05/21/2020: (2011: National Team Round, Problem 1)

The coronacation is giving me way too much free time, so I'm gonna start posting regularly.   

 Q: A triangle has three sides of integer lengths. What is the product of the smallest and largest possible values for the length of the unknown side if two of the sides measure 13 units and 16 units? 

A: 112

We use the Triangle Inequality Theorem, which states that:

If we are trying to find the least possible value for the 3rd side (x), then we know the other 2 sides are greater than it. That means that : 13 + x > 16, which simplifies to x > 3. Because the sides have integer lengths, the third side, x, is the next greatest integer after 3, which is 4.

4 is the smallest value that the 3rd side can be.

If we are trying to find the greatest possible value for the 3rd side (x), then we know the other 2 sides are less than it. That means that : 13 + 16 > x, which simplifies to 29 > x. Because the sides have integer lengths, the third side, x, is the greatest integer before 29, which is 28.

28 is the greatest value that the 3rd side can be.

The product of the values we found ( 28 and 4) is 112.