2020 AMC 10A 第 7 题

Question

25

个从

-10

到

14

的整数可以排列成一个

5

乘

5

的方阵，使每一行、每一列以及两条主对角线上的数之和都相同。这个公共和是多少？

The

25

integers from

-10

to

14,

inclusive, can be arranged to form a

5

-by-

5

square in which the sum of the numbers in each row, the sum of the numbers in each column, and the sum of the numbers along each of the main diagonals are all the same. What is the value of this common sum?

2

5

10

25

50

答案：C

知识点：幻方等差数列

难度评级：960

小提示：

五个行和加起来等于所有

25

个整数的总和。

The five row sums together equal the sum of all

25

integers

大提示：

计算从

-10

到

14

的等差数列和。

Compute the arithmetic-sequence sum from

-10

through

14

视频讲解：

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文字解答：

从

-10

到

14

的整数总和为

25\cdot\dfrac{-10+14}{2}=50

。若每行公共和为

S

，则五行总和为

50

，所以

5S=50

，从而

S=10

。正确答案是 C。

The sum of the integers from

-10

to

14

is

25\cdot\dfrac{-10+14}{2}=50

. If every row has common sum

S

, then the five row sums add to

50

, so

5S=50

and

S=10

. Thus, C is the correct answer.