JBMO 2013 Problem 4

Let $n$ be a positive integer. Two players, Alice and Bob, are playing the following game:

• Alice chooses $n$ real numbers; not necessarily distinct.
• Alice writes all pairwise sums on a sheet of paper and gives it to Bob. (There are $\frac{n(n-1)}{2}$ such sums; not necessarily distinct.)
• Bob wins if he finds correctly the initial $n$ numbers chosen by Alice with only one guess. Can Bob be sure to win for the following cases?

a. $n=5$ b. $n=6$ c. $n=8$

Justify your answer(s).

[For example, when $n=4$, Alice may choose the numbers 1, 5, 7, 9, which have the same pairwise sums as the numbers 2, 4, 6, 10, and hence Bob cannot be sure to win.]