JBMO 2013 Q4: A Game of Sums: Middle Secondary Mathematics Competition Question

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.]

[Junior Balkan MO 2013 Problem 4][Note that the original paper has 4 problems to be done in 4.5 hours]

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