I think assumption 1 part 2 is not correct! The correct assumption is: the square of length is twice the square of width. Now the Answer:

Define variable a to be the width of paper size A6.

Then its length would be 1.4142 x a (square root of 2) which is equal to width of paper size A5

subsequently, for sizes A4, A3, A2, A1 and finally for paper size A0:

The width would be 8a and its length 8a x 1.4142

As we have the area of A0 as one square meter or 1000000 mm, therefor:

8a x 8a x 1.4142 = 1000000

a = 105 mm

the width of paper size A6 is 105 mm

and its length is 105 x 1.4142 = 148 mm