MathDB

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.

Problem Statement