| HAMMERWARS 2025 |
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| Finished |
Thomas has generously provided $$$n$$$ lines in a plane! The lines are in regular position! That is, no two lines are collinear and no three lines intersect at a point.
The lines divide the plane into regions. Formally, points $$$A$$$ and $$$B$$$ are in the same region if and only if the segment $$$\overline{\rm AB}$$$ doesn't intersect a line. Two regions are adjacent if their boundaries overlap on a segment.
Arvind and Zhongtang are going to play a game on the regions. First, Arvind will select a region and color it red. Then turns will proceed as follows: Zhongtang will select a region adjacent to a red region and color it blue, then Arvind will select a region adjacent to a blue region and color it red. The last person who can play wins.
You want prime betting odds. Can you predict the outcome of the game with perfect play?
$$$n \\ a_1 \ b_1 \ c_1 \ d_1 \\ a_2 \ b_2 \ c_2 \ d_2 \\ \ldots \\ a_n \ b_n \ c_n \ d_n$$$
The $$$i$$$-th line passes through lattice points $$$(a_i,b_i)$$$ and $$$(c_i,d_i)$$$.
Constraints
$$$2 \le n \le 2 \cdot 10^5 \\ \max(|a_i|,|b_i|,|c_i|,|d_i|) \le 10^9$$$
"Arvind" or "Zhongtang" (without quotes)
20 0 2 11 1 3 5
Zhongtang
TODO: diagram for sample
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