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

(Solution)
m (Problem)
Line 1: Line 1:
 
== Problem ==
 
== Problem ==
Let <math>\displaystyle S</math> be a list of [[positive integer]]s - not necessarily [[distinct]] - in which the number <math>\displaystyle 68</math> appears. The [[arithmetic mean]] of the numbers in <math>\displaystyle S</math> is <math>\displaystyle 56</math>. However, if <math>\displaystyle 68</math> is removed, the arithmetic mean of the numbers is <math>\displaystyle 55</math>. What's the largest number that can appear in <math>\displaystyle S</math>?
+
Let <math>S</math> be a list of positive integers - not necessarily distinct - in which the number <math>68</math> appears. The arithmetic mean of the numbers in <math>S</math> is <math>56</math>. However, if <math>68</math> is removed, the arithmetic mean of the numbers is <math>55</math>. What's the largest number that can appear in <math>S</math>?
  
 
== Solution ==
 
== Solution ==

Revision as of 00:15, 22 June 2021

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

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 $660 - 11 = \boxed{649}$.

See also

1984 AIME (ProblemsAnswer KeyResources)
Preceded by
Problem 3
Followed by
Problem 5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions
Invalid username
Login to AoPS