JBMO 2013 Problem 1: Middle Secondary Mathematics Competition Question

Find all ordered pairs (a, b) of positive integers for which the numbers

(a³b + 1)/(a + 1) and (b³a + 1)/(b – 1)

are both positive integers.


From the Junior Balkan Mathematical Olympiad 2013

My apologies, but the above question has a typo! The actual question is the following.

Find all ordered pairs (a, b) of positive integers for which the numbers

(a³b – 1)/(a + 1) and (b³a + 1)/(b – 1)

are both positive integers.

Both questions have solutions, and similar methods, but the real question has fewer of them. Feel free to try both. Also, note that 0 is not a positive integer.

 

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