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Difference between revisions of "2012 AMC 10A Problems/Problem 25"

(Created page with "== Problem 25 == Real numbers <math>x</math>, <math>y</math>, and <math>z</math> are chosen independently and at random from the interval <math>[0,n]</math> for some positive in...")
 
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== Problem 25 ==
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== Problem ==
  
 
Real numbers <math>x</math>, <math>y</math>, and <math>z</math> are chosen independently and at random from the interval <math>[0,n]</math> for some positive integer <math>n</math>. The probability that no two of <math>x</math>, <math>y</math>, and <math>z</math> are within 1 unit of each other is greater than <math>\frac {1}{2}</math>. What is the smallest possible value of <math>n</math>?
 
Real numbers <math>x</math>, <math>y</math>, and <math>z</math> are chosen independently and at random from the interval <math>[0,n]</math> for some positive integer <math>n</math>. The probability that no two of <math>x</math>, <math>y</math>, and <math>z</math> are within 1 unit of each other is greater than <math>\frac {1}{2}</math>. What is the smallest possible value of <math>n</math>?
  
 
<math> \textbf{(A)}\ 7\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11 </math>
 
<math> \textbf{(A)}\ 7\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11 </math>
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==Solution==
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== See Also ==
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{{AMC10 box|year=2012|ab=A|num-b=24|after=Last Problem}}

Revision as of 15:42, 11 February 2012

Problem

Real numbers $x$, $y$, and $z$ are chosen independently and at random from the interval $[0,n]$ for some positive integer $n$. The probability that no two of $x$, $y$, and $z$ are within 1 unit of each other is greater than $\frac {1}{2}$. What is the smallest possible value of $n$?

$\textbf{(A)}\ 7\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11$

Solution

See Also

2012 AMC 10A (Problems • Answer Key • Resources)
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Problem 24 		Followed by
Last Problem
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All AMC 10 Problems and Solutions
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