2019 AMC 10B Problem 13

The mean is easy to write down, but the median depends on where $x$ lands in the list. So split into cases.

The mean is
$$\frac{4 + 6 + 8 + 17 + x}{5} = \frac{35 + x}{5}.$$

The median is the middle (third) number once the five numbers are in order.

1. $x$ is $6$ or less. For example, with $x = 1$ the list is $1, 4, 6, 8, 17$. The middle number is $6$. Set the mean equal to $6$: $35 + x = 30$, so $x = -5$. Is $-5$ at most $6$? Yes. Keep it.
2. $x$ is between $6$ and $8$. For example, with $x = 7$ the list is $4, 6, 7, 8, 17$. The middle number is $x$ itself. Set $\frac{35 + x}{5} = x$: $35 + x = 5x$, so $x = \frac{35}{4} = 8.75$. But $8.75$ is not between $6$ and $8$. Throw it out.
3. $x$ is $8$ or more. For example, with $x = 10$ the list is $4, 6, 8, 10, 17$. The middle number is $8$. Set the mean equal to $8$: $35 + x = 40$, so $x = 5$. But $5$ is not $8$ or more. Throw it out.

Check $x = -5$: the list $-5, 4, 6, 8, 17$ has median $6$ and mean $\frac{30}{5} = 6$. They match.

Only one value works, so the sum of all such $x$ is $-5$.

The answer is $\boxed{\textbf{(A)}\ -5}$.

When the rule for something (here, which number is the median) changes depending on $x$, you have to split into cases. Each case gives an equation, but its answer only counts if it actually lands in that case. Two of the three answers here fail that check, and the wrong choices $5$ and $\frac{35}{4}$ are exactly those failed answers.

The mean is $\frac{4 + 6 + 8 + 17 + x}{5} = \frac{35 + x}{5}$. The median depends on where $x$ sits relative to $6$ and $8$.

Case $x \le 6$: the ordered list is $x, 4, 6, 8, 17$ or $4, x, 6, 8, 17$, so the median is $6$. Then $\frac{35 + x}{5} = 6$ gives $x = -5$, which is indeed $\le 6$. Valid.

Case $6 < x < 8$: the median is $x$ itself. Then $\frac{35 + x}{5} = x$ gives $35 = 4x$, $x = \frac{35}{4} = 8.75$, which is not between $6$ and $8$. Rejected.

Case $x \ge 8$: the median is $8$. Then $\frac{35 + x}{5} = 8$ gives $x = 5$, which is not $\ge 8$. Rejected.

The only solution is $x = -5$, so the sum of all solutions is $-5$.

The answer is $\boxed{\textbf{(A)}\ -5}$.