All-Soviet-Union Mathematical Olympiad

Solution:

Extend the sides to form two rectangles. Let the sides of the octagon have length $a$, $b$, $c$, $d$, $e$, $f$, $g$, $h$. Then we can find the rectangle sides. For example, one of the rectangles has opposite sides $a + \frac{b + h}{\sqrt{2}}$ and $e + \frac{d + f}{\sqrt{2}}$. Hence either $a = e$ or $\sqrt{2} = \frac{b + h - d - f}{a - e}$. The root is irrational, so we must have $a = e$. Similarly for the other pairs of opposite sides.