Slovenija 2008

First note that 7 contestants won the award (one sixth of 42). The upper half consisted of 21 contestants, so $21-7=14$ got between 14 and 21 points. Each of the 7 contestants who won the award had to get at least 22 points, so together they had at least $22\cdot 7=154$ points. Let $x$ represent the number of contestants who got at least 25% but less than 50% of the points, that is at least 7 but not more than 13 points. The

remaining $42-21-x=21-x$ contestants achieved less than 6 points. What was the total number of points given to those that did not win the award? For those with at most 6 points the total was at most $6\cdot (21-x)$, and for those with at least 7 but less than 13 points the total was at most $13x$ points. Finally, we have to consider those 14 contestants from the upper half with at most 21 points. They account for at most $21\cdot 14=294$ points. This makes for at most $6\cdot (21-x)+13x+294=420+7x$ points. On the other hand, this number should be three times as great as the number of points given to those who won the award, which is $3\cdot 154=462$. Thus, we have $462\le 420+7x$, or, equivalently, $7x\ge 42$ and so $x\ge 6$. We conclude there exist at least 6 contestants such that each of them got at least 25 points but none of them got more than 50%.