MathDB

2 <= i <= 8 there is a number $a_i$, such that a_{i - 1} + a_{i + 1} <ka_i

Source: 2001 Moldova JBMO TST p6

February 20, 2021

inequalitiesalgebra

Problem Statement

Let the nonnegative numbers

a_1, a_2,... a_9

, where

a_1 = a_9 = 0

and let at least one of the numbers is nonzero. Denote the sentence

(P)

: '' For

2 \le i \le 8

there is a number

a_i

, such that

a_{i - 1} + a_{i + 1} <ka_i

”. a) Show that the sentence

(P)

is true for

k = 2

. b) Determine whether is the sentence

(P)

true for

k = \frac{19}{10}