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.

### Statistics:

 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