The twelve students in an olympiad class went out to play soccer every day after their math class, forming two teams of six players each and playing against each other. Each day they formed two different teams from those formed in previous days. By the end of the year, they found that every group of five students had played together on the same team exactly once. How many different teams were formed throughout the year?
For every group of five students, there is a unique team formed that contains them. Therefore, we count

$$
C_{12}^{5}=\frac{12 \times 11 \times 10 \times 9 \times 8}{5!}=792
$$

teams for each five students chosen. On the other hand, in each team of six players, there are $C_{6}^{5}=6$ ways to choose five players, that is, there are six groups of five players that led to the same team in our first count. Thus, the total number of teams formed is $792 / 6=132$.