Frankin B.P.Dan gives a convex $n$-gon $A_{1} \ldots A_{n}$. Let $P_{i}(i=1, \ldots, n)$ be a point on its boundary such that the line $A_{i} P_{i}$ divides its area in half. It is known that all points $P_{i}$ do not coincide with the vertices and lie on $k$ sides of the $n$-gon. What is a) the smallest; b) the largest possible value of $k$ for each given $n$?
a) Since the segments $A_{i} P_{i}$ divide the area of the polygon in half, any two of them intersect. Suppose point $P_{i}$ lies on side $A_{j} A_{j+1}$. Then points $P_{j}$ and $P_{j+1}$ lie on different sides of $A_{i}$, meaning there will always be three points lying on different sides. On the other hand, if two vertices of the polygon are vertices of an equilateral triangle, and all the others are located near its third vertex, then all points $P_{i}$ lie on three sides of the polygon.

For a regular $n$-gon with odd $n$, all $P_{i}$ lie on different sides.

Let $n=2 m$. Since segments $A_{m} P_{m}$ and $A_{2 m} P_{2 m}$ intersect, points $P_{m}$ and $P_{2 m}$ lie on the same side of diagonal $A_{m} A_{2 m}$. On the other side of this diagonal lie $m$ sides of the polygon, and point $P_{i}$ can only fall on these sides if the corresponding vertex $A_{i}$ lies between $P_{m}$ and $P_{2 m}$. But there are no more than $m-1$ such vertices, so there exists a side on which there are no points $P_{i}$.

Now consider an $n$-gon, the vertices $A_{1}, \ldots, A_{n-2}$ of which are vertices of a regular $(n-1)$-gon, and vertices $A_{n-1}, A_{n}$ are located near the remaining vertex of this $(n-1)$-gon. Then the points $P_{i}$ are located on all sides of the constructed polygon except $A_{n-1} A_{n}$.

## Answer

a) 3 ; b) $n-1$ for even $n$ and $n$ for odd.