IMC 2017 · Problem 10

Let $K$ be an equilateral triangle in the plane. Prove that for every $p > 0$ there exists an $\varepsilon > 0$ with the following property: If $n$ is a positive integer, and $T_1, \ldots, T_n$ are non-overlapping triangles inside $K$ such that each of them is homothetic to $K$ with a negative ratio, and

$$\sum_{\ell=1}^{n} \operatorname{area}(T_\ell) > \operatorname{area}(K) - \varepsilon,$$

then

$$\sum_{\ell=1}^{n} \operatorname{perimeter}(T_\ell) > p.$$