International Mathematical Olympiad 1998 — Official contest

A convex quadrilateral $ABCD$ has perpendicular diagonals, and the opposite sides $AB$ and $CD$ are not parallel. The perpendicular bisectors of the sides $AB$ and $CD$ meet at a unique point $P$ inside $ABCD$. Prove that the quadrilateral $ABCD$ is cyclic if and only if triangles $ABP$ and $CDP$ have equal areas.

In a competition, there are $a$ contestants and $b$ judges, where $b \ge 3$ is an odd integer. Each judge rates each contestant as either “pass” or “fail”. Suppose $k$ is a number such that for any two judges, their ratings coincide for at most $k$ contestants. Prove that $$
\frac ka \ge \frac{b-1}{2b}.
$$

For any positive integer $n$, let $\tau(n)$ denote the number of its positive divisors (including $1$ and itself). Determine all positive integers $m$ for which there exists a positive integer $n$ such that $$
\frac{\tau(n^{2})}{\tau(n)}=m.
$$

Determine all pairs $(x,y)$ of positive integers such that $x^{2}y+x+y$ is divisible by $xy^{2}+y+7$.

Let $I$ be the incenter of triangle $ABC$. Let the incircle of $ABC$ touch the sides $BC$, $CA$, and $AB$ at $K$, $L$, and $M$, respectively. The line through $B$ parallel to $MK$ meets the lines $LM$ and $LK$ at $R$ and $S$, respectively. Prove that angle $RIS$ is acute.

Consider all functions $f$ from the set $\mathbb N$ of positive integers into itself satisfying

$$f(n^2f(m))=mf(n)^2$$

for all $m,n\in\mathbb N$. Determine the least possible value of $f(1998)$.