2012 AMC 10A 第 8 题

Question

三个非负整数两两相加得到的和分别为

12

，

17

和

19

。这三个数中的中间那个数是多少？

The sums of three whole numbers taken in pairs are

12,

17,

and

19.

What is the middle number?

4

5

6

7

8

知识点：方程组

把三个两两和相加。

Add the three pair sums

三个两两和的总和是三个数总和的两倍。

The total of the three pair sums is twice the sum of the three numbers

设三个数为

a, b, c

，且

a \lt b \lt c

。由于三个和都不同，所以这三个数也互不相等。

题意给出

a + b = 12, a + c = 17\text{，}

以及

b + c = 19\text{。}

把三式相加，得到

2(a + b + c) = 48\text{。}

因此

a + b + c = 24\text{，}

再减去

a + c = 17\text{，}

得到

b = 7

。

所以正确答案是 D。

Let the three numbers be

a, b, c

where

a \lt b \lt c.

None of them are equal, since all three sums are different.

Then

a + b = 12, a + c = 17,

and

b + c = 19.

Adding all three equations together gives us

2(a + b + c) = 48.

Then

a + b + c = 24,

from which we can subtract

a + c = 17,

to get

b = 7.

Thus, D is the correct answer.

其他年份的第 8 题