1985 AIME Problems/Problem 8
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 ?
Explanation of the Question
Note: please read the explanation AFTER YOU HAVE TRIED reading the problem but couldn't understand.
For the question. Let's say that you have determined 7-tuple . Then you get the absolute values of the differences. Namely, Then is the greatest of the absolute values. So basically you are asked to find the 7-tuple with the smallest , and the rest would just be a piece of cake.
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 .
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