Partial solutions from members Bradley University community came from Kris Buckert, Mickey Lenisa.  Complete solutions came in from Aaron Kahn, Lou Cairoli, Iñigo Picaza.

Most submissions assumed that the band was stretched evenly in both directions in which case, yes, the midpoint wouldn't move.  However, this assumption is both unwarranted and unnecessary.

Place a coordinate system on the rubber band with its left- and right-most endpoints at -1 and +1.  Define a function s: [-1,1] ® R  as follows: s(x) is the final coordinate, after stretching, of the point whose initial coordinate was x.  Notice that the function s is continuous, and because the right end of the band is stretched right and the left end is stretched left we also have s(1) ³ 1 and s(-1) £ -1.  The function t(x) = s(x) - x is continuous and satisfies t(-1) £ 0 and t(1) ³ 0.  Therefore, by the Intermediate Value Theorem (which we all learned in calculus class!) there exists a point z with -1 £ z £ 1 satisfying t(z) = 0, that is, s(z) = z, which means that the point on the rubber band initially at coordinate z ends up there, too. 

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©2005 Alberto L. Delgado