Difference between revisions of "2022 AIME II Problems/Problem 5"
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<cmath>a - b = p_3 = a - b + b - c = p_1 + p_2</cmath> | <cmath>a - b = p_3 = a - b + b - c = p_1 + p_2</cmath> | ||
| − | Because <math>p_3 = p_1 + p_2</math>, | + | Because <math>p_3 = p_1 + p_2</math>, <math>p_1</math> or <math>p_2</math> must be <math>2</math>. Let <math>p_1 = 2</math>, then <math>p_2 \in \{ 3, 5, 11, 17 \}</math>. |
Once <math>a</math> is determined, <math>b</math> and <math>c</math> are determined. There are <math>18</math> values of <math>a</math> where <math>a+2 \le 20</math>. Therefore the answer is <math>18 \cdot 4 = \boxed{\textbf{72}}</math> | Once <math>a</math> is determined, <math>b</math> and <math>c</math> are determined. There are <math>18</math> values of <math>a</math> where <math>a+2 \le 20</math>. Therefore the answer is <math>18 \cdot 4 = \boxed{\textbf{72}}</math> | ||
Revision as of 04:00, 19 February 2022
Problem
Twenty distinct points are marked on a circle and labeled 
through 
in clockwise order. A line segment is drawn between every pair of points whose labels differ by a prime number. Find the number of triangles formed whose vertices are among the original points.
Solution 1
Let 
, 
, and 
be the vertex of a triangle that satisfies this problem, where 
.
![\[a - b = p_1\]](http://latex.artofproblemsolving.com/d/9/b/d9b17f219c67eac3a05e01e2c9affb4a247d92bb.png)
![\[b - c = p_2\]](http://latex.artofproblemsolving.com/3/a/e/3aeb70bc77061c82eda49e30bfce03731a9ca746.png)
![\[a - b = p_3 = a - b + b - c = p_1 + p_2\]](http://latex.artofproblemsolving.com/d/0/1/d018ec0c7373c400ed8d3b5aa1e0cfefdd09a013.png)
Because 
, 
or 
must be 
. Let 
, then 
.
Once is determined, and are determined. There are 
values of where 
. Therefore the answer is 
See Also
| 2022 AIME II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 4 |
Followed by Problem 6 | |
| 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. 
