Problem

Source: Serbia IMO TST 2024, P3

Tags: algebra



Let $S$ be the set of all convex cyclic heptagons in the plane. Define a function $f:S \rightarrow \mathbb{R}^+$, such that for any convex cyclic heptagon $ABCDEFG,$ $$f(ABCDEFG)=\frac{AC \cdot BD \cdot CE \cdot DF \cdot EG \cdot FA \cdot GB} {AB \cdot BC \cdot CD \cdot DE \cdot EF \cdot FG \cdot GA}. $$ a) Show that for any $M \in S$, $f(M) \geq f(\prod)$, where $\prod$ is a regular heptagon. b) If $f(M)=f(\prod)$, is it true that $M$ is a regular heptagon?