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Difference between revisions of "2006 AIME I Problems/Problem 14"

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== Problem ==
 
== Problem ==
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A tripod has three legs each of length 5 feet. When the tripod is set up, the angle between any pair of legs is equal to the angle between any other pair, and the top of the tripod is 4 feet from the ground In setting up the tripod, the lower 1 foot of one leg breaks off. Let <math> h </math> be the height in feet of the top of the tripod from the ground when the broken tripod is set up. Then <math> h </math> can be written in the form <math> \frac m{\sqrt{n}}, </math> where <math> m </math> and <math> n </math> are positive integers and <math> n </math> is not divisible by the square of any prime. Find <math> \lfloor m+\sqrt{n}\rfloor. </math> (The notation <math> \lfloor x\rfloor </math> denotes the greatest integer that is less than or equal to <math> x. </math>)
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Let <math> S_n </math> be the sum of the reciprocals of the non-zero digits of the integers from 1 to <math> 10^n </math> inclusive. Find the smallest positive integer n for which <math> S_n </math> is an integer.
 
  
 
== Solution ==
 
== Solution ==
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Let <math>K = \sum_{i=1}^{9}{\frac{1}{i}}</math>.  Examining the terms in <math>S_1</math>, we see that <math>S_1 = K + 1</math> since each digit <math>n</math> appears once and 1 appears an extra time.  Now consider writing out <math>S_2</math>.  Each term of <math>K</math> will appear 10 times in the units place and 10 times in the tens place (plus one extra 1 will appear), so <math>S_2 = 20K + 1</math>.
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In general, we will have that:
 
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<math>S_n = (n10^{n-1})K + 1</math>
 
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because each digit will appear <math>10^{n - 1}</math> times in each place in the numbers <math>1, 2, \ldots, 10^{n} - 1</math>, and there are <math>n</math> total places. 
 
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The denominator of <math>K</math> is <math>D = 2^3\cdot 3^2\cdot 5\cdot 7</math>.  For <math>S_n</math> to be an integer, <math>n10^{n-1}</math> must be divisible by <math>D</math>.  Since <math>10^{n-1}</math> only contains the factors 2 and 5 (but will contain enough of them when <math>n \geq 3</math>), we must choose <math>n</math> to be [[divisible]] by <math>3^2\cdot 7</math>.  Since we're looking for the smallest such <math>n</math>, the answer is <math>063</math>
 
  
 
== See also ==
 
== See also ==
 
{{AIME box|year=2006|n=I|num-b=13|num-a=15}}
 
{{AIME box|year=2006|n=I|num-b=13|num-a=15}}
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[[Category:Intermediate Geometry Problems]]

Revision as of 18:35, 25 September 2007

Problem

A tripod has three legs each of length 5 feet. When the tripod is set up, the angle between any pair of legs is equal to the angle between any other pair, and the top of the tripod is 4 feet from the ground In setting up the tripod, the lower 1 foot of one leg breaks off. Let $h$ be the height in feet of the top of the tripod from the ground when the broken tripod is set up. Then $h$ can be written in the form $\frac m{\sqrt{n}},$ where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $\lfloor m+\sqrt{n}\rfloor.$ (The notation $\lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $x.$)

Solution

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See also

2006 AIME I (Problems • Answer Key • Resources)
Preceded by
Problem 13 		Followed by
Problem 15
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions
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