Moving Average of sinusoid
The fundamental property of averaging filters (in swear words, a linear system) is that the output of a sine is a sine of the same frequency, albeit of zero amplitude. So:


Which non-trivial moving average weights would you need to find to get a filter that returns the same output as its input?


An averaging in phase with the sine: take a filter full of zeroes, except $K$ values, pairwise distant by a multiple of $40$, with coefficient values $1/K$. This way, you will average $K$ sine value of the same amplitude, yielding a value with the same amplitude.

Hence, you have a bunch of non trivial filters. However, they are not very interesting in practice, because they shall be tuned to the period of a sine sampled with particular sampling.