MathDB

Let

n>5

be an integer. There are

n

points in the plane, no three of them collinear. Each day, Tom erases one of the points, until there are three points left. On the

i

-th day, for

1<i<n-3

, before erasing that day's point, Tom writes down the positive integer

v(i)

such that the convex hull of the points at that moment has

v(i)

vertices. Finally, he writes down

v(n-2) = 3

. Find the greatest possible value that the expression

|v(1)-v(2)|+ |v(2)-v(3)| + \ldots + |v(n-3)-v(n-2)|

can obtain among all possible initial configurations of

n

points and all possible Tom's moves.
Remark. A convex hull of a finite set of points in the plane is the smallest convex polygon containing all the points of the set (inside it or on the boundary).
Ivan Novak, Namik Agić

2

Problems(2)