[b]p1.[/b] A boy has as many sisters as brothers. How ever, his sister has twice as many brothers as sisters. How many boys and girls are there in the family?


[b]p2.[/b] Solve each of the following problems.
(1) Find a pair of numbers with a sum of $11$ and a product of $24$.
(2) Find a pair of numbers with a sum of $40$ and a product of $400$.
(3) Find three consecutive numbers with a sum of $333$.
(4) Find two consecutive numbers with a product of $182$.


[b]p3.[/b] $2008$ integers are written on a piece of paper. It is known that the sum of any $100$ numbers is positive. Show that the sum of all numbers is positive.


[b]p4.[/b] Let $p$ and $q$ be prime numbers greater than $3$. Prove that $p^2 - q^2$ is divisible by $24$.


[b]p5.[/b] Four villages $A,B,C$, and $D$ are connected by trails as shown on the map.
[img]https://cdn.artofproblemsolving.com/attachments/4/9/33ecc416792dacba65930caa61adbae09b8296.png[/img]
On each path $A \to B  \to C$ and $B  \to C  \to D$ there are $10$ hills, on the path $A  \to B  \to D$ there are $22$ hills, on the path $A  \to D  \to B$ there are $45$ hills. A group of tourists starts from $A$ and wants to reach $D$. They choose the path with the minimal number of hills. What is the best path for them?


PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
1. Let \( x \) be the number of boys and \( y \) be the number of girls in the family. According to the problem, the boy has as many sisters as brothers, and the sister has twice as many brothers as sisters.

   - For the boy: The number of brothers is \( x - 1 \) (since he does not count himself), and the number of sisters is \( y \). Therefore, we have:
     \[
     x - 1 = y
     \]

   - For the sister: The number of brothers is \( x \), and the number of sisters is \( y - 1 \) (since she does not count herself). Therefore, we have:
     \[
     x = 2(y - 1)
     \]

2. We now have a system of two equations:
   \[
   x - 1 = y \quad \text{(1)}
   \]
   \[
   x = 2(y - 1) \quad \text{(2)}
   \]

3. Substitute equation (1) into equation (2):
   \[
   x = 2((x - 1) - 1)
   \]
   Simplify the equation:
   \[
   x = 2(x - 2)
   \]
   \[
   x = 2x - 4
   \]
   \[
   4 = x
   \]

4. Substitute \( x = 4 \) back into equation (1):
   \[
   4 - 1 = y
   \]
   \[
   y = 3
   \]

5. Therefore, the number of boys is \( x = 4 \) and the number of girls is \( y = 3 \). The total number of children in the family is:
   \[
   x + y = 4 + 3 = 7
   \]

The final answer is \( \boxed{7} \).