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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 country has the only national flag that is not a rectangle?

Nepal

Nepal's flag is two stacked pennants, and its exact geometry is written into the constitution as a construction procedure. Switzerland and the Vatican are square, which is unusual but still rectangular.

Which city is split between two countries and shares a single name?

Nicosia, between Cyprus and the north

Nicosia has been divided since 1974. Other split cities such as Frankfurt an der Oder and Slubice sit either side of a river under different names.

Which country's borders have changed least in the past 500 years?

Portugal

Portugal's mainland borders have been essentially fixed since 1297, making them among the oldest stable frontiers in Europe. Its overseas empire changed enormously; the metropolitan shape did not.

Which country has the most official languages?

Zimbabwe, with 16

Zimbabwe's 2013 constitution recognises 16, including Shona, Ndebele, Tonga and sign language. South Africa recognises 12 and Bolivia 37 at the national level by some readings.

How many national languages does Switzerland have?

Four

German, French, Italian and Romansh. Romansh is spoken by well under one percent of the population but retains national status.

Which two countries have square national flags?

Switzerland and Vatican City

Both are 1:1. Switzerland's civil ensign at sea is rectangular, because square flags do not behave well on ships.

Which country has the highest average elevation?

Bhutan

Bhutan averages over 3,000 m, higher than Nepal, because Nepal includes low southern plains while Bhutan is almost entirely mountainous.