2006 Cyprus MO/Lyceum/Problem 21

Problem

A convex polygon has $n$ sides and $740$ diagonals. Then $n$ equals

$\mathrm{(A)}\ 30\qquad\mathrm{(B)}\ 40\qquad\mathrm{(C)}\ 50\qquad\mathrm{(D)}\ 60\qquad\mathrm{(E)}\ \text{None of these}$

The number of diagonals in a polygon is $\frac{n(n-3)}{2}$ . In this case, $\frac{n(n-3)}{2}=740$ , so $n(n-3)=1480$ .

By solving the quadratic equation, we find $n = 40$ , so the answer is $\mathrm{B}$ .