OPM 2007

Let $[ABC]$ be a triangle and $X, Y$ and $Z$ points on the sides $[AB], [BC]$ and $[AC]$ , respectively. Prove that circumcircles of triangles $AXZ, BXY$ and $CYZ$ intersect at a point.

Rua do Antonio has $100$ houses numbered from $1$ to $100$ . Any house numbered with the difference between the numbers of two houses of the same color is a different color. Prove that on Rua do Antonio there are houses of at least five different colors.