Prove that if $n$ is a square-free positive integer, there are no coprime positive integers $x$ and $y$ such that $(x + y)^3$ divides $x^n+y^n$

Let $n\geq 4$ be an integer. Find the maximum possible area of an $n-gon$ inscribed in a unit cicle and having two perpendicular diagonals.

Let $M_1, M_2, . . ., M_{11}$ be $5-$element sets such that $M_i \cap M_j \neq {\O}$ for all $i, j \in \{1, . . ., 11\}$. Determine the minimum possible value of the greatest number of the given sets that have nonempty intersection.

Consider a tetrahedron$ A_1A_2A_3A_4$. A point $N$ is said to be a Servais point if its projections on the six edges of the tetrahedron lie in a plane $\alpha(N)$ (called Servais plane). Prove that if all the six points $Nij$ symmetric to a point $M$ with respect to the midpoints $Bij$ of the edges $A_iA_j$ are Servais points, then $M$ is contained in all Servais planes $\alpha(Nij )$