### International Mathematics Competition for University Students

July 31 – August 6 2017, Blagoevgrad, Bulgaria

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### Problem 10

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 $$\sum_{\ell=1}^n \textrm{area}(T_\ell) > \textrm{area}(K)-\varepsilon,$$ then $$\sum_{\ell=1}^n \textrm{perimeter}(T_\ell) > p.$$

Proposed by: Fedor Malyshev, Steklov Mathematical Institute and Ilya Bogdanov, Moscow Institute of Physics and Technology