Problem K. 691. (March 2021)

K. 691. \(\displaystyle ABCDEFGH\) is a regular octagon and its sides are 2 units long. Squares \(\displaystyle BCIM\) and \(\displaystyle FGKL\) are drawn on sides \(\displaystyle BC\) and \(\displaystyle GF\), on the inside. What is the area of the rectangle bounded by lines \(\displaystyle AH\), \(\displaystyle KL\), \(\displaystyle ED\) and \(\displaystyle IM\)?

(6 pont)

Deadline expired on April 12, 2021.

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

Megoldás: Állítsunk merőlegest \(\displaystyle D\)-ből \(\displaystyle CI\)-re. \(\displaystyle DCD'\angle 45^{\circ}\) . Így a Pitagorasz-tétel miatt \(\displaystyle DD' = D'C = \sqrt 2 \). A \(\displaystyle PQRS\) téglalap \(\displaystyle QR\) oldalának hossza \(\displaystyle 2\sqrt 2 +2\).

\(\displaystyle ID' = 2-\sqrt 2 \). \(\displaystyle SR = 2-2RD = 2-2(2-\sqrt 2 ) = 2\sqrt 2-2\).

A \(\displaystyle PQRS\) téglalap területe: \(\displaystyle (2\sqrt 2 +2)(2\sqrt 2 -2)=8-4=4\).

Statistics:

83 students sent a solution.
6 points:	Bacsek Emma Borbála, Baksa Anna, Barta Veronika, Biró Anna, Biró Róza, Boros Helga Dóra, Bottyán Márton Péter, Buday Noémi, Csóka Péter, Dancsák Dénes, Divényi Bernadett, Dukát Levente, Érdi Ferenc Vince, Ferencsik Zsombor, Fodor Gergely, Fórizs Borbála, Fórizs Emma, Gulyás Janka, Gyönki Dominik, Heim Flóra, Hochenburger Zoárd, Horváth 221 Zsóka, Jenei Ákos Zoltán, Kéki Edit, Klusóczki-Bogdándi Alma, Kornya Gergely Csaba, Kuba Nikoletta, Kurucz Kitti, Laczó Dávid, Laskai Botond, Lehoczky Örs Hunor, Lupkovics Lilla, Mayer Krisztián, Mészáros Ádám, Molnár Kristóf, Richlik Márton, Sándor Eszter, Schäffer Donát, Sebestyén József Tas, Simon Géza, Solymosi Csongor, Susán Henrik, Szabó Csenge, Szeibert Dominik, Yang Yu jie.
5 points:	23 students.
4 points:	1 student.
3 points:	3 students.
2 points:	1 student.
1 point:	3 students.
0 point:	4 students.
Not shown because of missing birth date or parental permission:	3 solutions.

Problems in Mathematics of KöMaL, March 2021