OMM 2018

Let $A$ and $B$ be two points on a line $\ell$ , $M$ the midpoint of $AB$ , and $X$ a point on segment $AB$ other than $M$ . Let $\Omega$ be a semicircle with diameter $AB$ . Consider a point $P$ on $\Omega$ and let $\Gamma$ be the circle through $P$ and $X$ that is tangent to $AB$ . Let $Q$ be the second intersection point of $\Omega$ and $\Gamma$ . The internal angle bisector of $\angle PXQ$ intersects $\Gamma$ at a point $R$ . Let $Y$ be a point on $\ell$ such that $RY$ is perpendicular to $\ell$ . Show that $MX > XY$

For each positive integer $m$ , we define $L_m$ as the figure that is obtained by overlapping two $1 \times m$ and $m \times 1$ rectangles in such a way that they coincide at the $1 \times 1$ square at their ends, as shown in the figure.

$L_1$ $L_2$ $L_3$ $L_4$ ...

Using some figures $L_{m_1}, L_{m_2}, \dots, L_{m_k}$ , we cover an $n \times n$ board completely, in such a way that the edges of the figure coincide with lines in the board. Among all possible coverings of the board, find the minimal possible value of $m_1 + m_2 + \dots + m_k$ .

Note: In covering the board, the figures may be rotated or reflected, and they may overlap or not be completely contained within the board.

A sequence $a_2, a_3, \dots, a_n$ of positive integers is said to be campechana, if for each $i$ such that $2 \leq i \leq n$ it holds that exactly $a_i$ terms of the sequence are relatively prime to $i$ . We say that the size of such a sequence is $n - 1$ . Let $m = p_1p_2 \dots p_k$ , where $p_1, p_2, \dots, p_k$ are pairwise distinct primes and $k \geq 2$ . Show that there exist at least two different campechana sequences of size $m$ .

Let $n\geq 2$ be an integer. For each $k$ -tuple of positive integers $a_1, a_2, \ldots, a_k$ such that $a_1+a_2+\cdots +a_k=n$ , consider the sums $S_i=1+2+\ldots +a_i$ for $1\leq i\leq k$ . Determine, in terms of $n$ , the maximum possible value of the product $S_1S_2\cdots S_k$ .

Proposed by Misael Pelayo

Let $n\geq 5$ an integer and consider a regular $n$ -gon. Initially, Nacho is situated in one of the vertices of the $n$ -gon, in which he puts a flag. He will start moving clockwise. First, he moves one position and puts another flag, then, two positions and puts another flag, etcetera, until he finally moves $n-1$ positions and puts a flag, in such a way that he puts $n$ flags in total. ¿For which values of $n$ , Nacho will have put a flag in each of the $n$ vertices?

Let $ABC$ be an acute-angled triangle with circumference $\Omega$ . Let the angle bisectors of $\angle B$ and $\angle C$ intersect $\Omega$ again at $M$ and $N$ . Let $I$ be the intersection point of these angle bisectors. Let $M'$ and $N'$ be the respective reflections of $M$ and $N$ in $AC$ and $AB$ . Prove that the center of the circle passing through $I$ , $M'$ , $N'$ lies on the altitude of triangle $ABC$ from $A$ .

Proposed by Victor Domínguez and Ariel García