6. Triangle $A B C$ is an equilateral triangle with side length 1 . Let $X_{0}, X_{1}, \ldots$ be an infinite sequence of points such that the following conditions hold:
- $X_{0}$ is the center of $A B C$
- For all $i \geq 0, X_{2 i+1}$ lies on segment $A B$ and $X_{2 i+2}$ lies on segment $A C$.
- For all $i \geq 0, \measuredangle X_{i} X_{i+1} X_{i+2}=90^{\circ}$.
- For all $i \geq 1, X_{i+2}$ lies in triangle $A X_{i} X_{i+1}$.
Find the maximum possible value of $\sum_{i=0}^{\infty}\left|X_{i} X_{i+1}\right|$, where $|P Q|$ is the length of line segment $P Q$.
Answer: $\frac{\sqrt{6}}{3}$ Let $Y$ be the foot of the perpendicular from $A$ to $X_{0} X_{1}$ : note that the sum we wish to minimize is simply $X_{0} Y+Y A$. However, it is not difficult to check (for example, by AMGM) that $A Y+Y X_{0} \geq \sqrt{2} A X_{0}=\sqrt{6} / 3$. This may be achieved by making $\angle Y X_{0} A=45^{\circ}$, so that $\angle A X_{1} X_{0}=105^{\circ}$.