HMMT Spring, 2021

Solution:

Let $f_1$ denote a $45^{\circ}$ counterclockwise rotation about point $A$ followed by a dilation centered at $A$ with scale factor $1/\sqrt{2}$. Similarly, let $f_2$ denote a $45^{\circ}$ clockwise rotation about point $A$ followed by a dilation centered at $A$ with scale factor $1/\sqrt{2}$. For any point $B$ in the plane, there exists a point $C$ on $\Gamma$ such that $\angle ABC = 90^{\circ}$ and $AB = BC$ if and only if $B$ lies on $f_1(\Gamma)$ or $f_2(\Gamma)$. Thus, such points $B$ and $C$ on $\Gamma$ exist if and only if $\Gamma$ intersects $f_1(\Gamma)$ or $f_2(\Gamma)$. So, the minimum possible value of $r$ occurs when $\Gamma$ is tangent to $f_1(\Gamma)$ and $f_2(\Gamma)$. This happens when $r/\sqrt{2} + r = 30/\sqrt{2}$, i.e., when $r = \frac{30}{\sqrt{2} + 1} = 30\sqrt{2} - 30$. Therefore, the minimum possible value of $\lfloor r \rfloor$ is $\lfloor 30\sqrt{2} - 30 \rfloor = 12$.