A convex polyhedron has at least three pentagonal faces. What is the minimum number of faces the polyhedron can have?
Solution. Let the number of faces of the polyhedron be $\ell$, the number of edges be $e$, and the number of vertices be $c$. According to Euler's theorem for polyhedra, we know that

$$
\ell + c = e + 2
$$

Every vertex is connected by at least 3 edges, and if we count the edges along the vertices, each edge will be counted twice, so $2e \geq 3c$. Express the number of faces from Euler's theorem and use the previous estimate:

$$
\ell = 2 + e - c \geq 2 + e - \frac{2}{3}e = 2 + \frac{1}{3}e
$$

The polyhedron has at least 3 pentagonal faces according to the problem. This means at least 12 edges, because any two pentagonal faces can share one edge. From this, it follows that $\ell \geq 6$. We also see that a polyhedron with 6 faces can have at most 12 edges. A polyhedron with 6 faces and at most 12 edges can have at most 8 vertices according to Euler's theorem. The 3 pentagonal faces have more vertices (any two pentagonal faces can share 2 vertices, so they have at least $15 - 3 \cdot 2 = 9$ vertices), so a polyhedron with 6 faces cannot have 3 pentagonal faces. Therefore, the polyhedron must have at least 7 faces.

![](https://cdn.mathpix.com/cropped/2024_05_02_05d0c183773354ded66bg-1.jpg?height=303&width=287&top_left_y=827&top_left_x=908)

A polyhedron with exactly 7 faces can be constructed, for example, by "cutting off" one vertex of a rectangular prism with a plane.

This solid has three rectangular, three pentagonal, and one triangular face.

Remarks. 1. Contestants provided several constructions in their solutions. In addition to the one mentioned above, the following was also characteristic: "Cut off" three corners of a tetrahedron, resulting in 1 hexagonal, 3 pentagonal, and 3 triangular faces.

2. The solutions at the link http://www.komal.hu/verseny/feladat.cgi?a=feladat\&f=B4794\&l=hu differ from these. Solutions to the problems are generally available on our website a few days after the deadline.