Problem

Source: II International Festival of Young Mathematicians Sozopol 2011, Theme for 10-12 grade

Tags: algebra, Fraction



The fraction $\frac{1}{p}$, where $p$ is a prime number coprime with 10, is presented as an infinite periodic fraction. Prove that, if the number of digits in the period is even, then the arithmetic mean of the digits in the period is equal to $\frac{9}{2}$.