Let N be a positive integer. Two players, Yuri and Zeta, play the following game:
= Zeta picks a value for N;
= Yuri chooses N real numbers, not necessarily distinct, and keeps them secret;
= Yuri then writes all pairwise sums on a sheet of paper and gives it to Zeta (there are n(n-1)/2 such sums, not necessarily distinct);
= Zeta wins if she finds correctly the initial N numbers chosen by Yuri with only one guess.
Can Zeta be sure to win for the following cases?
a) N = 4 b) N = 5 c) N = 6
Justify your answer(s).
[For example, when N = 3, Yuri may choose the numbers 1, 5, 9, which have the pairwise sums of 6, 10 and 14 and can easily be resolved by simple algebra.]
Note, this has been adapted from the Junior Balkan MO 2013, Problem 4.
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