As in most skyscrapers, windows cannot be opened more than just a little bit. This is not because of the safety in respect to accidental dropping of long uranium nails, and neither to prevent residents from launching origami airplanes. The real reason is building safety. This might sound pretty strange at first - one would think that opening a window would release the wind pressure buildup on the building, but in fact it's tangential wind force which is significant.
Such buildings' windows are very strong, because if an incident storm is strong enough to break a few - it can bring down the whole structure. The problem here is, of course, cleaning these windows after a dust storm, which usually takes months. So my windows open just a little bit, and from the wrong side (bottom). They are double windows here at 59th floor (for safety) so magnets wouldn't work.
Here's a list of ideas - most of which contributed by the APhO2007 team while visiting me recently (for future reference):
 Yoav's idea of a sophisticated mechanical construction
 Jonathan: start a small fire, and they will bring a huge water pump outside
 Me: Installing wipers (like windshield wipers in cars)
 Catching a few bugs to stick little brushes to them, and release them to crawl outside
 Hamutal suggested that cleaning a window of someone below me is easier than cleaning my own. Assaf's added an important insight that residents of the first floor will clean the windows of the top floor, and it's their problem how they do it.
 AssafS also suggested that we could clean the inner side VERY hard
 Amos: cool the room down with the air conditioner to let drops condensate on the other side, and then use a standard ultrasonic cleaner.
 Inverted pendulum mounted on the roof: would bump into walls with something soft, and once started, maintaining the motion should be as easy as in a normal pendulum.