AMC8 2026

AMC8 2026 · Q2

AMC8 2026 · Q2. It mainly tests Arithmetic misc.

In the array shown below, three 3s are surrounded by 2s, which are in turn surrounded by a border of 1s. What is the sum of the numbers in the array? \[\begin{array}{ccccccc} 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 2 & 2 & 2 & 2 & 2 & 1 \\ 1 & 2 & 3 & 3 & 3 & 2 & 1 \\ 1 & 2 & 2 & 2 & 2 & 2 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 \end{array}\]

在下图所示的数组中，三个3被2包围，2又被一圈1包围。数组中所有数字的和是多少？ \[\begin{array}{ccccccc} 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 2 & 2 & 2 & 2 & 2 & 1 \\ 1 & 2 & 3 & 3 & 3 & 2 & 1 \\ 1 & 2 & 2 & 2 & 2 & 2 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 \end{array}\]

(A) 49 49

(B) 51 51

(D) 55 55

(E) 57 57

Answer

Correct choice: (C)

正确答案：（C）

Solution

Notice that if we label the rows $A_1$, $A_2$, $A_3$, ..., $A_5$ top down, then, if we denote $S(A_n)$ to be the sum of the $A_n$ row, $A_1 = A_5$, $A_2 = A_4$, and $A_3 = A_3$ through symmetry. Thus the total sum is $2A_1 + 2A_2 + A_3$. Then, it is obvious that the sum of $A_1 = 7$, the sum of $A_2 = 2 + 2(5) = 12$, and the sum of $A_3 = 2(3) + 3(3) = 5(3) = 15$. The answer is then $2 \cdot 7 + 2 \cdot 12 + 15 = 14 + 15 + 24 = 29 + 24 = \boxed{\textbf{(C)}\ 53}$.

注意，如果我们从上到下将行命名为 $A_1, A_2, A_3, \ldots, A_5$，记 $S(A_n)$ 为第 $A_n$ 行元素的和，根据对称性有$A_1 = A_5$，$A_2 = A_4$，且 $A_3 = A_3$。因此，总和为 $2A_1 + 2A_2 + A_3$。显然，第 $A_1$ 行的和为 $7$，第 $A_2$ 行的和为 $2 + 2 \times 5 = 12$，第 $A_3$ 行的和为 $2 \times 3 + 3 \times 3 = 5 \times 3 = 15$。答案即为 $2 \times 7 + 2 \times 12 + 15 = 14 + 24 + 15 = 53$，即 $\boxed{\textbf{(C)}\ 53}$。

Topics

AR.015 Arithmetic misc