To the best of my knowledge, MP3 players send an electrical signal through the wires of the jack connected to it, this electrical signal causes the speakers to generate the corresponding sound in the air.
Although MP3 players cannot (normally) generate signals in the MHz range, I'm fairly certain most can generate signals up to at least 20 KHz.

If I was to correctly merge a sine wave of 20 KHz with an audio signal and then send this signal to a powerful amplifier (which could tolerate that frequency), could I transmit this signal?

I realise there are some problems with this; the MP3 player would need the audio to be amplified enough for the amplifier to take it, there might also be problems with frequency stability. But if you had to guess - would it work? (If not, can you explain what the problems would be? Would it at least work as an oscillator?)

 

Hello,

On 20 kHz you can not transmit an audio signal.
The lowest AM signals can be found in the LW band.
A famous one is the BBC on 198 kHz.

If you want to listen online to the radio signals that are available, have a look at this WebSDR site:
http://websdr.ewi.utwente.nl:8901/

Bertus