make a conjecture, is there a value of N for which there could be a triangle with sides of length N, 2N, and 3N?

Answer: Conjecture: There is no triangle with side lengths N, 2N, and 3N (where N is a positive real number)

Proof:

We prove this by contradiction: Suppose there was an N for which we can construct a triangle with side lengths N, 2N, and 3N. We then apply the triangle inequalities tests. It must hold that:

N + 2N > 3N

3N > 3N

3 > 3

which is False, for any value of N. This means that the original choice of N is not possible. Since the inequality is False for any value of N, there cannot be any triangle with the given side lengths, thus proving our conjecture.