Difference between revisions of "1984 AIME Problems/Problem 4"

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(Solution 2 (One Variable))
 
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== Problem ==
 
== Problem ==
{{empty}}
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Let <math>S</math> be a list of positive integers--not necessarily distinct--in which the number <math>68</math> appears. The average (arithmetic mean) of the numbers in <math>S</math> is <math>56</math>. However, if <math>68</math> is removed, the average of the remaining numbers drops to <math>55</math>. What is the largest number that can appear in <math>S</math>?
== Solution ==
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{{solution}}
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== Solution 1 (Two Variables) ==
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Suppose that <math>S</math> has <math>n</math> numbers other than the <math>68,</math> and the sum of these numbers is <math>s.</math>
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We are given that
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<cmath>\begin{align*}
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\frac{s+68}{n+1}&=56, \
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\frac{s}{n}&=55.
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\end{align*}</cmath>
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Clearing denominators, we have
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<cmath>\begin{align*}
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s+68&=56n+56, \
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s&=55n.
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\end{align*}</cmath>
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Subtracting the equations, we get <math>68=n+56,</math> from which <math>n=12.</math> It follows that <math>s=660.</math>
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The sum of the twelve remaining numbers in <math>S</math> is <math>660.</math> To maximize the largest number, we minimize the other eleven numbers: We can have eleven <math>1</math>s and one <math>660-11\cdot1=\boxed{649}.</math>
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~JBL (Solution)
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~MRENTHUSIASM (Reconstruction)
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== Solution 2 (One Variable) ==
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Suppose that <math>S</math> has <math>n</math> numbers other than the <math>68.</math> We have the following table:
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<cmath>\begin{array}{c|c|c|c}
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& & & \ [-2.5ex]
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& \textbf{Count} & \textbf{Arithmetic Mean} & \textbf{Sum} \
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\hline
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& & & \ [-2.5ex]
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\textbf{Initial} & n+1 & 56 & 56(n+1) \
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\hline
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& & & \ [-2.5ex]
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\textbf{Final} & n & 55 & 55n
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\end{array}</cmath>
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We are given that <cmath>56(n+1)-68=55n,</cmath> from which <math>n=12.</math> It follows that the sum of the remaining numbers in <math>S</math> is <math>55n=660.</math> We continue with the last paragraph of Solution 1 to get the answer <math>\boxed{649}.</math>
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~MRENTHUSIASM
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== Video Solution by OmegaLearn ==
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https://youtu.be/xqo0PgH-h8Y?t=82
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~ pi_is_3.14
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== See also ==
 
== See also ==
* [[1984 AIME Problems/Problem 3 | Previous problem]]
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{{AIME box|year=1984|num-b=3|num-a=5}}
* [[1984 AIME Problems/Problem 5 | Next problem]]
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* [[AIME Problems and Solutions]]
* [[1984 AIME Problems]]
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* [[American Invitational Mathematics Examination]]
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* [[Mathematics competition resources]]
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[[Category:Intermediate Algebra Problems]]

Latest revision as of 02:38, 16 January 2023

Problem

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

Solution 1 (Two Variables)

Suppose that $S$ has $n$ numbers other than the $68,$ 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 $n=12.$ It follows that $s=660.$

The sum of the twelve remaining numbers in $S$ 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)

Solution 2 (One Variable)

Suppose that $S$ has $n$ numbers other than the $68.$ We have the following table: \[\begin{array}{c|c|c|c} & & & \\ [-2.5ex] & \textbf{Count} & \textbf{Arithmetic Mean} & \textbf{Sum} \\ \hline & & & \\ [-2.5ex] \textbf{Initial} & n+1 & 56 & 56(n+1) \\ \hline & & & \\ [-2.5ex] \textbf{Final} & n & 55 & 55n \end{array}\] We are given that \[56(n+1)-68=55n,\] from which $n=12.$ It follows that the sum of the remaining numbers in $S$ is $55n=660.$ We continue with the last paragraph of Solution 1 to get the answer $\boxed{649}.$

~MRENTHUSIASM

Video Solution by OmegaLearn

https://youtu.be/xqo0PgH-h8Y?t=82

~ pi_is_3.14

See also

1984 AIME (ProblemsAnswer KeyResources)
Preceded by
Problem 3
Followed by
Problem 5
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All AIME Problems and Solutions