2022 AMC 12B 第 25 题
先试着解答 2022 AMC 12B 第 25 题，然后核对你的答案与精心整理的解答，解答来自 LIVE by Po-Shen Loh。你也可以参加完整限时模拟考试、查看全部 2022 AMC 12B 解答，或核对答案。
所有题目均经美国数学协会（MAA）官方合法授权使用。
25.
四个正六边形围绕一个边长为 的正方形，每个六边形都与该正方形共用一条边，如下图所示。所得外侧 边非凸多边形的面积可写成 ，其中 、 和 都是整数，且 不被任何质数的平方整除。求 。
Four regular hexagons surround a square with a side length each one sharing an edge with the square, as shown in the figure below. The area of the resulting -sided outer nonconvex polygon can be written as where and are integers and is not divisible by the square of any prime. What is
小提示：
每个六边形与正方形共用一条边并延伸到对侧；把正方形中心设为原点建立坐标。
Each hexagon shares an edge with the square and reaches across to the far side; set up coordinates with the square centered at the origin
大提示：
外边界有距离中心 的平边，以及相邻六边形边相交形成的凹点 。
The outer boundary has flat edges at distance and reflex notches where adjacent hexagons’ edges cross, at
解答：
将正方形中心置于原点，顶点为 。每个六边形与正方形共用一条边，并向正方形的对侧延伸。例如底边上的六边形，其远端上边从 到 。
外边界是一个 边形，其平边到中心的距离为 ，凸顶点包括 ，而相邻六边形的斜边在 及其对称点相交，形成四个凹入的顶点。
对这 个顶点使用鞋带公式，面积为 ，所以 ，，，从而 。
所以正确答案是 B。
Center the square at the origin with vertices Each hexagon shares one edge with the square and extends across to the opposite side; the hexagon on the bottom edge, for instance, has its far (top) edge from to
The outer boundary is a -gon with flat edges at distance from the center, convex vertices such as and four reflex notches where adjacent hexagons’ slanted edges meet, at and its symmetric images.
Applying the shoelace formula to these vertices gives area so and
Thus, the correct answer is B.
其他年份的第 25 题
- 1950 AMC 12
- 1951 AMC 12
- 1952 AMC 12
- 1953 AMC 12
- 1954 AMC 12
- 1955 AMC 12
- 1956 AMC 12
- 1957 AMC 12
- 1958 AMC 12
- 1959 AMC 12
- 1960 AMC 12
- 1961 AMC 12
- 1962 AMC 12
- 1963 AMC 12
- 1964 AMC 12
- 1965 AMC 12
- 1966 AMC 12
- 1967 AMC 12
- 1968 AMC 12
- 1969 AMC 12
- 1970 AMC 12
- 1971 AMC 12
- 1972 AMC 12
- 1973 AMC 12
- 1974 AMC 12
- 1975 AMC 12
- 1976 AMC 12
- 1977 AMC 12
- 1978 AMC 12
- 1979 AMC 12
- 1980 AMC 12
- 1981 AMC 12
- 1982 AMC 12
- 1983 AMC 12
- 1984 AMC 12
- 1985 AMC 12
- 1986 AMC 12
- 1987 AMC 12
- 1988 AMC 12
- 1989 AMC 12
- 1990 AMC 12
- 1991 AMC 12
- 1992 AMC 12
- 1993 AMC 12
- 1994 AMC 12
- 1995 AMC 12
- 1996 AMC 12
- 1997 AMC 12
- 1998 AMC 12
- 1999 AMC 12
- 2000 AMC 12
- 2001 AMC 12
- 2002 AMC 12A
- 2002 AMC 12B
- 2003 AMC 12A
- 2003 AMC 12B
- 2004 AMC 12A
- 2004 AMC 12B
- 2005 AMC 12A
- 2005 AMC 12B
- 2006 AMC 12A
- 2006 AMC 12B
- 2007 AMC 12A
- 2007 AMC 12B
- 2008 AMC 12A
- 2008 AMC 12B
- 2009 AMC 12A
- 2009 AMC 12B
- 2010 AMC 12A
- 2010 AMC 12B
- 2011 AMC 12A
- 2011 AMC 12B
- 2012 AMC 12A
- 2012 AMC 12B
- 2013 AMC 12A
- 2013 AMC 12B
- 2014 AMC 12A
- 2014 AMC 12B
- 2015 AMC 12A
- 2015 AMC 12B
- 2016 AMC 12A
- 2016 AMC 12B
- 2017 AMC 12A
- 2017 AMC 12B
- 2018 AMC 12A
- 2018 AMC 12B
- 2019 AMC 12A
- 2019 AMC 12B
- 2020 AMC 12A
- 2020 AMC 12B
- 2021 AMC 12A Spring
- 2021 AMC 12B Spring
- 2021 AMC 12A Fall
- 2021 AMC 12B Fall
- 2022 AMC 12A
- 2023 AMC 12A
- 2023 AMC 12B
- 2024 AMC 12A
- 2024 AMC 12B
- 2025 AMC 12A
- 2025 AMC 12B