Difference between revisions of "2004 AMC 12A Problems/Problem 13"
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Revision as of 19:18, 3 July 2013
Problem
Let 
be the set of points 
in the coordinate plane, where each of 
and 
may be 
, 
, or 
. How many distinct lines pass through at least two members of ?
Solution
Solution 1
Let's count them by cases:
- Case 1: The line is horizontal or vertical, clearly
. - Case 2: The line has slope
, with 
through 
and 
additional ones one unit above or below those. These total 
. - Case 3: The only remaining lines pass through two points, a vertex and a non-vertex point on the opposite side. Thus we have each vertex pairing up with two points on the two opposites sides, giving
lines.
.
, with 
and 
additional ones one unit above or below those. These total 
.
lines.These add up to 
.
Solution 2
There are 
ways to pick two points, but we've clearly overcounted all of the lines which pass through three points. In fact, each line which passes through three points will have been counted 
times, so we have to subtract 
for each of these lines. Quick counting yields 
horizontal, vertical, and diagonal lines, so the answer is 
distinct lines.
See also
| 2004 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 12 |
Followed by Problem 14 |
| 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 | |
| All AMC 12 Problems and Solutions | |
