AMC8 1999

AMC8 1999 · Q11

AMC8 1999 · Q11. It mainly tests Linear equations, Averages (mean).

Each of the five numbers 1,4,7,10, and 13 is placed in one of the five squares so that the sum of the three numbers in the horizontal row equals the sum of the three numbers in the vertical column. The largest possible value for the horizontal or vertical sum is

将五个数字1、4、7、10和13各放入五个方格中，使得水平行三个数字的和等于垂直列三个数字的和。水平或垂直和的最大可能值为

(A) 20 20

(B) 21 21

(D) 24 24

(E) 30 30

Answer

Correct choice: (D)

正确答案：（D）

Solution

The largest sum occurs when $13$ is placed in the center. This sum is $13 + 10 + 1 = 13 + 7 + 4 = \boxed{\text{(D)}\ 24}$. Note: Two other common sums, $18$ and $21$, are also possible. Since the horizontal sum equals the vertical sum, twice this sum will be the sum of the five numbers plus the number in the center. When the center number is $13$, the sum is the largest, \[[10 + 4 + 1 + 7 + 2(13)]=2S\\ 48=2S\\ S=\boxed{\text{(D)}\ 24}\] The other four numbers are divided into two pairs with equal sums.

最大和出现在13放在中心时。此和为$13 + 10 + 1 = 13 + 7 + 4 = \boxed{\text{(D)}\ 24}$。注意：其他常见和18和21也是可能的。由于水平和等于垂直和，两倍此和将是五个数字之和加上中心的数字。当中心数字为$13$时，和最大，\[[10 + 4 + 1 + 7 + 2(13)]=2S\\ 48=2S\\ S=\boxed{\text{(D)}\ 24}\] 其他四个数字分成两对，和相等。

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