Solution:
Let the circles be $a$, $b$, $c$, $d$, $e$. Let $A$ be a point common to $b$, $c$, $d$, $e$, let $B$ be a point common to $a$, $c$, $d$, $e$ and so on. If any two of $A$, $B$, $C$, $D$, $E$ coincide then the coincident point is on all 5 circles. Suppose they are all distinct. Then $A$, $B$, $C$ are on $d$ and $e$. Hence $d$ and $e$ coincide (3 points determine a circle). Hence $D$ is on all 5 circles.