Problem:
Let . Given that has digits and that its first (leftmost) digit is , how many elements of have as their leftmost digit?
Solution:
Note that has one more digit than , except in the case when starts with a . In the latter case, long division shows that starts with a and has the same number of digits as . Therefore, when the powers of from to are computed there are increases in the number of digits. Thus there must be instances when computing from does not increase the number of digits. Since does not have leading digit we can conclude that has a leading digit of exactly when there is no increase in the number of digits in computing from . It follows that of the numbers must start with the digit .
The problems on this page are the property of the MAA's American Mathematics Competitions