2001 AIME I Problems/Problem 10
From AoPSWiki
Problem
Let 
be the set of points whose coordinates 
and 
are integers that satisfy 
and 
Two distinct points are randomly chosen from 
The probability that the midpoint of the segment they determine also belongs to is 
where 
and 
are relatively prime positive integers. Find 
Solution
The distance between the 
, 
, and coordinates must be even so that the midpoint can have integer coordinates. Therefore,
- For , we have the possibilities
, 
, 
, 
, and 
, 
possibilities.
- For , we have the possibilities , , ,
, , , 
, and 
, 
possibilities.
- For , we have the possibilities , , , ,
, , 
, , 
, 
, 
, , and , 
possibilities.
However, we have 
cases where we have simply taken the same point twice, so we subtract those. Therefore, our answer is 
.
See also
| 2001 AIME I (Problems • Resources) | ||
| Preceded by Problem 9 | Followed by Problem 11 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
