1. In a qualification arm wrestling tournament, 896 athletes participate. A victory earns 1 point, a defeat earns 0 points. Before each round, pairs are formed by drawing lots among participants with an equal number of points (those who do not find a pair are awarded a point without playing). After the second defeat, an athlete is eliminated from the tournament. As soon as a sole leader is determined, the tournament ends, and all remaining athletes on the tournament are considered to have passed the qualification. Find the number of such athletes.
Answer: 10.

Solution. Note that $896=1024-128$. First, let's prove an auxiliary statement.

In a tournament with $2^{n}$ participants, for any $m \in\{1, \ldots, n\}$, after the $m$-th round, there will be $2^{n-m}$ athletes without any losses and $m \cdot 2^{n-m}$ athletes 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 paired. After the round, the number of athletes 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 have $(m+1) \cdot 2^{n-m-1}$. The remaining participants will leave the tournament. Thus, the inductive step is completed.

Suppose that parallel to ours, a second qualifying tournament with 128 participants is held under the same rules. We can treat these two tournaments as parts of one (with 1024 participants) up to the seventh round. We will pair only athletes from their own tournament in each round. This is possible because, by the above, the number of leaders and participants with one loss in the second and overall tournament will be even, and thus in the first one as well. We will record the remaining participants as pairs $(a, b)$, where $b$ is the number of leaders, and $a$ is the number of athletes with one loss. By the auxiliary statement, after the seventh round, in the overall tournament, there will be $(56,8)$ participants, and in the second one - $(7,1)$. Therefore, in our tournament, there will be $(49,7)$ athletes, and their number will change as follows:

$$
(49,7) \rightarrow(28,4) \rightarrow(16,2) \rightarrow(9,1)
$$

Thus, the tournament will end after the tenth round, and 10 athletes will qualify.