In the footsteps of one task

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4. In the circuit shown in figure 6, both switches $K$—$K$ of the first are similarly open, capacitor $C_1$ is discharged; capacitors $C_2$ and $C_3$ are charged and have the same charge $q$, the polarity of their plates is indicated in the diagram. Find the charge that appears on capacitor $C_1$ after both switches are open. The capacitance of all capacitors is the same.

5. On a horizontally located two rectangular iron lie a block of mass $M$ and a wedge of mass $m$ in contact with it, which can slide along the block (Fig. 7, top view). The right vertical edge of the block is beveled at an angle $\alpha$. The block can move along the iron, the side walls of which serve as guides. Normally, a force $F$ is applied to the left vertical face of the block. With what acceleration will the block begin to move? Neglect the friction between all contacting surfaces.

During the First World War, among French officers who came to the army as students, during the sometimes long intervals between battles, compositions on solving problems in elementary mathematics were common. The winner must find the correct and most interesting solution according to the general opinion.

Here is one such task:
Prove that a triangle is equilateral if its bissertrices are equal.
Many people tried to solve this problem, but not all found the right ways. Would you like to try your hand?

In 1939, in the second type of the 5th Moscow Mathematical Olympiad, participants were given the task: “Decompose the expression $a^{10} + a^5 + 1$ into rational integer factors.”

This problem can be easily solved if we use the theorem: The polynomial $1 + a^{3k+1} + a^{3p+2}$ is divisible without remainder by the polynomial $1 + a + a^2$ for any natural numbers $k$ and $p$.

Try to determine whether the theorem is true:
The polynomial $1 + a^{4m+1} + a^{4l+2} + a^{4k+3}$ is divisible without remainder by the polynomial $1 + 1 + a + a^2 + a^3$ for any natural numbers $k$, $l$ and $m$.