1985 AIME Problems/Problem 8
Problem
The sum of the following seven numbers is exactly 19: ,
,
,
,
,
,
. It is desired to replace each
by an integer approximation
,
, so that the sum of the
's is also 19 and so that
, the maximum of the "errors"
, the maximum absolute value of the difference, is as small as possible. For this minimum
, what is
?
Solution
If any of the approximations is less than 2 or more than 3, the error associated with that term will be larger than 1, so the largest error will be larger than 1. However, if all of the
are 2 or 3, the largest error will be less than 1. So in the best case, we write 19 as a sum of 7 numbers, each of which is 2 or 3. Then there must be five 3s and two 2s. It is clear that in the best appoximation, the two 2s will be used to approximate the two smallest of the
, so our approximations are
and
and the largest error is
, so the answer is
.