Let X be a two-digit number and Y be another two-digit number formed by interchanging the digits of X. If (X + Y) is the greatest two-digit number, what is the number of possible values of X?

Let X = 10a + b where a, b are digits, a ≥ 1. Then Y = 10b + a. X + Y = 11(a + b). For this to be the greatest two-digit number, 11(a + b) must equal 99, so a + b = 9. Also Y must be a two-digit number, so b ≥ 1. Pairs (a, b) with a + b = 9, a ≥ 1, b ≥ 1: (1,8), (2,7), (3,6), (4,5), (5,4), (6,3), (7,2), (8,1) — that's 8 values. So number of possible X = 8.