MathDB
JBMO 2013 Problem 4

Source: Proposed by Bulgaria

June 23, 2013
functionfactorialalgorithmlinear algebracombinatorics proposedcombinatorics

Problem Statement

Let nn be a positive integer. Two players, Alice and Bob, are playing the following game: - Alice chooses nn 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\frac{n(n-1)}{2} such sums; not necessarily distinct.) - Bob wins if he finds correctly the initial nn numbers chosen by Alice with only one guess. Can Bob be sure to win for the following cases?
a. n=5n=5 b. n=6n=6 c. n=8n=8
Justify your answer(s).
[For example, when n=4n=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.]
Back to ProblemsView on AoPS