CAPS 2020

Let $ABCD$ be a parallelogram whose diagonals meet at $P$ . Denote by $M$ the midpoint of $AB$ . Let $Q$ be a point such that $QA$ is tangent to the circumcircle of $MAD$ and $QB$ is tangent to the circumcircle of $MBC$ . Prove that points $Q,M,P$ are collinear.

(Patrik Bak, Slovakia)

Given a positive integer $n$ , we say that a real number $x$ is $n$ -good if there exist $n$ positive integers $a_1,...,a_n$ such that $$x=\frac{1}{a_1}+...+\frac{1}{a_n}.$$ Find all positive integers $k$ for which the following assertion is true: if $a,b$ are real numbers such that the closed interval $[a,b]$ contains infinitely many $2020$ -good numbers, then the interval $[a,b]$ contains at least one $k$ -good number.

(Josef Tkadlec, Czech Republic)

The numbers $1, 2,..., 2020$ are written on the blackboard. Venus and Serena play the following game. First, Venus connects by a line segment two numbers such that one of them divides the other. Then Serena connects by a line segment two numbers which has not been connected and such that one of them divides the other. Then Venus again and they continue until there is a triangle with one vertex in $2020$ , i.e. $2020$ is connected to two numbers that are connected with each other. The girl that has drawn the last line segment (completed the triangle) is the winner. Which of the girls has a winning strategy?

(Tomáš Bárta, Czech Republic)

Let $a$ be a given real number. Find all functions $f : R \to R$ such that $(x+y)(f(x)-f(y))=a(x-y)f(x+y)$ holds for all $x,y \in R$ .

(Walther Janous, Austria)

Let $n$ be a positive integer and let $d(n)$ denote the number of ordered pairs of positive integers $(x,y)$ such that $(x+1)^2-xy(2x-xy+2y)+(y+1)^2=n$ . Find the smallest positive integer $n$ satisfying $d(n) = 61$ .

(Patrik Bak, Slovakia)

Let $ABC$ be an acute triangle. Let $P$ be a point such that $PB$ and $PC$ are tangent to circumcircle of $ABC$ . Let $X$ and $Y$ be variable points on $AB$ and $AC$ , respectively, such that $\angle XPY = 2\angle BAC$ and $P$ lies in the interior of triangle $AXY$ . Let $Z$ be the reflection of $A$ across $XY$ . Prove that the circumcircle of $XYZ$ passes through a fixed point.

(Dominik Burek, Poland)