An example of such a state can be the State of Israel. They got their independence in 1948, and before that they were a part of the territory known as the British Mandate for Palestine. The new problems they had were mostly territorial and cultural since none of the Arabic states wanted to recognize them since they believe that the territory should belong to them.
Part 1 Choose any nation that became independent after 1945. Part 2 Read about how that country became independent, then research the main problems the new nation had to handle soon after independence.