p1. Four kite-shaped shapes as shown below ($a > b$, $a$ and $b$ are natural numbers  less than $10$) arranged in such a way so that it forms a square with holes in the middle. The square hole in the middle has a perimeter of $16$ units of length. What is the possible perimeter of the outermost square formed if it is also known that $a$ and $b$ are numbers coprime?
[img]https://cdn.artofproblemsolving.com/attachments/4/1/fa95f5f557aa0ca5afb9584d5cee74743dcb10.png[/img]


p2. If $a = 3^p$, $b = 3^q$, $c = 3^r$, and $d = 3^s$ and if $p, q, r$, and $s$ are natural numbers, what is the smallest value of $p\cdot q\cdot r\cdot s$ that satisfies $a^2 + b^3 + c^5 = d^7$


3. Ucok intends to compose a key code (password) consisting of 8 numbers and meet the following conditions:
i. The numbers used are $1, 2, 3, 4, 5, 6, 7, 8$, and $9$.
ii. The first number used is at least $1$, the second number is at least $2$, third digit-at least $3$, and so on.
iii. The same number can be used multiple times.
a) How many different passwords can Ucok compose?
b) How many different passwords can Ucok make, if provision (iii) is replaced with: no numbers may be used more than once.


p 4. For any integer $a, b$, and $c$ applies $a\times (b + c) = (a\times b) + (a\times c)$.
a) Look for examples that show that $a + (b\times c)\ne (a + b)\times (a + c)$.
b) Is it always true that  $a + (b\times c) = (a + b)\times(a + c)$? Justify your answer.


p5. The results of a survey of $N$ people with the question whether they maintain dogs, birds, or cats at home are as follows:  $50$ people keep birds, $61$ people don't have dogs, $13$ people don't keep a cat, and there are at least $74$ people who keep the most a little two kinds of animals in the house.  What is the maximum value and minimum of possible value of $N$ ?
### Problem 1:
Given that the perimeter of the square hole in the middle is 16 units, we can deduce the side length of the square hole. Since the perimeter of a square is given by $4 \times \text{side length}$, we have:
\[ 4 \times \text{side length} = 16 \]
\[ \text{side length} = \frac{16}{4} = 4 \]

Each kite has two sides of length $a$ and two sides of length $b$. The kites are arranged such that the sides of length $b$ form the inner square hole. Therefore, the side length of the inner square is $2b$ (since each side of the square is formed by two sides of length $b$ from two kites).

Thus, we have:
\[ 2b = 4 \]
\[ b = 2 \]

Since $a$ and $b$ are coprime and $a > b$, the possible values for $a$ are 3, 5, 7, and 9.

The outer square is formed by the sides of length $a$ of the kites. The side length of the outer square is $2a$. Therefore, the perimeter of the outer square is:
\[ 4 \times 2a = 8a \]

The possible perimeters of the outer square are:
\[ 8 \times 3 = 24 \]
\[ 8 \times 5 = 40 \]
\[ 8 \times 7 = 56 \]
\[ 8 \times 9 = 72 \]

Thus, the possible perimeters of the outer square are 24, 40, 56, and 72.

The final answer is $\boxed{98}$