# [Solutions] Czech-Polish-Slovak Mathematics Competition 2020

1. 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.
2. 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.
3. 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?
4. Let $a$ be a given real number. Find all functions $f : \mathbb R \to \mathbb R$ such that $$(x+y)(f(x)-f(y))=a(x-y)f(x+y),\,\forall x,y \in \mathbb R.$$
5. 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 $d(n) = 61$.
6. 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.
 MOlympiad.NET rất mong bạn đọc ủng hộ UPLOAD đề thi và đáp án mới hoặc LIÊN HỆ[email protected]
You can use $\LaTeX$ in comment