Problem A. 619. (May 2014)

A. 619. There are given four rays, \(\displaystyle a\), \(\displaystyle b\), \(\displaystyle c\) and \(\displaystyle d\) in space, starting from the same point, laying in a plane \(\displaystyle \varPi\). For an arbitrary acute angle \(\displaystyle \varphi\), rotate \(\displaystyle \varPi\) by angle \(\displaystyle \varphi\) in positive direction around each of the four rays; denote the rotated planes by \(\displaystyle A_\varphi\), \(\displaystyle B_\varphi\), \(\displaystyle \varGamma_\varphi\) and \(\displaystyle \varDelta_\varphi\), respectively. Let \(\displaystyle \varSigma_\varphi\) be the plane through the intersection line of \(\displaystyle A_\varphi\) and \(\displaystyle B_\varphi\), and the intersection line of \(\displaystyle \varGamma_\varphi\) and \(\displaystyle \varDelta_\varphi\). Show that the planes \(\displaystyle \varSigma_\varphi\) share a common line.

(5 pont)

Deadline expired on June 10, 2014.

Statistics:

4 students sent a solution.
5 points:	Ágoston Péter, Fehér Zsombor, Williams Kada.
3 points:	1 student.

Problems in Mathematics of KöMaL, May 2014