Round 3: Problem 2 Solution

Let’s call the maximum size a “sweet” set can have , and let us try to find an upper bound for n.

Assume we have a set of size with distinct positive integer elements. Let’s construct a set, say , with the elements of .

Note a few important points:

• All elements of are either {0, 1, 2}.
• If contains three of the same element, those three elements would sum to which would mean that those three elements could not sum to a prime.
• If contains one element each of {0, 1, 2}, then those three elements would sum to .
• All elements of the original set sum to at least 6 because we can have only distinct positive integers in .

Therefore, can contain at most 2 elements that are congruent mod 3 for at most 2 different possible remainders, meaning that our upper bound cannot be larger than 4 (otherwise at least one of the two conditions must fail). What is left is to see if there exists such a set of size 4, which indeed there are many, for example {1, 3, 7, 9}.