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inequality& sequence

Source: 2021 kmo P5

November 14, 2021
Sequenceinequalitiesalgebrapolynomial

Problem Statement

A real number sequence a1,⋯ ,a2021a_1, \cdots ,a_{2021} satisfies the below conditions. a1=1,a2=2,an+2=2an+12an+an+1(1≤n≤2019)a_1=1, a_2=2, a_{n+2}=\frac{2a_{n+1}^2}{a_n+a_{n+1}} (1\leq n \leq 2019) Let the minimum of a1,⋯ ,a2021a_1, \cdots ,a_{2021} be mm, and the maximum of a1,⋯ ,a2021a_1, \cdots ,a_{2021} be MM. Let a 2021 degree polynomial P(x):=(x−a1)(x−a2)⋯(x−a2021)P(x):=(x-a_1)(x-a_2) \cdots (x-a_{2021}) ∣P(x)∣|P(x)| is maximum in [m,M][m, M] when x=αx=\alpha. Show that 1<α<21<\alpha <2.
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