PL-OM 2022

Let $ABC$ be an acute triangle with $AB<AC$ . The angle bisector of $BAC$ intersects the side $BC$ and the circumcircle of $ABC$ at $D$ and $M\neq A$ , respectively. Points $X$ and $Y$ are chosen so that $MX \perp AB$ , $BX \perp MB$ , $MY \perp AC$ , and $CY \perp MC$ . Prove that the points $X,D,Y$ are collinear.

Let $m,n\ge 2$ be given integers. Prove that there exist positive integers $a_1<a_2<\ldots<a_m$ so that for any $1\le i<j\le m$ the number $\frac{a_j}{a_j-a_i}$ is an integer divisible by $n$ .

One has marked $n$ points on a circle and has drawn a certain number of chords whose endpoints are the marked points. It turned out that the following property is satisfied: whenever any $2021$ drawn chords are removed one can join any two marked points by a broken line composed of some of the remaining drawn chords. Prove that one can remove some of the drawn chords so that at most $2022n$ chords remain and the property described above is preserved.

Find all triples $(a,b,c)$ of real numbers satisfying the system $\begin{cases} a^3+b^2c=ac b^3+c^2a=ba c^3+a^2b=cb \end{cases}$

Let $ABC$ be a triangle satisfying $AB<AC$ . Let $M$ be the midpoint of $BC$ . A point $P$ lies on the segment $AB$ with $AP>PB$ . A point $Q$ lies on the segment $AC$ with $\angle MPA = \angle AQM$ . The perpendicular bisectors of $BC$ and $PQ$ intersect at $S$ . Prove that $\angle BAC + \angle QSP = \angle QMP$ .

A prime number $p$ and a positive integer $n$ are given. Prove that one can colour every one of the numbers $1,2,\ldots,p-1$ using one of the $2n$ colours so that for any $i=2,3,\ldots,n$ the sum of any $i$ numbers of the same colour is not divisible by $p$ .