Purple Comet 2016 HS Problem 10
Source:
April 12, 2018
Purple Comet
Problem Statement
Jeremy wrote all the three-digit integers from 100 to 999 on a blackboard. Then Allison erased each of the 2700 digits Jeremy wrote and replaced each digit with the square of that digit. Thus, Allison replaced every 1 with a 1, every 2 with a 4, every 3 with a 9, every 4 with a 16, and so forth. The proportion of all the digits Allison wrote that were ones is , where m and n are relatively prime positive integers. Find .