Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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# Problem C. 1046. (October 2010)

C. 1046. Let (n) denote the measure of the interior angles of a regular n-sided polygon. What is n if (n+3)-(n)=(n)-(n-2)?

(5 pont)

Deadline expired on November 10, 2010.

Sorry, the solution is available only in Hungarian. Google translation

Megoldás. $\displaystyle \alpha(n)= (n-2)\cdot \frac{180^\circ}{n}$, ezért a feltételt í­gy írhatjuk fel ($\displaystyle n\ge 3$):

$\displaystyle (n+1)\cdot \frac{180^\circ}{n+3}-(n-2)\cdot \frac{180^\circ}{n}=(n-2)\cdot \frac{180^\circ}{n}-(n-4)\cdot \frac{180^\circ}{n-2}.$

$\displaystyle 180^\circ$-kal való egyszerűsí­tés és rendezés után $\displaystyle \displaystyle{\frac{n+1}{n+3}+\frac{n-4}{n-2}=\frac{2(n-2)}{n}}$, majd $\displaystyle n(n^2-n-2+n^2-n-12)=2(n-2)(n^2+n-6)$, amiből $\displaystyle -14n=-16n+24$, ahonnan $\displaystyle n=12$.

### Statistics:

 327 students sent a solution. 5 points: 278 students. 4 points: 11 students. 3 points: 17 students. 2 points: 4 students. 1 point: 2 students. 0 point: 8 students. Unfair, not evaluated: 7 solutions.

Problems in Mathematics of KöMaL, October 2010