Difference between revisions of "2010 AIME II Problems/Problem 2"
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Revision as of 22:37, 4 July 2013
Problem 2
A point 
is chosen at random in the interior of a unit square 
. Let 
denote the distance from to the closest side of . The probability that 
is equal to 
, where 
and 
are relatively prime positive integers. Find 
.
Solution
Any point outside the square with side length 
that have the same center as the unit square and inside the square side length 
that have the same center as the unit square has .
![[asy] unitsize(1mm); defaultpen(linewidth(.8pt)); draw((0,0)--(0,30)--(30,30)--(30,0)--cycle); draw((6,6)--(6,24)--(24,24)--(24,6)--cycle); draw((10,10)--(10,20)--(20,20)--(20,10)--cycle); fill((6,6)--(6,24)--(24,24)--(24,6)--cycle,gray); fill((10,10)--(10,20)--(20,20)--(20,10)--cycle,white); [/asy]](http://latex.artofproblemsolving.com/2/f/8/2f82e5b678d789e6c2bfd670f46cff737577a21e.png)
Since the area of the unit square is 
, the probability of a point with is the area of the shaded region, which is the difference of the area of two squares
Thus, the answer is 
See also
| 2010 AIME II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 1 |
Followed by Problem 3 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
