[math-fun] RMS paradox

The root mean square (RMS) of a sine wave is always the peak value divided by the square root of 2. The same is true of the sums of multiple sine waves with different frequencies, phases, and amplitudes. But every repeating waveform is equal to the sum of sine waves with different frequencies, phases, and amplitudes. This includes the square wave, i.e. a function which always equals +X or -X, and never takes any other value. But obviously the RMS of that square wave is simply X, not X divided by the square root of 2. Explain. I learn a lot by coming up with such paradoxes then figuring out the solution.