KURSCHAK 2020

Let $n$ and $k$ be positive integers. Given $n$ closed discs in the plane such that no matter how we choose $k + 1$ of them, there are always two of the chosen discs that have no common point. Prove that the $n$ discs can be partitioned into at most $10k$ classes such that any two discs in the same class have no common point.

Find all functions $f\colon \mathbb{Q}\to \mathbb{R}_{\geq 0}$ such that for any two rational numbers $x$ and $y$ the following conditions hold

• $f(x+y)\leq f(x)+f(y)$ ,
• $f(xy)=f(x)f(y)$ ,
• $f(2)=1/2$ .

There are $N$ houses in a city. Every Christmas, Santa visits these $N$ houses in some order. Show that if $N$ is large enough, then it holds that for three consecutive years there are always are $13$ houses such that they have been visited in the same order for two years (out of the three consecutive years). Determine the smallest $N$ for which this holds.