Problem B. 4965. (May 2018)

B. 4965. For a vector \(\displaystyle \mathbf x\ne \mathbf 0\), let \(\displaystyle \mathbf e_{\mathbf x}=\frac{\mathbf x} {|\mathbf x|}\). A given plane \(\displaystyle \mathcal{S}\) is parallel, but not identical to the plane of a given (non-degenerate) triangle \(\displaystyle ABC\). Show that there exists a unique \(\displaystyle P\in \mathcal{S}\), such that the vector \(\displaystyle \mathbf e_{\overrightarrow {PA}}+\mathbf e_{\overrightarrow {PB}}+\mathbf e_{\overrightarrow {PC}}\) is perpendicular to \(\displaystyle \mathcal{S}\).

(6 pont)

Deadline expired on June 11, 2018.

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

.

Statistics:

7 students sent a solution.
6 points:	Gáspár Attila, Schrettner Jakab.
5 points:	Dobák Dániel.
3 points:	2 students.
1 point:	1 student.
0 point:	1 student.

Problems in Mathematics of KöMaL, May 2018