Me and a couple of friends have recently started solving Math problems in our breaks at school. We came along the Georg Mohr tournament, which is Denmark's first qualifications tournament for the IMO.
Thinking about joining this year, we took a look at the problems of the previous years, and came about the following problem:
The plane shape that appears, when you cut a cube with a plane, can never be:
The 5 possible answers were:
A pentagon, a line segment, a rectangle (that is not quadratic), a right angled triangle or an equilateral triangle.
We took a shot at it, and concluded that:
- A pentagon is possible (Cutting diagonally on a face away from the center of said face).
- A line segment is possible (Cutting as close to the bottom of a face as possible)
- A rectangle is possible (Cutting a straight line through the center of a face leaves a triangle on both sides)
- A right angled triangle is possible (Cut diagonally through the center of a face)
So we answered equilateral triangle.
After finishing the rest of the problems, we looked at the answer sheet, stating that the correct answer was a right angled triangle.
We went to Google to try to figure out why, and we found out how to produce the equilateral triangle, but are now left with the question of how the right angled triangle is not possible?
We think that we're interpreting the question wrong, so for any danes that wanna check our translation and/or interpretation, the original question is written as such:
Den plane figur, der fremkommer, når man skærer en terning med en plan, kan aldrig være:
So if I didn't make it clear, the question is why is it impossible to produce a right angled triangle in this way?
Thanks in advance!