, 2014

Using the given definition, $8 \odot 7=\sqrt{8+4(7)}=\sqrt{36}=6$. Since $16 \odot n=10$, then $\sqrt{16+4n}=10$ or $16+4n=100$ (by squaring both sides) or $4n=84$ and so $n=21$. We check that indeed $16 \odot n=16 \odot 21=\sqrt{16+4(21)}=\sqrt{100}=10$. We first determine the value inside the brackets: $9 \odot 18=\sqrt{9+4(18)}=\sqrt{81}=9$. So then $(9 \odot 18)\odot 10=9 \odot 10=\sqrt{9+4(10)}=\sqrt{49}=7$. Using the definition, $k \odot k=\sqrt{k+4k}=\sqrt{5k}$. So we are asked to solve the equation $\sqrt{5k}=k$. Squaring both sides we get, $5k=k^2$ and so $k^2-5k=0$ or $k(k-5)=0$, and so $k=0$ or $k=5$. Checking $k=0$, we obtain $k \odot k=0 \odot 0=\sqrt{0+4(0)}=\sqrt{0}=0=k$, as required. Checking $k=5$, we obtain $k \odot k=5 \odot 5=\sqrt{5+4(5)}=\sqrt{25}=5=k$, as required. Thus, the only possible solutions are $k=0$ and $k=5$.