06/08/2020 (2011: National Sprint Round, Problem 26)

Q: The formula for the area of any rectangle with the perimeter of 20 units can be written as A = m - nd², where d is the length of the diagonal and m and n are constants. What is the value of the product mn?

A: 25

Let's say that the side lengths of a rectangle are 5 and 5. The area is 5² = 25 and the diagonal is 5√2, so 25 = m - n(50).

Let's say that the side lengths of another rectangle are 6 and 4. The area is 6 * 4 = 24 and the diagonal is 2√13, so 24 = m - n(52)

Our system of equations is 25 = m - 50n and 24 = m - 52n. We subtract the second equation from the first equation and get 1 = 2n, so n = 1/2. That means 25 = m - (1/2)(50), so 25 = m - 25, so 50 = m.

If n = 1/2, and m = 50, then nm = 50 * 1/2 = 25.