AMC12 2019 B

AMC12 2019 B · Q7

AMC12 2019 B · Q7. It mainly tests Linear inequalities, Averages (mean).

What is the sum of all real numbers $x$ for which the median of the numbers 4, 6, 8, 17, and $x$ is equal to the mean of those five numbers?

对于所有实数 $x$，使得数字 4、6、8、17 和 $x$ 的中位数等于这五个数的平均数，它们的和是多少？

(A) -5 -5

(B) 0 0

(D) \frac{15}{4} \frac{15}{4}

(E) \frac{35}{4} \frac{35}{4}

Answer

Correct choice: (A)

正确答案：（A）

Solution

Answer (A): The mean of the given numbers is $\dfrac{4+6+8+17+x}{5}=\dfrac{x+35}{5}=\dfrac{x}{5}+7.$ The median depends on the value of $x$. If $x<6$, then the median is $6$. If the mean and median are equal, then $\dfrac{x}{5}+7=6$, which is equivalent to $x=-5$. If $6\le x\le 8$, then the median is $x$. If the mean and median are equal, then $\dfrac{x}{5}+7=x$, which is equivalent to $x=\dfrac{35}{4}$. But this is outside of the given range. If $x>8$, then the median is $8$. If the mean and median are equal then $\dfrac{x}{5}+7=8$, which is equivalent to $x=5$. Again this is outside of the given range. Therefore the only value of $x$ for which the mean equals the median is $-5$, so the requested sum is also $-5$.

答案（A）：给定数的平均数为 $\dfrac{4+6+8+17+x}{5}=\dfrac{x+35}{5}=\dfrac{x}{5}+7.$ 中位数取决于 $x$ 的取值。若 $x<6$，则中位数为 $6$。若平均数与中位数相等，则 $\dfrac{x}{5}+7=6$，等价于 $x=-5$。若 $6\le x\le 8$，则中位数为 $x$。若平均数与中位数相等，则 $\dfrac{x}{5}+7=x$，等价于 $x=\dfrac{35}{4}$。但这不在给定范围内。若 $x>8$，则中位数为 $8$。若平均数与中位数相等，则 $\dfrac{x}{5}+7=8$，等价于 $x=5$。同样不在给定范围内。因此，使平均数等于中位数的唯一 $x$ 值为 $-5$，所以所求的和也为 $-5$。

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