25.1. (England, 72). On the set $S$, a relation $\rightarrow$ is defined, which is satisfied for pairs of elements from the set $S$ and has the following properties: 1) for any distinct elements $a, b \in S$, exactly one of the relations $a \rightarrow b$ or $b \rightarrow a$ holds; 2) for any three distinct elements $a, b, c \in S$, the satisfaction of the relations $a \rightarrow b$ and $b \rightarrow c$ implies the satisfaction of the relation $c \rightarrow a$. What is the largest number of elements that the set $S$ can contain?
25.1. Suppose that in the set $S$ there are three elements $a, b, c$ such that $a \longrightarrow b$ and $a \longrightarrow c$. If $b \longrightarrow c$, then from $a \longrightarrow b$ and $b \longrightarrow c$ it follows that $c \longrightarrow a$, which contradicts the condition $a \longrightarrow c$. If $c \longrightarrow b$, then from $a \longrightarrow c$ and $c \longrightarrow b$ it follows that $b \longrightarrow a$, which contradicts the condition $a \longrightarrow b$. Similarly, it can be shown that there are no elements $a$, $d, e$ in $S$ such that $d \longrightarrow a$ and $e \longrightarrow a$. If the set $S$ contains 4 or more elements (one of which we denote by $a$), then from the first condition of the problem it follows that in $S$ there will be either elements $b$ and $c$ such that $a \longrightarrow b$ and $a \longrightarrow c$, or elements $d$ and $e$ such that $d \longrightarrow a$ and $e \longrightarrow a$. In both cases, we get a contradiction. Therefore, the set $S$ contains no more than three elements. An example of a set $S$ with three elements $a, b, c$ that satisfies both conditions of the problem can be: $a \rightarrow b, b \longrightarrow c, c \rightarrow a$.