I think all 6 are great problems. I am also so cynical that I am worried there's a good reason I don't know of that makes these not great problems. Anybody want to chime in?

sketches that I really hope aren't fakesolves, written before seeing any spoilers:

4: draw circle BCMN to see that the radical axis of the two problem-given circles go through A, chase angles and powers of points and you have all three wanted points perpendicular to that radical axis at the intersection other than W.

5: so

induct ; therefore , integer

then so

then is strictly increasing.

If then take huge integer , compare (from inducting to ) with (also inducting to ) for contradiction

Therefore so so by we're done

6: gg tooooo haaaaaard

(I noticed that an arithmetic progression works but then I also noticed works and I totally gave up listing labelings for 5)