11. For how many integer values of $k$ is $\sqrt{200-\sqrt{k}}$ also an integer?
A 11
B 13
C 15
D 17
E 20
Solution
C
The number $\sqrt{200-\sqrt{k}}$ is an integer if, and only if, the number $200-\sqrt{k}$ is a square.
Now $0 \leq 200-\sqrt{k} \leq 200$. There are 15 squares in this range, namely $n^{2}$ for integer values of $n$ with $0 \leq n \leq 14$.
We have that $200-\sqrt{k}=n^{2}$, if, and only if $k=\left(200-n^{2}\right)^{2}$.
Hence there are 15 integer values of $k$ for which $\sqrt{200-\sqrt{k}}$ is an integer, namely, $k=\left(200-n^{2}\right)^{2}$ for $0 \leq n \leq 14$.