Problem: (1995 AIME – #13) Let be the integer closest to . Find
Solution: So how can we best define “integer closest?” It basically means
Note we may use strict bounds because we know will not have fractional part .
So we can find bounds for such that .
We must have .
Expanding, we get
from which we get integral solutions (subtract lower bound from upper).
Thus the summation can be divided into separate cases based on the value of . Since there are terms and each has value , they sum to .
Since , we may only take this from to , giving
But this only accounts for
So we have remaining terms, all of which are , giving the new total of . QED.
Comment: At this point, we’re happy. Why? Because it’s an AIME problem and we got an integer answer! And the chances of us doing it wrong and getting an integer answer is not too high, so we are probably right.
Practice Problem: Find the smallest such that