MathDB

Arithmetic mean

Source: APMO 2002

April 8, 2006

floor functioninductionfunctionlogarithmsinequalitiescalculusderivative

Problem Statement

Let

a_1,a_2,a_3,\ldots,a_n

be a sequence of non-negative integers, where

n

is a positive integer. Let

A_n={a_1+a_2+\cdots+a_n\over n}\ .

Prove that

a_1!a_2!\ldots a_n!\ge\left(\lfloor A_n\rfloor !\right)^n

where

\lfloor A_n\rfloor

is the greatest integer less than or equal to

A_n

, and

a!=1\times 2\times\cdots\times a

for

a\ge 1

(and

0!=1

). When does equality hold?

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