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Problem 8. For a convex polygon $\mathcal{P}$, let $\mathcal{B}$ be the set of points on the boundary of $\mathcal{P}$. A function $f: \mathcal{B} \rightarrow \mathcal{B}$ is European if it satisfies the following properties:
(i) $f(f(X)) = X$ for all points $X \in \mathcal{B}$;
(ii) Line segments $Yf(Y)$ and $Zf(Z)$ have a common point strictly inside the polygon, for all points $Y, Z \in \mathcal{B}$.
What is the largest real number $c$ such that for any convex polygon $\mathcal{P}$ and European function $f$, there is a point $W \in \mathcal{B}$ such that the length of line segment $Wf(W)$ is at least $c$ times the perimeter of $\mathcal{P}$?
Solution 1Solution 2Solution 3Solution 4
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