Let $ Z $ be the set of all integers,
$ A = \{(x, y) \in Z \times Z : (x - 2)^2 + y^2 \leq 4\} $,
$ B = \{(x, y) \in Z \times Z : x^2 + y^2 \leq 4\} $ and
$ C = \{(x, y) \in Z \times Z : (x - 2)^2 + (y - 2)^2 \leq 4\} $
If the total number of relations from $ A \cap B $ to $ A \cap C $ is $ 2^p $, then the value of $ p $ is :

Question

If the set $R = {(a, b) : a + 5b = 42, a, b \in \mathbb{N}}$ has $m$ elements and $\sum_{n=1}^m (1 + i^n) = x + iy$, where $i = \sqrt{-1}$, then the value of $m + x + y$ is:

Answer 1

To solve this problem, we need to compute the set of integer points for the sets $A$, $B$, and $C$, and then find their intersections $A \cap B$ and $A \cap C$. Finally, we calculate the number of relations from $A \cap B$ to $A \cap C$.

**Step 1: Identify integer points in $A$**

The set $A = \{(x, y) \in Z \times Z : (x - 2)^2 + y^2 \leq 4\}$ represents the set of points within or on the boundary of a circle centered at $(2, 0)$ with radius 2. The integer solutions are:

$(2, 0)$
$(1, 0)$
$(3, 0)$
$(2, 1)$
$(2, -1)$
$(1, 1)$
$(1, -1)$
$(3, 1)$
$(3, -1)$

Thus, $|A| = 9$.

**Step 2: Identify integer points in $B$**

The set $B = \{(x, y) \in Z \times Z : x^2 + y^2 \leq 4\}$ represents points within or on the boundary of a circle centered at $(0, 0)$ with radius 2. The integer solutions are:

$(0, 0)$
$(1, 0)$
$(-1, 0)$
$(0, 1)$
$(0, -1)$
$(2, 0)$
$(-2, 0)$
$(0, 2)$
$(0, -2)$
$(1, 1)$
$(1, -1)$
$(-1, 1)$
$(-1, -1)$

Thus, $|B| = 13$.

**Step 3: Identify integer points in $C$**

The set $C = \{(x, y) \in Z \times Z : (x - 2)^2 + (y - 2)^2 \leq 4\}$ represents a circle centered at $(2, 2)$ with radius 2. The integer solutions are:

$(2, 2)$
$(1, 2)$
$(3, 2)$
$(2, 3)$
$(2, 1)$

Thus, $|C| = 5$.

**Step 4: Compute $A \cap B$ and $A \cap C$**

$A \cap B$: Intersection of points in $A$ and $B$.
Common points in $A$ and $B$ are: $(1, 0), (2, 0), (1, 1), (1, -1)$.
So, $|A \cap B| = 4$.
$A \cap C$: Intersection of points in $A$ and $C$.
Common points in $A$ and $C$ are: $(2, 2), (1, 2), (3, 2), (2, 1)$.
So, $|A \cap C| = 4$.

**Step 5: Calculate the number of relations from $A \cap B$ to $A \cap C$**

The total possible relations from one set with 4 elements to another set with 4 elements is $4^4 = 256$. Therefore, the total number of relations is $2^8 = 256$, implying $p = 8$.

Since the correct calculation should lead to the conclusion that total relations $= 2^m$, where $m$ was presented as the answer and here due to recalculated steps, it suggests $p = 25$ by misinterpretation of problem. Let's check landmark conditions.

**Conclusion**: After reconciling computation mismatch that triggered $A$ set initially for intersected points (as illustrated correctly), $2^8 = 256$ does answer option $p = 8$. Consider corrections where $|R| = 16$.

Answer 2