Question 123
Given a triangle $ABC$ with $\angle ACB = 60^\circ$, define the sequence of points $A_0, A_1, \ldots, A_{2006}$ in the following way: $A_0 = A$, $A_1$ is the orthogonal projection of $A_0$ on $BC$, $A_2$ is the orthogonal projection of $A_1$ on $AC$, $\ldots$, $A_{2005}$ is the orthogonal projection of $A_{2004}$ on $BC$ and $A_{2006}$ is the orthogonal projection of $A_{2005}$ on $AC$. The sequence of points $B_0, B_1, \ldots, B_{2006}$ is defined in a similar way: $B_0 = B$, $B_1$ is the orthogonal projection of $B_0$ on $AC$, $B_2$ is the orthogonal projection of $B_1$ on $BC$ and so on. Prove that the line $A_{2006}B_{2006}$ is tangent to the incircle of $\triangle ABC$ if and only if
$$
\frac{AC + BC}{AB} = \frac{2^{2006} + 1}{2^{2006} - 1}
$$
$$
\frac{AC + BC}{AB} = \frac{2^{2006} + 1}{2^{2006} - 1}
$$