MathDB

Sequence Gets Ratio’d

Source: EGMO 2023/1

April 16, 2023

EGMOEGMO 2023

Problem Statement

There are

n \ge 3

positive real numbers

a_1, a_2, \dots, a_n

. For each

1 \le i \le n

we let

b_i = \frac{a_{i-1} + a_{i+1}}{a_i}

(here we define

a_0

to be

a_n

and

a_{n+1}

to be

a_1

). Assume that for all

i

and

j

in the range

1

to

n

, we have

a_i \le a_j

if and only if

b_i \le b_j

. Prove that

a_1 = a_2 = \dots = a_n

.

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