2011 nines
Problem 318
Consider the real number .
When we calculate the even powers of we get:
When we calculate the even powers of we get:
It looks as if the number of consecutive nines at the beginning of the fractional part of these powers is non-decreasing.
In fact it can be proven that the fractional part of approaches for large .
In fact it can be proven that the fractional part of approaches for large .
Consider all real numbers of the form with and positive integers and , such that the fractional part of approaches for large .
Let be the number of consecutive nines at the beginning of the fractional part of .
Let be the minimal value of such that .
Find .