[b]p1.[/b] Find the value of the parameter $a$ for which the following system of equations does not have solutions:
$$ax + 2y = 1$$
$$2x + ay = 1$$


[b]p2.[/b] Find all solutions of the equation $\cos(2x) - 3 \sin(x) + 1 = 0$.


[b]p3.[/b] A circle of a radius $r$ is inscribed into a triangle. Tangent lines to this circle parallel to the sides of the triangle cut out three smaller triangles. The radiuses of the circles inscribed in these smaller triangles are equal to $1,2$ and $3$. Find $r$.


[b]p4.[/b] Does there exist an integer $k$ such that $\log_{10}(1 + 49367 \cdot k)$ is also an integer?


[b]p5.[/b] A plane is divided by $3015$ straight lines such that neither two of them are parallel and neither three of them intersect at one point. Prove that among the pieces of the plane obtained as a result of such division there are at least $2010$ triangular pieces.


PS. You should use hide for answers.
1. **Problem 1: Find the value of the parameter \( a \) for which the following system of equations does not have solutions:**
   \[
   ax + 2y = 1
   \]
   \[
   2x + ay = 1
   \]

   To determine when the system does not have solutions, we need to check the condition for the system to be inconsistent. This occurs when the determinant of the coefficient matrix is zero, i.e., the matrix is singular.

   The coefficient matrix is:
   \[
   \begin{pmatrix}
   a & 2 \\
   2 & a
   \end{pmatrix}
   \]

   The determinant of this matrix is:
   \[
   \text{det} = a \cdot a - 2 \cdot 2 = a^2 - 4
   \]

   For the system to be inconsistent, the determinant must be zero:
   \[
   a^2 - 4 = 0
   \]
   \[
   a^2 = 4
   \]
   \[
   a = \pm 2
   \]

   However, we need to check if both values of \( a \) make the system inconsistent. If \( a = 2 \), the system becomes:
   \[
   2x + 2y = 1
   \]
   \[
   2x + 2y = 1
   \]
   This system is consistent and has infinitely many solutions.

   If \( a = -2 \), the system becomes:
   \[
   -2x + 2y = 1
   \]
   \[
   2x - 2y = 1
   \]
   Adding these two equations, we get:
   \[
   0 = 2
   \]
   which is a contradiction. Therefore, the system is inconsistent for \( a = -2 \).

   Conclusion:
   \[
   a = -2
   \]

The final answer is \( \boxed{ a = -2 } \)