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Rationals and irrationals again

Source: Indian IMOTC 2005 Day 3 Problem 1

September 23, 2005
number theory unsolvednumber theory

Problem Statement

Let 0<a<b0 <a <b be two rational numbers. Let MM be a set of positive real numbers with the properties: (i) a∈Ma \in M and b∈Mb \in M; (ii) if xx ∈M\in M and y∈My \in M, then xy∈M\sqrt{xy} \in M. Let M∗M^*denote the set of all irrational numbers in MM. prove that every c,dc,d such that a<c<d<ba <c <d<b, M∗M^* contains an element mm with property c<m<dc<m<d
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