Školjka

Pronađi sve prirodne brojeve $n \geqslant 3$ za koje postoje realni brojevi $a_1, a_2, \dots a_{n + 2}$ takvi da je $a_{n + 1} = a_1$, $a_{n + 2} = a_2$ i $$a_ia_{i + 1} + 1 = a_{i + 2},$$za sve $i = 1, 2, \dots, n$.

Let $ABC$ be an acute-angled triangle with $AB > AC$ and circumcircle $\Gamma$. Let $M$ be the midpoint of the shorter arc $BC$ of $\Gamma$, and let $D$ be the intersection of the rays $AC$ and $BM$. Let $E \neq C$ be the intersection of the internal bisector of the angle $ACB$ and the circumcircle of the triangle $BDC$. Let us assume that $E$ is inside the triangle $ABC$ and there is an intersection $N$ of the line $DE$ and the circle $\Gamma$ such that $E$ is the midpoint of the segment $DN$.\\ Show that $N$ is the midpoint of the segment $I_B I_C$, where $I_B$ and $I_C$ are the excentres of $ABC$ opposite to $B$ and $C$, respectively.