The 2018 South African Mathematics Olympiad — Problem 6
The final round of the South African Mathematics Olympiad will be taking place on Thursday, 28 July 2019. I have been writing about some of the problems from the senior paper from 2018. A list of all of the problems can be found here.
Today we will look at the sixth and final problem from the 2018 South African Mathematics Olympiad:
Let be a positive integer, and let be distinct positive integers with . Construct an table where the entries of the -th row are for . Now follow a procedure where, in each step, two identical entries are removed from the table. This continues until there are no more identical entries in the table.
- Prove that at least three entries remain at the end of the procedure.
- Prove that there are infinitely many possible choices for and such that only three entries remain,
There are some heuristics that are often helpful when solving a problem, such as
- Looking at small cases:
This helps us to understand the problem and how the various pieces in the problem relate to each other.