What is the smallest natural number $k$ for which the following statement is true? "If a tetrahedron has $k$ edges with an angle of $60^{\circ}$, then the tetrahedron must be regular."
A tetrahedron is regular if every face is an equilateral triangle, and the dihedral angles are also equal. We will show that the smallest such $k$ number is seven. If $k=6$, it is possible for the tetrahedron to have two faces that are equilateral triangles, and these two faces, for example, enclose a $90^{\circ}$ angle. In this case, the tetrahedron is not regular.

If $k=7$, then by the pigeonhole principle, there are at least two faces with at least two angles of $60^{\circ}$, and in addition, there is at least one face with at least one angle of $60^{\circ}$. This latter face shares two sides with the previously mentioned congruent equilateral triangular faces, so it is isosceles, and since it has one $60^{\circ}$ angle, it will be equilateral. The fourth face is formed from the edges of the three previously described congruent equilateral triangles, so this face and the tetrahedron itself are - regular.

Breuer János (Budapest, Apáczai Csere J. Gymn., 10th grade) Csóka Endre (Debrecen, Fazekas M. Gymn., 8th grade) Reviczky Ádám János (Budapest, Szent István Gymn., 10th grade)