AMC10 2023 A

AMC10 2023 A · Q6

AMC10 2023 A · Q6. It mainly tests 3D geometry (volume), 3D geometry (surface area).

An integer is assigned to each vertex of a cube. The value of an edge is defined to be the sum of the values of the two vertices it touches, and the value of a face is defined to be the sum of the values of the four edges surrounding it. The value of the cube is defined as the sum of the values of its six faces. Suppose the sum of the integers assigned to the vertices is $21$. What is the value of the cube?

一个立方体的每个顶点都被赋上一个整数。边的值为它连接的两个顶点的值之和，面的值为围绕它的四个边的值之和。立方体的值为其六个面的值之和。假设赋给顶点的整数之和为$21$。立方体的值为多少？

(A) 42 42

(B) 63 63

(D) 126 126

(E) 252 252

Answer

Correct choice: (D)

正确答案：（D）

Solution

Each of the vertices is counted $3$ times because each vertex is shared by three different edges. Each of the edges is counted $2$ times because each edge is shared by two different faces. Since the sum of the integers assigned to all vertices is $21$, the final answer is $21\times3\times2=\boxed{\textbf{(D) } 126}$

每个顶点被计数$3$次，因为每个顶点被三个不同的边共享。每个边被计数$2$次，因为每个边被两个不同的面共享。由于所有顶点赋值的和为$21$，最终答案为$21\times3\times2=\boxed{\textbf{(D) } 126}$

Topics