SHARYGIN 2009 Problem 19

Given convex $n$ -gon $A_1\ldots A_n$ . Let $P_i$ ( $i = 1,\ldots , n$ ) be such points on its boundary that $A_iP_i$ bisects the area of polygon. All points $P_i$ don't coincide with any vertex and lie on $k$ sides of $n$ -gon. What is the maximal and the minimal value of $k$ for each given $n$ ?