MN-MO 2000

Let $\operatorname{rad}(k)$ denote the product of prime divisors of a natural number $k$ (define $\operatorname{rad}(1)=1$ ). A sequence $(a_n)$ is defined by setting $a_1$ arbitrarily, and $a_{n+1}=a_n+\operatorname{rad}(a_n)$ for $n\ge1$ . Prove that the sequence $(a_n)$ contains arithmetic progressions of arbitrary length.

Circles $\omega_1,\omega_2,\omega_3$ with centers $O_1,O_2,O_3$ , respectively, are externally tangent to each other. The circle $\omega_1$ touches $\omega_2$ at $P_1$ and $\omega_3$ at $P_2$ . For any point $A$ on $\omega_1$ , $A_1$ denotes the point symmetric to $A$ with respect to $O_1$ . Show that the intersection points of $AP_2$ with $\omega_3$ , $A_1P_3$ with $\omega_2$ , and $AP_3$ with $A_1P_2$ lie on a line.