P9827 [ICPC 2020 Shanghai R] Sky Garden

Prof. Du and Prof. Pang plan to build a sky garden near the city of Allin. In the garden, there will be a plant maze consisting of straight and circular roads. On the blueprint of the plant maze, Prof. Du draws $n$ circles indicating the circular roads. All of them have center $(0, 0)$. The radius of the $i$-th circle is $i$. Meanwhile, Prof. Pang draws $m$ lines on the blueprint indicating the straight roads. All of the lines pass through $(0, 0)$. Each circle is divided into $2m$ parts with equal lengths by these lines. Let $Q$ be the set of the $n+m$ roads. Let $P$ be the set of all intersections of two different roads in $Q$. Note that each circular road and each straight road have two intersections. For two different points $a\in P$ and $b\in P$, we define $dis(\{a, b\})$ to be the shortest distance one needs to walk from $a$ to $b$ along the roads. Please calculate the sum of $dis(\{a, b\})$ for all $\{a, b\}\subseteq P$.

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 $dis(p_1, p_2)=dis(p_2, p_3)=dis(p_3, p_4)=dis(p_1, p_4)=\frac{\pi}{2}$ $dis(p_1, p_5)=dis(p_2, p_5)=dis(p_3, p_5)=dis(p_4, p_5)=1$ $dis(p_1, p_3)=dis(p_2, p_4)=2$