## Task A-2.7.

In a basketball tournament, each team plays exactly twice against each of the other teams. A win brings 2 points, a loss 0 points, and there are no draws. Determine all natural numbers $n$ for which there exists a basketball tournament with $n$ teams where one team, the tournament winner, has 26 points, and exactly two teams have the lowest number of points, which is 20 points.
## Solution.

Let the total number of teams be denoted by $n$. Let $a$ be the number of teams that scored 24 points, and $b$ be the number of teams that scored 22 points. Then the total number of teams is

$$
n=3+a+b
$$

and the total number of points in all matches is

$$
2 n(n-1)=26+24 a+22 b+40
$$

Substituting $b=n-3-a$ from the first equation into the second, we get

$$
26+24 a+22 n-66-22 a+40=2 n(n-1), \quad \text { i.e. } \quad a=n(n-12)
$$

Since $a$ is a non-negative number, it follows that $n \geqslant 12$. 1 point

Substituting $a=n(n-12)$ into $n=3+a+b$, we can write

$$
b=(13-n) n-3,
$$

from which, due to the non-negativity of $b$, it follows that $n<13$.

Thus, $n=12$, $a=0$ and $b=9$.

This proves that if such a tournament exists, the number of teams must be 12. It is still necessary to show that there is indeed a tournament with 12 teams. We will do this by explicitly writing the outcomes of all matches in such a tournament.

Let the tournament winner be denoted by $A$, the last two teams by $X$ and $Y$, and the other nine teams by $B_{1}, B_{2}, \ldots, B_{9}$.

Let team $A$ win against teams $X$ and $Y$ twice and once against each team $B_{i}, i=1, \ldots, 9$.

In the set $\left\{X, Y, B_{1}, \ldots, B_{9}\right\}$, let each team win exactly once against every other team.

With these match outcomes, team $A$ has $9+2 \cdot 2=13$ wins, i.e., 26 points.

Teams $X$ and $Y$ have 10 wins each, i.e., 20 points (since they each won once against all opponents in the set $\left\{X, Y, B_{1}, \ldots, B_{9}\right\}$, which has 10 members).

Teams $B_{i}, i=1, \ldots, 9$ have 11 wins each, i.e., 22 points (since they each won once against all opponents in the set $\left\{X, Y, B_{1}, \ldots, B_{9}\right\}$ and once against team $A$).

Note: The student must notice that it is necessary to construct an example that will show that a tournament with 12 teams and the given point distribution is indeed possible, i.e., to explicitly state one such tournament. Constructing the example is worth 3 points.

Note: The example can be constructed in several different ways, and any correct example (which differs from the official one) should also be scored 3 points.

## SCHOOL/CITY COMPETITION IN MATHEMATICS

3rd grade - high school - A variant
January 4, 2015.

## IF THE STUDENT HAS A DIFFERENT APPROACH TO SOLVING THE PROBLEM, THE COMMITTEE IS OBLIGED TO SCORE AND EVALUATE THAT APPROACH IN AN APPROPRIATE MANNER.