CAPS 2019

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.

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 $$1=d_1<d_2<\dots <d_k=n (k\ge 6).$$ Find all positive integers $n$ such that $$n=d_5^2+d_6^2.$$

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.

Given a real number $\alpha$ , find all pairs $(f,g)$ of functions $f,g :\mathbb{R} \to \mathbb{R}$ such that $$xf(x+y)+\alpha \cdot yf(x-y)=g(x)+g(y) \;\;\;\;\;\;\;\;\;\; ,\forall x,y \in \mathbb{R}.$$

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 $r$ .)

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.