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N integers

Source: IMAR Test 2007

January 28, 2009
absolute valuenumber theory proposednumber theory

Problem Statement

Prove that N\geq 2n \minus{} 2 integers, of absolute value not higher than n>2 n > 2, and of absolute value of their sum S S less than n \minus{} 1, there exist some of sum 0. 0. Show that for |S| \equal{} n \minus{} 1 this is not anymore true, and neither for N \equal{} 2n \minus{} 3 (when even for |S| \equal{} 1 this is not anymore true).