MathDB

sum equality implies product-sum inequality

Source: China TST 1992, problem 2

June 27, 2005

inequalitiesalgebra unsolvedalgebra

Problem Statement

Let

n \geq 2, n \in \mathbb{N},

find the least positive real number

\lambda

such that for arbitrary

a_i \in \mathbb{R}

with

i = 1, 2, \ldots, n

and

b_i \in \left[0, \frac{1}{2}\right]

with

i = 1, 2, \ldots, n

, the following holds:

\sum^n_{i=1} a_i = \sum^n_{i=1} b_i = 1 \Rightarrow \prod^n_{i=1} a_i \leq \lambda \sum^n_{i=1} a_i b_i.

Back to Problems View on AoPS