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Source: Indian IMOTC 2013, Team Selection Test 3, Problem 3

July 30, 2013
inductionalgebra proposedalgebra

Problem Statement

Let h≥3h \ge 3 be an integer and XX the set of all positive integers that are greater than or equal to 2h2h. Let SS be a nonempty subset of XX such that the following two conditions hold:
[*]if a+b∈Sa + b \in S with a≥h,b≥ha \ge h, b \ge h, then ab∈Sab \in S;
[*]if ab∈Sab \in S with a≥h,b≥ha \ge h, b \ge h, then a+b∈Sa + b \in S. Prove that S=XS = X.
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