MathDB
Putnam 1995 A1

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June 29, 2014
Putnamcollege contests

Problem Statement

Let SS be a set of real numbers which is closed under multiplication (that is a,b∈S  ⟹  ab∈Sa,b\in S\implies ab\in S). Let T,U⊂ST,U\subset S such that T∩U=∅,T∪U=ST\cap U=\emptyset, T\cup U=S. Given that for any three elements a,b,ca,b,c in TT, not necessarily distinct, we have abc∈Tabc\in T and also if a,b,c∈Ua,b,c\in U, not necessarily distinct then abc∈Uabc\in U. Show at least one of TT and UU is closed under multiplication.
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