Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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Problem A. 678. (October 2016)

A. 678. The convex polyhedron \(\displaystyle \mathcal{K}\) has five vertices, \(\displaystyle A\), \(\displaystyle B\), \(\displaystyle C\), \(\displaystyle D\) and \(\displaystyle E\). The line segment \(\displaystyle DE\) intersects the plane of the triangle \(\displaystyle ABC\) in the interior of the triangle. Show that \(\displaystyle \mathcal{K}\) has an inscribed sphere – with being tangent to all six faces – if and only if the inscribed spheres of the terahedra \(\displaystyle ABCD\) and \(\displaystyle ABCE\) are tangent to each other.

(5 pont)

Deadline expired on November 10, 2016.


6 students sent a solution.
5 points:Baran Zsuzsanna, Bukva Balázs, Williams Kada.
4 points:Lajkó Kálmán.
2 points:1 student.
1 point:1 student.

Problems in Mathematics of KöMaL, October 2016