Suppose that $n$ is a positive integer number. Consider a regular polygon with $2n$ sides such that one of its largest diagonals is parallel to the $x$-axis. Find the smallest integer $d$ such that there is a polynomial $P$ of degree $d$ whose graph intersects all sides of the polygon on points other than vertices.

Proposed by Mohammad Ahmadi
1. **Assume the Setup:**
   Suppose we have a regular polygon with \(2n\) sides, and one of its largest diagonals lies on the \(x\)-axis. This means the polygon is symmetric about the \(x\)-axis.

2. **Intersection Points:**
   Let \(x_1 < x_2 < \cdots < x_{2n}\) be the \(x\)-coordinates of the intersection points of the polynomial \(P(x)\) with the sides of the polygon. 

3. **Non-Consecutive Intersection Points:**
   No three consecutive intersection points \(x_{i-1}, x_i, x_{i+1}\) can belong to the sides \(l_1, l_2, l_3\) on the same side of the polygon's mean diagonal (which lies on the \(x\)-axis). If they did, reflecting \(l_2\) across the \(x\)-axis would show that there is no intersection point on the reflected side, which is a contradiction.

4. **Pigeonhole Principle:**
   By the pigeonhole principle, there must be at least \(n\) indices \(i \le 2n\) such that \(x_i\) and \(x_{i+1}\) are on different sides of the \(x\)-axis. 

5. **Application of Rolle's Theorem:**
   For each such \(i\), by Rolle's theorem, \(P(x)\) must have a root in the interval \((x_i, x_{i+1})\). Therefore, \(P(x)\) has at least \(n\) distinct real roots, implying that \(\deg(P(x)) \ge n\).

6. **Constructing the Polynomial:**
   Consider vertical lines \(l_1, l_2, \ldots, l_{n+1}\) such that \(l_i\) passes through vertices \(A_i\) and \(A_{2n-i+2}\) of the polygon. Place points \(P_1, \ldots, P_{n+1}\) on these lines such that for each odd \(i\), \(y_{P_i} > y_{A_i}, y_{A_{2n-i+2}}\) and for each even \(i\), \(y_{P_i} < y_{A_i}, y_{A_{2n-i+2}}\).

7. **Unique Polynomial:**
   There is a unique polynomial \(P(x)\) of degree \(n\) that passes through these \(n+1\) points. By Rolle's theorem, this polynomial will intersect all sides of the polygon.

8. **Conclusion:**
   Therefore, the smallest integer \(d\) such that there is a polynomial \(P\) of degree \(d\) whose graph intersects all sides of the polygon at points other than the vertices is \(n\).

The final answer is \(\boxed{n}\)