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Sum of product of elements from infinite sets

Source: China Team Selection Test 2016 Test 2 Day 2 Q5

March 21, 2016
number theoryalgebra

Problem Statement

Does there exist two infinite positive integer sets S,TS,T, such that any positive integer nn can be uniquely expressed in the form n=s1t1+s2t2+…+sktkn=s_1t_1+s_2t_2+\ldots+s_kt_k ,where kk is a positive integer dependent on nn, s1<…<sks_1<\ldots<s_k are elements of SS, t1,…,tkt_1,\ldots, t_k are elements of TT?
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