[b]p1.[/b] How many real solutions does the following system of equations have? Justify your answer.
$$x + y = 3$$
$$3xy -z^2 = 9$$

[b]p2.[/b] After the first year the bank account of Mr. Money decreased by $25\%$, during the second year it increased by $20\%$, during the third year it decreased by $10\%$, and during the fourth year it increased by $20\%$. Does the account of Mr. Money increase or decrease during these four years and how much?


[b]p3.[/b] Two circles are internally tangent. A line passing through the center of the larger circle intersects it at the points $A$ and $D$. The same line intersects the smaller circle at the points $B$ and $C$. Given that $|AB| : |BC| : |CD| = 3 : 7 : 2$, find the ratio of the radiuses of the circles.


[b]p4.[/b] Find all integer solutions of the equation $\frac{1}{x}+\frac{1}{y}=\frac{1}{19}$


[b]p5.[/b] Is it possible to arrange the numbers $1, 2, . . . , 12$ along the circle so that the absolute value of the difference between any two numbers standing next to each other would be either $3$, or $4$, or $5$? Prove your answer.


[b]p6.[/b] Nine rectangles of the area $1$ sq. mile are located inside the large rectangle of the area $5$ sq. miles. Prove that at least two of the rectangles (internal rectangles of area $1$ sq. mile) overlap with an overlapping area greater than or equal to $\frac19$ sq. mile


PS. You should use hide for answers.
### Problem 1
We are given the system of equations:
\[ x + y = 3 \]
\[ 3xy - z^2 = 9 \]

1. From the first equation, solve for \( y \):
   \[ y = 3 - x \]

2. Substitute \( y = 3 - x \) into the second equation:
   \[ 3x(3 - x) - z^2 = 9 \]

3. Simplify the equation:
   \[ 9x - 3x^2 - z^2 = 9 \]

4. Rearrange the equation:
   \[ 3x^2 - 9x + z^2 = -9 \]
   \[ 3x^2 - 9x + z^2 + 9 = 0 \]
   \[ 3x^2 - 9x + (z^2 + 9) = 0 \]

5. Notice that the quadratic equation in \( x \) must have real roots. For a quadratic equation \( ax^2 + bx + c = 0 \) to have real roots, the discriminant must be non-negative:
   \[ \Delta = b^2 - 4ac \geq 0 \]

6. In our case:
   \[ a = 3, \quad b = -9, \quad c = z^2 + 9 \]
   \[ \Delta = (-9)^2 - 4 \cdot 3 \cdot (z^2 + 9) \]
   \[ \Delta = 81 - 12(z^2 + 9) \]
   \[ \Delta = 81 - 12z^2 - 108 \]
   \[ \Delta = -12z^2 - 27 \]

7. Since \( -12z^2 - 27 \) is always negative for all real \( z \), the discriminant is always negative. Therefore, the quadratic equation \( 3x^2 - 9x + (z^2 + 9) = 0 \) has no real roots.

8. Since there are no real roots for \( x \), there are no real solutions to the system of equations.

The final answer is \(\boxed{0}\)