[Solutions] Czech-Polish-Slovak Mathematics Competition 2019

  1. Let $\omega$ be a circle. Points $A$, $B$, $C$, $X$, $D$, $Y$ lie on $\omega$ in this order such that $BD$ is its diameter and $DX=DY=DP$ , where $P$ is the intersection of $AC$ and $BD$. Denote by $E,F$ the intersections of line $XP$ with lines $AB$, $BC$, respectively. Prove that points $B$, $E$, $F$, $Y$ lie on a single circle. 
  2. We consider positive integers $n$ having at least six positive divisors. Let the positive divisors of $n$ be arranged in a sequence $(d_i)_{1\le i\le k}$ with $$1=d_1<d_2<\dots <d_k=n\quad (k\ge 6).$$Find all positive integers $n$ such that $n=d_5^2+d_6^2$.
  3. A dissection of a convex polygon into finitely many triangles by segments is called a trilateration if no three vertices of the created triangles lie on a single line (vertices of some triangles might lie inside the polygon). We say that a trilateration is good if its segments can be replaced with one-way arrows in such a way that the arrows along every triangle of the trilateration form a cycle and the arrows along the whole convex polygon also form a cycle. Find all $n\ge 3$ such that the regular $n$-gon has a good trilateration. 
  4. Given a real number $\alpha$, find all pairs $(f,g)$ of functions $f,g :\mathbb{R} \to \mathbb{R}$ such that $$xf(x+y)+\alpha \cdot yf(x-y)=g(x)+g(y) ,\quad \forall x,y \in \mathbb{R}.$$
  5. Determine whether there exist $100$ disks $D_2,D_3,\ldots ,D_{101}$ in the plane such that the following conditions hold for all pairs $(a,b)$ of indices satisfying $2\le a< b\le 101$: If $a|b$ then $D_a$ is contained in $D_b$. If $\gcd (a,b)=1$ then $D_a$ and $D_b$ are disjoint.(A disk $D(O,r)$ is a set of points in the plane whose distance to a given point $O$ is at most a given positive real number $r$.) 
  6. Let $ABC$ be an acute triangle with $AB<AC$ and $\angle BAC=60^{\circ}$. Denote its altitudes by $AD$, $BE$, $CF$ and its orthocenter by $H$. Let $K$, $L$, $M$ be the midpoints of sides $BC$, $CA$, $AB$, respectively. Prove that the midpoints of segments $AH$, $DK$, $EL$, $FM$ lie on a single circle.

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