RU-ALLRUSSIAN 1964 Problem 4

Given n odd and a set of integers $\mathrm{a_1}$ , $\mathrm{a_2}$ , ..., $\mathrm{a_n}$ , derive a new set $(\mathrm{a_1} + \mathrm{a_2}) / 2$ , $(\mathrm{a_2} + \mathrm{a_3}) / 2$ , ..., $(\mathrm{a_{n - 1}} + \mathrm{a_n}) / 2$ , $(\mathrm{a_n} + \mathrm{a_1}) / 2$ . 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.