An intermediate value theorem for sequences with terms in a finite set
  We prove an intermediate value theorem of an arithmetical flavor, involving the consecutive averages of sequences with terms in a given finite set A. For every such set we completely characterize the numbers x ("intermediate values") with the property that the consecutive averages of every sequence with terms in A cannot increase from a value less than x to a value greater than x without taking the value x somewhere in between. 
Mihai Caragiu (Ohio Northern University), Laurence D. Robinson (Ohio   Northern University)