How can I reduce the force required to operate double-hung windows?
The wooden framed double hung sash design dates from a time when no elderly person was ever left alone long enough to open or close a window. Families were large and any oldster who wanted a window operated would simply whack the nearest youngster with a cane and croak out the appropriate orders. 

Today the demographic age curve is quite different and there are not enough window operators to go around. I expect sash window easy-open aftermarket add-on devices to appear on the market any day now, as soon as us boomers form a significant market. 

Meanwhile, I have two recalcitrant windows in my kitchen which are not only stiff but awkwardly placed above counters. Here is what I installed to operate them. 

For each window, I used four pulleys (about 1.25"), two anchors (I used screw eyes), and a long cord. I attached one pulley and one anchor to the window frame top, one pulley to the inner sash top, one pulley to the inner sash bottom, and one pulley and one anchor to the window frame bottom. I rigged the long cord according to this schematic diagram: 



Pulling down on the green section of the cord raises the window. Pulling down on the blue section, or up on the green section, lowers the window. The force advantage is 2 to 1. 

The pulleys are actually on the center line; they are offset in the diagram to show the rigging. Pulleys 3 and 5 are turned, and pulley 4 is shimmed out, to bring the blue and green cords out away from the sash frame. The red cords are almost touching the frames. 

(The cords are red, blue, and green only in the diagram. My actual cords are ecru.) 

You cannot use small pulleys for this project because moving the window involves turning all four of them. 

My wife and kids ridiculed this lashup when they first saw it, so I invited them to take up a collection and shop for other solutions. No takers so far.