A16456 | Triangle Construction

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.