Find all positive integers n such that the product $1! \cdot 2! \cdot \cdot \cdot \cdot n!$ is a perfect square

Find all strictly monotone functions $f : \mathbb{R} \to \mathbb{R}$ such that some polynomial $P(x, y)$ satisfies the equality $$f(x + y) = P(f(x), f(y))$$for all real numbers $x$ and $y$

In a football tournament $2n$ teams play in a round. Every round consists of $n$ pairs of teams that haven’t played with each other yet. Every round’s schedule is determined before the round is held. Find the minimal positive integer $k$ such that the following situation is possible: after $k$ rounds it’s impossible to schedule the next round.

Let $\Gamma$ be the incircle of a non-isosceles triangle $ABC$, $I$ be it’s incenter. Let $A_1, B_1, C_1$ be the tangency points of $\Gamma$ with the sides $BC, AC, AB$ respectively. Let $A_2 = \Gamma \cap AA_1$, $M = C_1B_1 \cap AI$, $P$ and $Q$ be the other (different from $A_1$ and $A_2$) intersection points of $\Gamma$ and $A_1M$, $A_2M$ respectively. Prove that $A$, $P$ and $Q$ are colinear.

Define $M_n = \{ 1, 2, \ldots , n \} $ for all positive integers $n$. A collection of $3$-element subsets of $M_n$ is said to be fine if for any colouring of elements of $M_n$ in two colours there is a subset of the collection all three elements of which are of the same colour. For each $n \geq 5$ find the minimal possible number of the $3$-element subsets of a fine collection

Prove that for coprime each positive integers $a, c$ there is a positive integer $b$ such that $c$ divides $\underbrace{b^{b^{b^{\ldots^b}}}}_\text{b times}-a$