1984 AMC 12 第 16 题

Question

函数

f(x)

满足

f(2+x)=f(2-x)

，其中

x

为任意实数。若方程

f(x)=0

恰有四个互不相同的实根，则这些根的和为

The function

f(x)

satisfies

f(2+x)=f(2-x)

for all real numbers

x.

If the equation

f(x)=0

has exactly four distinct real roots, then the sum of these roots is

0

2

4

6

8

该等式说明函数图像关于

x=2

对称

The equation makes the graph symmetric about

x=2

将每个根

2+r

与其对称根

2-r

配对

Pair each root

2+r

with its reflected root

2-r

该关系说明，每个根

2+r

都与根

2-r

配对。因此，四个互不相同的根组成两对，每对的和为

4

。四个根之和为

2\cdot4=8

。

所以正确答案是 E。

The relation shows that every root

2+r

is paired with

2-r.

Four distinct roots therefore form two such pairs, and each pair has sum

4.

The sum of all four roots is

2\cdot4=8.

Therefore, the correct answer is E.

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