2008 UNCO Math Contest II Problems/Problem 2

Problem

Let be a set of four positive integers. If pairs of distinct elements of are added, the following six sums are obtained: Determine the values of , and [Hint: there are two possibilities.]

Solution

The problem doesn't tell us that and are distinct integers, but we can prove it easily. If we assume that these four positive integers are all distinct, then we will obtain six distinct pair-wise sums, since the number of ways to choose two objects out of four total where order does not matter is . If any of the elements or are equal to another, at least one of these distinct pair-wise sums would disappear, because at least two of the sums would be identical. However, we can see that the pair-wise sums we are given are all distinct, and that there are 6 of them, meaning that elements and are all distinct.

Before we move on to considering cases, it would help to consider what information might help save time. If we assume that , it is obvious that and , as those are the minimum and maximum pair-wise sums. We can do better though, because we know that and , since those are the next smallest and next largest pair-wise sums.

With this knowledge, we can start considering cases. If and , we can use the equations we found to determine that and . Similarly, if and , and . Both of these solutions are valid, and are also the only solutions, because the only two possibilities for and were exhausted.

Thus, the answer is or