Problem:
The sum of the following seven numbers is exactly :
It is desired to replace each by an integer approximation , so that the sum of the 's is also , and so that , the maximum of the "errors" , is as small as possible. For this minimum , what is
Solution:
Two preliminary observations are needed:
(i) Each should be or , because for any scheme meeting this condition, ; while for any other choice of the .
(ii) There is only one way to sum seven integers, each of them or , and to obtain : two of them must be , while the other five must be .
In view of the above, and to make as small as possible, one must round down (to ) the two numbers with smallest decimal parts (i.e., and ), and round up (to ) the other five 's. Thereby one finds that
and that .
The problems on this page are the property of the MAA's American Mathematics Competitions