Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

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...")
 
Line 1: Line 1:
−
== Problem 25 ==
+
== 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>
 +
 +
==Solution==
 +
 +
== See Also ==
 +
 +
{{AMC10 box|year=2012|ab=A|num-b=24|after=Last Problem}}

Revision as of 16:42, 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)
Preceded by
Problem 24 		Followed by
Last Problem
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions
Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2012_AMC_10A_Problems/Problem_25&oldid=44676"