5) A regular polygon has $n$ sides and $4 n$ diagonals. What is the value of $n$?
(A) 8
(B) 9
(C) 10
(D) 11
(E) 12.
5) The answer is $(\mathbf{D})$.

Since in a convex polygon with $n$ sides, from each vertex there are $n-3$ diagonals, the total number of diagonals is $\frac{n(n-3)}{2}$. By setting $\frac{n(n-3)}{2}=4 n$, we get $n-3=8$, hence $n=11$.