SMO 2023 Junior Round 2

In a convex quadrilateral $ABCD$, the diagonals intersect at $O$, $M$ and $N$ are points on the segments $OA$ and $OD$ respectively. Suppose $MN$ is parallel to $AD$ and $NC$ is parallel to $AB$. Prove that $\angle ABM=\angle NCD$.

What is the maximum number of integers that can be chosen from $1,2,\ldots,99$ so that the chosen integers can be arranged in a circle with the property that the product of every pair of neighbouring integers is a 3-digit integer.

Define a domino to be a $1\times2$ rectangular block. A $2023\times2023$ square grid is filled with non-overlapping dominoes, leaving a single $1\times1$ gap. John then repeatedly slides dominoes into the gap; each domino is moved at most once. What is the maximum number of times that John could have moved a domino?

(Example: In the $3\times3$ grid shown below, John could move $2$ dominoes: $D$, followed by $A$.)

Two distinct 2-digit prime numbers $p,q$ can be written one after the other in 2 different ways to form two 4-digit numbers. For example, 11 and 13 yield 1113 and 1311. If the two 4-digit numbers formed are both divisible by the average value of $p$ and $q$, find all possible pairs $(p,q)$.

Find all positive integers $k$ such that there exists positive integers $a,b$ such that $a^2+4=(k^2-4)b^2$.