The IMO Shortlist
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Keywords  Content 

start  The problems here may look intimidating, but you don't need to come up with full mathematical proofs for the purposes of this puzzle. Computer algebra systems, bruteforce, geometry software like GeoGebra, extrapolation, and searching on the Internet will get you very far with getting answers to the problems. 
geometry  In the diagrams, the given country is not the only collinear triple! 
countries  The countries are not listed in alphabetical order. In fact, these are the most recent hosts of the International Math Olympiad, in chronological order. 
Each of the math problems gives you a way to convert the given country to a new country. Try looking at the IMO timeline for an idea of what data you could use. 

algebra  Try using the number of the IMO rather than the year as an input. 
combinatorics  Try using the year of the IMO (rather than the number) for this section. 
number theory  The previous sections used the first three columns of the IMO timeline page in order. The number theory section uses the fourth one. How can you convert a city to an ordered pair? 
extraction  Using the mechanics for each section, you should be able to arrange all 21 problems into a cycle. The cycle should start from G1. Notice the numbers are simply increasing within each subject, so only the subjects are relevant. 
extraction  It may seem odd that number theory abbreviated as NT instead of the more usual N. To understand why, think about "stranded" in the flavortext. 