IMO 2019 Shortlist A4

Let n ě 2 be a positive integer and a1, a2, ..., an be real numbers such that a1 a2 ¨¨¨ an “ 0. Define the set A by A “ p...

imomathematicsolympiadshortlistalgebra

IMO 2019 Shortlist A4

Category: Algebra

Problem

Let n ě 2 be a positive integer and a1, a2, ..., an be real numbers such that a1 a2 ¨¨¨ ` an “ 0. Define the set A by A “ pi,jq ˇ ˇ 1 ď i ă j ď n,|ai ´ aj| ě 1 ( . Prove that, if A is not empty, then ÿ pi,jqPA aiaj ă 0. (China)