1997 AMC 8 第 17 题
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所有题目均经美国数学协会（MAA）官方合法授权使用。
17.
一个立方体有八个顶点和十二条棱。连接两个不由一条棱相连的顶点的线段，如 ，称为对角线。线段 也是对角线。一个立方体有多少条对角线？
A cube has eight vertices (corners) and twelve edges. A segment, such as which joins two vertices not joined by an edge is called a diagonal. Segment is also a diagonal. How many diagonals does a cube have?
小提示：
每个面有两条面对角线，还有连接相对顶点的空间对角线
Each face has two diagonals, and opposite vertices give space diagonals
大提示：
分别数面对角线和空间对角线
Count face diagonals and space diagonals separately
解答：
每个面有两条对角线，连接这个面上两对相对顶点。
另外，对于每个顶点，都有一个与它在立方体内部相对的顶点。
这样空间对角线共有 条。
总对角线数为
所以正确答案是 E。
Each face has two diagonals connecting each of the two pairs of opposite vertices.
Also, for each vertex, there is one corresponding vertex that lies opposite it on the cube.
There are then interior space diagonals in the cube.
The total number of diagonals is then
Thus, E is the correct answer.
其他年份的第 17 题
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