AMC10 2019 B

Answer (A): The mean of the given numbers is $\frac{4+6+8+17+x}{5}=\frac{x+35}{5}=\frac{x}{5}+7.$ The median depends on the value of $x$. If $x<6$, then the median is $6$. If the mean and median are equal, then $\frac{x}{5}+7=6$, 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 $\frac{x}{5}+7=x$, which is equivalent to $x=\frac{35}{4}$. But this is outside of the given range. If $x>8$, then the median is $8$. If the mean and median are equal then $\frac{x}{5}+7=8$, 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 $-5$. $\frac{(n!)^{n+1}}{}$ is an integer because this expression counts the number of ways to separate $n^2$ objects into $n$ groups of size $n$ without regard to the ordering of the groups (which accounts for the extra factor of $n!$ in the denominator).