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arithmetic and geometric sequence inequality problems

Source: 2018 Saudi Arabia IMO TST II p2

July 28, 2020
geometric sequenceinequalitiesarithmetic sequencealgebra

Problem Statement

a) For integer n≥3n \ge 3, suppose that 0<a1<a2<...<an0 < a_1 < a_2 < ...< a_n is a arithmetic sequence and 0<b1<b2<...<bn0 < b_1 < b_2 < ... < b_n is a geometric sequence with a1=b1,an=bna_1 = b_1, a_n = b_n. Prove that a_k > b_k for all k=2,3,...,n−1k = 2,3,..., n -1. b) Prove that for every positive integer n≥3n \ge 3, there exist an integer arithmetic sequence (an)(a_n) and an integer geometric sequence (bn)(b_n) such that 0<b1<a1<b2<a2<...<bn<an0 < b_1 < a_1 < b_2 < a_2 < ... < b_n < a_n.
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