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x_1<y_1<x_2<y_2<...<x_n<y_n, where x_i geom. and y_i arithm. progression

Source: Singapore Open Math Olympiad 2017 2nd Round p4 SMO

March 26, 2020
algebrainequalitiesArithmetic Progressiongeometric progressionSequences

Problem Statement

Let n>3n > 3 be an integer. Prove that there exist positive integers x1,...,xnx_1,..., x_n in geometric progression and positive integers y1,...,yny_1,..., y_n in arithmetic progression such that x1<y1<x2<y2<...<xn<ynx_1<y_1<x_2<y_2<...<x_n<y_n
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