MathDB

1

Part of 2002 APMO

Problems(1)

Arithmetic mean

Source: APMO 2002

4/8/2006
Let a1,a2,a3,…,ana_1,a_2,a_3,\ldots,a_n be a sequence of non-negative integers, where nn is a positive integer. Let An=a1+a2+⋯+ann . A_n={a_1+a_2+\cdots+a_n\over n}\ . Prove that a1!a2!…an!≥(⌊An⌋!)n a_1!a_2!\ldots a_n!\ge\left(\lfloor A_n\rfloor !\right)^n where ⌊An⌋\lfloor A_n\rfloor is the greatest integer less than or equal to AnA_n, and a!=1×2×⋯×aa!=1\times 2\times\cdots\times a for a≥1a\ge 1(and 0!=10!=1). When does equality hold?
floor functioninductionfunctionlogarithmsinequalitiescalculusderivative