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A game on combinatorial number theory

Source: Argentina TST 2009

May 16, 2009
combinatorics unsolvedcombinatorics

Problem Statement

Let n≥3 n \geq 3 be an odd integer. We denote by [\minus{}n,n] the set of all integers greater or equal than \minus{}n and less or equal than n n. Player A A chooses an arbitrary positive integer k k, then player B B picks a subset of k k (distinct) elements from [\minus{}n,n]. Let this subset be S S. If all numbers in [\minus{}n,n] can be written as the sum of exactly n n distinct elements of S S, then player A A wins the game. If not, B B wins. Find the least value of k k such that player A A can always win the game.
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