Let $x$, $y$, $z$ be positive real numbers satisfying the following conditions:
$$
\begin{gathered}
\frac{1}{\sqrt{2}} \leq z \lt \frac{1}{2} \min \{x\sqrt{2},\ y\sqrt{3}\}, \\
x + z\sqrt{3} \geq \sqrt{6}, \\
y\sqrt{3} + z\sqrt{10} \geq 2\sqrt{5}.
\end{gathered}
$$
Find the greatest value of the expression:
$$
P(x, y, z) = \frac{1}{x^2} + \frac{2}{y^2} + \frac{3}{z^2}
$$
Consider the function $g(x) = \frac{2x}{1+x^2}$. Find all function $f(x)$ defined and continuous on the interval $(-1; 1)$ which satisfy the condition:
$$
(1 - x^2)f(g(x)) = (1 + x^2)^2 f(x)
$$
for every $x \in (-1; 1)$.
Let be given an integer $n \ge 1$. Consider a permutation $(a_1, a_2, ..., a_{2n})$ of the first $2n$ positive integers such that the numbers $|a_{i+1} - a_i|$ $(i = 1, 2, ..., 2n-1)$ are distinct each from others.
Prove that $a_1 - a_{2n} = n$ if and only if $1 \le a_{2k} \le n$ for every $k = 1, 2, ..., n$.
The positive real numbers $a$, $b$, $c$ satisfy the condition:
$$
21ab + 2bc + 8ca \le 12.
$$
Find the least value of the expression:
$$
P(a, b, c) = \frac{1}{a} + \frac{2}{b} + \frac{3}{c}.
$$
In plane, let be given two circles intersecting at $A$ and $B$ and let $PQ$ be a common tangent ($P$ and $Q$ are touching points). Let $S$ be the point of intersection of the tangents at $P$ and at $Q$ of the circumcircle of triangle $APQ$ and let $H$ be the reflection of $B$ in the line $PQ$. Prove that the points $A$, $S$, $H$ are collinear.
A club has $42$ participants. In each group of $31$ participants of this club, there is always an acquainted couple man - woman. Prove that in this club, we can select $12$ acquainted couples.
Let be given an integer $n \gt 1$ and an orthonormal system of coordinates $Oxyz$ in space. Let $T$ be the set of all points $P(x, y, z)$, the coordinates $x, y, z$ of which are integers satisfying the conditions $1 \le x, y, z \le n$. Colour some points in $T$ so that if the point $A(x_0, y_0, z_0)$ had been coloured then every point $B(x_1, y_1, z_1)$ with $x_1 \le x_0, y_1 \le y_0, z_1 \le z_0$ ($B \ne A$) could not be coloured. How many points at most can be coloured?
Let be given a positive integer $n$ and two coprime integers $a, b$ greater than $1$. Let $p, q$ be two odd divisors greater than $1$ of $a^{6n} + b^{6n}$. Find the remainder of the division of $p^{6n} + q^{6n}$ by $6 \cdot (12)^n$.
Let be given a sequence of positive integers $\{a_n\}$, $n = 1, 2, 3, \ldots$, satisfying the condition
$$
0 \lt a_{n+1} - a_n \le 2001
$$
for every $n = 1, 2, 3, \ldots$. Prove that there exists an infinite number of couples of positive integers $(p, q)$ such that $p \lt q$ and $a_p$ is a divisor of $a_q$.
For every couple of real numbers $(a, b)$, consider the sequence of numbers $\{x_n\}$, $n = 0, 1, 2, \dots$, defined by:
$$
x_0 = a \quad \text{and} \quad x_{n+1} = x_n + b \sin x_n \quad \text{for every } n = 0, 1, 2, \dots
$$
Prove that:
1) for every real number $a$, the sequence $\{x_n\}$ corresponding to $(a, b)$ has a finite limit when $n$ tends to infinity. Find this limit.
2) for every number $b \gt 2$, there exists a real number $a$ such that the sequence $\{x_n\}$ corresponding to $(a, b)$ does not have a finite limit when $n$ tends to infinity.
The sequence of integers $\{a_n\}$, $n = 0, 1, 2, \ldots$, is defined by:
$a_0 = 1$ and $a_n = a_{n-1} + a_{\lceil n/3 \rceil}$ for every $n = 1, 2, 3, \ldots$.
Prove that for every prime number $p \le 13$, there exists an infinite number of natural numbers $k$ such that $a_k$ is divisible by $p$.
([x] denotes the integral part of x).