CAPS 2006 Problem 6

Find out if there is a convex pentagon $A_1A_2A_3A_4A_5$ such that, for each $i = 1, \dots , 5$ , the lines $A_iA_{i+3}$ and $A_{i+1}A_{i+2}$ intersect at a point $B_i$ and the points $B_1,B_2,B_3,B_4,B_5$ are collinear. (Here $A_{i+5} = A_i$ .)