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Arithmetic and geometric progressions with integers

Source: Romanian IMO Team Selection Test TST 1999, problem 7

September 24, 2005
calculusintegrationarithmetic sequencegeometric sequencegeometric seriesalgebrabinomial theorem

Problem Statement

Prove that for any integer nn, n≥3n\geq 3, there exist nn positive integers a1,a2,…,ana_1,a_2,\ldots,a_n in arithmetic progression, and nn positive integers in geometric progression b1,b2,…,bnb_1,b_2,\ldots,b_n such that b1<a1<b2<a2<⋯<bn<an. b_1 < a_1 < b_2 < a_2 <\cdots < b_n < a_n . Give an example of two such progressions having at least five terms.
Mihai Baluna
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