2017 AMC 10A Problems/Problem 23
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
We can solve this by finding all the combinations, then subtracting the ones that are on the same line. There are 
points in all, from 
to 
, so 
is 
, which simplifies to 
.
Now we count the ones that are on the same line. We see that any three points chosen from and 
would be on the same line, so 
is 
, and there are rows, columns, and 
long diagonals, so that results in 
.
We can also count the ones with 
on a diagonal. That is 
, which is 4, and there are of those diagonals, so that results in 
.
We can count the ones with only 
on a diagonal, and there are diagonals like that, so that results in .
We can also count the ones with a slope of 
, , 
, or 
, with points in each. Note that there are such lines, for each slope, present in the grid. In total, this results in 
.
Finally, we subtract all the ones in a line from , so we have 
Solution 2 (extreme last-minute desperate guess)
We know there are points, and we need to pick . 
. But we notice that this includes colinear points. Answer choices 
and 
seem a bit too whole (perfect multiple of 10). 
seems unlikely, so it must be 
See Also
Video Solution:
https://www.youtube.com/watch?v=wfWsolGGfNY
https://www.youtube.com/watch?v=LCvDL-SMknI
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