Difference between revisions of "2006 Cyprus MO/Lyceum/Problem 21"

Latest revision as of 12:30, 26 April 2008

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 $\mathrm{B}$ .

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 ==Problem==
+A convex polygon has <math>n</math> sides and <math>740</math> diagonals. Then <math>n</math> equals
+<math>\mathrm{(A)}\ 30\qquad\mathrm{(B)}\ 40\qquad\mathrm{(C)}\ 50\qquad\mathrm{(D)}\ 60\qquad\mathrm{(E)}\ \text{None of these}</math>
 ==Solution==
-{{solution}}
+The number of diagonals in a polygon is <math>\frac{n(n-3)}{2}</math>. In this case, <math>\frac{n(n-3)}{2}=740</math>, so <math>n(n-3)=1480</math>.
+By solving the [[quadratic equation]], we find <math>n = 40</math>, so the answer is <math>\mathrm{B}</math>.
 ==See also==
-{{CYMO box|year=2006|l=Lyceum|num-b=21|num-a=22}}
+{{CYMO box|year=2006|l=Lyceum|num-b=20|num-a=22}}
+[[Category:Introductory Combinatorics Problems]]