10. A triangle has sides of lengths 20 and 19. If the triangle is not acute, how many possible integer lengths can the third side have?
Answer: 16
Solution: We begin by determining the possible integer lengths of the third side $s$. By the triangle inequality, $2 \leq s \leq 38$.
Now, we determine when the triangle is not acute. This means that the triangle is either a right triangle or an obtuse triangle. Since we are only considering integer lengths, notice that the third side will always be the shortest or the longest side. These are precisely the two cases to consider.
We use the Pythagorean Theorem to determine when the triangle is right or obtuse. If the two shorter sides of the triangle are $a$ and $b$ and the longest side is $c$, then the triangle is right or obtuse when $a^{2}+b^{2} \leq c^{2}$.
If the third side is the shortest, then by the above inequality we have that $s^{2} \leq 20^{2}-19^{2}$. This gives us $s^{2} \leq 39$. This in turn means that $2 \leq s \leq 6$ for this case, a total of 5 possible lengths of the third side.
If the third side is the longest, then by the above inequality we have that $19^{2}+20^{2} \leq s^{2}$. This gives us $s^{2} \geq 761$. This in turn means that $28 \leq s \leq 38$ for this case, or a total of 11 possible lengths of the third side.
Thus, there are $5+11=16$ possible lengths of the third side.