The USSR Olympiad Problem Book
#143. Which is larger, \(99^n + 100^n\) or \(101^n\) (where \(n\) is a natural number)?
Hint. Quick and dirty — evaluate
\begin{align*}
f(x) = \frac{99^x + 100^x}{101^x} = \left(\frac{99}{101}\right)^x + \left(\frac{100}{101}\right)^x
\end{align*}
using Python or WolframAlpha, or just with a logarithm table. Alternatively, expand \(101^n - 99^n\).