MathDB

Consider an arbitrary “figure”

F

(non convex polygon). A chord of

F

is defined to be a segment which lies entirely within

F

and whose ends are on its boundary.
(a) Does there always exist a chord of

F

that divides its area in half? (b) Prove that for any

F

there exists a chord such that the area of each of the two parts of

F

is not less than

1/3

of the area of

F

. (c) Can the number

1/3

in (b) be changed to a greater one?
(V Proizvolov)

Problem Statement