1. In a tennis tournament, 1152 schoolchildren are participating. 1 point is awarded for a win, and 0 points for a loss. Before each round, pairs are formed by drawing lots among participants with the same number of points (those who do not find a pair are awarded a point without playing). A player is eliminated after the second loss. The tournament continues as long as at least one pair of opponents can be formed. How many rounds will have to be played?
Answer: 14.

Solution. Note that $1152=1024+128$. Let's prove two statements first.

1) In a tournament with $2^{n}$ participants, for any $m \in\{1, \ldots, n\}$, after the $m$-th round, there will be $2^{n-m}$ participants without any losses and $m \cdot 2^{n-m}$ participants with one loss. We will use induction on $m$. For $m=1$, everything is obvious. Suppose for some $m<n$ the statement is true. Then before the $(m+1)$-th round, all participants will be divided into pairs. After the round, the number of participants without any losses will be halved, i.e., it will become $2^{n-m-1}$. One loss will be incurred by those who lost in the group of leaders (there are $2^{n-m-1}$ such people), and by those who already had one loss and won (by the induction hypothesis, there are $m \cdot 2^{n-m-1}$ such people). In total, we will get $(m+1) \cdot 2^{n-m-1}$. The rest of the participants will leave the tournament. Thus, the inductive step is completed.
2) We can divide the participants into two groups of 1024 and 128 participants and assume that for the first 10 rounds they play separate tournaments. Indeed, in each round, we will only pair participants from the same group. By 1), all participants in the first group can be paired. Therefore, if someone is left without a pair in the overall tournament, we can assume that this participant is from the second group. After 10 rounds, the results of the two groups are simply combined.

Now let's count the number of rounds. We will record the remaining participants as pairs $(a, b)$, where $b$ is the number of leaders, and $a$ is the number of participants with one loss. By 1), in the first group, after the tenth round, there will be $(10,1)$ participants. In the second group, after the seventh round, there will be $(7,1)$ participants, and their number will change as follows:

$$
(7,1) \rightarrow(4,1) \rightarrow(2,1) \rightarrow(1,1)
$$

In total, after the tenth round, there will be $(11,2)$ participants remaining in the tournament. Their further dynamics up to the fourteenth round are as follows:

$$
(11,2) \rightarrow(7,1) \rightarrow(4,1) \rightarrow(2,1) \rightarrow(1,1)
$$

After this, no more pairs can be formed, and the tournament ends.