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nb the area of the rectangle is fixed, but not its dimensions. so we want to maximize x(x+y) subject to the condition that x^2+y^2=1, since the optimal configuration will have the rectangles sharing (part of) a side.

i don't know a way of doing this with am-gm like techniques, so put x = cos t, y = sin t for t\in (0,\pi).

then we want to maximize cos t (cos t + sin t). i believe that you can write this as a single trigonometric function, but it's simple enough to differentiate and get cos 2t - sin 2t, which implies that 2t = pi/4 or t = pi/8. now all that remains is to check that this is optimal.

note that it's a little tricky to plug in pi/8 into cos t(cos t + sin t) unless you know the half angle formulas; it's probably better to use the double angle formulas to convert cos^2 t and cos t sin t into trigonometric functions in 2t.