Problem 1543

### Part (a)
1. Assume the contrary:
Suppose no two adjacent vertices of the 9-gon are of the same color. This means that if one vertex is white, the next must be black, and this pattern must continue around the entire 9-gon.

2. Assign colors to vertices:
Let's start by coloring vertex $A_1$ white. Then, by our assumption, $A_2$ must be black, $A_3$ must be white, and so on. This gives us the following pattern:
$$A_1 = \text{white}, \quad A_2 = \text{black}, \quad A_3 = \text{white}, \quad A_4 = \text{black}, \quad A_5 = \text{white}, \quad A_6 = \text{black}, \quad A_7 = \text{white}, \quad A_8 = \text{black}, \quad A_9 = \text{white}$$

3. Check the last vertex:
Notice that $A_9$ is white, and $A_1$ is also white. Since $A_1$ and $A_9$ are adjacent vertices, they are of the same color, which contradicts our initial assumption.

4. Conclusion:
Therefore, our assumption that no two adjacent vertices are of the same color must be false. Hence, there must be at least two adjacent vertices of the same color.

$\blacksquare$

### Part (b)
1. Coloring vertices:
Since each vertex can be either black or white, and there are 9 vertices, by the pigeonhole principle, at least 5 vertices must be of the same color (either black or white).

2. Consider the vertices of the same color:
Without loss of generality, assume that there are at least 5 white vertices. Label these vertices as $W_1, W_2, W_3, W_4, W_5$.

3. Forming an isosceles triangle:
We need to show that there exist three vertices among $W_1, W_2, W_3, W_4, W_5$ that form an isosceles triangle. Consider the distances between these vertices along the perimeter of the 9-gon. The possible distances (in terms of number of edges) between any two vertices are 1, 2, 3, 4, 5, 6, 7, and 8.

4. Using the pigeonhole principle:
Since there are 5 vertices and only 4 possible distances (1, 2, 3, 4) that are less than or equal to half the perimeter of the 9-gon, by the pigeonhole principle, at least two pairs of vertices must have the same distance. This means there are at least two pairs of vertices that are equidistant from each other.

5. Forming the isosceles triangle:
Let $W_i$ and $W_j$ be two vertices that are equidistant from $W_k$. Then, $W_i, W_j,$ and $W_k$ form an isosceles triangle with $W_k$ as the vertex where the two equal sides meet.

$\blacksquare$