Problem 1754

1. Understanding the Problem:
- We are given a grid plane and need to consider all possible broken lines starting at $(0,0)$, with vertices at integer points, and each segment either going up or to the right.
- For each broken line, we consider the corresponding "worm," which is the subset of the plane consisting of all cells that share at least one point with the broken line.
- We need to prove that the number of worms that can be divided into dominoes (rectangles $2 \times 1$ and $1 \times 2$) in exactly $n > 2$ different ways is equal to the number of positive integers less than $n$ and relatively prime to $n$.

2. Initial Observations:
- Each worm corresponds to a path from $(0,0)$ to some point $(a,b)$ on the grid.
- The problem can be approached by considering the tiling of the grid cells that the worm covers.

3. Recursive Approach:
- Suppose the last move of the broken line is rightward. If we tile the $2 \times 2$ square containing the last point with a horizontal domino, there is a unique way to tile the largest staircase of the skeleton such that the last vertex is in it.
- If we tile the $2 \times 2$ square containing the last vertex with a vertical domino, we can delete the last vertex and consider the remaining skeleton.

4. Recursion Formula:
- Let $f(\mathcal{S})$ be the number of ways to tile any skeleton $\mathcal{S}$ with dominoes.
- We get the recursion:
$$f(\mathcal{S}) = f(A_{\mathcal{S}}) + f(B_{\mathcal{S}})$$
where $A_{\mathcal{S}}$ is the remaining skeleton when we tile the last $2 \times 2$ square with a horizontal domino, and $B_{\mathcal{S}}$ is the remaining skeleton when we tile it with a vertical domino.

5. Tree Representation:
- We can represent the recursion as a tree where each node corresponds to a skeleton $\mathcal{S}$, and its children are $A_{\mathcal{S}}$ and $B_{\mathcal{S}}$.
- In the $k$-th row of the tree, we have all the $2^k$ paths of size $k$.

6. Labeling the Tree:
- For each skeleton $\mathcal{S}$ on the tree, label it with $f(\mathcal{S})$.
- We want to prove that each integer $n$ appears exactly $\varphi(n)$ times on the tree, where $\varphi(n)$ is the Euler's totient function.

7. Coprime Neighbors:
- From the recursion, two neighbors on the tree are coprime. This can be proved via induction.

8. Claim:
- For each vertex labeled with $n$, its parent is labeled with $m \in (\frac{n}{2}, n)$ and $\gcd(m, n) = 1$.
- Proof: Use induction. The base cases are done. The $\gcd$ condition is proved by the coprime neighbors property. Since $n - m$ comes before $m$ on the tree, $n - m < m \implies \frac{n}{2} < m$.

9. Counting Appearances:
- For each $m < n$ with $\gcd(m, n) = 1$, either $m$ or $n - m$ belongs to $(\frac{n}{2}, n)$. Thus, there are exactly $\frac{\varphi(n)}{2}$ numbers belonging to $(\frac{n}{2}, n)$ which are coprime to $n$.

10. Uniqueness of Neighbors:
- There aren't two pairs of neighbors on the same side of the tree that are equal. This can be proved using induction.

11. Final Argument:
- Using the above claims and properties, we can prove that for each vertex $v$ labeled with $l$, its parent is labeled with $l_0$ such that $\gcd(l, l_0) = 1$ and $l_0 \in (\frac{l}{2}, l)$, and its children are labeled with $l_1, l_2$ such that $\gcd(l, l_1) = \gcd(l, l_2) = 1$ and $l_1, l_2 \in (l, 2l)$.
- By induction, we can show that $n$ appears $\varphi(n)$ times on the tree.

$$\boxed{\text{The number of worms that can be divided into dominoes in exactly } n \text{ different ways is } \varphi(n).}$$