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Difference between revisions of "1989 AIME Problems/Problem 11"

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== Problem ==
 
== Problem ==
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A sample of 121 integers is given, each between 1 and 1000 inclusive, with repetitions allowed. The sample has a unique mode (most frequent value). Let <math>D^{}_{}</math> be the difference between the mode and the arithmetic mean of the sample. What is the largest possible value of <math>\lfloor D^{}_{}\rfloor</math>? (For real <math>x^{}_{}</math>, <math>\lfloor x^{}_{}\rfloor</math> is the greatest integer less than or equal to <math>x^{}_{}</math>.)
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A sample of 121 [[integer]]s is given, each between 1 and 1000 inclusive, with repetitions allowed. The sample has a unique [[mode]] (most frequent value). Let <math>D^{}_{}</math> be the difference between the mode and the [[arithmetic mean]] of the sample. What is the largest possible value of <math>\lfloor D^{}_{}\rfloor</math>? (For real <math>x^{}_{}</math>, <math>\lfloor x^{}_{}\rfloor</math> is the [[floor function|greatest integer]] less than or equal to <math>x^{}_{}</math>.)
  
 
== Solution ==
 
== Solution ==
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== See also ==
 
== See also ==
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* [[1989 AIME Problems/Problem 12|Next Problem]]
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{{AIME box|year=1989|num-b=10|num-a=12}}
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* [[1989 AIME Problems/Problem 10|Previous Problem]]
 
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* [[1989 AIME Problems]]
 

Revision as of 18:48, 7 March 2007

Problem

A sample of 121 integers is given, each between 1 and 1000 inclusive, with repetitions allowed. The sample has a unique mode (most frequent value). Let $D^{}_{}$ be the difference between the mode and the arithmetic mean of the sample. What is the largest possible value of $\lfloor D^{}_{}\rfloor$? (For real $x^{}_{}$, $\lfloor x^{}_{}\rfloor$ is the greatest integer less than or equal to $x^{}_{}$.)

Solution

It is obvious that there will be $n+1$ values equal to one and $n$ values each of $1000, 999, 998 \ldots$. It is fairly easy to find the maximum. Try $n=1$, which yields $924$, $n=2$, which yields $942$, $n=3$, which yields $947$, and $n=4$, which yields $944$. The maximum difference occurred at $n=3$, so the answer is $947$.

See also

1989 AIME (Problems • Answer Key • Resources)
Preceded by
Problem 10 		Followed by
Problem 12
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions
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