Find the number of 10-tuples of integers $a_1,a_2,...,a_{10})$ such that $|a_1| \leq 1$ and $a_1^2 + a_2^2 + \dots + a_{10}^2 - a_1a_2 - a_2a_3 - \dots -a_{10}a_1 = 2$
This is question 4 from the RMO 2013. I have a few issues with the proof provided.
- The assumption that $a_{11} = a_1$ is integral to the proof; how is this assumption justified? Isn't there a loss of generality?
- Why does the fact that $a_1 = -1,0,1$ justify the multiplication of 3 to the final solution, i.e $$\color{red}3\times45 \times 28$$
The use of $a_{11}$ is only a formal assumption to highlight the "circular" symmetry of the problem (as a confirmation of this, $a_{11}$ is no longer used in the proof after the initial definition).
As regards the multiplication to $3$, note that the combinations expressed by $45 \cdot 28$ only refer to the values of the differences $a_i -a_{i+1} $, and not to the values of $a_i $.