Question 11
Given $n$ odd and a set of integers $\{a_1\}$, $\{a_2\}$, ..., $\{a_n\}$, derive a new set $(\{a_1\} + \{a_2\}) / 2$, $(\{a_2\} + \{a_3\}) / 2$, ..., $(\{a_{n - 1}\} + \{a_n\}) / 2$, $(\{a_n\} + \{a_1\}) / 2$. However many times we repeat this process for a particular starting set we always get integers. Prove that all the numbers in the starting set are equal.
For example, if we started with $5, 9, 1$, we would get $7, 5, 3$, and then $6, 4, 5$, and then $5, 4, 5, 5.5$. The last set does not consist entirely of integers.
For example, if we started with $5, 9, 1$, we would get $7, 5, 3$, and then $6, 4, 5$, and then $5, 4, 5, 5.5$. The last set does not consist entirely of integers.