Problem K. 650. (February 2020)

K. 650. The side of the small square in the figure is 3 cm, the sides of the large rectangle have integer lengths, one being 2 cm longer than the other. The sides of the rectangle and the square are parallel, their centres coincide. The shaded region was made by extending the sides of the small square in one direction, and connecting the points where the sides of the large rectangle were reached. Is it possible for the area of the shaded region to be an even number (of cm\(\displaystyle {}^2\))?

(6 pont)

Deadline expired on March 10, 2020.

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

Megoldás. Jelölje a téglalap oldalait \(\displaystyle x\) és \(\displaystyle x+2\). A satírozott terület nagysága \(\displaystyle \frac{x(x+2)-9}{2}\), ha ez páros, akkor \(\displaystyle x(x+2)-9\)-nek 4-gyel oszthatónak kell lennie. Mivel \(\displaystyle x\) és \(\displaystyle x+2\) azonos paritású, ezért \(\displaystyle x\) páratlan kell legyen. \(\displaystyle x(x+2)-9=x^2+2x+1-10=(x+1)^2-10\). Ha \(\displaystyle x\) páratlan, akkor \(\displaystyle x+1\) páros, ekkor a négyzete osztható 4-gyel, de ha kivonunk belőle 10-et, akkor már nem osztható 4-gyel. Így nem lehet páros a satírozott terület nagysága.

Statistics:

124 students sent a solution.
6 points:	62 students.
5 points:	9 students.
4 points:	5 students.
3 points:	5 students.
2 points:	8 students.
1 point:	12 students.
0 point:	11 students.
Unfair, not evaluated:	6 solutionss.
Not shown because of missing birth date or parental permission:	6 solutions.

Problems in Mathematics of KöMaL, February 2020