SMO 2017 Senior Round 2

Let $n$ be a positive integer and $a_1,a_2,\ldots,a_{2n}$ be $2n$ distinct integers. Given that the equation $$|x-a_1||x-a_2|\cdots|x-a_{2n}|=(n!)^2$$ has an integer solution $x=m$, find $m$ in terms of $a_1,\ldots,a_{2n}$.

In the cyclic quadrilateral $ABCD$, the sides $AB,DC$ meet at $Q$, the sides $AD,BC$ meet at $P$, $M$ is mid point of $BD$. If $\angle APQ=90^\circ$, prove that $PM$ is perpendicular to $AB$.

There are $2017$ distinct points in the plane. For each pair of these points, construct the midpoint of the segment joining the pair of points. What is the minimum number of distinct midpoints among all possible ways of placing the points?

Find all functions $f:\mathbb Z^+\to\mathbb Z^+$ such that $f(k+1)>f(f(k))$ for all $k\ge1$, where $\mathbb Z^+$ is the set of positive integers.

Given $7$ distinct positive integers, prove that there is an infinite arithmetic progression of positive integers $a,a+d,a+2d,\ldots$ with $a\le d$ that contains exactly $3$ or $4$ of the $7$ given integers.