olymmath

USAMO 2009 Problem 2

  • Day 1
  • Answer

Let nn be a positive integer. Determine the size of the largest subset of {−n,−n+1,…,n−1,n}\{ -n, -n+1, \dots, n-1, n\} which does not contain three elements aa , bb , cc (not necessarily distinct) satisfying a+b+c=0a+b+c=0 .

Dataset text AI-MO AoPS. Quoted with source.

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Maximum size: n when n is even, and n+1 when n is odd. One extremal subset is all integers in {−n, …, n} except those with absolute value at most floor(n/2), i.e., {−n, …, −floor(n/2)−1, floor(n/2)+1, …, n}.

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