The problem is this: A set S consists of triangles whose sides have integer lengths less than 5, and no two elements of S are congruent or similar. What is the largest number of elements that S can have?
I got an answer of 9, but the solution on aops says: "Based on the wording of Problem 13 to specifically exclude triangles with zero area: "... triangle with positive area ...", the definition of a triangle in this test includes degenerate ones. That is, the triangle inequality is not strict."
They are then able to find three degenerate triangles, changing the answer to 12 which is also an answer choice. Can degenerate triangles be assumed to count as triangles in this way? Shouldn't they specify within the problem?
How is similarity for degenerate triangles determined? One may assume that a deg. triangle is a line segment, so they are all similar. Or are they different, in which case the side ratios have to be considered? How will MAA deal with this problem? Degenerate Triangles are triangles, after all, and if commonsense was sufficient for #13, the writers would not have written positive area, as that would be implied. EDIT: I looked up degenerate triangles, and it seems that the similarity depends on the RATIO of the sides: A-----B-----------C could be a degenerate triangle, but it is not the same as A--------B---------C, even though the lengths are the same. So therefore, 422 and 112 are similar, while the other 2 are not: there are 3 total, with 12 triangles total. However, MAA will likely accept both 12 and 9, as the question was poorly worded