Find all ( a , b , c ) (a,b,c)

Algebra Level 5

a , b , c a,b,c are positive integers such that

a b − c , b c − a , c a − b \large ab-c,\quad bc-a,\quad ca-b

are all powers of 2. Determine the total number of unordered triplets ( a , b , c ) (a,b,c) .

4 7 5 1 2 3 8 6

Munem Shahriar by Munem Shahriar , 18, Bangladesh

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1 solution

Patrick Corn
Oct 16, 2017

A small computer search yields ( 2 , 2 , 2 ) , ( 2 , 2 , 3 ) , ( 2 , 6 , 11 ) , ( 3 , 5 , 7 ) , (2,2,2), (2,2,3), (2,6,11), (3,5,7), but I have no idea whether these are the only solutions. Anyone?

Edit: this is a 2015 IMO problem, and seems extremely difficult! Here is what appears to be a full solution.

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