1984 AIME Problems/Problem 4

Problem

Let $\displaystyle S$ be a list of positive integers - not necessarily distinct - in which the number $\displaystyle 68$ appears. The arithmetic mean of the numbers in $\displaystyle S$ is $\displaystyle 56$ . However, if $\displaystyle 68$ is removed, the arithmetic mean of the numbers is $\displaystyle 55$ . What's the largest number that can appear in $\displaystyle S$ ?

Suppose $S$ has $n$ members other than 68, and the sum of these members is $s$ . Then we're given that $\frac{s + 68}{n + 1} = 56$ and $\frac{s}{n} = 55$ . Multiplying to clear denominators, we have $s + 68 = 56n + 56$ and $s = 55n$ so $68 = n + 56$ , $n = 12$ and $s = 12\cdot 55 = 660$ . Because the sum and number of the elements of $S$ are fixed, if we want to maximize the largest number in $S$ , we should take all but one member of $S$ to be as small as possible. Since all members of $S$ are positive integers, the smallest possible value of a member is 1. Thus the largest possible element is $660 - 11 = \boxed{649}$ .