Let $\displaystyle n \in \mathbb N$, $\displaystyle n \geq 2$. Determine $\displaystyle n$ sets $\displaystyle A_i$, $\displaystyle 1 \leq i \leq n$, from the plane, pairwise disjoint, such that:

(a) for every circle $\displaystyle \mathcal C$ from the plane and for every $\displaystyle i \in \left\{ 1,2,\ldots,n \right\}$ we have $\displaystyle A_i \cap \textrm{Int} \left( \mathcal C \right) \neq \phi$;

(b) for all lines $\displaystyle d$ from the plane and every $\displaystyle i \in \left\{ 1,2,\ldots,n \right\}$, the projection of $\displaystyle A_i$ on $\displaystyle d$ is not $\displaystyle d$.
To solve this problem, we need to construct \( n \) sets \( A_i \) in the plane that satisfy the given conditions. Let's break down the solution step by step.

1. **Define the Set \( S \)**:
   Let \( S \) be the set of points in the plane where both coordinates are rational numbers. Formally,
   \[
   S = \{ (x, y) \in \mathbb{R}^2 \mid x \in \mathbb{Q}, y \in \mathbb{Q} \}
   \]
   This set \( S \) is countable and dense in the plane.

2. **Choose Linearly Independent Reals**:
   Select \( n \) real numbers \( a_1, a_2, \ldots, a_n \) that are linearly independent over the rationals. This means that no non-trivial rational linear combination of these numbers is zero.

3. **Construct the Sets \( A_i \)**:
   Define the sets \( A_i \) as follows:
   \[
   A_i = S + (a_i, a_i) = \{ (x + a_i, y + a_i) \mid (x, y) \in S \}
   \]
   Each set \( A_i \) is a translation of the set \( S \) by the vector \( (a_i, a_i) \).

4. **Verify Condition (a)**:
   For any circle \( \mathcal{C} \) in the plane and any \( i \in \{1, 2, \ldots, n\} \), we need to show that \( A_i \cap \text{Int}(\mathcal{C}) \neq \emptyset \). Since \( S \) is dense in the plane, for any circle \( \mathcal{C} \), there exists a point \( (x, y) \in S \) inside \( \mathcal{C} \). Therefore, \( (x + a_i, y + a_i) \in A_i \) will also be inside \( \mathcal{C} \), ensuring that \( A_i \cap \text{Int}(\mathcal{C}) \neq \emptyset \).

5. **Verify Condition (b)**:
   For any line \( d \) in the plane and any \( i \in \{1, 2, \ldots, n\} \), the projection of \( A_i \) on \( d \) should not be the entire line \( d \). Since \( S \) is countable, its projection on any line \( d \) is also countable. The translation by \( (a_i, a_i) \) does not change the countability. Therefore, the projection of \( A_i \) on \( d \) is countable and cannot cover the entire line \( d \), which is uncountable.

Thus, the sets \( A_i \) constructed as above satisfy both conditions (a) and (b).

The final answer is \( \boxed{ A_i = S + (a_i, a_i) } \) where \( S = \{ (x, y) \in \mathbb{R}^2 \mid x \in \mathbb{Q}, y \in \mathbb{Q} \} \) and \( a_1, a_2, \ldots, a_n \) are linearly independent reals over the rationals.