Eighteenth STARS OF MATHEMATICS Competition

Let $A_i{A}_j{A}_k$ be a triangle of maximal area. Let $A_i^{\prime }$ be the reflection of $A_i$ across the midpoint of the side $A_j{A}_k$; the points $A_j^{\prime }$ and $A_k^{\prime }$ are defined similarly. Note that at most one of these three reflections can be a vertex of the polygon — otherwise, the polygon would have three collinear vertices, contradicting convexity.

We will prove that the triangle $A_i^{\prime }{A}_j^{\prime }{A}_k^{\prime }$ covers the polygon $A_1{A}_2...A_n$. By the preceding, the cover is strict, so $[A_i^{\prime }{A}_j^{\prime }{A}_k^{\prime }]>[A_1{A}_2...A_n]$. As $[A_i^{\prime }{A}_j^{\prime }{A}_k^{\prime }]=4[A_i{A}_j{A}_k]$, the conclusion follows.

To prove the covering claim above, suppose, if possible, some vertex $A_{\ell }$ lies outside the triangle $A_i^{\prime }{A}_j^{\prime }{A}_k^{\prime }$. Let $\mathrm{dist}(X,YZ)$ denote the Euclidean distance of the point $X$ to line $YZ$. Note that at least one of the three inequalities below holds: $$\begin{array}{rl}\mathrm{dist}(A_{\ell },A_j{A}_k)& >\mathrm{dist}(A_i,A_j{A}_k),\\ \mathrm{dist}(A_{\ell },A_k{A}_i)& >\mathrm{dist}(A_j,A_k{A}_i),\\ \mathrm{dist}(A_{\ell },A_i{A}_j)& >\mathrm{dist}(A_k,A_i{A}_j).\end{array}$$ Hence at least one of the triangles $A_{\ell }{A}_j{A}_k$, $A_{\ell }{A}_k{A}_i$, $A_{\ell }{A}_i{A}_j$ has an area (strictly) greater than that of $A_i{A}_j{A}_k$, contradicting the choice of this latter. This ends the proof.