Let $n$ and $p$ be positive integers, with $p>3$ prime, such that: i) $n\mid p-3;$ ii) $p\mid (n+1)^3-1.$ Show that $pn+1$ is the cube of an integer.

We say that a real $a\geq-1$ is philosophical if there exists a sequence $\epsilon_1,\epsilon_2,\dots$, with $\epsilon_i \in\{-1,1\}$ for all $i\geq1$, such that the sequence $a_1,a_2,a_3,\dots$, with $a_1=a$, satisfies $$a_{n+1}=\epsilon_{n}\sqrt{a_{n}+1},\forall n\geq1$$and is periodic. Find all philosophical numbers.

On a board are written some positive reals (not necessarily distinct). For every two numbers in the frame $a$ and $b$ distinct such that $$\frac{1}{2}<\frac{a}{b}<2,$$an allowed operation is to delete $a$ and $b$ and write $2a-b$ and $2b-a$ in their place. Show that we can do the operation only a finite number of times.

Let $ABC$ be a triangle, $I$ its incenter and $\omega$ its incircle. Let $D$,$E$ and $F$ be the points of tangency of $\omega$ with $BC$,$AC$ and $AB$, respectively and $M$,$N$ and $P$ be the midpoints of $BC$, $AC$ and $AB$. Let $D'$ be the second intersection of $DI$ with $\omega$, $Q$ the intersection of $DI$ with $EF$ and $U \ne Q$ be the intersection of $(AD'Q)$ with $(DMQ)$. Suppose that $U$ lies on the circumcircle of $BDF$. Prove that $PN, AM, UF$ concur.

Let $p,q$ prime numbers such that $$p+q \mid p^3-q^3$$Show that $p=q$.

Let $ABC$ be a triangle and $\omega$ its incircle. $\omega$ touches $AC,AB$ at $E,F$, respectively. Let $P$ be a point on $EF$. Let $\omega_1=(BFP), \omega_2=(CEP)$. The parallel line through $P$ to $BC$ intersects $\omega_1,\omega_2$ at $X,Y$, respectively. Show that $BX=CY$.

Let $m$ and $n$ be positive integers. In Philand, the Kingdom of Olymphics, with $m$ cities, and the Kingdom of Mathematicians for Fun, with $n$ cities, fight a battle in rounds. Some cities in the country are connected by roads, so that it is possible to travel through all the cities via the roads. In each round of the battle, if all cities neighboring, that is, connected directly by a road, a city in one of the kingdoms are from the other kingdom, that city is conquered in the next round and switches to the other kingdom. Knowing that between the first and second round, at least one city is not conquered, show that at some point the battle must end, i.e., no city can be captured by another kingdom.

Let $a_1,a_2,\dots$ be a sequence of integers satisfying $a_1=2$ and: $$a_n=\begin{cases}a_{n-1}+1, & \text{ if }n\ne a_k \text{ for some }k=1,2,\dots,n-1; \\ a_{n-1}+2, & \text{ if } n=a_k \text{ for some }k=1,2,\dots,n-1. \end{cases}$$Find the value of $a_{2022!}$.