1. Find the biggest integer that is less than $ \frac{2 + \cos\frac{\pi}{5}}{3} + \frac{3 + \sin(\frac{\pi}{5} - \frac{\pi}{2})}{2} $.
2. Natural numbers $ a_1,...,a_n $ make together a strictly rising arithmetical progression. Find all the possible $ n $, given that $ n $ is odd, $ n > 1 $ and: $$ \sum_{k=1}^n a_k = 2024 $$
3. Solve the following inequation: $ \log_{x+3}(x^2 - 7x + 12) \leq 2 $.
4. Solve the following equation: $$ \frac{\tan3x + \tan x}{1 + tan3x \space tan x} = tan4x \space tan2x $$
5. The circle that is inscribed into the triangle ABC, is touching AC in point D. AD = $ 2 + \sqrt{3} $, CD = $ \sqrt{3} $. Find the $ \angle $CAB, given that it is twice smaller than $\angle$ACB
6. $a$, $b$, $c$ are numbers which satisfy the following term: $a+b+c = 1$. Find the least possible value of: $$ \frac{a+1}{a-1}\cdot\frac{b+1}{b-1}\cdot\frac{c+1}{c-1} $$
7. Plane $\pi$ is perpendicular to the edge SA of a perfect triangular pyramid ABCS with apex S and base ABC. SA is divided by $\pi$ in a ratio of 1 : 2 (counting from S) and SB is divided by $\pi$ exactly in half. Find the angle between plane $\pi$ and base plane (ABC) of the pyramid.
If you solved everything, you can return to the blissful ignorance.