Find all real numbers $x$ such that exactly one of the four numbers $x-\sqrt 2$, $x-\dfrac{1}{x}$, $x+\dfrac{1}{x}$ and $x^2+2\sqrt{2}$ is not an integer.

Uri must paint some integers from $1$ to $2022$ (inclusive) in red, such that none of the differences between two red numbers is a prime number. Determine the maximum number of numbers Uri can paint red. Note 1: The difference between two distinct numbers is the subtraction of the larger minus the smaller. Note 2: $1$ is not a prime number.

Let $A$, $X$, $Y$ be three non-aligned points of the plane. Construct with a straightedge and compass a square $ABCD$ such that one of its vertices is $A$ and also $X$ is on the line determined by $B$ and $C$ and $Y$ are on the line determined by $C$ and $D$.

Determine the smallest positive integer $n$ that is equal to the sum of $11$ consecutive positive integers, the sum of $12$ consecutive positive integers and the sum of $13$ consecutive positive integers.

Determine all positive integers that cannot be written as $\dfrac{a}{b}+\dfrac{a+1}{b+1}$, where $a$ and $b$ are positive integers.

In a hockey tournament, there is an odd number $n$ of teams. Each team plays exactly one match against each of the other teams. In this tournament, each team receives $2$ points for a win, $1$ point for a draw, and $0$ points for a loss. At the end of the tournament, it was observed that all the points obtained by the $n$ teams were different. For each $n$, determine the maximum possible number of draws that could have occurred in this tournament.