CEMC Euclid, 2013

The expression $\sqrt{113+x}$ is an integer whenever $113+x$ is a perfect square.

To find the smallest positive integer $x$ for which $113+x$ is a perfect square, we find the smallest perfect square larger than 113.

Since $10^2=100$ and $11^2=121$, then this perfect square must be 121.

Therefore, $113+x=121$ or $x=8$.
The average of 3 and 11 is $\frac{1}{2}(3+11)=7$. Thus $a=7$.

Using this with the given information, we see that the average of 7 and $b$ is 11.

Therefore, $\frac{1}{2}(7+b)=11$ or $7+b=22$ and so $b=15$.

(Alternatively, we could note that since 7 is 4 less than 11 (the average), then $b$ must be 4 more than 11, so $b=11+4=15$.)
Let $c$ be Charlie’s age in years and $b$ be Bella’s age in years.

From the first sentence, $c=30+b$.

From the second sentence, $c=6b$.

Combining these, we obtain $6b=30+b$ or $5b=30$, and so $b=6$.

Since $c=30+b$, then $c=36$, and so Charlie’s age is 36.