2006 Cyprus MO/Lyceum/Problem 21

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Problem

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

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

Solution

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 $\mathrm{B}$.

See also

2006 Cyprus MO, Lyceum (Problems)
Preceded by
Problem 20
Followed by
Problem 22
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