Problem C. 891. (March 2007)

C. 891. What maximum number of sides may a convex polygon have in which the interior angles form an arithmetic progression wit a common difference of d=1^o?

(5 pont)

Deadline expired on April 16, 2007.

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

Megoldás. Az n oldalú konvex sokszög legkisebb szöge legyen $\alpha$ (a fokok kiírásától innentől kezdve eltekintünk), legnagyobb szöge ekkor $\alpha$ +n-1, belső szögeinek az összege pedig

$\frac{(2\alpha+n-1)n}{2}=(n-2)180.$

Innen

$\alpha=\frac{-n^2+361n-720}{2n}.$

Tudjuk, hogy a sokszög legnagyobb szöge is kisebb 180-nál, tehát

$\alpha$ +n-1<180,

amiből

n²-n-720<0.

Emiatt n legfeljebb 27, ekkor a legkisebb szög $\alpha$ =153 2/3, a legnagyobb szög pedig $\alpha$ +26=189 2/3.

Statistics:

214 students sent a solution.

5 points: 92 students.

4 points: 17 students.

3 points: 18 students.

2 points: 19 students.

1 point: 24 students.

0 point: 31 students.

Unfair, not evaluated: 13 solutionss.

Problems in Mathematics of KöMaL, March 2007

214 students sent a solution.
5 points:	92 students.
4 points:	17 students.
3 points:	18 students.
2 points:	19 students.
1 point:	24 students.
0 point:	31 students.
Unfair, not evaluated:	13 solutionss.

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