KURSCHAK 2009

Let $n,k$ be arbitrary positive integers. We fill the entries of an $n\times k$ array with integers such that all the $n$ rows contain the integers $1,2,\dots,k$ in some order. Add up the numbers in all $k$ columns - let $S$ be the largest of these sums. What is the minimal value of $S$ ?

Find all positive integer pairs $(a,b)$ for which the set of positive integers can be partitioned into sets $H_1$ and $H_2$ such that neither $a$ nor $b$ can be represented as the difference of two numbers in $H_i$ for $i=1,2$ .

Find all functions $f:\mathbb{Z}\to \mathbb{Q}$ with the following properties: if $f(x)<c<f(y)$ for some rational $c$ , then $f$ takes on the value of $c$ , and $$f(x)+f(y)+f(z)=f(x)f(y)f(z)$$ whenever $x+y+z=0$ .