Filtering without using any transforms
You can achieve this result by using two combs filters : https://en.wikipedia.org/wiki/Comb_filter

Put simply, the comb filter consists of adding a delayed version of the signal to itself, causing destructive or constructive interference. 



For instance, with $K = 20$ and a negative gain value after the delay line, you can significantly decrease or suppress harmonics that are even multiples of 20: $\sin(2\pi40t)$ and $\sin(2\pi60t)$, which leaves you with only
$ y(t) = \sin(2\pi10t) + \sin(2\pi50t)$.

Apply the same method one more time with $K = 100$ and you will end up with $y(t)=\sin(2\pi10t)$, although scaled and delayed due to the filters. The same method works for each of the four harmonics.

It might not be the most straightforward way to achieve this result, but it only uses simple filter structures. Another approach could be to use notch filters to remove the unwanted harmonics from your signal.