MathDB

AM-GM Jensen Bait

Source: UMD 2023 Part II Problem 5

November 30, 2023

inequalitiesAM-GMUMD

Problem Statement

Let

0 \le a_1 \le a_2 \le \dots \le a_n \le 1

n

real numbers with

n \ge 2

. Assume

a_1 + a_2 + \dots + a_n \ge n-1

. Prove that

a_2a_3\dots a_n \ge \left( 1 - \frac 1n \right)^{n-1}