MK-MO 2014 Problem 3

Let $k_1, k_2$ and $k_3$ be three circles with centers $O_1, O_2$ and $O_3$ respectively, such that no center is inside of the other two circles. Circles $k_1$ and $k_2$ intersect at $A$ and $P$ , circles $k_1$ and $k_3$ intersect and $C$ and $P$ , circles $k_2$ and $k_3$ intersect at $B$ and $P$ . Let $X$ be a point on $k_1$ such that the line $XA$ intersects $k_2$ at $Y$ and the line $XC$ intersects $k_3$ at $Z$ , such that $Y$ is nor inside $k_1$ nor inside $k_3$ and $Z$ is nor inside $k_1$ nor inside $k_2$ .

a) Prove that $\triangle XYZ$ is simular to $\triangle O_1O_2O_3$ b) Prove that the $P_{\triangle XYZ} \le 4P_{\triangle O_1O_2O_3}$ . Is it possible to reach equation? $*Note:$ P$ denotes the area of a triangle*