MEMO 2017

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.

Let $ABC$ be an acute-angled triangle with $AB \neq AC$ , circumcentre $O$ and circumcircle $\Gamma$ . Let the tangents to $\Gamma$ at $B$ and $C$ meet each other at $D$ , and let the line $AO$ intersect $BC$ at $E$ . Denote the midpoint of $BC$ by $M$ and let $AM$ meet $\Gamma$ again at $N \neq A$ . Finally, let $F \neq A$ be a point on $\Gamma$ such that $A, M, E$ and $F$ are concyclic. Prove that $FN$ bisects the segment $MD$ .

Determine all integers $n \geq 2$ such that there exists a permutation $x_0, x_1, \ldots, x_{n - 1}$ of the numbers $0, 1, \ldots, n - 1$ with the property that the $n$ numbers $$x_0, \hspace{0.3cm} x_0 + x_1, \hspace{0.3cm} \ldots, \hspace{0.3cm} x_0 + x_1 + \ldots + x_{n - 1}$$ are pairwise distinct modulo $n$ .

For an integer $n \geq 3$ we define the sequence $\alpha_1, \alpha_2, \ldots, \alpha_k$ as the sequence of exponents in the prime factorization of $n! = p_1^{\alpha_1}p_2^{\alpha_2} \ldots p_k^{\alpha_k}$ , where $p_1 < p_2 < \ldots < p_k$ are primes. Determine all integers $n \geq 3$ for which $\alpha_1, \alpha_2, \ldots, \alpha_k$ is a geometric progression.