2024 AMC 10B Problems/Problem 14
- The following problem is from both the 2024 AMC 10B #14 and 2024 AMC 12B #9, so both problems redirect to this page.
Problem
A dartboard is the region B in the coordinate plane consisting of points 
such that 
. A target T is the region where 
. A dart is thrown at a random point in B. The probability that the dart lands in T can be expressed as 
, where 
and 
are relatively prime positive integers. What is 
?
Solution 1
![\[|x|+|y| \le 8\]](http://latex.artofproblemsolving.com/b/7/4/b741da773c0a2ef60cdab17e06fe48bf5bfcb015.png)
Inequalities of this form are well-known and correspond to a square in space with centre at origin and vertices at 
, 
, 
, 
.
The diagonal length of this square is clearly 
, so it has an area of
![\[\frac{1}{2} \cdot 16 \cdot 16 = 128\]](http://latex.artofproblemsolving.com/5/a/6/5a66fdc982311cb313225cf592a68b0cc98e45f3.png)
Now,
![\[(x^2 + y^2 - 25)^2 \le 49\]](http://latex.artofproblemsolving.com/0/a/5/0a51bc9364a32d91a5f2fc59c5d48fd738cfd599.png)
Converting to polar form,
![\[r^2 - 25 \le 7\]](http://latex.artofproblemsolving.com/3/b/9/3b91cca2301fb0aedef897e40e9629e61ce56793.png)
![\[r \le \sqrt32\]](http://latex.artofproblemsolving.com/6/0/4/604bd659dd4deb8abbde8ab226116397f6f7c2ab.png)
And
![\[r^2 - 25 \ge -7\]](http://latex.artofproblemsolving.com/9/3/e/93ef67e4111cc02f9f3359ac9c8b7a30dc0b6f32.png)
![\[r \ge \sqrt18\]](http://latex.artofproblemsolving.com/2/e/e/2ee76229284f6265f790d8025eb98aae2aa8d5d8.png)
This corresponds to a ring in space with outer radius 
and inner radius 
.
Note that the outer circle is inscribed within the square, meaning it completely lies within the square.
Our probability, then, is ![\[\frac {\pi(32 - 18)}{128}\]](http://latex.artofproblemsolving.com/4/d/d/4dd17c0c0443f3e9bfffb61b71024776ad4b9956.png)
![\[= \frac{7\pi}{64}\]](http://latex.artofproblemsolving.com/8/c/c/8cc1ae0e0723cadb3aae8e9a0ff8cebf1fd3d822.png)
and 
![\[m + n = 71\]](http://latex.artofproblemsolving.com/d/f/7/df72dee09c2097d0e0eb6d98a8740c067caedeae.png)
So 
See also
| 2024 AMC 10B (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 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AMC 10 Problems and Solutions | ||
| 2024 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 8 |
Followed by Problem 10 |
| 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 12 Problems and Solutions | |
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