06/18/2020 (2011: National Target Round, Problem 6)

Q: If a and b are nonzero unequal real numbers and , what is the sum of all possible values for ?

A: 3

We can substitute a value for b and add together the terms:

1 : (a - 1)/a = 1/(a - 1) → a² - 2a + 1 = a → a² - 3a + 1 = 0 → a = (3 ± √(9 - 4))/2 → a = (3 + √(5))/2 and (3 - √(5))/2

You add (((3 + √(5))/2)/1) and (((3 - √(5))/2)/1) and you get (3 + √(5))/2 + (3 - √(5))/2 = 6/2 = 3.

That means the sum of all the possible values for a/b is 3.