Difference between revisions of "1984 AIME Problems/Problem 4"
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== Problem == | == Problem == | ||
| − | Let <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 
be a list of positive integers - not necessarily distinct - in which the number 
appears. The arithmetic mean of the numbers in is 
. However, if is removed, the arithmetic mean of the numbers is 
. What's the largest number that can appear in ?
Solution
Suppose has 
members other than 68, and the sum of these members is 
. Then we're given that 
and 
. Multiplying to clear denominators, we have
and
so
,
and
.
Because the sum and number of the elements of are fixed, if we want to maximize the largest number in , we should take all but one member of to be as small as possible. Since all members of are positive integers, the smallest possible value of a member is 1. Thus the largest possible element is 
.
See also
| 1984 AIME (Problems • Answer Key • Resources) | ||
| 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 | ||
