Junior Mathematical Olympiad

Let the sides be $13$, $x$, and $2x$.

By the triangle inequality, the sum of the lengths of any two sides must be greater than the third side.

So, we have:

1. $13+x>2x$ 2. $13+2x>x$ 3. $x+2x>13$

Let's solve each inequality:

1. $13+x>2x⟹13>x$ 2. $13+2x>x⟹13+x>0$ (which is always true for $x>0$) 3. $x+2x>13⟹3x>13⟹x>\frac{13}{3}$

Since $x$ is an integer, $x\ge 5$.

Also, from (1), $x<13$.

So the possible integer values for $x$ are $5,6,7,8,9,10,11,12$.

There are $8$ possible values.

Answer: D) 8