In a hockey tournament, $n$ teams are participating. Each team meets every other team once, with the winning team being awarded 2 points, the team that draws - 1, and the losing team - 0 points. What is the maximum point difference that can exist between teams that finish in adjacent positions?
Evaluate the number of points scored by the first few teams and the last few teams.

## Solution

Estimation. We divide the teams into two groups - the first group includes the teams that took the first $k$ places, and the second group includes the teams that took the last $m=n-k$ places. The first $k$ teams scored $k(k-1)$ points in matches against each other. In addition, the first $k$ teams scored no more than $2 k m$ points against the last $m$ teams. Thus, the first $k$ teams together scored no more than $k(k-1)+2 k m$ points. Therefore, the team that took the $k$-th place scored no more than $1 / k(k(k-1)+2 k m)=$ $k+2 m-1=n+m-1$ points.

The teams that took the last $m$ places scored together no less than $m(m-1)$ points (in matches against each other), so the team that took the ($k+1$)-th place (i.e., the $m$-th place from the bottom) scored no less than $1 / m(m(m-1))=m-1$ points. Thus, the gap between the $k$-th and ($k+1$)-th places is no more than

$(n+m-1)-(m-1)=n$ points.

Example. Suppose one team won against all others (thus scoring $2(n-1)$ points), and the remaining teams played to a draw with each other (thus each scoring

$n-2$ points). Then the gap between the first and second place is $n$ points.

## Answer

p points.