JBMO 2013, Problem 4: Middle Secondary Mathematics Competition Question

This is probably my favourite question in this year's JBMO. The hint at the end of the question was printed as part of the question; working through the algebra of the counter-example gives a big clue as to what is going on!


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 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.]

From the Junior Balkan Mathematical Olympiad 2013
[This actually is Problem 4; made an error in the previous post which now has the correct title but a misleading URL.]

 

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