2023-2024 ICPC, Asia Jakarta Regional Contest (Online Mirror, Unrated, ICPC Rules, Teams Preferred) |
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You are given a regular $$$N$$$-sided polygon. Label one arbitrary side as side $$$1$$$, then label the next sides in clockwise order as side $$$2$$$, $$$3$$$, $$$\dots$$$, $$$N$$$. There are $$$A_i$$$ special points on side $$$i$$$. These points are positioned such that side $$$i$$$ is divided into $$$A_i + 1$$$ segments with equal length.
For instance, suppose that you have a regular $$$4$$$-sided polygon, i.e., a square. The following illustration shows how the special points are located within each side when $$$A = [3, 1, 4, 6]$$$. The uppermost side is labelled as side $$$1$$$.
You want to create as many non-degenerate triangles as possible while satisfying the following requirements. Each triangle consists of $$$3$$$ distinct special points (not necessarily from different sides) as its corners. Each special point can only become the corner of at most $$$1$$$ triangle. All triangles must not intersect with each other.
Determine the maximum number of non-degenerate triangles that you can create.
A triangle is non-degenerate if it has a positive area.
The first line consists of an integer $$$N$$$ ($$$3 \leq N \leq 200\,000$$$).
The following line consists of $$$N$$$ integers $$$A_i$$$ ($$$1 \leq A_i \leq 2 \cdot 10^9$$$).
Output a single integer representing the maximum number of non-degenerate triangles that you can create.
4 3 1 4 6
4
6 1 2 1 2 1 2
3
3 1 1 1
1
Explanation for the sample input/output #1
One possible construction which achieves maximum number of non-degenerate triangles can be seen in the following illustration.
Explanation for the sample input/output #2
One possible construction which achieves maximum number of non-degenerate triangles can be seen in the following illustration.
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