2017 AMC 10A Problems/Problem 23
Contents
[hide]Problem
How many triangles with positive area have all their vertices at points in the coordinate plane, where and are integers between and , inclusive?
Solution 1
There are a total of sets of three points. However, some of them form degenerate triangles (i.e., they have area of 0) if the three points are collinear. There are a total of 12 lines that go through 5 points (5 vertical, 5 horizontal, 2 diagonal), which contributes degenerate triangles, 4 lines that go through exactly 4 points, which contributes degenerate triangles, and 16 lines that go through exactly three points, which contributes degenerate triangles. Subtracting these degenerate triangles, we get an answer of .
Solution 2
We can find out that the least number of digits the number is , with 's
See Also
2017 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 22 |
Followed by Problem 24 | |
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