Difference between revisions of "1981 AHSME Problems/Problem 22"
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If <math>a = -1</math>, then <math>1 \leq i, i-1, i-2, i-3 \leq 4</math>. In this case, only <math>i = 4</math> is valid. | If <math>a = -1</math>, then <math>1 \leq i, i-1, i-2, i-3 \leq 4</math>. In this case, only <math>i = 4</math> is valid. | ||
| − | Therefore, <math>(a, i) \in {(1, 1), (0, 1), (0, 2), (0, 3), (0, 4), (-1, 4)}</math>. By similar logic, <math>(b, j)</math> and <math>(c, k)</math> must also be in this set. | + | Therefore, <math>(a, i) \in \{(1, 1), (0, 1), (0, 2), (0, 3), (0, 4), (-1, 4)\}</math>. By similar logic, <math>(b, j)</math> and <math>(c, k)</math> must also be in this set. |
If <math>a = b = c = 0</math>, then all four points are <math>(i, j, k)</math>, so at least one of <math>a, b, c</math> must be nonzero. Therefore, there are <math>6^3 - 4^3 = 216 - 64 = 152</math> choices for <math>(i, j, k, a, b, c)</math>. However, each line can be determined by two different values of <math>(i, j, k, a, b, c)</math>, as the line can be traced in two different directions. For instance, <math>(i, j, k, a, b, c) = (4, 1, 3, -1, 1, 0)</math> and <math>(i, j, k, a, b, c) = (1, 4, 3, 1, -1, 0)</math> determine the line containing <math>(4, 1, 3), (3, 2, 3), (2, 3, 3), (1, 4, 3)</math>. Therefore, there are <math>\frac{152}{2} = \boxed{(\textbf{D})\ 76}</math> such lines. | If <math>a = b = c = 0</math>, then all four points are <math>(i, j, k)</math>, so at least one of <math>a, b, c</math> must be nonzero. Therefore, there are <math>6^3 - 4^3 = 216 - 64 = 152</math> choices for <math>(i, j, k, a, b, c)</math>. However, each line can be determined by two different values of <math>(i, j, k, a, b, c)</math>, as the line can be traced in two different directions. For instance, <math>(i, j, k, a, b, c) = (4, 1, 3, -1, 1, 0)</math> and <math>(i, j, k, a, b, c) = (1, 4, 3, 1, -1, 0)</math> determine the line containing <math>(4, 1, 3), (3, 2, 3), (2, 3, 3), (1, 4, 3)</math>. Therefore, there are <math>\frac{152}{2} = \boxed{(\textbf{D})\ 76}</math> such lines. | ||
Latest revision as of 13:32, 20 August 2025
Problem 22
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
Let be the first point whose coordinates are positive integers at most 
that a line passes through when being traced in a certain direction. Then the next three lattice points the line passes through must be in the form 
, 
, and 
, where 
are integers.
Note that if 
, 
, which is too large. Therefore 
, and by similar logic 
and 
. Also, if 
, 
, which is too small. Therefore, 
, and by similar logic 
and 
. So 
.
If 
, then 
. In this case, only 
is valid.
If 
, then 
. In this case, 
are all valid.
If 
, then 
. In this case, only 
is valid.
Therefore, 
. By similar logic, 
and 
must also be in this set.
If 
, then all four points are , so at least one of must be nonzero. Therefore, there are 
choices for 
. However, each line can be determined by two different values of , as the line can be traced in two different directions. For instance, 
and 
determine the line containing 
. Therefore, there are 
such lines.
See also
| 1981 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 21 |
Followed by Problem 23 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
| All AHSME Problems and Solutions | ||
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. 
