8-5. In a tournament, 6 teams $P, Q, R, S, T$ and $U$ participate, and each team must play against every other team exactly once. Each day, they are divided into 3 pairs, and all three matches are played simultaneously. The "Sports" channel has chosen which match it will broadcast each day:

$$
\begin{array}{c|c|c|c|c}
1 \text { day } & 2 \text { day } & 3 \text { day } & 4 \text { day } & 5 \text { day } \\
\hline P-Q & R-S & P-T & T-U & P-R
\end{array}
$$

On which day can teams $S$ and $U$ play against each other? Mark all possible options.
Answer. Only in the 1st.

Solution. Let's look at team $P$: on the 1st, 3rd, and 5th days, it will play against teams $Q, T$, and $R$. Therefore, in the remaining two days, it must play against teams $S$ and $U$. Since on the 2nd day $S$ plays against $R$, $P$ has no choice but to play against $U$ on the 2nd day, and against $S$ on the 4th day.

$$
\begin{array}{l|l|l|l|l}
1 \text { day } & 2 \text { day } & 3 \text { day } & 4 \text { day } & 5 \text { day } \\
\hline P-Q & R-S & P-T & T-U & P-R \\
& P-U & & P-S &
\end{array}
$$

| 1 day | 2 day | 3 day | 4 day | 5 day |
| :---: | :---: | :---: | :---: | :---: |
| $P-Q$ | $R-S$  $P-U$ | $P-T$ | $T-U$  $P-S$ | $P-R$ |
|  | $T-Q$ |  | $Q-R$ |  |

![](https://cdn.mathpix.com/cropped/2024_05_06_1d29d5cb430346a58eb5g-17.jpg?height=66&width=1382&top_left_y=1829&top_left_x=89)

Similarly, let's look at team $T$. On the 5th day, it must play either against $R$ or $S$; $R$ is already playing against $P$, so $T$ will play against $S$. Therefore, the match $T-R$ must take place on the 1st day:

| 1 day | 2 day | 3 day | 4 day | 5 day |
| :---: | :---: | :---: | :---: | :---: |
| $P-Q$ | $R-S$ | $P-T$ | $T-U$ | $P-R$ |
| $T-R$ | $P-U$ |  | $P-S$ | $T-S$ |
|  | $T-Q$ |  | $Q-R$ |  |

Thus, the remaining match on the 1st day is the match $S-U$, which is what we were looking for.

Comment. The situation described in the problem is indeed possible:

| 1 day | 2 day | 3 day | 4 day | 5 day |
| :---: | :---: | :---: | :---: | :---: |
| $P-Q$ | $R-S$ | $P-T$ | $T-U$ | $P-R$ |
| $T-R$ | $P-U$ | $Q-S$ | $P-S$ | $T-S$ |
| $S-U$ | $T-Q$ | $R-U$ | $Q-R$ | $Q-U$ |