I am going to begin a new series called 1 STEP 1 Day. This title is a pun: it means that I will be posting one interesting STEP question per say, yet it also implies "A journey of a thousand miles begins with a single step." All the questions come from the book Advanced Problems in Mathematics written by Dr. Stephen Siklos which is under CC BY 4.0.
The question today comes from 1993 Paper I. The original question is shown above.
(i) Find all sets of positive integers , and that satisfy the equation
(ii) Determine the sets of positive integers , and that satisfy the inequality
This is a simple yet elegant problem. Noticing that both the equation and the inequality are symmetric between , and might help.
Solution. (i) WLOG, let . Consider different values of . If , it is impossible. If , consider the value of b. By trial and error, can only be 3 or 4, which gives solution sets and . Now, if , only could work. cannot be greater than 4 since otherwise at least one of and would be .
(ii) We may use basically the same method as in (i). Of course, all the solutions satisfying (i) will do, so we shall not consider them here. WLOG, let . Consider different values of . If , any values of and will do, giving as a set of solutions. If , consider the value of b. If , can pick any values, so is another solution set. If , by trial and error, only and will work. Now, if , one can easily show that no more solutions exist.