69th Belarusian Mathematical Olympiad

A circle $\omega$ of radius $1$ is given. A collection $T$ of triangles is called good, if the following conditions hold: (i) each triangle from $T$ is inscribed in $\omega$; (ii) no two triangles from $T$ have a common interior point. Determine all positive real numbers $t$ such that, for each positive integer $n$ there exist a good collection of $n$ triangles, each of perimeter greater than $t$.