1984 AIME Problems/Problem 4

Problem

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

Solution 1 (Two Variables)

Suppose that $S$ has $n$ numbers other than $68,$ and the sum of these numbers is $s.$

We are given that $\begin{align*} \frac{s+68}{n+1}&=56, \\ \frac{s}{n}&=55. \end{align*}$ Clearing denominators, we have $\begin{align*} s+68&=56n+56, \\ s&=55n. \end{align*}$ Subtracting the equations, we get $68=n+56,$ from which $n=12$ and $s=660.$

The sum of the twelve remaining numbers is $660.$ To maximize the largest number, we minimize the other eleven numbers: We can have eleven $1$ s and one $660-11\cdot1=\boxed{649}.$

~JBL (Solution)

~MRENTHUSIASM (Reconstruction)

1984 AIME (Problems • Answer Key • Resources)
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1984 AIME Problems/Problem 4

Problem

Solution 1 (Two Variables)

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