MathDB

1

Part of 2023 EGMO

Problems(1)

Sequence Gets Ratio’d

Source: EGMO 2023/1

4/16/2023
There are n≥3n \ge 3 positive real numbers a1,a2,…,ana_1, a_2, \dots, a_n. For each 1≤i≤n1 \le i \le n we let bi=ai−1+ai+1aib_i = \frac{a_{i-1} + a_{i+1}}{a_i} (here we define a0a_0 to be ana_n and an+1a_{n+1} to be a1a_1). Assume that for all ii and jj in the range 11 to nn, we have ai≤aja_i \le a_j if and only if bi≤bjb_i \le b_j. Prove that a1=a2=⋯=ana_1 = a_2 = \dots = a_n.
EGMOEGMO 2023