\begin{tabular}{l} 1. В кружке у каждого ровно однн друт и ровно однн враг. Докажнте, что кружок можно \\ разделить на два кружка так, чтобы ни в каком из двух кружков не было ни друзей, ни врагов. \\ \hline\end{tabular}

To solve this problem, we can use graph theory. Let's represent the circle as a graph where each person is a vertex and each friendship or enmity is an edge. Since each person has exactly one friend and one enemy, the graph is regular with degree 2. Here's a step-by-step proof: 1. **Graph Representation**: Let's denote the circle as a set of vertices {A, B, C, D, E, F} where each vertex represents a person. Since each person has exactly one friend and one enemy, the graph will have edges between each pair of vertices. 2. **Coloring the Graph**: We can color the graph using two colors, say red and blue, such that no two adjacent vertices (friends or enemies) have the same color. This is possible because the graph is regular of degree 2, meaning each vertex has exactly two neighbors. 3. **Dividing the Circle**: Now, let's divide the circle into two parts based on the colors. Let's say we have two sets of vertices: {A, C, E} (red) and {B, D, F} (blue). 4. **Checking the Conditions**: In the set {A, C, E}, no two vertices are friends or enemies because they have different colors. Similarly, in the set {B, D, F}, no two vertices are friends or enemies. This satisfies the condition that in neither of the two circles is there a friend or an enemy. 5. **Conclusion**: Therefore, we have successfully divided the circle into two circles such that in neither circle are there any friends or enemies. This proof demonstrates that it is always possible to divide a circle where each person has exactly one friend and one enemy into two circles with the desired properties.