Consider a non-zero real number $a$ such that $\{a\} + \left\{\frac{1}{a}\right\}=1$, where $\{x\}$ denotes the fractional part of $x$. Prove that for any positive integer $n$, $\{a^n\} + \left\{\frac{1}{a^n}\right\}= 1$.
Source: 2011 Saudi Arabia BMO TST 1.4 - Balkan MO
Tags: algebra, factorial
Consider a non-zero real number $a$ such that $\{a\} + \left\{\frac{1}{a}\right\}=1$, where $\{x\}$ denotes the fractional part of $x$. Prove that for any positive integer $n$, $\{a^n\} + \left\{\frac{1}{a^n}\right\}= 1$.