2017 AMC 10 A
Complete problem set with solutions and individual problem pages
How many triangles with positive area have all their vertices at points in the coordinate plane, where and are integers between and , inclusive? (2017 AMC 10A Problem, Question#23)
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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 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. There are of them, so that results in . Finally, we subtract all the ones in a line from , so we have .
