BG-MO 2022

A white equilateral triangle $T$ with side length $2022$ is divided into equilateral triangles with side $1$ (cells) by lines parallel to the sides of $T$ . We'll call two cells $\textit{adjacent}$ if they have a common vertex. Ivan colours some of the cells in black. Without knowing which cells are black, Peter chooses a set $S$ of cells and Ivan tells him the parity of the number of black cells in $S$ . After knowing this, Peter is able to determine the parity of the number of $\textit{adjacent}$ cells of different colours. Find all possible cardinalities of $S$ such that this is always possible independent of how Ivan chooses to colour the cells.

Let $ABC$ be an acute triangle and $M$ be the midpoint of $AB$ . A circle through the points $B$ and $C$ intersects the segments $CM$ and $BM$ at points $P$ and $Q$ respectively. Point $K$ is symmetric to $P$ with respect to point $M$ . The circumcircles of $\triangle AKM$ and $\triangle CQM$ intersect for the second time at $X$ . The circumcircles of $\triangle AMC$ and $\triangle KMQ$ intersect for the second time at $Y$ . The segments $BP$ and $CQ$ intersect at point $T$ . Prove that the line $MT$ is tangent to the circumcircle of $\triangle MXY$ .

Let $x>y>2022$ be positive integers such that $xy+x+y$ is a perfect square. Is it possible for every positive integer $z$ from the interval $[x+3y+1,3x+y+1]$ the numbers $x+y+z$ and $x^2+xy+y^2$ not to be coprime?

Let $n\geq 4$ be a positive integer and $x_{1},x_{2},\ldots ,x_{n},x_{n+1},x_{n+2}$ be real numbers such that $x_{n+1}=x_{1}$ and $x_{n+2}=x_{2}$ . If there exists an $a>0$ such that $$x_{i}^2=a+x_{i+1}x_{i+2} \forall 1\leq i\leq n$$ then prove that at least $2$ of the numbers $x_{1},x_{2},\ldots ,x_{n}$ are negative.

Let $ABC$ be an isosceles triangle with $AB=4$ , $BC=CA=6$ . On the segment $AB$ consecutively lie points $X_{1},X_{2},X_{3},\ldots$ such that the lengths of the segments $AX_{1},X_{1}X_{2},X_{2}X_{3},\ldots$ form an infinite geometric progression with starting value $3$ and common ratio $\frac{1}{4}$ . On the segment $CB$ consecutively lie points $Y_{1},Y_{2},Y_{3},\ldots$ such that the lengths of the segments $CY_{1},Y_{1}Y_{2},Y_{2}Y_{3},\ldots$ form an infinite geometric progression with starting value $3$ and common ratio $\frac{1}{2}$ . On the segment $AC$ consecutively lie points $Z_{1},Z_{2},Z_{3},\ldots$ such that the lengths of the segments $AZ_{1},Z_{1}Z_{2},Z_{2}Z_{3},\ldots$ form an infinite geometric progression with starting value $3$ and common ratio $\frac{1}{2}$ . Find all triplets of positive integers $(a,b,c)$ such that the segments $AY_{a}$ , $BZ_{b}$ and $CX_{c}$ are concurrent.