We are given a triangle $ABC$ in a plane $P$. To any line $D$, not parallel to any side of the triangle, we associate the barycenter $G_D$ of the set of intersection points of $D$ with the sides of $\triangle ABC$. The object of this problem is determining the set $\mathfrak F$ of points $G_D$ when $D$ varies. (a) If $D$ goes over all lines parallel to a given line $\delta$, prove that $G_D$ describes a line $\Delta_\delta$. (b) Assume $\triangle ABC$ is equilateral. Prove that all lines $\Delta_\delta$ are tangent to the same circle as $\delta$ varies, and describe the set $\mathfrak F$. (c) If $ABC$ is an arbitrary triangle, prove that one can find a plane $P$ and an equilateral triangle $A'B'C'$ whose orthogonal projection onto $P$ is $\triangle ABC$, and describe the set $\mathfrak F$ in the general case.

Study the convergence of a sequence defined by $u_0\ge0$ and $u_{n+1}=\sqrt{u_n}+\frac1{n+1}$ for all $n\in\mathbb N_0$.

Consider three circles in the plane $\Gamma_1,\Gamma_2,\Gamma_3$ of radii $R$ passing through a point $O$, and denote by $\mathfrak D$ the set of points of the plane which belong to at least two of these circles. Find the position of $\Gamma_1,\Gamma_2,\Gamma_3$ for which the area of $\mathfrak D$ is the minimum possible. Justify your answer.

Suppose $A_1,A_2,A_3,B_1,B_2,B_3$ are points in the plane such that for each $i,j\in\{1,2,3\}$ it holds that $A_iB_j=i+j$. What can be said about these six points?

Let $f$ be a bijection from $\mathbb N$ to itself. Prove that one can always find three natural number $a,b,c$ such that $a<b<c$ and $f(a)+f(c)=2f(b)$.