# 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 byProblem 20 Followed byProblem 22 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30
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