2001 AIME I Problems/Problem 10
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
Solutions
Solution 1
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
.
Solution 2
There are points in total. We group the points by parity of each individual coordinate -- that is, if
is even or odd,
is even or odd, and
is even or odd. Note that to have something that works, the two points must have this same type of classification (otherwise, if one doesn't match, the resulting sum for the coordinates will be odd at that particular spot).
There are EEEs (the first position denotes the parity of
the second
and the third
),
EEOs,
EOEs,
OEEs,
EOOs,
OEOs,
OOEs, and
OOOs. Doing a sanity check,
which is the total number of points.
Now, we can see that there are to choose two EEEs (respective to order),
ways to choose two EEOs, and so on. Therefore, we get
ways to choose two points where order matters. There are
total ways to do this, so we get a final answer of
for our answer of
Solution by Ilikeapos
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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