I booked a flight on the 19th of November, but my Russian visa expires on the 22th of November.
Would there be any problems to board my flight and enter the country?
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As per rules, there should not be any issues entering the country with valid visa (if you are in good terms with previous travel and stay)
I have entered the US with only 3 days available on my visa.
Border security will definitely ask questions about return flight or extension of visa, that you should have a SOLID answer (may be a return flight ticket, visa extension application, or similar stuff).