, 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%.