MathDB

1

Part of 1974 Czech and Slovak Olympiad III A

Problems(1)

Inequality with geometric mean

Source: Czech and Slovak Olympiad 1974, National Round, Problem 1

7/19/2024
Let (ak)k=1∞\left(a_k\right)_{k=1}^\infty be a sequence of positive numbers such that ak−1ak+1≥ak2a_{k-1}a_{k+1}\ge a_k^2 for all k>1.k>1. For n≥1n\ge1 denote bn=(a1a2⋯an)1/n.b_n=\left(a_1a_2\cdots a_n\right)^{1/n}. Show that also the inequality bn−1bn+1≥bn2b_{n-1}b_{n+1}\ge b_n^2 holds for every n>1.n>1.
inequalitiesmeanalgebra