2001 AIME I Problems/Problem 10
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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 • Answer Key • Resources) | ||
Preceded by Problem 9 |
Followed by Problem 11 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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