Difference between revisions of "1981 AHSME Problems/Problem 22"
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− | Restating the problem, we seek all the lines that will pass through (<math>i</math>, <math>j</math>, <math>k</math>), (<math>i + a</math>, <math>j + b</math>, <math>k + c</math>), (<math>i + 2a</math>, <math>j + 2b</math>, <math>k + 2c</math>), and (<math>i + 3a</math>, <math>j + 3b</math>, <math>k + 3c</math>), such that <math>a,b,c</math> are integers, and all of our points are between 1 and 4, inclusive. With this constraint in mind, we realize that for each coordinate, we have three choices: | + | Restating the problem, we seek all the lines that will pass through (<math>i</math>, <math>j</math>, <math>k</math>), (<math>i + a</math>, <math>j + b</math>, <math>k + c</math>), (<math>i + 2a</math>, <math>j + 2b</math>, <math>k + 2c</math>), and (<math>i + 3a</math>, <math>j + 3b</math>, <math>k + 3c</math>), such that <math>i,j,k</math> are positive integers, <math>a,b,c</math> are integers, and all of our points are between 1 and 4, inclusive. With this constraint in mind, we realize that for each coordinate, we have three choices: |
# Set <math>a/b/c</math> to <math>0</math>. This then allows us to set the corresponding <math>i,j,k</math> to any number from <math>1</math> to <math>4</math>, inclusive. | # Set <math>a/b/c</math> to <math>0</math>. This then allows us to set the corresponding <math>i,j,k</math> to any number from <math>1</math> to <math>4</math>, inclusive. | ||
# Set <math>a/b/c</math> to <math>1</math>. This forces us to set the corresponding <math>i/j/k</math> to <math>1</math>. | # Set <math>a/b/c</math> to <math>1</math>. This forces us to set the corresponding <math>i/j/k</math> to <math>1</math>. |
Revision as of 19:45, 9 September 2024
Problem
How many lines in a three dimensional rectangular coordinate system pass through four distinct points of the form , where
,
, and
are positive integers not exceeding four?
Solution 1(casework)
Restating the problem, we seek all the lines that will pass through (,
,
), (
,
,
), (
,
,
), and (
,
,
), such that
are positive integers,
are integers, and all of our points are between 1 and 4, inclusive. With this constraint in mind, we realize that for each coordinate, we have three choices:
- Set
to
. This then allows us to set the corresponding
to any number from
to
, inclusive.
- Set
to
. This forces us to set the corresponding
to
.
- Set
to
. This forces us to set the corresponding
to
.
Note that options 2 and 3 will give us the same coordinates if we mirror the assignments of each coordinate. Also note that we cannot set all three coordinates to not change, as that would be a point.
All of this gives us ways to assign each coordinate, for a total of
. We then must subtract the ways to get a point (
ways per coordinate, for a total of
). This leaves us with
. Finally, we divide by
to account for mirror assignments giving us the same coordinate, for a final answer of
.
(This was my first solution, apologies if it is bad).