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Part of 1995 Putnam

Problems(2)

Putnam 1995 A1

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6/29/2014
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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Putnam 1995 B1

Source:

7/1/2014
For a partition π\pi of {1,2,3,4,5,6,7,8,9}\{1, 2, 3, 4, 5, 6, 7, 8, 9\}, let π(x)\pi(x) be the number of elements in the part containing xx. Prove that for any two partitions π\pi and π′\pi^{\prime}, there are two distinct numbers xx and yy in {1,2,3,4,5,6,7,8,9}\{1, 2, 3, 4, 5, 6, 7, 8, 9\} such that π(x)=π(y)\pi(x) = \pi(y) and \pi^{\prime}(x) = \pi^{\prime}(y).
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