Problem B. 4785. (March 2016)

B. 4785. \(\displaystyle \mathcal{G}\) is a given sphere in the space. For any line \(\displaystyle e\) that has no common point with \(\displaystyle \mathcal{G}\), define the line \(\displaystyle f\) as the conjugate of \(\displaystyle e\) with respect to \(\displaystyle \mathcal{G}\) if \(\displaystyle f\) joins the points of tangency on the two planes tangent to \(\displaystyle G\) passing through \(\displaystyle e\). Show that two lines of the space passing \(\displaystyle \mathcal{G}\) are skew if and only if their conjugates with respect to \(\displaystyle \mathcal{G}\) are skew.

(5 pont)

Deadline expired on April 11, 2016.

Statistics:

26 students sent a solution.
5 points:	Baran Zsuzsanna, Döbröntei Dávid Bence, Fuisz Gábor, Gáspár Attila, Glasznova Maja, Kerekes Anna, Lajkó Kálmán, Matolcsi Dávid, Polgár Márton, Sudár Ákos, Tiszay Ádám, Zólomy Kristóf.
4 points:	Bukva Balázs, Csahók Tímea, Cseh Kristóf, Hansel Soma, Tóth Viktor.
3 points:	2 students.
2 points:	1 student.
1 point:	2 students.
0 point:	4 students.

Problems in Mathematics of KöMaL, March 2016