We say that a rectangle and a triangle are similar, if they have the same area and the same perimeter. Let $P$ be a rectangle for which the ratio of the longer to the shorter side is at least $\lambda -1+\sqrt{\lambda (\lambda -2)}$ where $\lambda =\frac{3\sqrt{3}}{2}$. Prove that there exists a tringle that is similar to $P$.
Problem
Source: XI International Festival of Young Mathematicians Sozopol 2022, Theme for 11-12 grade
Tags: algebra