A correct solution came from Bradley student Will Cragoe.  Other correct solutions were submitted by John Stamper, Bill Webb, Bryan Fluhrer, Juan Carlos Marivela, Iñigo Picaza, Siu Por Lam, Nancy Scharzkopf, A. Teitelman, Dan Dima, Lou Cairoli.

The minimum value of 224² is attained whenever all the a_i are equal.

Distribute the product to get .

Rewrite the summands as .  Since the function satisfies  f (x) = f (1 / x) and increases without bound as x goes to infinity, it follows that  f achieves a minimum at x = 1.  So and achieves its minimum when a_i = a_j.   Since there are such summands, the maximum value is 224 + 224·223 = 224². 

The result can also be verified by an application of the Arithmetic/Geometric Property or by induction.   

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©2005 Alberto L. Delgado