RU-ALLRUSSIAN 1981 Problem 17

A convex polygon is drawn inside the unit circle. Someone makes a copy by starting with one vertex and then drawing each side successively. He copies the angle between each side and the previous side accurately, but makes an error in the length of each side of up to a factor $1 \pm \mathbf{p}$ . As a result the last side ends up a distance $\mathbf{d}$ from the starting point. Show that $\mathbf{d} < 4 \mathbf{p}$ .