II OBM

Question

Show that for any positive integer n > 2 we can find n distinct positive integers such that the sum of their reciprocals is 1.

Accepted Answer

We have 12 + 13 + 16 = 1. Now we have 12 + 14 + 18 + + 12^n + 12^n = 1, replace the second 12^n by (12 + 13 + 16) 12^n to get n + 3 terms (n 1): 12 + 14 + 18 + + 12^n+1 + 13 2^n + 13 2^n+1 = 1. For example, we have 12 + 14 + 16 + 112 = 1 and 12 + 14 + 18 + 112 + 124 = 1.