New York City
I'm looking to travel to South America in the next year or so -- when are the flights cheapest to buy and on what airline?
Flights are extremely cheap during the fall months - September, October, November. I've found mid- September to be the absolute cheapest, and then prices go up from there. New York to Colombia is a good route. Sometimes you can find cheap flights NY to Peru, and NY to Rio de Janeiro.
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For internation flights, try booking about 12 weeks in advanced. This is when fares are usually cheapest. Google Flights is a great tool for price/airline comparison.