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This question already has an answer here:
Is there a place to find all (or nearly all) flights going from a certain airport during a certain time period?
If the "departure announcement board" at the airport was much longer (covering months rather than hours), that would work as an answer for me.
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
There is a chance you find your answer in the 'supposedly duplicate' question someone has linked to in the comments.
Failing this, the only way that I have found an answer to this is by actually visiting the airports website. Most major ones will have a section that displays a list of all the airlines that fly to-from that airport. Most of these lists will also contain a list of destinations directly reachable using these airlines, unless the list is super-huge!
See for example Gatwick Airport: http://www.gatwickairport.com/flights/destinations/
If looking for low cost companies have a look at http://www.flycheapo.com/
You can choose either the destination city or country and than you get a list of countries / airports that have connections to your destination. You can check if your city or any other city in your country is included.
The logic in not exactly the same you are requesting but in the end you get the same answer.
Many search engines have option to choose Country (Any airport) as destination.
As for given time period, the only typical options I've seen are: ±X days (usually 3), month overview, whole year overview. I've not seen search engine which would allow you defining custom search periods.
Example:
From any airport in London to any destination in Italy in March 2013 (via SkyScanner)
