km. Monaco's border with France is about 6 km, the next shortest.

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.