Question

用边长为 $1$ 个单位的正六边形庭院砖围出一个花园，砖块边对边摆放，每一边有 $n$ 块。图示为 $n = 5$ 时围绕花园的砖块路径。

若 $n = 202$ ，则路径围成的花园面积（不包括路径本身）为 $m\left(\sqrt{3}/2\right)$ 平方单位，其中 $m$ 是正整数。求 $m$ 除以 $1000$ 的余数。

Patio blocks that are regular hexagons $1$ unit on a side are used to outline a garden by placing the blocks edge to edge with $n$ on each side. The diagram indicates the path of blocks around the garden when $n = 5.$

If $n = 202,$ then the area of the garden enclosed by the path, not including the path itself, is $m\left(\sqrt{3}/2\right)$ square units, where $m$ is a positive integer. Find the remainder when $m$ is divided by $1000.$

答案

答案：803

知识点：正多边形求和面积分割

难度评级：2340

解答：

由路径围成的花园本身是一个单位六边形阵列，每边有 $n - 1$ 个六边形。从中心向外按 $6, 12, \ldots$ 个六边形的环来计数，它包含 $\begin{aligned} &1 + 6 + 12 + \cdots + 6(n-2) \\ &= 1 + 3(n-2)(n-1) \end{aligned}$ 个小六边形。取 $n = 202$ ，得到 $1 + 3 \cdot 200 \cdot 201 = 120601$ 个。

每个单位六边形由 $6$ 个边长为 $1$ 的等边三角形组成，所以面积为 $6 \cdot \frac{\sqrt{3}}{4} = 3 \cdot \frac{\sqrt{3}}{2}$ 。因此花园面积是 $3 \cdot 120601 = 361803$ 倍的 $\frac{\sqrt{3}}{2}$ ，所以 $m = 361803$ ，除以 $1000$ 的余数为 $803$ 。

The garden enclosed by the path is itself a hexagonal arrangement of unit hexagons with $n - 1$ on each side. Counting from the center outward in rings of $6, 12, \ldots$ hexagons, it contains $\begin{aligned} &1 + 6 + 12 + \cdots + 6(n-2) \\ &= 1 + 3(n-2)(n-1) \end{aligned}$ blocks, which for $n = 202$ is $1 + 3 \cdot 200 \cdot 201 = 120601.$

Each unit hexagon consists of $6$ equilateral triangles of side $1,$ so its area is $6 \cdot \frac{\sqrt{3}}{4} = 3 \cdot \frac{\sqrt{3}}{2}.$ The garden's area is therefore $3 \cdot 120601 = 361803$ times $\frac{\sqrt{3}}{2},$ so $m = 361803,$ and the remainder upon division by $1000$ is $803.$

2002 AIME II 第 4 题

答案：803

难度评级：2340

解答：

其他年份的第 4 题