The six-pointed star in the figure is regular: all interior angles of the small triangles are equal. Each of the thirteen marked points is assigned a color, green or red. Prove that there are always three points of the same color, which are the vertices of an equilateral triangle.

Let $a,b,c,d$ be four positive real numbers. If they satisfy $$a+b+\frac{1}{ab}=c+d+\frac{1}{cd}\quad\text{and}\quad\frac1a+\frac1b+ab=\frac1c+\frac1d+cd$$ then prove that at least two of the values $a,b,c,d$ are equal.

Let $ABC$ be a triangle, with $AB<AC$, and let $\Gamma$ be its circumcircle. Let $D$, $E$ and $F$ be the tangency points of the incircle with $BC$, $CA$ and $AB$ respectively. Let $R$ be the point in $EF$ such that $DR$ is an altitude in the triangle $DEF$, and let $S$ be the intersection of the external bisector of $\angle BAC$ with $\Gamma$. Prove that $AR$ and $SD$ intersect on $\Gamma$.

Let $P$ be a point in the plane. Prove that it is possible to draw three rays with origin in $P$ with the following property: for every circle with radius $r$ containing $P$ in its interior, if $P_1$, $P_2$ and $P_3$ are the intersection points of the three rays with the circle, then $$|PP_1|+|PP_2|+|PP_3|\leq 3r.$$

Given is a simple graph $G$ with $2022$ vertices, such that for any subset $S$ of vertices (including the set of all vertices), there is a vertex $v$ with $deg_{S}(v) \leq 100$. Find $\chi(G)$ and the maximal number of edges $G$ can have.

Find all triples $(x,y,z)$ of positive integers, with $z>1$, satisfying simultaneously that $$x\text{ divides }y+1,\quad y\text{ divides }z-1,\quad z\text{ divides }x^2+1.$$