AMC10 2012 B

AMC10 2012 B · Q8

AMC10 2012 B · Q8. It mainly tests Linear inequalities, Absolute value.

What is the sum of all integer solutions to $1 < (x - 2)^2 < 25$?

求 $1 < (x - 2)^2 < 25$ 的所有整数解的和。

(A) 10 10

(B) 12 12

(D) 19 19

(E) 25 25

Answer

Correct choice: (B)

正确答案：（B）

Solution

Answer (B): If $x-2>0$, then the given inequality is equivalent to $1<x-2<5$, or $3<x<7$. The integer solutions in this case are $4,5,$ and $6$. If $x-2<0$, then the given inequality is equivalent to $-5<x-2<-1$, or $-3<x<1$. The integer solutions in this case are $-2,-1,$ and $0$. The sum of all integer solutions is $12$.

答案（B）：如果 $x-2>0$，则所给不等式等价于 $1<x-2<5$，即 $3<x<7$。此时整数解为 $4,5,6$。如果 $x-2<0$，则所给不等式等价于 $-5<x-2<-1$，即 $-3<x<1$。此时整数解为 $-2,-1,0$。所有整数解的和为 $12$。

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