(CZS) (a) Find all natural numbers $n$ such that the number $2^n-1$ is divisible by $7$.
(b) Prove that for all natural numbers $n$ the number $2^n+1$ is not divisible by $7$.
(YUG) The incircle is inscribed in a triangle $ABC$ with sides $a,b,c$. Three tangents to the incircle are drawn, each of which is parallel to one side of the triangle $ABC$. These tangents form three smaller triangles (internal to $\triangle ABC$) with the sides of $\triangle ABC$. In each of these triangles an incircle is inscribed. Determine the sum of areas of all four incircles.
(HUN) Each of $17$ students talked with every other student. They all talked about three different topics. Each pair of students talked about one topic. Prove that there are three students that talked about the same topic among themselves.
(ROM) Five points are given in the plane. Among the lines that connect these five points, no two coincide and no two are parallel or perpendicular. Through each point we construct an altitude to each of the other lines. What is the maximal number of intersection points of these altitudes (excluding the initial five points)?
(POL) Given a tetrahedron $ABCD$, let $D_1$ be the centroid of the triangle $ABC$ and let $A_1,B_1,C_1$ be the intersection points of the lines parallel to $DD_1$ and passing through the points $A,B,C$ with the opposite faces of the tetrahedron. Prove that the volume of the tetrahedron $ABCD$ is one-third the volume of the tetrahedron $A_1B_1C_1D_1$. Does the result remain true if the point $D_1$ is replaced with any point inside the triangle $ABC$?