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algebra problem

Source: Netherlands TST for IMO 2017 day 3 problem 2

February 1, 2018
Sequencesalgebra

Problem Statement

let a1,a2,...ana_1,a_2,...a_n a sequence of real numbers such that a1+....+an=0a_1+....+a_n=0. define bi=a1+a2+....aib_i=a_1+a_2+....a_i for all 1≤i≤n1 \leq i \leq n .suppose bi(aj+1−ai+1)≥0b_i(a_{j+1}-a_{i+1}) \geq 0 for all 1≤i≤j≤n−11 \leq i \leq j \leq n-1. Show that max⁡1≤l≤n∣al∣≥max⁡1≤m≤n∣bm∣\max_{1 \leq l \leq n} |a_l| \geq \max_{1 \leq m \leq n} |b_m|
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