Mock AIME 4 2006-2007 Problems/Problem 1
Problem
Albert starts to make a list, in increasing order, of the positive integers that have a first digit of 1. He writes 
but by the 1,000th digit he (finally) realizes that the list would contain an infinite number of elements. Find the three-digit number formed by the last three digits he wrote (the 998th, 999th, and 1000th digits, in that order).
Solution
It is clear that his list begins with 1 one-digit integer, 10 two-digits integers, and 100 three-digit integers, making a total of 
digits.
So he needs another 
digits before he stops. He can accomplish this by writing 169 four-digit numbers for a total of 
digits. The last of these 169 four-digit numbers is 1168, so the next three digits will be 
.
