Question 6
Let $n\ge5$ be an integer. Find the largest integer $k$, as a function of $n$, for which there exists a convex $n$-gon $A_1A_2\cdots A_n$ with exactly $k$ of the quadrilaterals $A_iA_{i+1}A_{i+2}A_{i+3}$ having an inscribed circle. Indices are taken modulo $n$.