# Difference between revisions of "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 $660-11\cdot1=\boxed{649}.$

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## Solution 2 (One Variable)

Suppose that $S$ has $n$ numbers other than $68.$ We have the following table: $$\begin{array}{c|c|c|c} & & & \\ [-2.5ex] & \textbf{Count} & \textbf{Arithmetic Mean} & \textbf{Sum} \\ \hline & & & \\ [-2.5ex] \textbf{Initial} & n+1 & 56 & 56(n+1) \\ \hline & & & \\ [-2.5ex] \textbf{Final} & n & 55 & 55n \end{array}$$

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