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Omni-MATH / 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 …

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problem

Let n>5n>5 be an integer. There are nn 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 ii-th day, for 1<i<n−31<i<n-3, before erasing that day's point, Tom writes down the positive integer v(i)v(i) such that the convex hull of the points at that moment has v(i)v(i) vertices. Finally, he writes down v(n−2)=3v(n-2) = 3. Find the greatest possible value that the expression |v(1)−v(2)|+|v(2)−v(3)|+…+|v(n−3)−v(n−2)||v(1)-v(2)|+ |v(2)-v(3)| + \ldots + |v(n-3)-v(n-2)| can obtain among all possible initial configurations of nn points and all possible Tom's moves.
Plain-text mathematical notation (without MathML)
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)|+…+|v(n−3)−v(n−2)|
can obtain among all possible initial configurations of n points and all possible Tom's moves.
Original LaTeX notation
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.

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Source and history

task_id
13101ef5-5106-5135-adef-1528028a493f
task_key
test--13101ef5-5106-5135-adef-1528028a493f
task_revision_id
1
config / split
/ test

Official source

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