Bitonality refers to the use of two different key signatures at the same time. Bartok was known to use bitonality, and from the image you've attached, this exercise appears to utilize bitonality. The first system (the one on top that is fully visible in your image) appears to be in
A melodic minor. The melodic minor scale uses a ♮6 and a ♮7 rather than a flatted 6 and 7:
In more traditional classical music contexts,
A melodic minor strictly applies to the ascending form of the scale. Whenever the scale is used in a descending fashion, one would use
A natural minor scale (♭6 and ♭7):
Accordingly, more traditional pieces would notate this key signature with no sharps and flats, thereby indicating the key signature of
A natural minor. Then, whenever an ascending form of the scale was used, sharps would be written in front of the F and G, thus achieving the melodic minor.
Bartok, however, is breaking from this more traditional approach. He is notating the song as
A melodic minor rather than
A natural minor, and in the bit of the song that your image shows, he is not using the natural minor scale at all. Moreover, the second system (which is only partially shown) appears to be in an altogether different key (perhaps
E maj or
C♯ min), which is yet another departure from more traditional music. These deviations are intentional, and so is the non-standard notation. Once, when talking about a different of his bitonal songs,
Bartok said this:
this half-serious, half-jesting procedure was used to demonstrate the absurdity of key signatures in certain kinds of contemporary music.
That quote can provide a useful context for understanding and interpreting pieces like the one you're playing, which deviate from the "rules" we learn in traditional music theory courses.